A THEORETICAL STUDY of LOW SIDE LOBE ANTENNA ARRAYS By

A THEORETICAL STUDY OF LOW

SIDE LOBE ANTENNA ARRAYS

Brian Robert Gladman

A thesis submitted to the University of London for Examination for the degree of Doctor of Philosophy (September 1975)•

Work carried out at the Admiralty Surface Weapons Establishment.

-1- ABSTRACT

This thesis presents theoretical work undertaken in support of a research programme on low side lobe antenna arrays. In particular it presents results on the effects of mutual coupling on the performance of microwave arrays that consist of a number of identical radiating elements fed by a wide-band power dividing network. Low side lobe distributions are reviewed and the effects of various types of distributional error are derived. A simplified but representative array geometry is described for which a detailed analysis of mutual coupling is possible. Results are given for the element patterns in a finite array which clearly show the influence of mutual coupling and also the distortion in the patterns for elements close to the array edges. Array patterns obtained from low side lobe distributions are given.

In studying the effects of inter-action between the array and its feed network, a set of 'aperture modes' are discovered that are orthogonal and uncoupled. These modes have a number of interesting properties and are shown to provide a pattern synthesis technique that can take account of the effects of mutual coupling between the array elements. The synthesis of multiple beam networks for producing these modes is covered. A number of potential wide-band feed networks are considered using both directional couplers and shunt coaxial junctions for power division. Their performance is considered both in isolation and in the presence of array effects which can distort the power distribution provided by the feed. Practical results are given for a low power array of 18 elements designed for a side lobe level of -40 decibels over a 20% band.

-2- CONTENTS

Page

ABSTRACT 2 ACKNOWLEDGEMENTS 4 NOTATION 5

CHAPTER ONE DISTRIBUTIONS AND ERRORS 6 TWO MUTUAL COUPLING MIECTS THREE SYNTHESIS WITH MUTUAL COUPLING 98 FOUR ARRAY FEED NETWORKS 142 FIVE A PRACTICAL ARRAY 178 SIX CONCLUSIONS 203

REFERENCES 210

APPENDIX ONE CHEBYSHEV ARRAY DESIGN 215 TWO TAYLOR LINE SOURCE DESIGN 228 THREE THE EVALUATION OF THE FIELD INTEGRAL 247 FOUR COMPUTER PROGRAMS FOR ARRAY SOLUTION 257 FIVE PROGRAMS FOR MATRIX EIGENVALUES AND 282 EIGENVECTORS SIX IMPEDANCE TRANSFORMER AND 287 DIRECTIONAL COUPLER SYNTHESIS SEVEN PROGRAMS FOR SHUNT JUNCTION NETWORK ANALYSIS 300

-3- ACKNOWLEDGEMENTS

I would like to thank Professor J Brown and Mr H Page of Imperial College, University of London for their help during this work. Also Professor J Croney of the Admiralty Surface Weapons Establishment (ASWE) who has been a constant source of inspiration and without whose unshakable support and encouragement this work would never have been completed. Thanks are also due to Dr H Salt of ASWE for the many stimulating discussions that preceded the concepts of chapter three, to J Wyatt and F Gregory for their help with the practical work and to Mrs Tandy for typing this thesis. Finally I would like to thank the Director of ASWE for his permission to undertake and publish this work . 0

NOTATION

The numbering of figures, equations and tables is self contained within each chapter but references are numbered across the thesis as a whole. Shorthand notation of F and R followed by a number is used when referring to figures and references respectively.

In the main symbols are introduced as required. The symbol i is used throughout for the square root of minus one. Due to an oversight it has also been used as a subscript in chapter three in particular. While this dual usage is unfortunate, it should not cause any confusion in practice as it is always clear from the context which form is in use (the only equation in which both usages occur simultaneously is Eqn. 2.1 of chapter three). The notations ishl and 1 ch1 are used for hyperbolic sines and cosines. Both 'e' and 'exp ( )' are used for the exponential function,

1 121/ is used for natural logarithm and Tr has its normal meaning (v = 3.1415926535898 ....). The function sgn (x) is +1 if x is positive and -1 if x is negative.

0 -5- CHAPTER ONE - DISTRIBUTIONS AND ERRORS

• -6- 1. The Requirement for Low Side Lobes The design of radar systems for ship defence is heavily influenced by the ability of an attacker to interfere with or 'Jam' their normal operation by radiating a noise like signal within the band. in which they function. In this 'contest' the radar designer is at a severe disadvantage in that his signal, the radar echo, varies inversely with the fourth power of the target range whilst that from the jammer suffers only inverse square law attenuation. With this advantage the jammer can often inject sufficient power into the side lobes of the radar antenna to prevent target detection over large azimuth sectors or even over all azimuth angles if the antenna side lobes are poor. Whilst it is unlikely that low side lobe antenna design can completely eliminate this problem, the degradation can be restricted to a small sector (or sectors) by providing antennae in which the side lobe magnitudes fall rapidly with angle from the main beam.

This volume presents the authors contribution to a research programme on low side lobe design at the Admiralty Surface Weapons Establishment. It is a largely theoretical study of the factors that limit the side lobe performance (and band width) of linear arrays of identical elements.

2. Performance Criteria for Aperture Distributions Antenna beam width and directive gain are among the more important characteristics of antennae. As is well known, they are functions of the size and shape of the antenna and also of the distribution of electromagnetic energy across the aperture. In comparing 'aperture distributions' it is helpful to normalise beam width and gain parameters in such a way that the influence of antenna size is taken out thus leaving values that characterise the performance of the distributions alone. In addition, as both continuous 'line source' and discrete 'array' distributions will be considered, it is important, if possible, to ensure that the performance parameters for these two classes are both consistent and directly comparable. This section serves to introduce these parameters and a number of formulae commonly used in line source and linear array analysis.

Dealing first with the continuous line source distribution, the relationship between the aperture distribution and its pattern function is normally expressed in the form:

E(U) = g(x) exp (iUx) dx 2.1 -1 • -7- where g(x) is the distribution on the standard interval -1 to 1 and U is defined by:

U = ITZ = 17-Dsine/X ... 2.2 where D is the actual length of the aperture, A is the wavelength and 0 is the angle of the far field point from the normal to the line source (measured in a plane containing both the point and the line source). Related to the above are the inverse Fourier transform: 00 g(x) - f E(U) exp (-iUx) dU ... 2.3

••C0 and Parsevals theorem which is a mathematical statement of the conservation of energy: 00 1E(U)12 dU = Ig(xW dx ... 2.4 -00

F1 shows the familiar sin (U)/U pattern obtained when g(x) is a constant. It is plotted in decibels as a function of the parametei. Dsine/A. The latter provides a convenient normalised measure of beam shape since the antenna pattern plotted as a function of this parameter depends only upon the form of the aperture distribution. The quantity DsinO/A. (ie z) measured at a particular level below the peak of the mainbeam will be called the normalised beam width (nbw) at the specified level. The most common measurement levels are at -3 dB (ie half power) and at the first zero, the values for the uniform distribution at these points being 0.))13 and unity respectively. The actual angular beam associated with an nbw value of Z is given by: width 19b b -1 (Zip A/D) ... 2.5 eb = 2 sin 0

and if D/A is large, approximately by:

115 Z A/D degrees ... 2.6 b b

The ratio of the normalised beam width of a given pattern to that of the uniform aperture pattern (measured at the same level) is an alternative and often used relative beam width measure that is known as the beam broadening_ factor (bbf).

-8- V

nuoung ;e xecianx onn JO uoT4ounj w ow uorwaxma TiTmouTs eta jo tappitmeg pormemoN JOKOd nim zi

—cram: .::n:. • ....-. -7H----1-7-- PM aw p...... ::::: O M an MIMINI MINEMINMENSWIMINEIMM•IMMIMILE ■• dhi_ 1 lliflt, :Ii_ .,...... milliiii...... /t.:L.V - Im0411=133Z3e UTZ SAMIUMMI gia...... =1...... -nra ...... mmr Tm -...ramii g:ITIMLffro;.' Men il.Clen=MWD: r,===r- a r: no: ...... :::asum meg ,1110Laim ' ffilar.--M= MMMM=MM:1---11 mmIll i • mir.rtal ,m6-1-- : E 9 S I I '.. MIL...r....u.... __:=33::: ...... :1-1.11 4....:.IgiN MEM 61111 ME NM MMEMPME ,...... ,-- miteeharxiln . g....:amilMPAR ingni NAMPIESTw -1..ANWM"I ---.1 •13,Firign- hhip ...... rall"1"11111111111011 IN NEN Nill um keilsn' Fauteldi RETIONMINII IMI r isINELa affL.: =F2--"iimmm- Mgr 3" JIMMENFOR ill Emmegmlimw,•=pfh... nr, smi re Effi.....„....-===..,=. ;Jai .111.1 -um• .."--Mti— 42:11Mallaid mmulsmarn numnr,– mimil ..„...... um.= .....nurJr. at-- .,11, va-. c 71.1 ill:174 :I ...... W.6.151.r...... wril" rmr---n-rJrM=3:rivairri-

044114403V PoWuTxtrat Ato40;Tuil sta ;o uaw4141 en TAI The directive gain of an antenna is defined as the ratio of the peak power density in the far field pattern to the average value. For the line source with a broadside main beam (ie in a direction normal to the line source) this definition gives the value:

2 1E(o)1 G - 77D/X ... 2.7 x 27rD 1E(u)1 2 au -7rD/A.

Clearly the effects of aperture length and distribution on this gain value cannot be separated. However, the moderate to large apertures with low side lobe distributions, the limits on the integral in Eqn 2.7 can be extended to infinity with little error giving:

Aa'w 711E(0)1 2 ... 2.8 f 1E(U)12 dU

This value is known as the aperture efficiency (77) as it depends only on the form of the aperture distribution. Using Eqn 2.1 and Eqn 2.4 gives the aperture efficiency in terms of the distribution as: 1 g(x) dx 1 2 A.G

2 f 1g(x)1s dx ... 2.9 -±

The aperture efficiency has a value of unity when g(x) is uniform, all other distributions having a lesser value. Normalised beam width and aperture efficiency will be used to describe the performance of line source distributions.

Before considering arrays, a method of synthesis for continuous distributions will be covered. The method was first described by Woodward (Ri) and is based on the expansion of the aperture distribution in sine and cosine modes. Consider a distribution of the for= CO

g(x) = dm exp [-i(mg+E)x] ... 2.10

m--Co

-10--

Using Eqn 2.1 the far field pattern can be evaluated as:

d sin(U-mgr-c) E(U) = 2 m U -mg -E. ... 2.11

which immediately gives:

2dm = E(mg+c) ... 2.12

Using this expression Eqn 2.10 and Eqn 2.11 above can be written in the for=

g(x) = E(mr+c) exp [-i(ma+c)x] ... 2.13

and

00 U E(U) = E(mg-1-E) sin U-my-e Oil, 2 0 14

M= Oa

Eqn 2.13 relates the aperture distribution to the far field pattern and can be used to find a distribution that will give an approximation to a specified pattern. The second equation provides a method of pattern interpolation. If the pattern function is even with E(-U) = E(U) and E takes the values 0 and Y/2, then Eqn 2.13 can be put in the forms:

00 g(x) = (cm(2) Emir) cos (mnx) ... 2.15 m=0

Co g(x) = 1:E(my+Y/2) cos [(2m4.1)/rx/2] ... 2.16 m=0

These expressions will prove useful later in dealing with the Taylor line source distribution.

Linear, uniformly spaced arrays of elements will now be considered. An array of N elements, each with identical individual patterns f(0) has a pattern given by: N-1

F(0) = f(8) an exp i2gnd sin0/4] ... 2.17 n=0 where d is the element spacing and an is the excitation amplitude of the n'th element. The second factor can be written as:

N-1

E(u) = an exp (inu) ... 2.18

n=0 where u = 2ffd sine/X. It is known as the array factor and is responsible for most of the directive properties of the array. The inverse transform and the equivalent of Parsevals theorem are:

1 an = 2g E(u) exp(-Inu) du ... 2.19

-7T and:

1E(u)i 2 du = 2g ... 2.20

n=0 respectively. Associating a total length of (N-1)d with the array is somewhat arbitrary as the actual length will depend upon the form of the elements. Nevertheless, in the absece of information about the latter, it is a logical choice and will be used in what follows. It yields the relationships: 0 U = (N-1) u/2 ... 2.21

Z = (N-1) u/2g = (N-1) dsine/A ... 2.22

between the array variable u and the variables U and Z used previously for continuous line sources and related to the total length of the aperture. Thus adopting the normalised beam width measure for arrays as (N-1)dsinO/X will enable direct comparison with line source distributions.

The directive gain associated with the array factor is easily derived as:

-12- IE(0)i2 G 2gdp. ... 2.23 J 271 4/1E(u)12 du

As for continuous apertures, an approximation must be made to obtain separation of length and distribution effects. In this case the limits of integration are taken as tir which will give an exact value for an array spacing d of half a wavelength. For spacings not approaching a wavelength, the error in using these fixed limits will be small if E(u) has low side lobes. This yields:

XG 2g1 E(0)12 ... 2.24 2d - g 1E(u)1 2 du

N-1 an12 n=0 N-1 ... 2.25 2 1 1 a I / l n=0 This expression has a maximum value of N when all an are equal and a lesser value otherwise. To achieve compatibility with the normalised gain definition for the line source, array aperture efficiency will be defined as: N-1 12 a XG n g I , J _ n=0 17 = 2Nd N-1 ... 2.26 2 N ylanI ri=0

which will have a maximum value of unity for the uniformly excited array and a smaller value for all other distributions. Comparing Eqn 2.9 with Eqn 2.26 suggests that the above definition of aperture efficiency is equivalent to taking the total aperture length 15 equal to Nd in contrast to (N-1)d used for beam width comparison. For the microwave aperture the former choice is possibly more realistic as it takes some account of the element size which will maks the array length longer than (N -1)d, the centre to centre spacing a —13— of the end elements. The use of Nd for beam width comparison would result in normalised beam width values larger than those presented here by a factor NAN-1). This tends to suggest that the beam widths presented will be optimistically narrow when the number of elements N is small. This is in fact the case and explains how, in some of the figures that follow, some distributions appear to produce beams that are narrower than achieved by a uniform distribution on the same aperture. In spite of this disadvantage, the length (N-1)d has been used for beam width measurement because it avoids the factor N/(N-1) in many formuli and also simplifies the graphical presentation of some results. In computing array beam widths the 'aperture' should thus be interpreted as the distance between the centres of the end elements.

3. The Binomial Distribution This array distribution is of interest because it yields a far field pattern without side lobes. The amplitude of the n'th element is:

(N-1): ... 3.1 an - n! (N-n-1):

with N elements labelled 0 to N-1. The corresponding array factor is easily obtained using Eqn. 2.18 as:

E(u) = [cos(42)]N-1 3.2

The nbw value is derived by substituting for u in terms of Z using Eqn. 2.22 and inverting to give:

Z ... 3.3 b - (N-1)iT cos-1 [101/(N-1)]

where p is the level at which the nbw is to be measured (0.707 ... for the -3 dB point). The -3 dB nbw is plotted as a function of the number of elements N in F2, the point marked on the vertical axis being the corresponding value for the uniformly illuminated aperture. For moderate to large numbers of elements the beam width of the binomial distribution is substantially larger than that of the uniform distribution. In fact when N is large, Eqn 3.3 can be solved approximately as:

V (N-1 ) 1Pdbl Z . N -4- 00 ... 3.4 b 6.547

where pal) is the level for beam width measurement in decibels. With the nbw -14- • •

plan ogatong sT4 ;o umouni tt see uowurpnaut AmmAtotio en jo troTanuog posnimaoN JOKOd ;Tim on ta

squemen ;o Javan& eta ;o uopzuna e se uoTvaqTaleTa inmona .114 ;o 4ou•ToT;;a eanlatiody eqy Ed

•

varying as yr the angular beam width will be proportional to 1/VN. Thus for a fixed element spacing the angular beam width will vary inversely with the square root of the length in contrast to the normal inverse length behaviour.

The aperture efficiency of the binomial distribution can be obtained from Eqn. 2.26 and Eqn. 3.1 as:

1 2.4 6 (2N-2) = N 1.3 5 ... 3.5 • with the limiting value given by:

n 177/7-E ... 3.6

The aperture efficiency is plotted as a function of N in F2. As can be seen there is a rapid drop in efficiency with increasing N.

In summary F2 and F3 show that, while this distribution does not have side lobes, it has a poor performance in respect of both beam width and gain. Of course, the formation of side lobes is not in itself important but rather the decay in pattern magnitude with angle from the main beam. In studying other distributions it will be apparent that the binomial distribution is poor in this respect also.

4. The Chebyshev Distribution This distribution was first constructed by Dolph (R2) who demonstrated that, for a fixed number of elements, it achieves the minimum attainable beam width for a given side lobe level and vice versa. For element spacings of half a wavelength or more the Chebyshev array factor is constructed as:

E(u) = TN-1[s cos (u/2)]/R 0641 4.1 and a Chebyshev parameter s is chosen such that:

R = TN-1(s) ... 4.2

Where R is the ratio of the main beam to side lobe amplitudes (R is 10 for -20 dB side lobes and 100 for -40 dB) and the Tm(x) are Chebyshev polynomials defined by:

-1 Tm(x) = ch, m ch (x)] 1 5 x< 00 ... 4.3 -1 = cos Lin cos (x)] 0 x < ... 4.4 squoloom JO avoleaN inn Jo uorpouni V OW

umaacrplaTa Aorioitiomp ogl Jo wprpouvog pvorromaoli aoftod rpm otix 9,11

110

4 Tom o,/ octoions eta Jo uapouni v ov unpulTapqa AoloAttoR0 Jo TI.10Tomootg PosTromjcli oJez KU, cal

eh being shorthand notation for cosh. This pattern construction gives a main beam and a number of equal level side lobes. The nbw is obtained by inverting the equation:

TN-1[ s cos IITZ /(N-1)fl = pR ... 4.5

to give using the above equations: •

-1 y N-1 -1 ch[ ch Z.b — cos (pR)/(N-1)] pR 1 ... 4.6 Ch{ ch (R) /(N-1)]

-1 loos [cos(pR)/(N-1)] = cos-1 1 ; pR 1 ... 4.7 Ch. [ Ch (R) AN-1)]

Por low side lobes, R is large, and it is interesting to enquire what occurs as R goes to infinity. Taking the limit for Eqn 4.6 gives:

cos-i[p1/(N-1)] Zb -* N-1 ; R 00 ... 4.8

indicating an nbw identical to that of the binomial array. In fact it i5 easy to show that the Chebyshev distribution reduces to the binomial distribution as the side lobe level is decreased to zero. As the Chebyshev array is known to offer an optimum performance this limit appears to suggest that in achieving low side lobes the poor performance of the binomial distribution must be accepted. In practice this is not the case for, as F4 and F5 show, the limiting value given above is approached only slowly. For example, whilst the 25 element binomial array has a -3 dB nbw which is nearly 3 times that of the uniformly illuminated aperture (seen), the equivalent factor for the Chebyshev array is less than 2 for a design side lobe level as low as -100 dB: It is clear from FL. and F5 which show the nbw as a function of side lobe level, that in practical terms the Chebyshev array is greatly superior to the binomial array.

As the number of elements (N) is increased the Chebyshev array factor expressed in terms of U, the continuous aperture variable, approaches the limiting form:

-18-

E(U) = ch[4J (ch-i R.)2 - U2I R U oh R ... 4.9

-1 cos ["U2 - (ch R)21 / R U ch R 4.10 a functional form which is often referred to as the 'ideal Chebyshev pattern'. The corresponding limiting values for the nbw are easily obtained as:

-1 „\I (ch R)2 - (ch p102 ;pR 1 ... 4.11

(c13.-41W + (cos-ipP)2 ... 4.12

Plotting the nbw as a function of the number of elements as in F6 shows that these limiting values are rapidly approached and that, for arrays of 20 or more elements, the nbw values they give will be quite accurate for all practical side lobe levels. Equation 4.11 can be further approximated to yield the half power beam width and beam broadening factor as:

0.9 at - 3 dB ... 4.13 Zb 1Rdb1 4.5

bbf 0.203IRdbl + at 4.5 - 3 dB ... 4.14 where Rdb is the side lobe level in decibels. For arrays of 20 or more elements these expressions are quite accurate for side lobe levels below -20 dB.

There does not appear to be any simple expressions for the aperture efficiency of the Chebyshev array. The values presented here have been obtained using the computer programs given in Appendix 1. In F7 the efficiency is plotted as a function of the number of elements. With the exception of the '-20 dB' curve where the power 'supplied' to side lobes is clearly high enough to degrade the efficiency, it can be seen that convergence to a nearly constant value is fairly rapid. This again shows that for practically low side lobe levels the Chebyshev and binomial distributions are very different. An approximation for the aperture efficiency of the large Chebyshev array can be obtained by using the limiting forms of the pattern functions given in Eqn. 4.9 and Eqn. 4.10 to evaluate Eqn. 2,23. This gives: -20-

2R2 (N -1)/N ... 4.15 1 (N-1)+ ch R (Li R) + ]

where Li(x) is the modified Struve function of order one. When R is large further approximation yields:

- ... 4.16 1,n(2R) 3 R 4" 16 in (2R)]

and finally when N is large, but not so large that the total side lobe power becomes significant: 7 n=j1:722y . • • 4.17 5.22 ... 4.18

In spite of the large number of approximations made in obtaining the latter formuli, they are accurate within a few per cent for arrays of 20 or more elements with side lobe levels of -30 dB or lower. The 'large array' aperture efficiency of Chebyshev distribution is plotted in F8.

Figures F4 to F8 give sufficient information to derive the approximate performance of a specific Chebyshev array in terms of beam width, side lobe level and efficiency. Appendix 1 contains tables that give more accurate values of beam width and aperture efficiency as functions of the number of a elements and the design side lobe level. It also contains details of methods for generating the element amplitudes of the Chebyshev array.

As the Chebyshev distribution provides minimum beam width for a given side lobe level it is, with one qualification, a good choice for low side lobe array design. Its one disadvantage lies in its equal side lobes, there being no decay with angle from the main beam. However the element pattern will provide some fall off in level and, in any event, the side lobe level can always be chosen such that it will be dominated by the effects of the amplitude and phase errors that will be present in any practical implementation.

-21- 5. The Taylor Distribution and Fundamental Limitations on the Decay of Side Lobes with Angle The factorisation of the far field pattern of an array into the product of an element pattern and an array ~actor can only be achieved if all elements possess identical patterns in the array environment. In practice this will not be the case since mutual coupling effects will result in element patterns that vary with position in the array. Thus array analysis is an approximation to the actual behaviour of such an aperture which might in practice be misleading when low side lobe aspects are being considered since the small differences in element patterns assumed to be identical could clearly alter the side lobe behaviour. In any event, a study of the array factor alone cannot provide the complete picture since the element pattern contains important properties with respect to the ultimate decay of side lobe level. An alternative but essentially equivalent viewpoint is that all practical apertures are continuous and cannot be represented exactly in terms of an array of identical 'point sources'.

Limitations on the form of the space factor for a continuous line source have been covered in an excellent paper by Taylor (R3) and more recently by Rhodes (R4-R6). Taylor shows that the decay of the remote side lobes depends upon the behaviour of the distribution in the form:

••• 5.1 where hex) is non zero and finite at x = ± 1. Distributions with an a value of zero are of the familiar 'pedestal' type while those for a = 1 and 2 have simple and quadratic zeros at their edges respectively. He then proves that the pattern function E(U) of a line source aperture with the above distribution must have the following characteristics:

(1) E(U) must be an entire function of U. That is, analytic at all finite points in the complex plane. While not a required character istic, the patterns considered here will also be even functions of U.

(2) The remote side lobes of E(U) must decay as \uj-(a+1).

(3) If the zero pairs of E(U)(the zero at U = U and that at U =:-U are z Z treated as a pair) are labelled from unit~ upwards and ordered with respect to distance' from the origin, then the remote zerOB IfiUSt lie on the real axis, the n'th zero pair approaching the positions ± (n+a/2)~ ~s n tends to infinity.

-22- In the absence of these conditions the 'ideal Chebyshev pattern' might well be considered to be the optimum line source pattern. However its zeros are at the positions:

••• 5.2 un = and its remote side lobes are equal in amplitude, this behavioUr corresponding to an a value of -1. With this value of a, the distribution given by Eqn. 5.1 is unbounded at the aperture edges and is therefore not physically attainable.

Faced with this difficulty Taylor devised a method of approximating the ideal pa,ttern as closely as desired. He considered only pedestal distributions but for our purposes the higher order distributions are also of interest and his method extended to these values will now be briefly outlined. Consider the entire function Et(U) with the zeros:

-1 + (ch R)2 1 ~ n ~ n ••• 5.3

where n is a design parameter known as the transiti'on point and a, the dilation factor, is given by:

a = ••• 5.5 ~1(- n-~1)2 ~ 2 + (-1)2ch R '

When n is less than n the zeros of the function Et(U) are the same as those of Eo(U), the ideal Chebyshev pattern 'stretched' by the factor a :

••• 5.6 when n is equal to -n the zero of this dilated pattern is at the position (n+a/2)~. Beyond this point the zeros of the ideal pattern are moved to the points (n+a/2)~ to correspond with the required asymptotic behaviour for large n. The Taylor pattern Et(U) when constructed in this manner approximates the behaviour of the dilated ideal Chebyshev pattern Eo (U) below U = (rw.a/2)1T and exhibits side lobe decay above this point appropriate to the chosen a value. Appendix 2 contains a more detailed account of the method together with computer programs and tables that give the dilation factor (a), the beam width and aper ture efficiency for the Taylor distribution.

-23-

Figure F9 illustrates a Taylor space factor for a design side lobe level of -20 dB. Whilst such side lobes would be too high in practice it is easier to demonstrate the general behaviour of the distribution at this level. The pattern shown in F9 has a low transition point at ; = 2 (the second zero), _2 the three a values of 0, 1 and 2 giving side lobe decay rates of U U and _3 U respectively. The very marked difference in these rates of fall off is evident in F9. Equally evident is the price that has been paid in terms of beam width which is much increased for the a=1 and 2 distributions. Reference to the tables at the end of this chapter shows that the aperture efficiency is • also degraded being 0.95, 0.81 and 0.70 respectively for a equal to 0, 1 and 2. Figure 10 illustrates the choice of a higher transition point at the sixth zero (n = 6), the different behaviour of the patterns above and below this point being evident. While the gain in decay rate with a is not so large it is still significant and has been achieved at only small expense in terms of beam width and aperture efficiency. The characteristics illustrated in F9 and F10 are typical, the choice of the two parameters ; and a giving flexible control of the form of the pattern within the specified side lobe level constraint. Low transition points and especially the higher a values, while inefficient in beam width and gain terms, offer rapid rates of side lobe decay and conversely the high n values yield Chebyshev like performance with near optimum efficiency. The Taylor aperture distribution is most easily derived from the pattern function using the Woodward synthesis technique described earlier in Section 2. It is particularly convenient because the zeros of the Taylor pattern at and above the transition point lie on the sampling points in Eqn.2.15 or Eqn. 2.16 with the result that the aperture distribution can be expressed as the sum of a finite number of cosine modes:

n+a 2-1

g(x) = (e/2) Et (mIT) cos (mvx) ; a even ... 5.7 Ln= s

;+(a71)/2 -1

g(x) = E t[(2m4.1)7r/2] cos[(2m71-1)vx/2] ; a odd ... 5.8

M=0

Taylor aperture distributions for -20 dB side lobes, a values of 0, 1 and 2 and transition points of 2, 6, 10 and 14 are shown in F11'and F12. They are illustrated for one half of the aperture, the pedestal, linear zero and quadratic zero behaviour being clearly evident. For the higher transition points, these distributions for the different a values are quite similar except -28- near the edge where rapid variations are developing because the pattern is approaching the non realisable Chebyshev form. In contrast, when the transition point is low, the different edge behaviour significantly alters the form of the distribution over the whole aperture. Low transition point Taylor distribution must be of fairly simple form since, from Eqn. 5.7 and Eqn. 5.8 they are composed of a few low order cosine terms. As an example, the -L0 dB side lobe pedestal distribution with a transition point at n = 6 is given by:

g(x) = 0.125 + 0.875 cos2(vx/2) -0.011 cos (217x) + 0.005 cos (31rx)

-0.002 cos (Om) ... 5.9

which is very similar in form to the well known a + b cos2(rx/2) family of low side lobe distributions.

In summary therefore the Taylor distribution, by choice of the two parameters n and a and the side lobe level, offers a range of pattern functions. A comparison of the beam width and efficiency tables for the Chebyshev and the a = 0 Taylor distributions given in the appendices shows that, when the latter has a high transition point, they have very similar performance. Lowering the transition point provides a progressive lowering of the far out side lobes with consequent increase in beam width and lowering of aperture efficiency. The higher a values can provide substantial gains in the rates of side lobe decay at the expense of beam width and gain.

6. The Use of a Continuous Distribution on an Array In discussing any continuous line source distributions, the effects of using a limited number of elements to approximate its shape should be considered. In

• using an array of N elements with the continuous distribution g(x), it is convenient to place the elements at the points:

x n = (2n-N+1)/N 0 n N-1 ... 6.1

such that they are evenly spaced on the aperture with a half space at either end. The array factor is therefore given by:

N-1 E (U) = N [(2n-N+1)/N] exp (2inU/N) ... 6.2

which is obtained from Eqn. 2.18 with u = 2v(D/N)sinO/A = 2U/N.

• -29-

Now using Eqn. 2.13 in Eqn. 6.2 gives:

00 E (U) = N E(m7r+E) exp mff+c)[(2n-N-1-1)/N] } x LJM= — co N-1

x exp [2in(U-m7r-E)/N] ... 6.3 n=6

where the summation order has been interchanged. Performing the second summation and normalising gives: 00 E (U) = E(m7r+f) sin (U-m7r-e) ... N N sin [(U-mv-c)/N1 6.4 M= — co

Using the expansion: 00 (-)1 cosec(x) = x - 17r 6.5 =-00

Eqn. 6.4 can be written as:

... 6.6 17r

co 00 ( _)1(N-1 E(m714e) sin [(U-1NW) - m7r-c] mv-e] ... 6.7 1=-00

and finally using Eqn. 2.14:

00 (N-1)1 E (U) = (-) E(U -1N7r) N ... 6.8 •

which expresses the resulting array factor as an infinite sum of continuous aperture patterns which are shifted progressively by an amount NIT in U. If N is large enough and E(U) is a well behaved line source pattern then the array (U) will closely approximate E(U) over the range IUI IrD/A. which factor EN corresponds to the pattern formed in space. However array cost will increase with the number of elements and it is'therefore norMal to reduce N as much as possible. Equation 6.8 shows that IEN(U)I is periodic in U with a main beam at U = 0 and equal secondary beams at U = ± N7r, ± 2N7r etc. The requirement -30--

• that these secondary beams should not appear in real space leads to the well known criteria for the avoidance of grating lobes (with a main beam at broadside):

N > D/A ... 6.9

d = D/N < A 6.10 which sets a lower limit on the choice of the number of array elements N. While this condition guarantees no grating lobes, it does not necessarily give an array factor that closely matches the pattern of the continuous aperture. Firstly, if N is only slightly greater than D/A , then the side lobe structure associated with the grating lobes at U = ± Nir will appear at wide angles in real space. Secondly, even if N is substantially larger than D/A, the 'overlap' of the adjacent continuous pattern components of the array factor as demonstrated by Eqn. 6.8 may result in increased side lobe levels in the latter if the former do not decay rapidly with the increasing angle. However it should be remembered that the array pattern is the product of the array factor and the element pattern and, as the latter will normally lead to a reduction in side lobe magnitudes at wide angles, it may be quite possible to allow the array factor to diverge significantly above the 'model' continuous pattern at wide angles whilst maintaining a total antenna pattern whose side lobes decay with angle. Nevertheless this consideration is relevant to the use of a Taylor line source distribution on an array aperture.

The Taylor pattern, by design, has little side lobe decay below its transition point at U = (n+o/2)17 and it is therefore prudent in its use on an array to choose the number of elements N in such a way that the adjacent continuous patterns in Eqn. 6.8 cannot add below their transition points. It is easy to show that this requires that:

N 271 + a ... 6.11

or equivalently:

R 4 [(N-a)/2] ... 6.12 a limit on the choice of the transition point for a Taylor pattern when used on an array with N elements. It should perhaps be emphasised that these conditions do not guarantee that the array factor side lobes will be below -31-

the design level although in practice they will be very close (in fact, if the conditions are met, it can be shown that the array side lobes are senerally better or worse than those of the 'model' Taylor pattern depending upon whether the edge exponeng a is odd or even). Equally under some circumstances'an array that violates Eqn. 6.11 and Eqn. 6.12 can nevertheless meet or exceed the design side lobe requirement. In essence the above conditions could be termed 'safe'.

Figure F13 shows the effect of varying amounts of 'overlap' on a 12 element array used with 'a -20 dB pedestal (a = 0) Taylor distribution. The lower pattern for n = 7 does not meet the above condition and gives side lobes rising above the design level. The centre pattern just meets the condition but does not quite achieve -20 dB side lobes. However the beam width is 0-3% narrower than the equivalent Chebyshev array and the aperture efficiency is slightly higher. The upper pattern meets the condition with something in hand and achieves -1969 dB side lobes with small angular decay in level and has a beam width 067% greater than the Chebyshev array. It is interesting to note that, if a is even and the above condition (Eqn. 6.11) is met, the 'discrete' Taylor array can be shown to have an aperture efficiency which is identical to that of the 'model' distribution. Further, with a = 0 and n = N/2, a comparison of the aperture efficiency tables for Taylor and Chebyshev designs shows the former to have a very slight advantage. In summary, the 'discrete' Taylor distribution with a = 0 and n = N/2 is extremely similar in performance to the Chebyshev array. Reducing n below N/2 gives some side lobe decay with angle and beam broadening on the array but in practical terms the gains at the lower design side lobe levels are very small unless ; is very much less than N/2 with the implication that such results can only be obtained on arrays with many elements. • The use of the a = 1 and a = 2 Taylor distributions on arrays does offer the prospect of achieving good side lobe decay rates when ; is less than (N-a)/2 since in these cases the decay of the continuous aperture pattern with angle is much faster and only one or two terms in Eqn. 6.4 contribute significantly to the summation. Clearly beam width and gain will suffer and it is quite likely that these distributions will be more difficult to implement. Neverthe- less they should be considered if side lobe decay with angle is of prime importance.

• -33-

7. The Effects of Am litude and Phase Errors on Side Lobe Performance Little of what has been described so far will be of much practical value if we cannot predict and control the affects of errors. In any practical implementation that is making good use of technology to obtain low side lobes, it will be the errors that determine the performance ultimately achieved. Two types of errors can be distinguished, those that are systematic and hopefully correctable by specific changes in design and those that are random. A good appreciation of the effects of the former type of error can be obtained by considering an amplitude or phase ripple on a linear array aperture. Considering first the effects of a shall phase ripple of the form:

96(x) = V.2 Or cos(tx) 41 40 0 7.1 the pattern will be given by

E(U) = fg(x) exp [112 Or cos (tx)] exp [tUk] dx ... 7.2 -1 which is easily reduced to the form:

E (U) = Eo (U) + /.0/1ri [E0 (U+t) + E0 0-'0 7.3 where E0(U) is the unperturbed pattern. Thus 'error lobes' will appear at U = ± t at a level given by:

Sc = 20 log Or - 3 dB ... 7.4

= 20 log Oa - 38 dB 7.5 where 0r and 0a are the r.m.s. values of the phase errors in radians and degrees respectively. This relationship is plotted under the heading 'correlated errors' in F14 and immediately demonstrates the difficulty of achieving very low side lobes. In the same fashion a distribution with a fractional amplitude error /2- e cos (tx) has a pattern given by:

E(U) = g(x)[14-1T e cos (tx)] exp (iUx) dx ... 7.6 -1

= 30(u) + ern [E0(U+t) Eo -t)] 7.7 -35- the 'error lobe' levels being given by:

Sc = 20 log e - 3 dB ... 7.8 where e is the r.m.s. fractional error. This is shown as the 'correlated error' line in F15 and again demonstrates just how important it is to avoid any sort of periodic error on the aperture distribution (the decibel scale in F15 is approximate only although the error is small below 1 dB).

The effects of random errors have been covered in some detail by Ruze (R7), Schanda (R8) and Bramley (R.9). Their results are fairly complex because of their generality, a much simpler development being more than adequate for our purposes. If there are amplitude and phase errors An and Sri respectively on the n'th element of an N element array the pattern will be of the form:

N-1 E(u) = Tan (1441,1) exp (i611) exp (inu) ... 7.9 n=0 where a is the n'th element's amplitude. If the phase and amplitude errors are small this approximates to:

N-1 N-1 E (u) = an exp (inu) ) an An exp (inu) n=0

N-1 i)' a 611 ... n exp (inu) 7.10 • n=0

The first term represents the unperturbed pattern, the second the third terms being the 'error patterns' caused by the amplitude and phase errors respectively.

Computing the power within one pattern period in u space gives:

J P(u) du = f E(u) E* (u) du -IT

+ /or An n=0 n o

N-1

+ 27T 62 +27T ... 7.11 n=0 n=0

The first term represents the power in the unperturbed pattern, the remaining terms resulting from the amplitude and phase errors. Assuming that the weighted average amplitude error as represented by the second term is zero, Eqn. 7.11 gives the average power of a period in u space as:

N-1

P (u) = 1%12 [ I + 11 7 ] )2/ 0411. 7.12 n=0

where: INT-1

\2_,J lani2 6121 = n=0 ... 7.13 N-1 • lanI 2 n=0

the power weighted mean square phase error, and

N-1 fan12,2,

2 n=0 A = ... 7.14 N-1 ,,2 Ean n=0 • -37-

the power weighted mean square amplitude error. Now the main beam power density P0(0) is equal to Go P0(u) where Go is the array gain, the subscript denoting the unperturbed pattern. Thus:

7 P(u) 02 Po (0 — Go + T Ga ... 7.15

The first term is the a7erage power density of the unperturbed pattern relative to the main beam density, the second and third terms being the average power density resulting from amplitude and phase errors respectively. Relating the gain to the numhi-r of elements and the aperture efficiency gives the average 'error pattern' level as:

÷ 77 N ... 7.16

Since the error pattern will have some sort of side lobe structure, it will have peaks rising above this mean level. In judging patterns it is normal to use the average of the peaks of the side lobes. Although the ratio of the 'mean peak' to 'mean'level given above clearly depends upon the exact side lobe structure, a factor of 2 will not be far out giving the result:

S = [ r 77N ... 7.17 for the average side lobe levels resulting from random errors. This result, separated into its amplitude and phase components, is shown under 'random errors' in F14 and F15 for an aperture efficiency value of unity (ie setting n = i in Eqn. 7.17). The reduced efficiency of typical low sidelobe (listributions will raise the curves by about 1.5 dB. As would be expected, the levels lie below those resulting from correlated errors especially when the number of • elements is large. Nevertheless, the low side lobe revirement places fairly exacting limits on the tolerable levels of random errors. A study of the more rigourous results given by Ruze, Schanda and Bramley leads to essentially the same result at small angles from broadside but does predict some fall off of error side lobe levels with angle:

r- (8 1+ -71 (U = vir exp [-(U/N)2 7.18 Sr 77 N

-38-

• This result applies to a continuous rather than an array aperture, N being interpreted as the number of correlation intervals in the aperture (ie the aperture width D divided by the correlation length c, an estimate of the distance over which errors are largely correlated). Also the mean square errors A2 and 62 used by these Authors are amplitude weighted rather than power weighted as in Eqn. 7.13 and Eqn. 7.14..

Estimates of performance derived from Eqn. 7.17 have proved to be realistic for several arrays on which the near field errors have been measured.

8. Conclusions The conclusions to be drawn from the results presented in this Chapter are fairly straightforward. Firstly it is clear that errors will be a significant problem in achieving low side lobes. Periodic errors must be avoided at all costs and even 'random' errors must be closely controlled if side lobe levels of -40 dB or below are being considered. Secondly, of a number of possible distributions the discrete Taylor, by choice of sidelobe level, transition point and edge coefficient (a), offers the most flexible control of array pattern.

In addition to these conclusions the results highlight the need to gain under- standing in other areas. As already discussed, the element patterns in an array will not in general be identical, there being a variation with position in the array as a result of mutual coupling effects. It is clear that the use of an array distribution on this practical array aperture will not yield the ideal result suggested so far (ie array factor x element pattern) but rather a distorted pattern. The extent of the variations in element patterns and their influence on array performance requires further study. Another effect of mutual coupling is to make the 'match' of the array elements a function of position (the term 'match' is used here in a loose sense since the fields reflected from the aperture are a combination of reflected field resulting from element mismatch and the coupled fields resulting from the excitation of other array elements). These reflected fields can have a significant effect on the performance of the power dividing network feeding the array as this is normally designed on the basis of matched output ports. The array far field pattern will not be degraded if all elements have the same mismatch since in this case the reflected fields are such that all the reflected power will return to the input port of the feed network (this argument is based upon the array being used as a transmitter and assumes that the feed produces an in-phase distribution). Different

-39- mismatches however will result in the reflected power appearing at other network ports thus altering the power distribution from that for which the network was designed. Thus the interaction of the network ana the aperture must be studied. Then of course there is network design itself. In general the network to feed an array with a specified distribution will not be unique. In practice performance will vary for different designs and it may be possible to obtain an optimum solution or, failing this, some indication of which design philosophies are likely to achieve a better performance. These areas are • among those studied in the chapters that follow. CHAPTER TWO - MUTUAL COUPLD(G EFFECTS

• -41- 1. The Configuration Analysed Over recent years the interest in electronic scanning has resulted in an extensive literature on mutual coupling effects in phased arrays (see R10- R12 for example). The use of the digital computer has fostered considerable advance in the theoretical analysis of infinite array geometries in particular and it is now true to say that the effects of mutual coupling on phased array performance in respect of the scan limitations it imposes are fairly well understood. In view of the extent of this effort it is perhaps surprising that (for microwave arrays at least) the influence of mutual coupling on other aspects of array performance such as band width and side lobe level when unscanned has received little attention. In this Chapter a simplified array geometry is analysed in an attempt to gain insight into the ways in which mutual coupling effects will influence low side lobe design.

The array geometry chosen for study must be simple enough to allow a fairly precise theoretical analysis and yet at the same time be capable of exhibiting all the essential characteristics of practical configurations that will generally be more complicated in form. The finite linear array of uniformly spaced waveguide elements feeding an infinite parallel plate region of the same width as illustrated in Fl meets these requirements well and, in addition, is of practical importance in its own right. Propagation is confined between two infinite parallel xz planes with separation b, the space between them being split into two regions by the )cy aperture plane at z = O. The array will be analysed as a transmitter with a finite number of waveguides of uniform width a and pitch d occupying the region for z < 0 and radiating into the 'free space' region for z > O. All bounding surfaces are assumed to be perfectly conducting. This geometry has two characteristics that simplify its theoretical analysis. First, it is essentially two dimensional in nature and can be analysed in terms of the solutions for the structure shown in F2 which has the same x z cross-section but is infinite in the y direction. Thus solutions for this geometry with y and t variations of the form exp[t(wt-ky)] and exp [i(wt+ky)] can be combined to form solutions exp(iwt) cos (k/) and exp (iwt) sin (k y) which, with k = qir/b and q integer, are solutions for the waveguide array (w is the angular frequency, t the time variable, ky the wave number is the y direction and the other quantities as indicated in Fl and F2).

Thus we need only consider solutions for the parallel plate geometry of F2 with y and t variations of the form exp[i (wt-kyy)] in order to solve the

> x

Fl. THE WAVEGUIDE ARRAY CONFIGURATION

-43-

F2. THE PARALLEL PLATE ARRAY GEOMETRY waveguide array problem. The second useful simplification that results from the rectangular geometry is the separation of what is generally one complex solution into two simpler uncoupled solutions that are respectively transverse electric (TE or H type) and transverse magnetic (TM or E type) to the y direction. Before deriving these solutions some general notation will be explained. The following fairly standard notation will be used:

Y, z rectangular co-ordinates as shown in Fl and F2.

t time variable.

angular frequency.

electric field, subscript denoting a rectangular component.

H magnetic field with subscript as above.

free space impedance 40A0 where e0 and po are the permittivity and permeability of free space respectively.

1 the square root of -1.

k with a subscript x,y or z denoting a rectangular component of the wave number (propagation constant).

ko w -477to the free space wave number with k2 + k2 + le = k:T Y z

k without a subscript will denote the transverse wave number VI kE -k2]

To the free space wavelength (note that ko = 21TN).

the transverse wavelength for propagation in the xz plane • = 2w/X). For the TM solution X and X10 are equal but for the TE solution X is given by -7\dV1-(X0/2b)2 . It is X rather than 'Xo that is fundamental to the field propagation between the parallel plates and for this reason the term 'wavelength' without qualification will refer to X rather than -Ao. Element widths and spacings in the diagrams at the end of this chapter are expressed in terms of X. Additional notation will be introduced as requirei. In the analysis of' the next two sections a variation of the form exp [1 (wt - k y)] will be assumed for all field components and will be omitted from the fi3ld equations. -4-5- 0

The parallel plate array of F2 will be analysed in the transmit sense with the parallel plate waveguides being excited by waves travelling towards the aperture from z = - 00

Sections 2 to 6 inclusive present details of the analysis and can be skipped if of no interest. Transverse electric (DIE) and transverse magnetic (TM) excitations are often referred to and are perhaps easier to visualise in terms of aperture electric field direction which is respectively parallel and perpendicular to the array axis (ie the x axis). That is with the array axis horizontal TE and TM excitation correspond to horizontally and vertically polarised excitation respectively.

2. The Transverse Electric Solution Fields TE to the y axis in which Ey = 0 can be derived from Hy using:

all - 1 nko _.....x Inki El Ex E - - k2 3A ' z k2 ax 2.1 • 1 a H = x k2 Hz = k2 ay a z

with Hy satisfying the equation:

a2H a2H 73714 + + =0 2.2

where k is the transverse wave number in the xz plane. Field expansions for the parallel plate (z < 0) and free space regions (z > 0) will now be given and the solution derived by matching these fields in the aperture plane (z = 0). In

0 the free space region solutions of Eqn. 2.2 of the form:

exp [- 1 (kx x kz z)]

will be used where:

k2 k2 = k2 2.3

and the condition that the fields remain finite as z 00 and x - t 00 reqiiires that

klc real, - 00 < kx co ... 2.1k. • kz either +ve real or -ve imaginary 2.5

The most general TE field for z > 0 is therefore given by:

Hy = f H(kx) exp (k 41.0 2.6 . xx + kzz)j dkx -00

From Eqn. 2.1 Ex is derived as: •

Co E = DISD3 f k H(kx) exp [- i (kxx + k x z zz)j dkx ... 2.7 -00

The function H(kx) represents an angular plane wave spectrum. The fields in the aperture plane (z = 0) are therefore given by:

00 yo = f H(kx) exp (- 7 kxx) dkx ... 2.8 -00 • 00 = LS° k H (k ) exp (- k x) dk Exo k2 z x x x ... 2.9 ""*C0

Considering now the p'th parallel plate waveguide with its centre at x = x . Solutions of Eqn. 2.2 for H that satisfy the boundary conditions are of the form:

cos [mv(x-xp + a/2)/aj LA exp (- 7 k z) - B exp (ik z) mn zn ... 2.10 where:

k2 kzm2 + (Im7r/a)2 = ... 2.11 andmis integer and in the range 0 .‹...m00. AandBare the incii9nt and reflected mode amplitudes respectively. Thus the field within the p'th parallel plate region can be written as a sum of modes:

Aro exp (- ik zmz) cos [mir(x-x + a/2),/aj n HYP = -Bmp exp ( 1,1‹ mmz)J m -47- ... 2.12

•

and from Eqn 2.1 E is derived as: xP A exp (-ik z) cos [Mw(x-x +a/2)/a] kk mP m xp zm + B exp (ik z)] • mp zm ... 2.13

where m ranges from zero to infinity. The wave number kzm must be +ve real or -ve imaginary and Amp, the incident model amplitude for the m'th mode is the p'th parallel plate waveguide, is non zero only when kzm is real. The reflected model amplitude Bmp can be non zero for all modes. The aperture plane fields are obtained by setting z = 0 in Eqn 2,12 and Eqn. 2.13 to give:

Hypo =(Amp - Bmp ) cos [mw(x-x +A/2)/a] ... 2.14

Expo ... 2.15 (Amp + Bmp ) kzm cos [mw(x-x +A/2)/al

Thus there are two different representations of the aperture fields (Eqn.2.8 with Eqn. 2.9 and Eqn. 2.14 with Eqn. 2.15) which must in fact be consistent. The Fourier integral in Eqn 2.9 can be inverted to give:

CO 277.0:0k2 f dx ... 2.16 kz H( X) = Exo exp (ikZ)

MP 00

Now the electric field in the aperture plane is only non zero over the waveguide apertures so that Eqn. 2.16 reduces to a summation of integrals over each waveguide aperture. Thus using Eqn. 2.15 to represent these aperture fields gives:

x +A/2 H(k ) = kz x Amp + Bnip )kzm if x -A/2

cos[mr(x-xp+a/2)/a] exp (ikxx)] dx 2.17

which on changing the order of summation and integration and integrating gives:

k exp (ik )sin[(ka+mw)/2] 1 x x xp k H(k ) = A + B )k z x 7T Amp nip zm - (mw/a)e 10,m ... 2.18 • -4.8-

where the summation is over all modes (m) and all waveguides (p). Now the Fourier series of Eqn. 2.14 can be inverted to show that for the M'th mode in the P'th waveguide region: x +a/2

(1+ ) (Ale- )a/2 = Hypo cos [Mff(x-xp+a/2)/a] dx ... 2.19 P-81/2

where 8M is the Kronecker delta function which is unity when M is zero and zero otherwise. Substituting for Hypo from Eqn. 2.8 and changing the order of integration gives: xp+a/2 (1+6m)(Amp-Bmp)a/2 = f li(kx)rf exp (-ikxx) cos [M7r(x-xp+a/2)]d): dx xp-a/2 ... 2.20 which on performing the inner integration becomes: , . -M k exp (-ik x )sin [(kxa+Mff)/2] (1+8m)(Amp-Bmp)a/2 = 2t f 110

Finally substituting for H(kx) in E2.21 from E2.18 yields:

)a/2 = - 111 im (1+8m)(A p- 77. (Amp(A +B mp )k zm P/m

kx2sin [(kxa+mff)/2]sin [(kxa+Mir)/2] exp [-ikx(xp-xp)] x dkx [lc.; -(mv/a)2]X2 -(M77/a)2] kz ... 2.22

which after rearrangement yields:

(1+80( / [(A mp+B mp)k zma] [IH(m, p; M, P) + 8 mM8 pP k p,m zM a

mp = 2(ka)2(1+8m) A 2.23

where 8mn is unity if m is equal to n and zero otherwise and:

-4-9- • IH (m, p; M, P) = [1 +(-)111441Ih (xp-xp mg/ka, Mg/ka)

-(-)m Ih (xp-x-a, mir/ka, Mg/ka)

-(-)M Ih (xp-xp+a, mg/ka, Mg/ka) ... 2.24

• with: 00 Ih (x, p, v) 1 f t2 exp (-ikxt) dt ... 2.25 g -03 (t2-p2)(t2-v2)1/1-t

42 [F I(x2o2p) - v2 I(x,o,v)] ; g / v

... 2.26

a [gx,0,0 [pI(x,o,p)]] ; p

and:

= 1 I exp [-ik(xt+z 11:1;7)1 I(x,z,p) at ... 2.27 (t2-ti2 00 )

This integral is more complex than needed here as z is always zero in Eqn. 2.26. However it is needed with non zero z in the computation of the array near fields and as its evaluation is fairly difficult and lengthy this has been placed in Appendix 3. Equation 2.23 is an infinite matrix equation for the reflected model amplitudes Bmp in terms of the incident model excitations represented by Amp on the right hand side. To clarify the meaning of the subscripts, the capital letters P and M indicate which waveguide array element and mode is being actively excited while lower case subscripts indicate waveguide modes that are generated parasitically by this particular excitation. Discussion of its solution will be deferred until the transverse magnetic solution has been derived.

3. The Transverse Magnetic Solution This derivation closely parallels that for the TE solution above and will therefore be given in an abbreviated form. Fields TM to the y axis are obtained from Ey using:

-50- •

ikb aE H x — az Hz ... 3.1

d2E a2E 1 y I y, E = E = x 777 a ya x z k aya z

with E satisfying the same equation as H for the TE solution. The TM field representations in the free space region are:

E( x)exp [-i(kxx+kzz)] dkx ... 3.2

and:

H = - x kz E(kx ) exp [-i(kx x+kz z)] dkx ... 3.3

with kx and kz satisfying Eqn. 2.3, Eqn. 2.4 and Eqn 2.5. Thus the aperture field representations are: 00 E'o= E(kx) exp (-ikxx) dk Y T x ... 3.4 -00

k E(k ) exp (-ikx x Hxo = z x x) dk ... 3.5

The model expansions for the fields in the p'th waveguide are:

'-Crop exp (-ik zmz) sin [my(x-x +a/2)/a] EYP = + D exp (i k mp zm ... 3.6

kb 1C p exp (-ikzmz) H = -772 sin [m77-(x-x m xp 4 p+a/2)/a] k 77 ouL -D exp (ik z)J m mp zm ... 3.7

with m in this case an integer in the range 1 m -51-

• These equations yield the aperture field representations:

Eypo = (C mp +D mp) sin [ma(x-x +a/2)/a] ... 3.8 m

) -D ) k sin [mv(x-x+a/2)/a] • Hoxp nk2 mp mp zm ... 3.9 m

Inverting Eqn. 3.4. yields

00 1 ... 3.10 E(kx) = 2v f Eyo exp (ikxx) dx -00

Substituting for Eyo from Eqn. 3.8 into Eqn. 3.10 and integrating gives:

• :(c 41) ) mim exp (ikx xp )sin [(kxa+mv)/21 E(kx ) = 4Lta mp mp - (mv/a)'

... 3.11

Inverting Eqn. 3.9 gives: xp+a/2 2 kzm(1-6m)(Cmp-Dmp)a - Hxpo sin [Mv(x-xp+a/2)/a]dx xp-a/2 ... 3.12

from Eqn. and integrating gives: Substituting for Hxpo 3.7

CO 41111/1. k E(k ) i-M (1-814) kzm_a(C MP-D MP )= a z x

exp(-ikxxp) sin [(kxa+Mv)/2] dkx k7c - (Mv/a)2 ... 3.13

-52-

Substituting Eqn. 3.11 into Eqn. 3.13 gives:

(1-8 a (c (C +D ) m im-M M) kzM MP -DMP ) = a mP mP p5ra

00 I kz sin[ (kxa+mff),/2] sin [ (Isca+Mu)/2] exp [-ikx(xp-xp)] dkx -- [k:- (mv/a)2] [k: - (Mv/a)2 1 • ... 3.14 which after rearrangement yields:

(C +D ) IE(m,p;M,P) + 8mm 8pp (1-8m) k mp mp, zma-) = 2kzMa (1-8m) Cmp p,m

... 3.15 with:

IE(m,p;M,P) = [1+(-)m+M] Ie(xp-xp , mff/ka, Mff/ka)

-(-)m Ie(xp-xp-a , mff/ka, Mff/ka)

-(-)M Ie(xp-xp+a , Mff/ka) ... 3.16 and

pv117-77 exp (-ikxt) dt ... 3.17 Ie(x,p,v) = (t2„42)(t2„2)

P2) I(x30,P) - (1-v2) I(x$0,v) ; uv

[ a ... = (1-11 2) 57 [III(x,o,p)]-(1+1/2) I(x,o,4) ; 3.18 with I(x,z,p) as given in Eqn. 2.27. Equation 3.15 is an infinite matrix equation for the reflected mode amplitudes Dmp in terms of the incident mode amplitudes Cmp on the right hand side.

-53- •

4. The Radiated Fields of the Array For prescribed incident waveguide mode amplitudes Eqn 2.23 and Eqn. 3.15 can be used to calculate the reflected mode amplitudes. To complete the solution we need a method of computing the radiated fields of the array from the known solution in the waveguide region. For the TE solution this is achieved by substituting from Eqn. 2.18 into Eqn. 2.6 to give: 0 H = (Amp+Bmp)kzma] y 2ka m [Ix (x-xp +a/2,z,my/ka) P1ra

-(-)m I (x-x -a/2,z,m7r/ka)] x p ... 4.1

where: 00 exp [ k(xt+z 11.7--t2)] t dt Ix x,z,P) = - ... 4.2 (t2.- p2)17T2 01000

•

_ 1 a I (x, k ax ... 4.3

The other field components can be obtained from Eqn. 4.1 by using Eqn. 2.1. This yields expressions similar to Eqn. 4.1, the main difference being the introduction of higher differentials of the field integral I(x,z,p). Using these equations the fields over the whole space region can be calculated. However the ability to calculate the radiated fields of the array at all ranges is bought at the expense of considerable complexity and it would obviously be useful if possible to obtain simpler expressions for the radiated fields at great distances from the array. These can in fact be derived by calculating - the limiting behaviour of the field integral and its derivatives when x and/or z is large but a somewhat simpler approach will be adopted here. Introducing the polar co-ordinates (r,oP) of a point in the space region and making the substitution:

k = k cos8 kz = k sine ... 4.4

allows Eqn 2.6 to be written:

H = k f H(k cost)) exp [-ikr cos (0-C sine de ... 4.5

-54- where the path of integration is now the contour 0 in the complex 8 plane as shown in F3. With e replaced by the complex variable 6 + I y the exponten- tial term in Eqn. It_.3 becomes:

exp [ikr ch y cos (e-95)] exp [-kr sh y sin (6-0)] ... 4.6 when r is very large the magnitude of this exponential will either be very large or -rery small depending upon whether sh y sin (0-0) is negative or posLive respectively. The regions of large magnitude are shown bounded by thick lines in F3. Thus on the contour O the only significant integrand values when r is large occur when sh y sin (6-0) is very small. Thus the main contribution to the integral comes from a small region about the point 0=0 on the real axis and as r increases the size of this region in which the integrand is of significant value decreases. Over this small region the slowly varying components of the integrand will be nearly constant and can be factored out of the integral to give: 0+6

H k sink H sos0) f exp [-Ekr cos (6-55)]d6 r ---> os ... 4.7

where 6' is small. Making the substitution:

6 = 0 + V27/7 6 exp (ta) • • • 1-1-o8 where e is the new integration variable and a is a constant yields:

)117-1.; 12 exp (-Ia) k sine H (k cos0) exp [t(a-kr)] exp [1 e2 exp (2ta)] de H f -471 rc exp (ta) ... 4.9 where, as c is small, the cosine function has been expanded into the first two terms of its power series. Now, the integrand decays most rapidly when a = and, as no poles intervene, the present contour can be rotated into this direction without changing the value of the integral. This reduces the integral in Eqn. 4..9 to the form: 11:77. I . f exp (-e2) de ... 4.10 -1./kr/2 do -55- which rapidly approaches 1/7"- as r increases. Equation 4.9 therefore finally becomes:

2rr H -> „\I exp (ITA-kr)] k c H (k cosc6) ; r -->

... 4.11 showing as we would expect that the far field pattern is directly related to the angular plane wave spectrum H(kx). Thus using Eqn. 2.18 and Eqn. 4.12 the far field pattern of the array can be calculated from the known solution for the waveguide region.

As the analysis for the TM case is so similar the results will be quoted without derivation. The electric field is given by:

_ L (Cmp +Dmp ) m7r I (x-x +a/2, z, mw/ka) E 2ka z p POn

m (x-x -a/2, z, mir/ka) /4-.12 -(-) Iz

where:

I g) _ exp [-Ik(xt+7,477)] z (t2 4 2) +,81 3 -co e os 1

1 31 (x, z, g) 4.14- k az

with the limiting form:

2w — exp [t(w/4-kr)] k sin0 E (k cos0) ; r -> 00 kr • . . . 4.15

5. ' The Evaluation of the Coupling Integral As can be seen from Eqn. 2.23 to Eqn. 2.27 and Eqn. 3.15 to Eqn. 3.18 the evaluation of the functtons:

a and [I/1(x, o )] I (x,) au

is central to the solution of the array problem posed in this chapter; -57- • (I(x,z,p) and I(x,o,p) will be referred to as the field and coupling integrals respectively because of their roles in the analysis). This section reviews the theoretical development and numerical evaluation of these functions as well as their behaviour since more difficulties were encountered in this area than in the remainder of the analysis combined._

• The field integral is derived in Appendix 1 and yields the coupling integral as:

a. g < 1 ; a = cos-lp ; 0 < a < 77/2

2 I(x,o, cos a) = sin 2a (1-200 cos (kx cos a)

+2 ( -)n sin 2na [H2(kx)

- iS2n (kx),477- + I T2n (kx)/77- ] j ... 5.1

b. p > 1 ; a ch g ; 0 < a 00

21 I(x, o, ch a) = sh-2 2a sin (ikxlch a) - g (kx ch a)

00 + 21.1.(-)n exp (-2na) - i S2(kx)/ir] j n= ... 5.2

e is 1 if n is zero and 2 otherwise and : where n

H1(x) is Hankels' function of the second kind.

s (x) is Schlafli's polynomial (R13, section 9.3). n

T (x) is a function introduced by Watson in the decomposition of n the Bessel function I (x) of the second kind (R13, section 10.6)

f(x) are the auxiliary functions for sine and cosine g(x) integrals (Pt14 Page 232).

• -58- Differentiation of Eqn. 5.1 and Eqn. 5.2 yields:

a [gI(x,o,g] g2I(x,g) + kx H1 (kx) am kx K (x, o,g) s" 5'3 (1 - 112) where:

• a. g < 1 ; a = cos t g ; 0 < a < 77/2

1 K(lt o cos a)= sin a (1-200 sin (kx cos a)

00 (-)n sin [(2n+1)a][H2111_1(kx)

- I S2n+1(kx)/77- + i T2n+1(kx)///1 J ... 5.4 b. u> 1 ; a . =ch i g ; 0 < a <00

IC(x, o, ch a) = sh f (kx ch a) - i co § (kx ch a) sgn(kx) 1 a ir

+ 2 (-)n exp [-(2n+1)a][H +1

I (1

As Eqn. 5.4 and Eqn. 5.5 are similar to Eqn 5.1 and Eqn. 5.2 only the evaluation of the latter will be considered in detail.

When first evaluated, the summation in Eqn. 5.1 was in the form:

sin (2na) [H2n(kx) - S (kx)/771 _:(*)n 2l Yn=1 with the real and imaginary parts: 00

(-)n sin 2na J2n (kx) ... 5.7 n=1

-59- • and:

- (-)n sin 2na [Y2 (kx) + San(kx),41 ... 5.8 n=i

Summation of Eqn 5.7 poses no problems as Jn(x), the Bessel function of the first kind, decays rapidly in magnitude when n is greater than x with the result that the sum can be truncated when .n is a little greater than kx/2 with little error.

Unfortunately the same is not true of Eqn 5.8 as it can be shown that when n* x :

Y2n (x) + S2 (x) - lAur 5.9

thus making the summation of Eqn. 5.8 similar to:

00 n sin 2na ... 5.10 (-) n n=1

which is very poorly convergent and does in fact represent a discontinuous function. After a number of unsuccessful attempts to improve this situation the following decomposition described by Watson (R13, section 10.6) was studied:

2 Yn(x) + Sn (x) = [in x - to 2 + W] Jn(x)

+ (x)/ir 5.11 Un(x)/w- Tn

where Tn(x) and n(x) have the expansions: 00

(x) J (x)] ... 5.12 Tn(x) = ' m1 [ n+2m n-2m m=1

00 +2m (x)(-)m 0 .(x) ... 5.13 n(x) = sn n in(n a+ n+2m /, m=1

with so = 0 , sn = 1+ 1/2 + + 1/n otherwise and W = 0.57721566..0 • -6o- •

Eulers constant. From Eqn. 5.13 it is seen that n(x) will decay rapidly in magnitude when n is greater than x in the same fashion as n(x) (sn will make n(x) larger than J n(x) but it 'will not alter the convergence property as it varies only very slowly with n). Thus Eqn. 5.8 can be written:

00 )11 sin 2na [[zm (kx/2) + /i] - U2n(kx)] •

00 (-)n sin 2n a T2n(kx) ... 5.14 n=1 with the first series exhibiting good convergence properties. Surprisingly, the second summation which must contain the poorly convergent component of Eqn. 5.8 is analytically summable. Using Eqn. 5.12 some tedious algebra shows that: co n 0 sin 2na. T (x) = a cos (x cos a) ; lal < g/2 (-) 2n n=1 ... 5.15 and: n -) sin (2n+1 )a T (x) = a sin (x cos la I

Thus using Eqn. 5.14 and Eqn. 5.15 , Eqn 5.8 can be transformed into a series with good convergence. This transformation has been used in obtaining both Eqn. 5.1 and Eqn. 5.4 which are in a convenient form for numerical evaluation. It is worth noting that Eqn. 5.15 and Eqn. 5.16 appear to be new results. •

A second problem that occurred, this time in the derivation of Eqn. 5.2 is most easily illustrated by an example. In this derivation integrals of the form

00 exp (-kr shy) ' dy ...5.17 1 + exp (2y-2a) 0

occur. A considerable effort went into the evaluation of this type of integral before a suitable solution was found, the principal problem being that a

-61- •

single series cannot represent the denominator over the whole range of the integrand (eg in Eqn. 5.17 different expansions are required for the, ranges y < a and y > a). Eventually it was discovered that the identity:

1 1 -sh y ch y ... 5.18 1 + exp (2y-2a) 1 + exp (-2y-2a) = ch2 a + sh2 y • offers a solution. Applying it for example to the integral of Eqn. 5.17 gives:

iexp (-kr sh y) dy exp (-krt) tdt 1 + exp (-2y-2a) f 0112 a t2 ... 5.19 0 0

where the substitution t = shy has been used in the second integral. A further transformation u = t ch a shows the second integral in Eqn. 5.19 to be related to g(x), the auxiliary function for the sine and cosine integrals giving finally: • I exp (-kr sh y) dy g (kr eh a) 1 + exp (-2y-2a) • 5.20 0 A single series expansion suffices for the denominator of this integral over the whole interval and provides a suitable series expansion for the integral. This technique also provides the expansions:

CO = e (-)n[ `Y Ci (x) cos x + Si (x) sin -x 2 n 2n(x) + S2n(x)A7] n=0 • 5.21

• and:

01(x) sin x - Si (x) cos x = (-)n[Y2n+1 (x) S2n+1 (x)/n] n=0 ▪ 5.22

which again appear to be new results (St(x) and Ci(x) are the sine and cosine integrals as defined in R14, page 231).

Proceeding now to the techniques used in the numerical evaluation of the coupling integral, the summation in Eqn 5.1 is most easy evaluated in the form: -62- •

CO 2 [1- 1 [in (kx/2) + 6]] ( -)n sin 2na J n=1 oo ( -)n sin 2na U2n(kx) ... 5.23 n=1 • obtained using Eqn. 5.11 and reviving the function Jn(x) and Un(x) of orders from zero upwards (although Eqn. 5.1 requires only even order functions, Eve 5.4_ requires the odd orders). In previous work on the summation of Bessel function expansions (R15, page 266) I have derived simple expressions for the truncation point requited to achieve a specified' accuracy. When the order n is greater than the argument x the Bessel function Jn(x) decreases rapidly in magnitude, being of the order of d 10 when:

1/3 n= x + 1.81 (xd2)

This is quite accurate when x is greater than unity and can clearly be used to determine the truncation point for the first series in Eqn. 5.23. In fact it also provides the truncation point for the second series as the behaviour of Un(x) and Jn(x) in the transition region is similar. The Bessel functions Jn(x) are easily obtained using the backward recurrence technique as implemented in R15. Computation using recurrence relations is very efficient and, having obtained suitable starting values, it is tempting to use the relationship:

0 u (x) (x) = _ x j (w.) n-1 - x Un (x) + Un+1 ... 5.25 to compute the Un(x). This does in fact work well when n*is below x in both the forward and backward directions (ie n increasing or n decreasing) but when n is greater than x the results quickly become useless. The reason is that Eon. 5.25 is satisfied by any function of the form:

Un(x) + a J.a(x) + b Yn(x) ... 5.26 with a and b constants. When n is greater than x either Jn(x) (with n decreasing) or Yn(x) (with n increasing) grows very rapidly in magnitude

-63- • with n and any component of these functions in the values being computed will eventually 'swamp' the desired solution. In practice such components are bound to be introduced by finite computational accuracy thus making Eqn. 5.25 unusable when n is greater than x. In this region, where fortunately only a few values are required, Eqn.5.13 is used. The two functions of lowest order produced are then used as starting values for

• the use of Eqn. 5.25 in the backward direction to generate the n(x) for n less than x.

Care is also required in the calculation of the sum in Eqn. 5.2 which, when separated into real and imaginary parts, is:

00 co en (-)n exp(-2na)J2n(kx) - en(-)nexp(-2na)[Y2n(kx) + S2n(kx)/7/1

n=0 n=0 ... 5.27

The real part presents no difficulty but there are problems in generating the functions needed for the imaginary part. As shown in F4 (for n=5) the function n(x) has a singularity at the origin but, because of an equal and opposite singularity in Sn(x),777- , the function n(x) + Sn(x)bris well behaved. Thus when x is less than n (or from the summation point of view when n is greater than x) n(x) and Sn(x) are individually large but their sum is small. As a result any attempt to compute Yn(x) and Sn(x),477- separately with subsequent addition will result in severe cancellation with loss of accuracy and possibly meaningless results. However using Eqn. 5.11 the sum can be put in the form;

• 00 i[Lni (kN/2) + K] );en(-)n exp (-2na) J2n(kx)

n=0

00 00 en(-) nexp( -2na) U2n(kx) + :7en(-)nexp(-2na) T2n(kx) n=0 n=0

5.28

which does not suffer from cancellation effects. Summation of the first two series is relatively straightforward with convergence being principally

• -64- COEFFICIENTS FOR SICBX)/X FOR B = 10.0 COEFFICIENTS FOR CI(BX)-LN(BX)-GAMMA FOR B = 10.0 0 7.19054 44641 86755 27720 0 4.35573 28401 86671 99688 2 - 3.09639 02330 35253 59096 2 - 0.99130 54507 66227 71347 4 1.86747 49899 50150 69539 4 0.47813 89115 91557 81295 6 - 0.98958 01249 93803 49481 6 - 0.39677 97837 89969 75557 8 0.35259 77297 38274 27911 8 0.22173 60166 52709 28677 10 - 0. 8319 65791 84050 02705 10 - 0. 7232 07666 26724 21276 12 0. 1367 64.426 45763 899 33 12 0. 1512 45785 88351 03178 14 - 0. 164 82077 06713 92554 14 - 0. 220 28647 60069 51385 16 . 0. ,15 17526 45003 42023 16 0. 23 70167 36165 35472 18 - 0. 1 10263 70186 05156 18 - 0. 1 96594 68030 79253 20 0. 6487 44769 22433 20 0. 12979 35768 37951 22 - 0. 315 55485 66705 22 - 0. 699 19993 06297 24 0. 12 90740 51847 24 0. 31 34898 47860 26 - 0. 45034 56110 26 - 0. 1 18892 83006 28 0. 1356 52904 28 0. 3865 99619 30 - 0. 35 64315 30 - 0. 109 01987 32 0. 82428 32 0. 2 69246 34 - 0. 1690 34 - 0. 5873 36 0. 30 36 0 • 114 38 - 0. 38 - 0 • 1 40 0. F5. CHEBYSHEV COEFFICIENTS FOR SI(() AND CI(X)

• •

COEFFICIENTS FOR F(B/X)/X FOR B = 10.0 COEFFICIENTS FOR G(B/X) FOR B = 10.0 0 0.19814 81541 77794 19643 0 0. 960 71408 96328 18716 2 - 0. 90 34105 49037 97470 2 0. 474 21763 65839 14437 4 0. 2 13564 12639 80477 4 - 0. 5 91707 78725 12348 6 - 0. 10630 18199 10234 6 0. 20814 67751 90378 8 0. 828 19014 30160 8 - 0. 1288 04297 17290 10 - 0. 87 41420 21649 10 0. 114 83003 22022 12 0. 11 51018 79359 12 - 0. 13 25470 17511 14 - 0. 1 79524 92040 14 0. 1 85728 92581 16 0. 32032 31454 16 - 0. 30289 53824 18 - 0. 6379 58593 18 0. 5584 08928 20 0. 1392 80772 20 - 0. 1139 48352 1 22 - 0. 328 80305 ' 22 0. 253 34376 CON 24 0. 83 04209 24 - 0• 60 629 64 26 - 0. 22 248 39 26 0. 15 46948 28 0. 6 27996 28 - 0. 4 17581 30 - 0. 1 85704 30 0. 1 18508 32 0. 57260 32 - 0. 35175 34 - 0. 18336 34 0. 10871 36 0. 6077 36 - 0. 3486 38 - 0. 2079 38 0. 1155 40 0. 732 40 - 0. 395 42 - 0. 264 42 0. 139 44 0. 98 44 - 0. 50 46 - 0. 37 46 0. 18 48 0. 14 48 - 0. 7 50 - 0. 5 50 0. 2 52 0. 2 52 - 0. 1 54 - 0. 54 0.

F6. CHEBYSHEV COEFFICIENTS FOR F(X) AND G(X)

determined by the exponential functions when a is not small or by the established behaviour of J (x) and U (x) when it is. As T (x) does n n n not decrease rapidly when n is greater than x the third series only converges well from medium to large a values, many terms being needed when a is small. Remembering that p = ch a this means that there is a small range of p values 1 < p < 1 + 8 where the series summation is not practical. Providing for the summation of up to 150 terms however -8 • gives an accuracy of 10-18 outside the interval 1 < p < 1.01 and 10 outside of the interval 1 < p < 1.001. In any event it will be seen later that the behaviour of I(x,o,p) for p just above unity is dominated by a singularity of the real part at x = 1+ and it is therefore hardly necessary to calculate the imaginary part to high accuracy. The restrictions on the techniques for generating the Tn(x) functions are the (x) functions described earlier. Thus Eqn. 5.12 is used same as for the Un when n is greater than x and the recurrence relationship:

(x) + Tn+1(x) = [cos2(n/r/2) - Jn(k)1 ... 5.29 Tn_1(x) xn + Tn

is used otherwise.

Finally in the computation of Eqn. 5.2 and Eqn. 5.5 the functions f(x) and g(x) are required. High accuracy Chebyshev expansions for these functions were developed by the methods given by Clenshaw in R16. The expansion coefficients, shown in F5 and F6, required the development of multiple precision routines for their generation.

The typical behaviour of the coupling integral I(x,o,p) as a function of p is shown in F7 where its discontinuous nature at p = 1 is evident. +B ) and (C +D ) are related to the electric field in Noting that (Amp mp mp mp the aperture and that (A -B ) and (C -D ) are related to the magnetic mp mp mp mp field, both Eqn. 2.23 and Eqn. 3.15 can be written in the general form:

\LI Y(m,p;M,P) Vmp = I MP ... 5.30

p,m

where V is a mode voltage (electric field) and I a mode current mp (magnetic field) with Y(m,p;M,P) related to the coupling integral through Eqn. 2.24 to Eqn. 2.27 or Eqn. 3.16 to Eqn. 3.18. In addition,

-68- these equations show that u is always given a value of the form mir/ka which will be less than unity for propagating modes and greater than unity for evanescent modes. Thus the coupling integral is a component of the 'mutual admittance' function Y(m,p;M,P) and the parameter m is related to the nature of the waveguide mode, the point m=1 being the dividing line between propagating and evanescent behaviour. The physical reason for the form of I(x,o,u) is now evident. In the propagating region below g=1 the real part of the coupling integral generally • predominates, the contribution to the mutual admittance therefore being mostly resistive in nature. Just before the mode in question passes from propagation to evanescence however a large reactive admittance contribution develops. When the mode is evanescent with g above unity the admittance contribution is seen to be generally reactive with a large resistive contribution occurring only when the mode is 'almost propagating'. The derivative function is shown in F8 and is seen to have similar behaviour

6. The Solution of the Matrix Equations Only the TE solution will be discussed since, except for minor details, the TM case is identical. For convenience the basic equations Eqn.2.23 and Eqn. 2.24_ will be requoted:

mp a [111 (m,Mp;AI s s (1440(ka ) [(A, (A +B mn mM pP k a zm P, m

2(ka)2(14411) Amp ... 6.1 and: • IH(m,p;M,P) = [1+(-)"11] ih(xp-xp, mir/ka, Mir/ka)

-(-)m Ih(xp-xp-a, InP/ka, Wka)

-(-)M Ih(xp-xp+a, mv/ka, Wka) ... 6.2 where Ih(x,g,v) can be shown from Eqn. 2.25 to be an even function of x, of g and of v and invariant with respect to interchange of g and u. From Eqn. 6.2 and, the; properties of Ih (x,m,v) the following relationships can be obtained. -69- • •

M COEFFICIENT MATRIX SOLUTION EXCITATION Pit II •••• 0 1 1 1 2 S(0) S(1) S(2) S(3) 3 0 1 1 2 2 5(1) S(0) S(1) S(2) •■•■• 3 ■■■•••■ -- -I -- 0 1 1 o I 1 1 3 g(2) i g(1) S(0) i S(1) 2 I I I 3 - _ _ 0 I I I I 1 i I 1 I i 1 4 ; 5(3) g(2) : §(1) 1 S(0) ■ 1 1 I 3 1 1 1 1 . , 1 1 I _ _ m = 0 1 2 301230 .123012 3 1 1 2 3 4 1 2 3 4 1 p= 1 2 3 4 EXCITED ELEMENT NUMBER

F9. THE FORM OF THE MATRIX EQUATION TH (M, P ;m, P) = IH (m, p ;M, P ) ... 6.3

IH(M, p P) = TH (m, P ;M, p )

Also for the uniformly spaced array the element spacing is given by:

• -xp = (P-p) d ... 6.5 showing that for such arrays the coupling coefficients IH(m,p;M,P) depend upon P-p rather than upon p and P separately. This is very useful as it reduces the N(N-1)/2 +1 independent combinations of p and P on the non uniformly spaced array to only N for the uniformly spaced array (in this argument all elements are assumed identical. The fact that coupling coefficients for negative spacings can be derived from these for the equivalent positive spacings using Eqn. 6.3 has also been used). When these factors are taken into account a matrix equation of the form shown in F9 results. This figure corresponds to a 4 element array with the IF lowest modes on each element taken into account. In theory an infinite number of modes are excited on each element but, except for some rather special cases, a practical solution is only possible when the number of modes taken into account is restricted (typically by ignoring all those above a particular order). The coefficient matrix is partitioned into the submatrices S(0), S(1), S(2) etc; with S(P-p) containing the cross- coupling coefficients between the modes on elements with spacings of (P-p)d. Thus for example S(0) contains the coupling coefficients between modes on the same element and S(1) between modes on adjacent elements with the 'driven' element (P) at a higher x position than the passive element (p). When the driven element is below the passive element giving P-p negative Eqn. 6.3 shows that the mode coupling submatrix is the transpose of that (ie m and M inter- • changed) for the equivalent positive spacing p-P (ie with p and P interchanged). This is illustrated in F9 where a tilde (-) is used to denote matrix transposition. The repetition of the same submatrix along forward diagonals is a direct consequence of the coefficient dependence on P-p as already discussed. This form is known as a block Toeplitz matrix.

Although the analysis is capable of handling multimode'excitation, in considering the waveguide modes incident on the array it will be assumed that the array elements support only one propagating mode. With this restriction the array excitations of most immediate practical interest are those for which this mode

-71- • is excited in each element in turn with zero incident excitation of other elements. This is the excitation shown on the right in F9 where only non zero entries in the matrix of excitation vectors have been shown. 'On solving the matrix equation, the solution vector after some manipulation gives the reflected mode amplitudes for all elements and in particular the reflected amplitudes for the propagating modes. Using the latter the element reflection coefficient and the power coupled to other array elements can be determined and from these the radiation efficiency of the element. Finally, as described earlier, the radiation pattern of the element can be obtained from the solution for the waveguide modes. Thus, with a complete knowledge of array behaviour when each element in turn is excited, we can by using the superposition principle determine its characteristics for any incident distribution.

A number of techniques are available for solving the matrix equation for the array. Gaussian elimination followed by back substitution was considered first and has in fact been used (see R17 for a description of the method). This is generally more efficient than matrix inversion when, as in this case, the number of right hand side vectors is much smaller than matrix order. In addition, noting that S(0) = S(0), F9 shows the coefficient matrix to be symmetric and enables further efficiency improvements for Gaussian elimination. The subroutine used was developed in connection with earlier work on mutual coupling on cylindrical arrays. Only matrix elements on and above the matrix diagonal are required and are packed in a compact form which nearly halves the storage requirement for the coefficient matrix. The cost of inverting a 50 x 50 complex symmetric matrix with this routine is about 50p.

While matrix inversion is generally less efficient than elimination, powerful techniques have recently been developed for the inversion of matrioes with block Toeplitz structure OR 18). With M waveguide modes in each of N elements it should be possible using these techniques to obtain a solution in a time proportional to M3 N2 in comparison with M2 N3 (M/6 + 1) for a solution using symmetric Gaussian elimination. Thus the latter takes N(1/6 + 1/M) times as long as the former, a factor lying somewhere between N/2 and N/3 for typical M values. As in the present work only relatively- small arrays have been studied, the block Toeplitz method has not been implemented but there is little doubt that it will enable the solution developed here to be economically applied to large arrays.

As indicated earlier only a limited number of waveguide modes can be included in the practical solution. In order to determine a 'safe' minimum number of -72- modes, a number of array geometries were solved with varying numbers of modes. The results indicated that the amplitudes of the propagating modes in the solution which are of prime interest are not very sensitive to the number of evanscent modes included and that the use of 4 modes provided a reasonable compromise between accuracy and computing cost.

7. Element Efficient The preceding sections have illustrated the derivaticn of the matrix equations that provide a solution for the parallel plate array. The techniques derived have been implemented in the computer programs in Appendix 4, the remainder of this chapter being devoted to the presentation of results obtained using these programs.

Element efficiency will be defined as the fraction of the power incident on an element that is radiated when all other elements in the array are terminated in matched loads. The efficiency of a single element feeding a parallel plate region with transverse electric excitation is shown in F10 as a function of element width (a/k). As noted earlier the waveguide widths and spacings in all the diagrams that follow are expressed in terms of A, the wavelength applicable to propagation between parallel plates (see the definition of terms in Section 1). As can be seen in F10 the element efficiency increases rapidly with element width, being above 0.95 when the element width is greater than 0.4X. Above a/X equal to one, the third waveguide mode (LSE 21) is able to propagate and carries energy away from the aperture in addition to that reflected in the fundamental mode (LSE 01). This accounts for the sudden drop in efficiency above this point. While the LSE 11 mode can propagate above a/X equal to 0.5, the symmetry of the configuration is such that it is not excited. The single element efficiency for transverse magnetic excitation is shown in F11. In this case the fundamental LSM 10 mode cannot propagate below a/X

equal to 0.5 (note that in longitudinal section nomenclature for modes being • used here LSE 10 and LSM 10 are alternative designations of the same mode as are LSE 01 and LSM 01). Above cut off, element efficiency rises extremely - rapidly, being better than 0.99 above a/k equal to 0.7X. The LSM 20 mode that cuts on at a/X equal to unity is not excited because of symmetry and does not therefore degrade the efficiency.

Placing an element in an array can be expected to degrade the efficiency as some energy will be coupled into the propagating modes in adjacent elements. Presentation of values is difficult as the efficiency will vary with element

-73-

_111111111114190•MMINNI mum EMMEN MENEMEMEffirliEraitiriNIME .-1lowszAwm ME.AMMEEmokis -lithitdid" .t. v-saRL-nommaimistamm 131:911= NS Lam Sp•mr. .113111 M O: lllll amici ==:::: -n...•• an.a si ipso: gmanutmlismmulmnsmunmadnimugopmili gmise :I alai .3 gly" i rrmarri Hp Rip,mmanmummummimmallumnumm0 inq imunmsenuni 3-1"ni •.• 31 • • • u• mmsemulau m.:..mamemm muithig : 1113-1 l "---.1 ourilmrnw k ll NI! 11 r 3 nit": sphi RI nni iiis l.... OP11 111,,! Ham ; " pliarc=ild MI" ILI Hi lib poinidgnig:;rsitilLITEIHELLINNErnsEs:■AMILLEI •• • .

milas--::is. , ..• , ...... ara---3 h kiiiihniFirsins.... ::::...... : miLl--elld crimmiiiiiiiiiiiiiiiiiiEl N. IFILuk' tubs:MEL HEMEr"i Ni: pg—idirnMENINIEral t...... 51 mar-ar ras...... • : ...... - - "bun WHIM 1111119ME minlug: iaur:pmei:r umm:iiimilibidemirmamme klmmientamm-r me nunnunnim Ing lilt emag lishilispimammipmmummantaumummapemn diffikermammummu I 1 miimog iiiiiiimimummumammumumnammmmiiiminuniummammemmo : memninnielliffilirnilliiffiMMUMMENISINIMMIMMIIIIIM I . .--;'UMW IIMM11111:19 riElf•M•M •••■••••••••••••111•1•11•• 1. WII. Mail Likwiiihis Aliminnumbpamunimammin.a mmmiUMPHilit mnumii 1 .1. liSi.—::: s :-- 9..: .1. 11hdamilii mblippiirim NFU -dmemmurmil ..:.=-:: .. • • ... • __.....hilliffillnkillill .. ■••■• 1 . 1 + Now Nimammarenaralr III •14 Ed IM • ...... • ..... Immilm mempiammunliiii •.. n..... e. au. Lifi . nusur ..-any- Fmalimumpqnsammusamiginiijih useRM Limm9L, -11...0aillIMMIII611111111 %IR lir uniummnupas ..... iiihatLage 111111111:113 13IIIIIII: IOW

width, element spacing, the number of elemcns in the array and also with position in the array. For this reason only one fairly representative result will be presented, that chosen being a seven elem,:ni array with clement pitch equal to element width. F12 and F13 show the efficiencies of the centre and edge elements in such an array for TE excitation. Note that in this case energy is wasted in the LSE 11 mode which propagates above a/X equal to • 0.5. The effect is more significant for the centre element as the mode is generated principally in the element or elements immediately adjacent to that being driven as a result of their asymmetric, excitation by the: latter. These results are'markedly different to those of the single element and suggest that mutual coupling effects will be significant for TE excitation. With seven elements the array is large enough'to make its edge and centre element efficiencies representative of those for elements at and removed from the edge of much larger arrays (with the same element width and pitch). In contrast to the TE result, coupling effects for TM excitation prove to be small with array element efficiencies little different to that presented in F11 for a single element. For this reason they will, not be covered further.

Before leaving the subject of element efficiency a few specific comments are worthwhile. Firstly, whilst array elements can be and often are of such a size that they support more than one propagating mode in the vicinity of the aperture, the network feeding them is almost without exception capable of propagating in only a single mode. Thus higher order modes generated at the aperture and returning towards the feed can only propagate for a limited distance before becoming 'cut off'. At or before reaching this point, the energy in the higher order modes must either be simply reflected, transformed into a lower mode returning towards the aperture or transformed into the

S fundamental mode returning towards the feed. Only the latter reduces the efficiency and only in the worst case of all higher mode energy being transformed into the 'reflected' fundamental mode will the degradation be as high as suggested here. To avoid the difficulties of having to account for these higher order propagating mode effects, the model will generally be used with element widths that support only a single propagating mode.

A second point worth emphasising is that a low elemeeit efficiency does not necessarily imply that the efficiency of the array as a whole is poor. This results because, when all elements are excited, it is quite possible for the vector stimulation cf the reflected fields from all excited elements to be small although individually they are not so. Thus, for example, when excited • -76- individually with all other elakents match terminated the elements of a seven, element TE array with a = d = 0.1X have an arrage voltage reflection coefficient of 0.71 but when excited simultaneously and equally in amplitude and phase the average reflection coefficient is 0.20. For this reason element efficiency is not really a very useful concept. Thia subject will be taken up again in the next chapter.

8. Element Patterns Array distributions are normally based on the supposition that all array elements have the same element pattern and it is therefore of interest to discover to what extent this holds in practice. The element patterns presented here are often knawn as active element patterns and are those obtained by feeding the element in question with all other array elements terminated in matched loads. It is these patterns that are important in determining the behaviour of the array when all elements are excited. Element power patterns for TE excitation on 1, 2, 4 and 8 element arrays with element spacings of 0.4, 0.6, 0.8 and 1.0 wavelength are illustrated in F14 to F17. Each figure covers a particular element spacing and consists of four diagrams labelled (a) to (d). In each set (a) presents single and 2 element arrays, (b) the 4 element array and (c) and (d) the 8 element array. The curves are marked with the element number and also the number of array elements in brackets, 3(8) for example being the third element in an eight element array. Only patterns for elements on the lower half of each array are illustrated as elements on opposite sides of, and equidistant from, the array centreline have mirror image patterns. Angles in the diagrams are measured from the array normal with positive angle corresponding to positive x values and the patterns are normalised to unit incident power on the element in question (ie the area under the curve with angle in radians gives the radiated power with unit incident power). The patterns are shown for elements lying on the negative angle half of the array. The element width for all the results shown is 0.4X. •

Looking first at F14 which corresponds to a spacing of 0.4-X, F14 (a) shows immediately that the element patterns of the two element array are quite different to that of the single element. As might be expected, the power density at +90 degrees is lower than at -90 degrees for element one of the two element array as the wave travelling towards the former direction passes over element two which 'extracts' some power. Differences in the element patterns became more marked on progressing to 4 elements in F14 (b) and 8 elements in F14 (c) and (d). Patterns for the 8 element array in particular are exhibiting a more detailed structure and while elements 3, 4, 5 and 6 are

-77- 0

beginning to show same similarities in general shape, the end elements (ie 1 shown and 8) diverge significantly from the rest. Looking now at the 8 element array with a spacing of 0.6X as illustrated in F15 (c) and (d), it is clear that some definite characteristics are emerging. Firstly, all the element patterns are showing some fairly rapid variations in amplitude that are more characteristic of array length than element width. These ripples are clearly the result of • coupling and parasitic excitation of the whole array even though only one element is excited. Secondly, the patterns are tending to 'signore up' with rapid drops in level at t L5 degrees. While the centre 4 elements are quite similar the outer elements are again exhibiting markedly different patterns. Progressing to an element spacing of 0.8X shown in F16 yields still more directive element patterns with a pronounced central peak and a lower skirt region. The central peak must be the result of parasitic array excitation that is constructive in the forward direction, On increasing the spacing to 1.0X shown in F17 the central peak is replaced by a dip. While in this case the parasitically excited elements radiate constructively in their own right they prove to be out of phase with the field of the driven element and as a result a dip is produced in the forward. direction.

It is worth noting that the steep sides of the element patterns in evidence in particular for the eight element array occur at an angle that corresponds to a main beam position when an endfire beam is just forming. Thus with angle /9 measured from array normal and a main beam at A), the first order grating lobes form at angles given by:

sin # = sin. A) t Vd ... 8.1

which gives the main beam position for endfire grating lobes as:

-1 = t sin (1 -X/d) ... 8.2

For array element pitches of 0.6 and 0.8X this yields 00 as 41.8 and 14.5 degrees respectively which correspond reasonably well with the steep sides of the element patterns in F15(d) and. F16 (d) respectively.

As noted earlier mutual coupling effects are smaller for TM excitation. This is again demonstrated by F18 which shows element patterns on 1,2,4 and 8 element arrays with TM excitation and element widths and spacings of 0.6 and 0.7X respectively. Although pattern differences are less pronounced there is evidence of some 'squaring up' and skirt formation in F18 (d) in particular. 0 -88- Summarising then, the results presented here suggest that for the TE excited array at least there is little justification for assuming that all elements have identical element patterns. As would be expected the two end elements show the most deviation from a common pattern with the differences being less pronounced but still evident for elements one removed from the array edges. However the results do suggest that elements two or more removed from the array edges (ie Numbers 3 or more and N-2 or less) have patterns that are certainly becoming both fairly symmetric and similar in their major characteristics at least. Thus, if element pattern variations prove to be troublesome, • it shoult be possible by adding two match terminated parasitic elements at each end of the array, to produce an array environment in which element pattern variations are much reduced. It seems unlikely that any such devices will be needed for the TM array.

9. Array Patterns The determination of element patterns in the array environment makes it possible to compute array patterns with the effects of mutual coupling taken into account. Most results presented here will be for eight element arrays with Chebyshev distributions. The latter have been used because their equal side lobe behaviour in the ideal case makes it easy to assess the extent to which the pattern has been modified by mutual coupling effects. Figure F19 shows the results of using a -40 decibel Chebyshev distribution on arrays with eleMent widths of 0.4.k and spacings of 0.6, 0.7 and 0.8X. The electric field vector is parallel to the array axis (TE or horizontally polarised excitation). In view of the element pattern variations presented in the previous section these results are remarkably close to the ideal result, the side lobe degradations for 0.6, 0.7 and 0.8X being 1.0, 1.7 and 3.0 decibels respectively at worst. For the largest spacing the side lobe structure associated with the first grating lobe is in evidence at wide angles. The results for a -50 decibel Chebyshev distribution shown in F20 are equally encouraging (and surprising) with side • lobe level degradations of 2.5, 2.0 and 7.5 decibels for the three spacings. For 0.8X spacing the first grating lobe is beginning to show at endfire and the -50 dB side lobes have been considerably degraded, the two effects making if a fairly poor result. However for spacings of 0.6 and 0.7X the resultant patterns are close to the ideal result.

When the electric field vector is perpendicular to the array axis (TM or vertically polarised excitation) there is only the slightest degradation of worst side lobe level as is shown in F21 for a -50 dB distribution. In fact the directive nature of the element pattern for TM excitation (see F18) results -89-

•

in considerable reduction of the outer side lobes and also provides a significant suppression of the grating lobe at endfire for the 0.8X spacing.

In the previous section it was suggested that the results for TE arrays might well be improved by having extra passive elements ateach end of the array. While the results of this section indicate that there is little need for this device it is interesting to determine if it achieves any improvement in performance. The upper diagram in F22 shows the results of applying an 8 element Chebyshev distribution to a 12 element array with 2 passive elements at each end. The element width and spacing in use are 0.4 and o.ax respectively making this result directly comparable with that presented in the lower diagram in F20. Discounting the wide angle grating lobe structure which we could not expect to remove, it is seen that there is a definite improvement in the near in side lobes of between 4. and 10 decibels. However the lower diagrams in F22 illustrate the excellent results that can be achieved by actively feeding 10 and all 12 elements of the same array. The last result is clearly the best of the trio in terms of both overall side lobe level and beam width and tends to suggest that the use of passive elements to improve side lobe level, as suggested, while feasible, does not make the most effective use of the extra elements. Nevertheless it is feed networks that are expensive rather than elements as such and the device may therefore find some application.

Before concluding this section it is worth considering how arrays that exhibit wide variations in active element patterns from element to element can provide array patterns so close to the ideal result based on an array factor multiplied by a constant element pattern. Taking the TE case as an example, the reason for this is most easily understood if the angular component of the array far field pattern as given by Eqn. 4.11 and Eqn. 2.18 is written in the form: •

G(0) = ksin0 H (kcos0)

cosh) ... 9.1 mP) P m(0) exp (ik x

111,13

where:

p (A) = Olkmala [sin J(kacosc6-traffi/21 (-)nisinr(kacos1-thir)/21j m.`" [(kacosOimg/2 [ (Icacos0-00/2] ... 9.2 -94-

Remembering that the Amp only exist for m e o, Eqn 9.1 gives:

G(0) = P0(95) (A.op+Bop) exp (Ikxpcos0)

9pexp (ikx cosh)

0 P2(0) B2p exp (ikx cos0) + ... 9.3 P

where the first summation represents the contribution to the array pattern from the incident and reflected fields for the lowest (propagating) mode and the other terms result from the excitation of higher order evanescent waveguide modes. It is evident from Eqn. 9.3 that the pattern contribution from each mode is in the classical form of an element pattern %(95) mu]tipled by an array factor in which the element amplitudes are the excitations at each element of the mode in question. For most array excitations it is generally reasonable to assume that the higher order mode excitation is small compared to that for the propagating mode and in this case the 'zero mode' term gives a good approximation to the array pattern:

G(0) - Po (t)(Lop+Bop) exp (Ikxpeos0) ... 9.4

G(0) - P0(0) exp (ikxpcos0) / p

+ p0(0) Bo, exp (Ikxpcos0) ... 9.5

The first term in Ecin 9.5 is the desired array pattern, the second term being the pattern deviation that results from the 'reflected' fields for the lowest mode. It is now clear from Eqn. 9.4 that any array for which the excitation of higher order modes can safely be neglected will exhibit a performance characterised by an array factor and a common element pattern. The reflected at an element is the result of reflection at the element and coupling field Bop from other array elements and can be envisaged as a sort of weighted average of the element excitation and that of its neighbours. Thus we can expect the across the array to follow the variation of the reflected amplitudes Bop

• -95- form of the incident distribution A with some smoothing off t variations in op amplitude and phase from element to element. Thus when the incident distribution only varies slowly across the array aperture, we can expect the 'reflected' ) to be very similar in form with perhaps a slightly distribution (ie the Bop reduced taper. As a result, the distribution A op +Bop upon which the pattern of the array depends will be very nearly proportional to the incident distribution Aop and will provide a similar pattern function.

That this behaviour is consistent with wide variations in active element patterns on the array is easily demonstrated by using Eqn. 9.4 to approximate the q'th element pattern as:

• Igg(0)1 = 1P0(0)1 exp [ik (xp-xq) cos0] I... 9.6

which is obtained by setting all incident amplitudes to zero except oqA which is set to unity. The summation in this expression represents the contribution to the active element pattern that is caused by parasitic excitation of other array elements. For elements removed from the array edges this parasitic S excitation will occur on both sides of the driven element and as a result the element patterns will be fairly symmetric and similar. In contrast, since edge elements only have parasitically excited elements on one side, we would expect their active patterns to be distinctly different from central elements and also to exhibit a marked asymmetry. Thus active element patterns vary because of the varying parasitic excitation of other elements and not because of any intrinsic change in the radiation pattern of the elements themselves with position. In effect we can take account of this parasitic excitation in either of two equivalent ways. The first and most used method is to account for it in the element pattern of the driven element leading to the concept of the element pattern in the array environment as described above. A second and equally S valid approach when higher mode radiation can be neglected is to attribute an identical 'passive' element pattern Po(0) to all elements and to calculate the excitation- of each element as the sum of the self induced and all parasitically induced fields. In simple terms it can be accounted for either by modifying the element patterns or the array excitation function. Finally it is perhaps worth noting that the analysis of this chapter can easily be used to derive an incident distribution A that will yield a given 'incident plus reflected' QP distribution A +B . Thus the A could be chosen such that the distribution op op op A +B gives low side lobes rather than the distribution A alone. Since op op op Eqn. 9.4 suggests that in the absence of higher mode effects it is the former that determines the array pattern, this could provide a technique for adjusting the incident distribution to account for the effects of coupling. • -96- 10. Conclusions The results presented in this chapter suggest the following conclusions:

a. If the correct incident distribution is achieved and the number of array elemenLs and their spacing are chosen to avoid end.fire grating lobe effects, then mutual coupling is likely to be of lesser importance than errors in degrading design side lobe levels.

b. Large variations of active element patterns in an array are not necessarily an impediment to low side lobe design.

c. Patterns predicted on the basis of an array factor and a common element pattern are generally close to the result obtained using the more exact analysis presented here.

d. When higher order mode effects are small it should be possible to modify the incident array distribution to account for the effects of mutual coupling.

It should be noted that (a) assumes that the specified distribution has been achieved. The difficulty of designing feed networks to do this will depend upon the magnitude of effects that result from aperture reflected (and coupled) fields returning into, and being redistributed by, the feed network. It may well be that these secondary effects of mutual coupling are more troublesomethau the 'forward' coupling effects covered in this chapter (the term 'forward' is used to distinguish coupling and re-radiation occurring on the aperture itself as distinct from any effects resulting from the interaction of reflected fields and the feed network).

-97- • CHAPTER THREE - SYNTHESIS WITH IVIVAL COUPLING

• -98-

1. Feed Network Considerations Matrices and in particular the scattering matrix representation of network behaviour will be used extensively in this chapter and it is assumed that the reader is familiar with these methods. Column vectors will be denoted by underlined lower case letters and matrices by capitals. The conjugate, the transpose and the conjugate transpose of a matrix or vector will be denoted by the superscripts *, t and + respectively. Thus for an n+1 port network the relationship between the amplitudes ao, al ... an of waves entering ports zero to n and the amplitudes bo, b1 bn of waves leaving:

bo = so 0 ao + so 1 al + • • • . + son an

bi = sio a0 + si 1 a1 + • • sin an

b o + s al + + snn a n = no a ni n

is expressed in matrix form as:

b = S a 1.2

In array feed networks designed to produce a single beam from n elements it is often convenient to make a distinction between the input/output port of the network and those remaining ports connected to the array elements. Labelling the input/output port zero and the array ports 1 to n, the required distinction can be made by partitioning the scattering matrix thus:

[ bo [ soo i Eol I. ... 1.3 c, bi S b 0 I 011 [ a

where al = [a1, a2 an] and bi = b2 bn]

(to save space column vectors will be written in row form and enclosed in square brackets. Row vectors will be enclosed in round brackets). All networks considered in this chapter will be reciprocal and will therefore satisfy the relationship:

St -99- ... 1.4 gli r VV VVVVVVV

FEED NETWORK

col i ba

Fl. A NETWORK FEEDING AN ARRAY ▪•

They will also be passive requiring that the total power input is greater than or equal to the power output. Thus:

a a b+ b = a+ S+ S a

it a+ (I - S+ S) a o, a arbitrary 1.6

Equality with a particular a means that the network is lossless for that particular input distribution, while equality for arbitrary a giving:

S+ S = I

means that the network is lossless for any input distribution. As is usual it is assumed that the wave amplitudes at any port are so normalised that their modulus squared gives the power entering (ats)or leaving (hts) the network at the port. S

Consider a reciprocal network feeding an array as shown in F1. With suitable partitioning as above its scattering matrix can be written.

[ to tt ao [21 ... 1.8 t 1 T al - -

Now it is normal to design such networks on the basis of matched array ports (ie al = o) and to ensure that under these conditions the input port (zero) is matched and all input power appears at the array ports. • Setting al = o in Eqn. 1.8 gives:

bo = to ao ; b I = t ao ••. 1.9

and the power constraints therefore require that:

to = o and t+ t = 1 1.10

Thus the normally designed feed network has a scattering matrix of the form:

-101-

bo tt with t+ t = 1 ... 1.11 bi t T al [ao

where the network has been designed to produce a transmit distribution t at the array ports without loss.

• Before considering what happens when the array ports are not matched it is useful to consider the behaviour of the network under 'receive' conditions with the array ports fed and the input port matched (ie ao = o). If under such conditions the array ports are fed with the distribution t* then Eqn. 1.11 shows that t bo = t t* = (t+ t)* = 1 1.12

bi = T t* ... 1.13

Now the input power is:

P.in tt t* = (t+ I)* = ... 1.14.

and the output power is:

P = 1b0 1 2 + bi bi ... 1.15 out 1 + bt bi ... 1.16 and since the former must be greater than or equal to the latter it is clear that vector bi must be zero which gives:

Tt* = o 000 1017 •

It is clear that for the receive distribution t* all the power incident on the array ports appears at the feed port. Thus a network designed to losslessly transmit a distribution t will losslessly receive the distribution t*. This can be viewed as the network analoLue of the conjugate match theorem for maximum power transfer.

Summarising, the scattering matrix of the normally designed feed network is:

-102-

... 1.18 - t 1 T b'J [ aa:

where:

• t+ t = 1 and T t* = o ... 1.19

t and T = T as a result of reciprocity. As indicated earlier, under matched conditions, it provides an array distribution given by the vector t

Considering now the practical situation when the array ports are not matched, the array distribution with unit power at the input port is obtained from Eqn. 1.18 as:

1 = t + T al ... 1.20

and clearly depends on the reflected and cross-coupled fields al returning from the array aperture. Now, the work of chapter two allows us to compute an 'array scattering matrix' S relating the fields al returning from the array elements to the incident fields bi:

al = S bi ... 1.21

Eliminating al from Eqn. 1.20 and Eqn 1.21 gives:

t = (I - T S) bi ... 1.22

and:

3. = (I T S)-1 t ...1.23

which relate the 'matched' and'unmatched' distributions t and bl (I is the identity matrix). Knowing S and T, a matched excitation can be found that yields a given distribution bi under unmatched conditions. However the matrix T depends upon the construction of the feed network which is dictated at least partially by the distribution t which it is designed to produce. The complex nature of the dependence of T upon t makes this approach extremely difficult if not impossible.

-103- It is natural to ask if there are any situations for which the Matched and unmatched excitations are identical. From Eqn. 1.22 bi and t are equal only if:

TSt = o ... 1.24

The solutions t = o and St = o are trivial (from Eqn 1.21 St is the reflection from the array aperture, St = o corresponding to the matched condition) leaving the possibilities T = o or T (St) = o with both T and St non zero. Feed networks with T = o are realisable but containlossy elements and are as a result more difficult and expensive than lossless networks. They are covered in the next chapter. Remembering that Tt* = o it is clear that T (St) will be zero for excitations t that satisfy the relationship:

St = p t* ... 1.25 where p is an arbitrary scalar quantity. Networks designed to provide excitations that satisfy Eqn 1.25 will provide the same distributions when connected to the array as they do when their array ports are matched. This behaviour is a result of the fact that a network designed to losslessly 'transmit' a distribution t will losslessly 'receive' the distribution t*. Thus, if the aperture cross couplings are such that the 'reflected' fields are proportional to t* when the incident fields are t, then all reflected energy will return to the input port of the feed network and cannot therefore change the incident distribution in any way. It is clear from Eqn 1.25 that if such distributions exist they are functions of array scattering matrix S and therefore upon the array configuration and cannot be chosen at will.

2. The Aperture Modes While Eqn. 1.25 is tantalisingly close to the classical eigenvalue equation there is at first sight no obvious way of solving it. Note however that the phase of either p or t is arbitrary for if pt and ti are a solution of Eqn. 1.25 then:

et a) = '(ple-2-14() ‘(Lle(a)* ... 2.1

-27.a ta which shows that p. e and t4 e are also solutions where a is arbitrary. We can therefore specify that p is real without loss of generality. Letting S = U + EV and t = x + and expanding Eqn 1.25 gives:

-104- 0 Ux Viz = px

Vx + Uy = -Az 6.0 2.2

with p real. Eqn. 2.2 is equivalent to:

U -v x x = p ... 2.3 -V i -U

which is a conventional eigenvalue equation of order 2N (N is the number of elements in the array and also therefore the order of the matrix S). While Eqn. 2.3 has 2N eigenvalues and vectors, simple manipulation shows that if p and [x y] is a solution then -p and [y. -x] is also a solution. From the point of view of Eqn. 1.25 these two solutions are equivalent as one can be derived from the other by a choice of the arbitrary phase a noted above. In fact Eqn. 1.25 can be reduced to a conventional eigenvalue equation in another way. Multiplying Eqn. 1.25 by S+, the conjugate transpose of S, gives:

eSt =p S+ t*

= p(St t)*

= p (S t)*

= p (P 2)* 6400 2.4

t since S = S Thus:

es t = Ipl't ... 2.5

an eigenvalue equation of order N. The matrix S+Sis Hermitian (ie A = A+) and, as a result, exactly N distinct orthogonal eigenvectors are guaranteed. Eqn. 2.5 has been used for solution rather than Eqn. 2.3 as it is of order N rather than 2N and is therefore almost certain to provide a more efficient solution in spite of the need for complex arithmetic. The calculation of the eigenvalues and vectors of Hermitian matrices is fairly well documented (R17 and 19) and will not be covered here. The program developed uses Jacobi's method and is given in Appendix 5. S -105-

For an N element array the solutions of Eqn. 2.5 are the excitations for which a 'matched' feed design can be pursued with a guarantee that the distribution it provides will not be altered by the reflected and coupled fields returning from the aperture. The form of these distributions and their radiation patterns are determined entirely by the array element reflection and cross- coupling coefficients which specify the array scattering matrix S and unless at we can construct an array for which the latter have been prespecified it is clear that the above distributions cannot be chosen at will. As any such prespecification is not possible at present, we can only proceed as far as the analysis and study of the above distributions for array geometries of interest, in the hope that some may be found with the required low sidelobe properties. To illustrate the form of these modes some theoretical results for the geometry analysed in the last chapter will now be presented. Modal radiation patterns rather than the modes themselves will be given as these are of more immediate interest and significance.

Figures F2, F3 and F4 illustrate modal radiation patterns for an E plane parallel plate array of eight elements (A is element width, D element spacing) at three different spacings. The modes fall into two sets, those which have patterns that are even (full line) about the array normal (sum modes) and those which are odd (difference modes). The latter are of lesser interest in the work that follows and are only shown on P2 and F5. The figures given with each pattern are corresponding p values. The fields breflected from the array are S t when the incident fields are t and the reflected power is given by:

b+ b = S+ S t

4 = I p 1 2 t+ t ... 2.6

using Eqn. 2.5. As t+ t is the incident power, 1p12 is the power reflection coefficient of the array for the corresponding mode, p being the voltage reflection coefficient. Figures F2, F3 and F4 show that the aperture mode patterns take the form of a series of overlaping 'multiple beams' with their main beams interlaced. The element spacing of 0.4X in F2 corresponds to walls of zero thickness between elements. For such an array the walls have little effect on a uniform, in phase, distribution, the configuration being nearly equivalent to a much larger single waveguide element for such an excitation. Thus we would expect the axial beam produced by this distribution to have a low reflection coefficient as is illustrated in F2. • -111- While this beam is not exactly uniform, the distribution varies only slowly from one element to the next and the same considerations apply. Beam reflection coefficients increase with 'main beam' distance from the array normal and the highest angle beam has a very high value as its phasing corresponds to beam formation in imaginary space. For F3 and F4 with element spacings of 0.6 and 0.8X all beams are formed in real space and the beam reflection coefficients do not reach such high values. • Results for the 8 element E plane parallel plate array are shown in F5 and F6. Figure F5 corresponds to an array with thin walls and shows the 4 even modes in detail and the main beams of the 4 odd modes. For this array the beam reflection coefficients decrease with main beam distance from the array normal. This is to be expected as the waveguide modes in this plane can be viewed as two plane waves reflected backwards and forwards between the waveguide walls. When the phasing is such that these waves from adjacent elements match up, the composite wave is radiated with little reflection. -1 / The angle of these waves from the normal is sin (X/2a), 56.44 degrees when a = 0.6X, which is seen to be close to the position of the beam with the smallest reflection coefficient. Increasing the spacing to 0.8X as shown in F6 produces more directive patterns as would be expected and also destroys the monotonically decreasing nature of the modal reflection coefficient p as a function of main beam position.

For the results shown in F2 and F6 it is seen that all the aperture modes except one have two 'main beams' of equal amplitude and either equal or opposite phase placed symmetrically with respect to the array normal (the mode patterns are either even or odd, only half of each pattern being shown in the diagrams). Thus for low side lobe design only the broadside beam is of interest and over many samples its side lobe level has been seen to vary from -8 to -27 decibels. It is clear from these results that this mode does not generally have a pattern with low side lobe levels.

While the aperture modes do not provide a solution of the low side lobe feed problem which led to their investigation, it turns out that they have a number of interesting and important properties that can be exploited in their own right. In particular they provide a technique of pattern synthesis which takes account of the effects of mutual coupling. This appears to be the first such method reported for aperture type radiations (Cheng in a December 1971 paper, Reference 20, states that he is not aware of any work in this area). This work is covered iv the remainder of this chapter. -112- • •

3. Aperture Mode Properties For convenience Eqn.-1.25 and Eqn. 2.5 are repeated. here:,

t = p t* 3.1

s+ s t = Ip12 t ... 3.2 • where p is the voltage reflection coefficient of the aperture mode. Now Eqn. 3.2 only gives realisable values of p if the eigenvalues of the matrix S are real and positive. That this is true for all eigenvalues can be shownasfollows..Supposethatet ande.are eigenvalues of S+ S with tt and t. g as their respective eigenvectors. Thus:

S+ s t. =e. t. ... 3.3 4.1 ""I,

and:

S+ S t. = e. t. 3.4. -g ...

Now multiplying Eqn. 3.3 by 1* gives:

tt s+ S t. e. t+ t. ... 3.5 -g c -/

and multiplying tz by the conjugate transpose of Eqn 3.4 gives:

t.+ S+ S t. = e.* t. t. • 3.6 -g -/

From Eqn 3.5 and Eqn. 3.6 we see that: • (6. - 6.) t = o ... 3.7 t -0 -t

Setting 1 and j equal in this equation shows that

6. -^ 6. 3.8

since -0' t. is non zero for any non trivial eigenvector. Thus the eigenvalues of S+ S must be real. Equation 3.7 also shows that if two eigenvalues are distinct (ie e. c.) then:

t. t. = o ... 3.9 -.3 -6 -113-

That is, their eigenvectors are orthogonal. It can be shown that even when two eigenvalues are not distinct, their eigenvectors can be chosen in such a way as to satisfy Eqn 3.9. In what follows the eigenvectors will be assumed to satisfy

t = S. ti t j ... 3.10

which is identical to Eqn 3.9 with the additional requirement that individual • eigenvectors be normalised to unity ti(S.the Kronecker delta function, is unity if i equals j and zero otherwise). Multiplying Eqn. 3.3 by ti gives:

e . = tt S+ S t. (St.)+ (St.) 3.11

giving e i as positive or zero since any vector multiplied by its conjugate transpose is greater than or equal to zero. Thus all the eigenvalues of SI" S are positive real or zero. Also from the constraint for passive networks, Eqn. 1.6, we have

ti (1 - s+ s) t i > o . 3.12

giving: e. < 1 ... 3.13

Thus all of the N eigenvalues of the matrix SI- S satisfy:

o S e. < 3.14-

and therefore give realisable values of I P I 2 in Eqn 3.2. Thus Eqn. 3.1 has exactly N solutions:

= p. t. ; 1, 2 ... N ... 3.15 S t.t t

with Ip7 1 < 1 and:

tt t = S.. —.3 —4 to 3.16

The eigenvectors form a complete set and an arbitrary excitation t can therefore be expanded in the form:

-114- •

.. 3.17

multiplying by t+ and using Eqn 3.16 shows immediately that the a i are given by:

a. = tt t ... 3.18 t —t

Now let the eigenvector excitation ti on the array produce a far field pattern Et (e ) and let the arbitrary excitation t represented by Eqn. 3.17 have a pattern BM. Now the total incident power is given by:

19* t+ t . a3 t . a. t. —3 t —t 1=1

a. a. t t. 1, 0 —0 —I, i=1 j=1 N N a. 6 . t a. t j ... 3.19 =1 j=1

Now the reflected vector b is given by:

= S t N 3.20 a.t p.t t. i=1

using Eqn 3.15 and Eqn 3.17, and the total reflected power is: N N b+ b = a p t* [ fX p. t*. i i -...i t t j=1 1=1

* a. a. ... 3.21 Locop. p . 6%.to i =1 i=1

as before. Now the radiated field E(6) is given by:

-115-

E (e ) = )) a.L E.t ( ... 3.22 1=1 and the total radiated power is:

(e) * (e) ae rad f E E

ot,ta4:j. (a) t 3 (0) de ... 3.23 t=1 j=1

Now, as there are no losses in the system, the power radiated must be the difference between the incident and reflected power. Thus from Eqn. 3.19 and Eqn. 3.21 N N ---) P - a. a* (1 - pi ) S. rad - )1 )1 t t tj 3.24 1=1 j=1

CoMbining Eqn. 3.23 and Eqn. 3.24 gives: N N ) * / a.a [kl - p. p*.) S. - E.(6) E* (6) de] . ... 3.25 t t j Li t J =1 j=

The integrations of the patterns in the above are performed over all space. Now t, and therefore the coefficients a.are completely arbitrary and t ' Eqn. 3.25 therefore shows that: E. (e) (e) ae (1 - pt citi ... 3.26 f This states that the patterns of the eigenvectors are orthogonal: I Et(6) (e) de = 0 3 i / ... 3,27 f 1E, (0)12 de = 1 - Ip1 1 2 ;

Thus the fraction of the incident power radiated for the ith beam is (1 - Ipt 1 2), giving each such beam a unique radiation efficiency. The 'radiation efficiency' of an arbitrary excitation given by Eqn. 3.17 is easily obtained from Eqn. 3.19 and Eqp. 3.24. as: -116- • 214014 4UOMOTa SA 24UOTOW900 UOT4OOTOE epow eaulaody

TT MINE1111111111 ■1 1 1 [$ : 111 I I I III Pr" I LTIN . I IIMMUMNIIIMMIll igall:E.;„7-71111"";" ''''' Bilimmirdiffluignum " MIEW

• N

) (1 - 1/J1 1 2)1,2.12 77 t-1 ... 3.28

which is clearly a weighted sum of the beam efficiences. It is easy to show that this radiation efficiency must lic between the minimum and maximum efficiency attained by any of the aperture mode patterns as given by (1- IPtI2). From these results it is clear that the concept of aperture mode efficiency is more useful than that of element efficiency developed in chapter two.

The aperture mode reflection coefficients (Ipi l) for an eight element E plane

parallel plate array with elements of width 0.4X. are shown as a function of element spacing in F7. The results show that this configuration gives all mode radiation efficiencies better than 0.75 for spacings between 0.55 and 0.8X, the range of most practical importance.

4. Pattern Synthesis From Eqn. 3.17 and Eqn. 3.22 the inpident array excitation:

... 4.1 t = y a/ t. I=1

gives a radiation pattern:

E(e) = Xa. E.t (e) ... 4.2

• with notation as in the previous section. The problem posed here is to determine the values of the coefficients a. in such a way that E(6) approximates a desired pattern P(e). The function:

IP(e) - E(8) I2 ae ... /4.3 I will be taken as a measure of the error in such an approximation and the at will be chosen if possible in such a way as to minimise the value of Q. Using Eqn. 4.2, Eqn. 4.3 can be written:

-118-

•

N Q = f [ PO) - Zat Ei (e).} i=1 N [ P*(e)- a.t E*. (e)i ae ... 4.4 t.1

* • Now consider the function F(z,z ) of the complex variable z = x iy. Clearly:

8F _ aF aF ax - az az* r

aF .t aF . aF 8a t ... 4.6

if the variables z and z* are treated as independent. Setting these derivatives to zero to determine the extreme values of F gives:

aF az - az* = 0 ... 4.7

and, if F is real, either of these is sufficient as one is simply the conjugate of the other. Applying this to Q in Eqn. 4.4 and differentiating with respect to a* to determine a minimum of the function Q gives:

r[P(e) - az E. (0] El: (0) de = o ... 4.8

1=1

Using Eqn 3.26, this yields:

1 ai = 1 ... 4.9 _ 1,02i,j, IP(0) E*. (6) de

giving the a values for a 'least squares' approximation to the pattern P(e). Having obtained the a values, the distribution to obtain this approximate pattern is given by Eqn.4.1. Clearly the process takes account of mutual coupling through the use of the aperture modes which are computed from the array scattering matrix.

Figures F8 and F9 illustrate the use of this synthesis technique for the eight element E plane parallel plate array analysed in chapter two. The 'model' patterns P(e) for F8 and F9 are -40 and -50 decibel Chebyshev patterns respectively, each figure giving results for three spacings (the 0 -119-

element width is 0.4X). These two figures can be compared directly with figures F19 and F20 of chapter two which show the results obtained using the unmodified Chebyshev distributions on arrays identical to those for F8 and F9 respectively. A comparison of the -40 decibel results in F19 (Ch.2) and F8 shows the side lobe level of the synthesised pattern to be between 1.5 and 3.0 dB lower than that of the unmodified Chebyshev distribution. In the presence of side lobe jamming this would provide a range improvement of between 20 and 40% and is therefore quite significant. Comparison of the -50 dB • patterns of F20 (Ch.2) and. F9 shows that in this case the synthesised pattern for a spacing of 0.6X shows a 2.0 dB improvement in side lobe level while that for 0'7X is little different in level although its general behaviour is certainly better. The synthesised pattern for a spacing of 0.8X would probably be judged inferior to its unmodified counterpart as it exhibits main beam distortion but it is clear that neither pattern is practical as a result of endfire effects.

The following tables show the unmodified and synthesised distributions for the illustrated patterns. The distributions are symmetric and only excitations for one half of the array are shown.. Figures in the tables are amplitudes with phases in brackets. TABLE 3.1 40 DECIBEL DESIGN, 8 ELEMENT ARRAY

ELEMENT MODEL 0.6 ?. SYN 0.7 X SYN 0.8 X SYN

EDGE 1 0.078 (0.0) 0.079 (-1.5) 0-078 (-3.1 0.071 2 0.222 (0.0) 0.218 (-0.3) 0-221 (-1.0 0.207 ( -3.3 3 0.403 (0.0) 0.405 (-0.9 0.401 ( -0.1 0.399 ( -0.7 CENTRE 4 0.531 (0.0) 0.531 ( 0.0) 0.533 ( 0-0 0.541 ( 0.0

TABLE 3.2 50 DECIBEL DESIGN, 8 ELEMENT .tFRAY •

ELEMENT MODEL 0.6 X SYN 0.7 X SYN 0.8 X SYN

EDGE 1 0.052 (0.0) 0.051 (-2.6) 0.050 (-5.8) 0.046 (-27.8) 2 0.193 (0.0) 0.190 (-0.7) 0.191 (-1.7) 0.162 ( -3.5) 3 0.396 (0.1 0.396 (-0.3) 0.395 (-01 0.383 ( -0.2) CENTRE 4 0.551 (0.0 0.552 ( 0.0)0. 0.552 ( c.0 0.570 ( 0.0)

These tables show that optimisation only slightly changes the distribution at element spacings of 0.6 and 0.7X but has more effect at a spacing of

-122- •

Results for a 12 element E plane parallel plate array are shown in F10 and F11. The former is directly comparable with the lower pattern in F22 (Ch. 2) which shows the model 50 dB pattern. Synthesis raises the first side lobe by 1- 5 dB but lowers all others by 1- 5 to 2- 0 -aB. It, would probably be judged superior irl performance. F11 shows norme.l and synthesised patterns with a 60 dB Chebyshev model and in this case the synthetic result is clearly superior.

While the synthesis technique developed here has only been demonstrated on low side lobe patterns it is evident that it is perfectly general and can be applied to obtain approximations to any desired pattern. It is also clear that it can be applied to any array for which the array scattering matrix and the spatial element patterns (amplitude and phase) are known.

5. Network Synthesis A study of the paper by Seymour Stein on cross coupling in multiple beam antennae (R21) demonstrates many similarities between his work and that presented here. In particular he describes a set of 'canonical,waves' and a network which, if realisable, is oapable of exciting them in an uncoupled manner. These waves prove to be identical to the aperture modes derived here and the network, with suitable excitations at its uncoup~ed ports, could therefore generate a 'least squares' approximation to any given pattern. The question of network realisation which was left unanswered by Stein is the subject of this and the following sections of this chapter. While it is not directly relevant to low side lobe antenna design, it is of general interest and is a logioal extension of the work presented here on pattern synthesis.

Consider a network that connects M input ports to N output ports. SuCh a net work, if it is lossless, reciprocal and has matched input ports when its out put ports are matched, (end vice versa) is sui table for providin~ M radiated 'multiple beams' from an N element arraJT. With incident and reflected fields at the input ports represented by !!. and !? and those at the output ports by ~' and ~/, the match conditions speoified above require a network soattering matrix of the form:

••• 5.1 [-=-]b l = [--~~--~:--]T : a [-=-]a l - I N -

-125-

where 0 and 0 N are null matrices of order M and N respectively. As the network is lossless Eqn. 1.7 must apply giving:

T T+ =IN ; T+ T IM ... 5.2

where and. IN are unit matrices of order M and N respectively. In comyonent form the diagonal elements of these two equations are: M . . t*. = 1 N ... 5.3 z ttj tj I = 1, j=1 N

t.. = 1 j = 1, M 10** to • ... 5.4 E=1

Now, from Eqn. 5.3: N = N ... 5.5 i=1 j=1

and from Eqn. 5.4:

ttjI 2 = M ... 5.6

As these summations are identical it is clear that the equations of Eqn. 5.2 can only be simultaneously true if M is equal to N. Thus a network that meets the above requirements must have the same number of input and output ports. These networks will be called 'all-pass networks' and have scattering matrices of the form: •

-a- ... 5.7

T 0 a'

with

T+ T 5.8

-126- T is known as the 'transfer matrix' of the network, its columns being the distributions obtained at the output ports when the input ports are excited individually. Connecting such a network to an N element array provides N multiple beams from the array. The fact that T is unitary (ie T+ T = I) With columns that are orthogonal shows that a lossless network call only produce beams with orthogonal distributions. This has been known for some time, there being an extensive literature on the subject of multiple beam arrays. Using Eqn. 5.7 it is easy to show that the connection of an all- pass network to an array with an aperture scattering matrix S provides an N port multiple beam antenna with an input port scattering matrix S' given by:

t S' = T S T ... 5.9

The columns of S' give the fields returning from the network when the input ports (and therefore the multiple beams) are excited individually.

In section 3 it was shown that the N aperture modes satisfy:

Pt it = 1,.2 ... N ... 5.10 with: t+ t. = S. ... 5. j —t tj 11

Constructing a matrix T:

T 1 $ t $ S. $ t ) ... 5.12 = Ct —2 3 ' N whose columns are the aperture mode distributions, Eqn. 5.11 is equivalent to:

T+ T = I ... 5.13

This is identical to Eqn. 5.8 and shows that the aperture 'mode' patterns can be generated using a lossless all-pass network. Equation 5.10 can be written

ST = T* D ... 5.14

t where D is a diagonal matrix of the pt values. Pre-multiplying by T and using Eqn. 5.13 shows that: -127-

t D = T S T ... 5.15 which, being in the form of Eon. 5.9, gives the scattering matrix at the input ports of the multiple beam array. As D is a diagonal matrix, Eqn. 5.15 shows that the excitation of any single beam port gives a reflection at the excited port only with no fields coupled to the other input ports of the multiple beam system. While there are infinitely many orthogonal beam combinations, the N aperture mode distributions form the only set that are both orthogonal and decoupled.

From the work of the previous section it can be seen that an array together with a multiple beam network generating the aperture mode excitations can provide a 'least scuares' approximation to an arbitrary pattern. The following sections present techniques for the synthesis of networks with specified transfer matrices.

6. Network Synthesis Using Directional Couplers Synthesis is based upon the cascade connection of simple all-pass networks. It is straightforward to show that two all pass networks with transfer ma'rices

T1 and T2 placed in series produce an all-pass network with a transfer matrix

T1 T2 (with the network T1 nearer the output ports of the network). Thus synthesis is achieved by factorising the overall transfer matrix T in the form:

T = Ti T2 TM-1 TM ... 6.1

where T1, T2 etc. are the transfer matrices of simple all pass networks.

Perhaps the phase shift network illustrated in F12 is the most elementary N'th

order all-pass network. (ie N input and N output ports). By inspection its • transfer matrix is:

I exP (i01) o aI 1 1 = o exP(.102) O a2 ' 1

exp(toN) aN I

... 6.2 -128- A 02

F12. PHASE SHIFTING ALL-PASS NETWORK

1 L-1 L L+1 N-1 N

F13. A SIMPLE COUPLER ALL-PASS NETWORK

-129-

b' ... 6.3

a diagonal matrix containing N complex phase shift terms. Another fairly simple N'th order all-pass network is shown in F13. Arms 2 to / - 1 and / + 1 to N pass through unchanged while arms 1 and / are connected by a simple directional coupler. In this case the transfer matrix has the form:

.10

1 d o o o i c o a1 o 1 0 1 0 1/2 i a2 1 i I b/3 1 o 1 1 a3 1 i O o , i I 1 o i I 1 o 1 i c o a ... 6.4 bi i o d / 1 o 1 1

1 0 1 o 0 0 0 o aN

where 02 + d2 = 1 with c and d real and 1 as usual being the square root of minus one (the coupler representation used here is not the most general possible but is sufficient for our purposes. The parameters c and a are the amplitude coupling factors for the coupler). The factors of T in Eqn. 6.1 will be of one of the forms. given in Eqn. 6.2 and Eqn. 6.4.. Since all constituent networks have all-pass characteristics with transfer matrices Tm that satisfy VI Tm = I, Eqn 6.1 is easily reduced to:

T+ T+ T 3- Tit Tt T = I 6.5 M M-1 • This demonstrates that the matrices Ti to TM are row transformations that reduce the network transfer matrix to the identity matrix.

The first step in the synthesis is the formation of:

T` = Ti T ... 6.6

such that the first element of the first column of T' is real and the remaining elements in this column are positive imaginary. This factorisation is illustrated in F14(a), T1 being a phase shift network. A second factor that -130- • T1

F14 (a) FIRST NETWORK REMOVAL

T2 Ti

0 0 0 •

F14 (b) SECOND NETWORK REMOVAL

FURTHER NETWORKS I I 111 o • , . o O

TN+ 1 T3 T2

F14 (c) N'th NETWORK REMOVAL

- 1 31- • represents s simple coupler connecting arms 1 and 2 is now removed from T' to form T" given by:

T" = T2 T' ... 6.7 where T" has a zero in the second row of the first column. Noting the form of T:. as the conjugate transpose of the transfer matrix in Eqn. 6.2 with 7, equal to two, T" is given by:

T" d -ic t'12

-ic d o 21 t'22 0 0 6.8 0

- o o

Thus: • t" = -ic t11 + d t'21 21

(-c til + d I V211) ... 6.9 where the latter results from the form of the first column of T' as derived above. With c2 + d2 equal to unity, tS, is zero if:

c = ... 6.10 / 1211 2 Clearly c is always less than unity so it is always possible to provide a coupler that will introduce the zero element into T". The factorisation so far • is shown in F14 (b). The significance of the zero value of tS1 is that for the first orthogonal distribution no power is required at the second output port of T". In an exactly analogous manner a coupler between arms 1 and 3 represented by T3 will transform T" into T"'

T'" = T 6.11 with the third row of the first column zero. It is easy to show that the second row zero is not disturbed. After N-1 zero introductions corresponding to the remcvol cf N-1 couplers from T we have:

-132-

1 2 3 ..... - N-1 N 4- OUT

I ONIMMI MUM" 1

2 3 N-1 N 4--ZIN

F15. THE COUPLERS REDRAWN

1 2 3 4 4- OUT 90.0° .184.1° .1841° •0.0° IN

0.473 ■0.427. 0.623 0.881 0.904 0-782 -90.0° ■180.0° ■0.0°

0.623 0.427 ■ '2 0.782 0.904 406.4° .0.0° 0.579 3 0.815 .0-0°

F16. FORTH ORDER NETWORK EXAMPLE

-133- or

F17. RE-ENTRANT COUPLING NETWORK

•

F18. RESONANT RING COUPLER

-134- • 4- 4- N T2 T T = 0 6.12 T(N) I I 0 (N) where T is a transfer matrix of order N-1. The zeroes on the first column have been introduced, the unity diagonal element and the first row zeroes occurring as a result of the unitary nature of the matrices concerned. This factorisation is illustrated in F14 (c). For the first orthogonal beam no power is required at ports 2 to N, port one therefore being the network port for exciting the first beam. The factorisation so far is clearer when the couplers removed to date are redrawn as in P15. Repeating the above process to reduce the transfer matrix of order N-1 to N-2 and so on will decompose the whole matrix into the form illustrated in F16 for a 4'th order system. This network corresponds to the orthogonal beams listed in the table below.

ELEMENT BEAM 1 BEAM 2 _ BEAM 3 BEAM 4. r 1 0.62309 ( 90.0) 0.33430 r0.1 0.40945 270.1 0.57650 (190.1 2 0.33430 274.1 0.62309 94.1 0.57650 97.7 0'40945 (187'7 3 0.33430 274.1 0.62309 94.1 0.57650 277.7 0.40945 ( 7'7) 4 0.62309 ,90.0 0 33430 (90.0) 0.40945 90.0) 0 57650 ( 0.0)

While an algorithm equivalent to the process described here was discovered by Dr E Shaw in 1967 (R22) while vorking at ASWE, its equivalence, to matrix factorisation was first suggested to the author by J P Shelton in 1969 (private communication). It provides a formal method for designing multiple beam networks, an N beam network consisting of N(N-1)/2 couplers and N(N4.1)/2 phase shifters.

7. Other Network Reductions As a unitary matrix is specified by N(N-1)/2 complex numbers together with N phase terms it is clear that the network synthesis described above is canonical in that reduction with fewer couplers or phase shifters is not normally posssible. While the networks to be described use more than the minimum number of couplers, the sub-networks have some interesting properties. Consider a network consisting of two identical rows of N-1 couplers connected as illustrated in F17. It is evident that it has all-pass characteristics and letting the parameters of the coupler on the i'th cross arm be ci and di it is tedious but straightforward to show that the elements of the transfer matrix are given by:

-135- tip Sid - (1 + s) w. w. 1. ... 7.1

where s = exp (14) and:

w. = c ; i v. 1 d1 d2 . dc ; i = 2, ..., N-1 a a aN ; i = N 2 -1 ... 7.2 (the use of i both as a subscript and as the square root of minus one should not cause any confusion). This is equivalent to:

T = I - (1 + s) w n 41.• 0•10, 7.3 where w is a real column vector with components given Eqn. 7.2 (since w is + t. real w w ). Now using Eqn. 7.2 it is easy to show that:

+ W W = ... 7.4 (this results from the fact that e + d! . 1 for all couplers). Using I 2 Eqn. 7.3 and Eqn. 7.4 shows that:

T n n = I - (1 -s s*) w w+ I ... 7.5 sinceSs* = 1, which verifies that the network has all-pass characteristics. This network can be used in the reduction of a general all pass network as follows. Fitst the network transfer matrix T is transformed by removing a phase shift network to a transfer matrix T':

+ T' = T T ... 7.6 such that the elements of the first column of T' are all real. Next a

network of the type discussed here is removed to yield T" given by: •

T" = [I - (1 + s*)w w+] T' 7.7 in such a way that T" is simpler than T'. With t and e as the first columns of T'and T" respectively, Eqn. 7.7 shows that: e . [I - (1 + s*)w if] t ... 7.8

giving: (1 + s*)(w+t) w = t - e ... 7.9

-136-

Also from Eqn. 7.5 and Eqn. 7.8 t and e must satisfy: + + e e = t t ... 7.10 for Eqn. 7.8 to hold. Equation 7.9 shows that for t to be transformed into + e, w must be in the direction of t - e. Since w w = 1, w must be of the form:

(t - e) w _ 7.11 -1/(t e)±(t - e)

Substitution into Eqn. 7.9 shows that Eqn. 7.8 will hold if (1 - e+ t)"4- s* ... 7.12 (1 - e t)

Since w and t are real, Eqn.,7.11 shows that e must also be real, and from Eqn. 7.12 s is therefore unity. Thus we can choose any real e of unit length (ie e+ e = 1) for the first column of T" and use Eqn. 7.11 to determine a vector w such that the transformation T: given by:

7.13 T2 [I - 2 w w+r/. transforms T' into Ts'. In particular choosing e as the first column of the n'th order identity matrix yields:

+ 0 0 T"T2 Ts = 1 0 ... 7.14 (N-1) T

0 • which constitutes an order reduction. Clearly the process can now be repeated using lower order networks and will lead to complete network decomposition. The form of the network is illustrated in F19 for a four beam system. The coupler values required to provide a sub-network with a given w are easily determined using Eqn. 7.2.

The reduction networks used in this section and illustrated in F17 with the transfer matrix:

T I - (1 - s) w w+ ... 7.15 -138- a have an interesting property that is illustrated by the equations:

[I - (1 + s)w ] W = - S W 000 7 0 16

[1 - S)W M74. ] X = x, if w7fx = o ... 7.17

Thus w is an eigenvector of T with eigenvalue -s and any x orthogonal to w is an eigenvector with an eigenvalue of I. In other words all distributions orthogonal to w pass through the network unchanged whilst any component of w undergoes a phase shift represented by -s. The network is in a sense 'resonant' for a particular distribution and passive otherwise, a fact that suggests that there may be a truly resonant network with the same behaviour. The ring coupler structure shown in P18 in which the i'th coupler has the i and d as usual and the phase change between couplers around parameters c 1 the ring is a multiple of 2y has a transfer matrix with elements:

t.i = - ci di+1 dN di d2 di ci/A i

= (di - di d2 /di) / A i=j

c. / A - a.j d.j+1 dj+2 d.t-1 t i>... 7.18

with:

A = 1 - ds d2 •••• aN 7.19

Now if all the couplers in the ring are reduced to zero in a manner described by:

c =ywt. y o ... 7.20 •

with the W. such that:

1 ... 7.21 i=1

then it is easy to show that:

t.. = 8.t j- 2 w.t w. ... 7.22

and therefore that:

T = I - 2 w w+ ... 7.23 -139- which is of the required form. This 'resonant coupler' is essentially a resonant cavity that is tuned to a specific incident distribution. The resonant coupler is of some interest in the feeding of circular antenna arrays and was studied by the author in this context in 1968. Clearly any practical implementation of a ring of couplers can only approximate to the above behaviour as the couplers must have finite though small values. In addition it is evident that such a network would have an extremely narrow bandwidth. The circular array work failed as a result of the impracticality • of the resonant coupler and perhaps the most interesting aspect of the work of this section is the discovery of an exact wide band network equivalent of the resonant coupler in the form illustrated in F17.

8. Discussion The networks for exciting the aperture modes are of prime interest when an array has to provide a variable radiation pattern. When a single pre-determined distribution is required the multiple beam network is clearly inappropriate because of feed complexity since in addition to the network itself, a power divider is required to feed its input ports. The fact that the network ports are decoupled is of no real advantage since the power divider will itself introduce coupling.

In contrast the use of the aperture modes for pattern synthesis is of practical importance as it has generally resulted in an improved performance. In this context it is of interest to compare the aperture modes with the uniform amplitude, phase tilted plane waves used in the Woodward synthesis technique (R1). The most obvious difference is that the aperture modes are even and odd functions whereas the Woodward modes are not. However the latter can be converted to an even/odd representation without sacrificing any of their fundamental properties (ie orthogonality and completeness) by taking sums and differences of oppositely phased uniform modes. In contrast the aperture mode representation cannot be transformed into a 'Woodward' system without sacrificing the 'uncoupled' property. The even and odd aperture modes are unique in that they are the only model set that are orthogonal, complete and uncoupled and in this sense they provide a more fundamental aperture field representation than the Woodward modes. This conclusion is in close agreement with the recent theoretical work of Rhodes on the realisability of planar apertures (R5 and R25). Briefly he demonstrates that the tangential electric fields in the aperture of a planar antenna must vanish in a specific manner at the edges of the aperture and that, as a result, pedestal distributions of the Woodward type are not physically realisable. He demonstrates -140- that sum and difference combinations of Woodward. modes can meet the edge conditions and therefore form valid field representations. It is particularly interesting that the same conclusion should result from the work of this thesis since the theory for the horizontally polarised parallel plate array allows non-zero edge fields and yet still predicts non degenerate even and odd mode representations. This also results from the analysis of cross coupling for w an array built by H Salt (R2)+) which lends practical backing to the fundamental nature of the evep/odd mode representation.

• CHAPTER FOUR — FEED NETIORICS

fs —142—

1. General Considerations In this chapter a number of distinct types of array feed networks are considered with an emphasis on wide band operation. While in many areas the term 'wide-band' would imply multi-octave operation, in the context of the work to which this contributes, a bandwidth of 1.5 to 1 is considered wide (bandwidth in this chapter will be expressed as highest operating frequency divided by lowest operating frequency). Using the notation described in section 1 of the previous chapter, the operation of a feed network can be described in terms of the scattering matrix representation:

bo o 1 tt ao SO. loi 1 {ti T [a,

where: t+t = 1 and Tt* = a ... 1.2 t and T = T as a result of reciprocity. When used to feed an array of matched elements, this network will yield the distribution t and its input port will be matched. If the array elements are not matched, the array scattering matrix being S, then the distribution b1 achieved is given by:

-a = (I - TS) it ... 1.3

The distributions t and bi will be equal if:

TS b1 = TS t = o ••• with the possibilities t = o, St = ol TS t = o or T = o. Of these the first two are trivial and the third has been covered in chapter three. The last condition, T = o, corresponds to a network for which there being: is no coupling between the array ports, the scattering matrix SN 1 t o 1 -t SN = t+ t =1 ... 1.5 [ o

In this case the product S;1- SN which would yield the identity matrix for a lossless network is:

1 + S = SN N 0 -t - -143-

which demonstrates that this type of network is not lossless in general although no loss occurs when it feeds a matched array. For the unmatched array, 'reflected' power is absorbed in lossy elements or appears at the network feed port. As it does not reappear at the array ports'of the network, the distribution remains unaltered as expected. Networks of this type, can be constructed using directional couplers or hybrid junctions of some forms. From a theoretical viewpoint these networks are ideal as arrays feeds; they do however have a number of practical disadvantages that will be discussed later.

If a totally lossless feed network is required, it is easy to show by forming S144. SN that T and t are related by:

* t ... T+ T = I - t t 1.7

and since:

* t (I - t*tt)+ = I - t t ... 1.8

t t (I - ett)l- (I —tt _ ) = I - tt ... 1.9 4

T+ T can be written in the form:

1+ T = (I - t*tt)+ V-1- V (I - ett) ... 1.10

where V is any unitary matrix (ie V+ V = I). Thus factorising Eqn. 1.10 gives:

T = V (I - t*tt)

as the general form for T. In fact this form is not very restrictive as it • really only expresses the constraint that all the rows of T are orthogonal / * to the vector t' (ie Tt = o). Montgomery, Dicke and Purcell (R25, p 151 onwards) provide some general results for the frequency dependence of a lossless junction. Briefly, for distributions of incident waves that satisfy:

S = exp (ii) a* ... 1.12 N a

they demonstrate that:

+ + a S S' a - 2 -11JV ... 1.13 NN -101 - where S' is the derivative of S with respect to frequency andW is the N stored energy in the junction. Using SN as in Eqn. 1.1 and letting the j'th component of the array distribution t to be It .I exp (i0 ) equations 1.12 and 1.13 give after considerable manipulation:

o ... 1.14 2_J itj12 dw a constraint on the rates of phase change with frequency at the array ports. The limitations imposed by Eqn. 1.14 do not appear particularly useful in predicting the performance of array feeds. Since T is not zero for the lossless feed network the distribution bi obtained when feeding an unmatched array is given by Eqn. 1.3 and will generally depart from the desired excitation t. The practicality of such feed networks clearly depends upon the magnitude of this departure, a topic that will be covered in later sections.

2. Directional Coupler Feed Networks There is a considerable volume of literature on coupled TEM transmission line (U sectional couplers (11.26 to 29 for example) and in this section their use as components in microwave array feeds will be discussed. The layout of symmetric and asymmetric couplers are shown in F1, each consisting of a number of discrete, uniform coupled transmission line sections. All sections are identical in length, being a quarter wavelength at the centre of the operating band corresponding to a 90 degree phase delay. For a given number of sections the asymetric coupler provides a better performance either in terms of bandwidth or coupling variation within band, but the symmetric coupler has the property that the relative phase of its outputs is always 90 degrees. While this is often very desirable, it is not generally required in the design of single beam array feeds and may in fact be an embarrassment • since extra components may be required to equalise the phase of the output ports if it is present. For this reason only the asymmetric coupler has been considered in detail. Levy (R25) gives the method of designing these couplers with a Chebyshev coupling variation in the operating band and a computer program has been written to implement his design technique. The method and the program are given in Appendix 6. The amplitude response of these couplers is extremely good but some care is needed in achieving good phase tracking of the output arms. This is illustrated in F2 which shows the output phases of a two section -3 dB coupler as a function of section phase delay (equal to 1111/2f0 where f and fo are the frequency and the mid-band

-146-

frequency respectively). The direct line phase varies'from 0 to 360 degrees and in practice is not greatly different from the phase delay of the equivalent length of uncoupled line (ie n line sections for an n section coupler). The coupled arm phase varies from +90 to -90 degrees and is in phase with the input wave at centre band. Thus the phase between the output arms at mid- band is 180 degrees (90 n degrees for a n section coupler) and for equal output phases a two section line length has to be added to the coupled arm. The dashed curve shows the resulting phase delay on the coupled arm and as can be seen the two output phases are in fairly close agreement over the operating band of the coupler as marked by the vertical lines. In general an n section coupler will need an n section length of line added to the coupled arm and in the following it is assumed that this compensation is applied to all couplers as illustrated in F3.

In some feed networks there are more couplers in some input to output paths than in others and in such networks wide-band operation can only be obtained if the extra delay of these couplers is offset by adding equivalent line lengths to the other arms. In this type of network the phase errors result from the deviations of the behaviour of the coupler outputs from that of the compensating line lengths. Thus for example on the two section coupler of F2 it would be the deviations from the phase of a two section line length that are significant. These errors are plotted in F4 together with those for one and three section couplers, the design bandwidth being 1.5 as for F2. It is clear from this diagram that the phase deviations on the coupled arm are the more significant.

Peed networks using directional couplers will now be illustrated using a -4.0 dB Chebyshev amplitude distribution on a 16 element array as a standard and designing for a bandwidth of 1.5 to 1. As most array distributions are symmetric about the array centre line, corporate feed networks are normally designed as two identical halves, and each feeding one half of the array. Figure F5 illustrates a fairly common form of 'binary split' type of feed network together with the phase errors produced at the lower end of the design band when constructed with couplers having 1, 2 and '3 sections. At the upper band edge the phase errors are of the same magnitude but of opposite sign. As the array distribution being studied decreases steadily from centre to edge, the construction illustrated, with coupled arms outermost, ensures that all coupling values are less than -3 dB. This will not always occur with this construction as some high order Chebyshev distributions do in fact increase in value in the vicinity of the edge; nevertheless the coupling values can clearly be chosen to be less than -3 dB by suitable network -149-

arrangement. This is desirable as couplers tighter than -3 dB are difficult to construct. The phase errors demonstrated in F5 are fairly large and, perhaps more significantly, are irregular in nature and therefore likely to cause severe degradation in side lobe level. This is the case as is demonstrated by F6 which gives the array pattern for the 3 section coupler network at the edge of the design band (ie 0.8 or 1.2 fo) and shows increases in side lobe level of more than 10 decibels. A different design approach is illustrated in F7. This is often referred to as a serial design. Note that in order to keep all coupling values below -3 dB's the construction technique changes from one in which the array elements are connected to the coupled arm to one in which they are connected to the direct arm. The phase errors are again fairly severe resulting in high side lobes as shown in F8 which shows the pattern for the three section coupler design. lieferring back to F7 it can be seen that the phase across the array has a fairly regular performance for the elements that are connected to coupled arms and also for those connected to'direct arms, the irregularity being the result of a 'design' change half way along the network. It is evident from this diagram that an array in which all elements are fed from the coupled arms of the couplers will have a better phase performance than the present design, the problem being that this requires impractically tight coupling values towards the end of the array. This limitation can be overcome however at the expense of a slight loss in gain as illustrated in F9. Here by allowing some power to be absorbed in a load, a uniform design is achieved with reasonable coupler values. The 5% power in the load shown reduces the gain by about 0.25 decibels, the largest coupling value being -3.67 dB. The phase variation across the array aperture is greatly reduced (note the change of ordinate scale) and is now regular in form. As would be expected, the array pattern for this design using triple section couplers and shown in F10 is a great improvement over the previous designs, the only side lobe grossly effected being the first which has merged into the side of the main beam. The remaining side lobes are virtually unchanged. If more power is dissipated in the load the situation improves further as illustrated in F11 for 2C$ load power and a resulting gain reduction of about 1 dB. In practice the improvement is not large enough to 'trade' against what is now an appreciable gain loss. In addition in high power applications the design of a load to handle 20% of the power would certainly cause difficulties. Using this 'series' design technique even the network constructed using single section couplers gives side lobes removed from the main beam at the design level as is illustrated in F12 for a load power of 5%. For many applications the main beam distortion would not be of any concern and in these cases the simplicity of the serial design with single section couplers is attractive. -153- In summary there is little doubt that, unless a completely lossless directional coupler feed is required, the last design approach is superior. For wide-band operation at low side lobe levels, the alternative designs will need networks for phase correction and these are difficult to design as, in general, both phase and rate of change of phase with frequency need to be controlled.

From a practical viewpoint, networks constructed from directional couplers have a number of disadvantages that tend to reduce their attractiveness:

a. In practice the forth port of all directional couplers has to be terminated in a load and this can add to the cost, weight and complexity of the feed.

b. The provision of accurate directional couplers requires the main- tenance of close tolerances both in components and in their relative positioning.

c. The transmission lines ih directional couplers are often very closely spaced and this can cause difficulty if the network has to be capable of high power handling.

As a result it is worthwhile examining the performance of other types of feed network.

3. Performance Analysis for Shunt Connected Coaxial Networks The shunt coaxial line junction, in which the inners (and separately the outers) of a number of coaxial lines meet at a point, is a simple and effective device for the division and distribution of microwave power. To obtain the desired power division and a good match, the impedance that each arm presents at the junction needs- to be controlled and this can generally be achieved using multiple section: quarter-wave impedance transformers. The resulting networks consist of a number of interconnected lines of varying characteristic impedance and it is desirable to have a computer program for analysing the performance of the complete network given its specification. r The development of such a program is briefly described here, the program and further details being given in Appendix 7. The network is specified in terms of its nodes and the lines that interconnect them, the nodes being of two types. External nodes are the input/output ports of the network and are constrained to have only one line 'connecting them to the remainder of the network. Allowing more than one line to be connected to an output port does not increase the generality of the program since this situation can always be represented by adding a zero length line between the desired multiple line port and a 'conceptual' single line port. An additional complication of allowing multiple line connection at input/output ports arises because in this situation there is no natural specification of the 'port characteristic

impedance to which the port wave amplitudes cap be referred. With the simple node this can be taken as the line characteristic impedance of the single connecting line.

Internal nodes are those junctions internal to the network, the multiplicity of the node being defined as the number of lines joined at the node. In a network consisting of L lines and N ports there 2L unknowns, two for each line, 2N of which are also external variables, N wave amplitudes incident on the network ports (vector a) and N 'reflected wave amplitudes (vector b). For internal nodes of multiplicity K (ie K lines) there are K-1 equations relating line voltages and 1 relating line currents giving exactly one equation for each line at the node. External nodes do not give equations and as a result lines entirely internal to the network yield two unknowns and two equations (one for each end) whereas lines at external nodes yield two unknowns and only one equation, giving 2L unknowns and 2L-N equations in total. In matrix form, this can be represented in the form:

N A B a 4 = o 3.1 b 2(L-N) D 1 F

N -) .4-4-2(L-N) where the submatrices A to F are known from the configuration of the network, the vectors a and b take their usual meaning and the vector c contains all the unknown wave amplitudes internal to the network. After some simple manipulation the scattering matrix can be derived as:

= (B CF 1E)-1 - CF-1D) *110 3.2

The problem was originally reduced in this form as it required only matrix routines that had already been written. However a complication arose because in the above technique an arbitrary choice has to be made for the N equations that lie above the horizontal partition and this can lead to a matrix F that is ill-conditioned and subject to large errors in its inversion. In practice it was found that this effect was quite common and therefore of high 'nuisance value'. To overcome this difficulty, a program was written using standard Guassian elimination on the complete 2L-N by 2N matrix to reduce it to the form: -155- S A! I -1N 0 a

= 0 b ... 3.3 D I / \ 0-12JJ-N)

where I and N I2(L-N) are identity matrices of order N and 2(L-N) respectively and A' and D' are transformations of the matrices A and D. This • gives:

b = A' a and c = Di a ... 3.4 which identifies Alas the network scattering matrix and D' as the matrix which relates the wave amplitudes internal to the network to those incident upon it. During elimination, rows are moved to maintain good conditioning and no problems arise. The network analysis program is given in Appendix 7 with some futher description. It has been used to derive the results presented in the following sections.

4. Performance of Lossless Shunt Connected Coaxial Networks With notation and partitioning as used previously, the feed network scattering matrix is written:

t too I t S N ... 4.1 t T

t where T = T as a result of reciprocity and t is the array port distribution obtained when feeding matched elements. Losslessness imposes the constraints:

t+t = - itoor

* T t = - ton

t T÷T I - t* t

It is normal to design for a matched feed at mid-band and if this is the case then Eqn. 4.2 and Eqn. 4.3 reduce to:

-156-

F13 A Typical Node in a Shunt Connected Coaxial Line Network

1P14 A Tree Structured Feed Network

-157-

t+ t = 1 and T t* = o ... 4.5

at this frequency. As already mentioned, the feed networks considered in this section will make considerable use of multiple section quarter wave impedance transformers. While in principle line sections that are not part of these transformers can be of any length, some simplification occurs if all uniform line sections within the network are multiples of a quarter wave in length at mid-band. This can be demonstrated by considering a single • network node as illustrated in F13 where J lines meet. There are two unknown wave amplitudes on each line and for convenience in this analysis, waves converging on the node being considered will be defined at the remote ends of the lines (ie at the adjacent nodes) and those leaving the node will be defined at the node itself. This is illustrated in F13. In terms of these variables the current equation at the node can be written: J [a. exp (-ifil) b.] ITT= 0 ... 4.6 0 0 3

• wherelinejhasacharacteristicadmittancey,Equally the node voltage V can be expressed in terms of the line variables on each line in turn giving J equations:

[a. + exp (--1,61) b.] /1/77= V = 1, 2 .., J 0 0 0 ... 4.7

ninainatingeachb.from Eqn. 4.6 using Eqn. 4.7 gives:

2 7 a TT = yj ... 4.8 3 - 3 •

and eliminating V from the equations of Eqn. 4.7 and simplifying yields the set:

j = 1, 2 .., J jk IL ak exp (-1'61) b = ... 4.9

wnere R which is purely real, is given by: 'jk

-158- •

2 1777-- R - a k - 5 jk J jk ... 4.10 L Y/ 1=1

Now, as all line sections are multiples of a quarter wavelength, the whole network can be analysed in terms of line sections of a quarter wavelength (longer lines can be reduced to a number of quarter wave lines by introducing nodes) and thus:

exp (-WO = exp [-illf/2f0]

= exp (.;.iirAf/2f0) ... 4,11

where f and fo have their usual meanings and Lf is the deviation in frequency from mid-band (ie Of = f - f0). Denoting the term 77Af/2f0 as 0 for convenience and letting superscript dashes and double dashes denote the real and imaginary parts of quantities, the real and imaginary parts of the equation set Eqn.4.9 are: J R. a'k + sine b' - cos0 = o j = 1, 2, k=1 4.12 J 11,k alc + cos0 lo'i + sin0 11 = o ; j = 1, 2, ) i t) ..., J. k=1 .„. 4.13

Now assume a solution for the complete network exists at a frequency

deviation Of with quantities a. a". b/. and b". Then, as far as the 0 J J 0 node being considered is concerned, either of the two solutions:

(1) , a". , ... 4.14

, , ... 4.15 0 J J J

are equally consistent and form valid solutions at this node. In complex notation these solutions are:

• -159-

(1) -Af ... 4.16

(2) -Af , b* . 4.17

Thus as far as the node being considered is concerned, going from Af to -Af in frequency, conjugating variables defined at this node and taking the negative conjugates of variables at adjoining nodes provides a valid solution (equation set 1). Thus if the considered node solution at -Af • is equation set 1 in which 'self' variables are conjugated, then all adjoining nodes must obey equation set 2 in which 'self' variables are 'negative conjugated'. Equally all nodes adjoining them must lie in equation set 1 for consistency since equations of the form of Eqn. 4.12 and Eqn. 4.13 must apply to them also. Thus to convert a solution at a frequency fo + Af to one at fo - Af , it must be possible to group the network nodes into two non-intersecting sets such that the nodes of one set have only nodes of the other set directly adjacent to them and vice versa. If such a division is possible, then on transforming from 1'0 + Af to fo - Of , the amplitudes on one node set are conjugated and those on the other set take on negative conjugate values. The choice of which set is which is arbitrary since any solution remains a solution if all wave amplitudes are multiplied by the same constant and choosing this constant as -1 converts one choice into the other. Thus, provided the node grouping described above is possible, then the solutions at fo + if and fo - Af are simply related.

It is immediately evident that tree structured networks that do not contain loops (F14) meet the node grouping requirements and in fact loops are permissible if they have circumferences that are even multiples of a quarter wavelength at centre band. The node grouping cannot be achieved however if loops exist which contain an odd number of quarter wave sections. In • practice the majority of feed networks for microwave arrays are tree structured and all networks Considered in this section are of this type. Thus nodes fall into two groups for which wave amplitudes leaving the nodes satisfy:

(1) a (e. - Af) = a* (f + Af) ; a (fo) real ... 4.18 (2) a (f0 Af) = -a* (f0 + Af) ;a (f0) imaginary

Now the wave amplitudes entering nodes are defined at adjacent nodes, the amplitudes Y. defined at the nodes themselves being given by: 0 -160- •

exp (-i(.91) bj

exp (--18) bj ... 4.19

For group 1 nodes:

• Vj (f0 + Af) exp (-iS) bj(f + Af) ... 4.20

(f - Af) = exp (i8) bj (f0 - Af)

+i exp (i8) 134; (f0 + Af)

b"(f0 + Af) 1.04, 4..21

since if the node considered is group 1 then the b j are defined at a group 2 node. A similar analysis can be performed for the wave amplitudes entering a group two node to show that:

(2) (f - Af) . + Af) ... 4.22 J ° j (fo

and in summary all wave amplitudes entering and leaving a group 1 node and defined with respect to a reference at the node satisfy:

(1) a (f0 Af) = a* (f0 + Af) a (f0) real ... 4.23

(1)b (f0 - Af) = b* (f0 + Af) ; b (f0) real ... 4.24

while all amplitudes entering and leaving group 2 nodes obey: git

(2)a (fo Af) -a* (fo + Af) ; a (f0) imaginary ... 4.25

(2) b (f0 Af) -b* (fo + Af) ; b (f0) imaginary ... 4.26

Thus at centre band all quantities at group 1 nodes are purely real while all those at group two are purely imaginary.

Now feeds for broadside radiating arrays must produce excitations that are in phase and it is therefore clear that all output nodes of the feed network

• -161-

Or4

• must be in one node group or the other since the presence of both groups would give some 90° phase shifted element excitations at Mid-band. The network input port (input and output here are defined for the array in the transmitting mode) can lie in either group as its phase is not important relative to the output ports but by suitable choice of reference plane in the input line it can be chosen to lie in the same node group. When this i6 done, the above analysis shows that the network scattering matrix satisfies the constraint:

S (f - Of) = S ... 4.27 N 0 N (f and SN (f0 ) is real.

Thus in summary, tree structured, shunt junction coaxial feed networks constructed entirely from lines whose lengths at mid-band are mulitples of a quarter wavelength and designed to produce an in phase array distribution have scattering matrices that satisfy Eqn. 4.27. In practice this is useful as it means that network performance need only be studied and controlled over one half of the total design band in the knowledge that the performance over the other half is simply related. The networks considered in the rest of this section will be of this type and performance will be considered for frequencies below ft:, only.

A common type of coaxial feed network uses the type of junction illustrated in F15. Transformers in the output arms adjust the impedances presented at the junction in such a way that the input line is matched and the desired power division is obtained. In the design considered here, the input line is always at the same standard impedance. As was done for the directional coupler feeds, a 16 element array with a -40 dB Chebyshev distribution will be used to illustrate the design. The bandwidth will be 1.5 to 1 and, as usual, only one half of the feed network will be illustrated. This type of network is illustrated in F16, the triangles in the feed representing the impedance transformers. This diagram also shows the phase errors at the low end of the band when the network is implemented with 1, 2 and 3 section quarter wave impedance transformers. The computer program used to design these transformers is given in Appendix 6 along with that for directional couplers. The phase errors decrease fairly substantially as more sections are used and are fortunately 'quadratic' in form and therefore not too serious. As already discussed, the phase errors at the upper end of the band will be simply those illustrated with a sign reversal.

-16)+- •

The array patterns expected from these networks are shown in F17, array mutual coupling effects not being included. As would be expected from the form of the error, the patterns for two and three section transformers are very good. The single section tranformer network shows considerable distortion which results more from a degradation of the amplitude distribution that is significant only for this single section implement ion. The three section performance is certainly acceptable and the two section network may also be so. However it must be remembered that mutual coupling effects must be added to the degradation already present and this may make it necessary to accept only a small deviation from design side lobe level as a result of network effects in isolation.

The second network considered uses the junction illustrated in F18. The vertical arms are of constant impedance and directly feed the array elements, the right hand arm feeding all remaining array elements. Assuming that the right hand junction has been designed by the process to be described, the impedance presented by the two output arms in parallel at the junction will be known. Now the power division required at the left hand junction is known, as is the impedance of the vertical arm, these facts allowing the computation of the impedance that the horizontal output arm must present at the junction. Thus a quarter wave impedance transformer can be inserted in the horizontal arm as illustrated to provide for conversion between the two calculated impedance values. The array structure is shown in F19 as are the phase errors for construction with 1, 2 and 3 section transformers. The distribution is the normal -40 dB Chebyshev on 16 elements, one half of the complete feed being shown with the phase errors at the lower band edge. Note the large reduction in error on going from one to two section transformers. The errors are again quadratic, but smoother in form and reduced in magnitude in comparison with those of the 'parallel' network. As in the case of the latter, the 'serial' network implemented with single section transformers gives a significant distortion of the amplitude distribution away from the centre of the design band which shows in the array patterns which are given in F20. Array coupling effects are omitted and as expected from the phase error results, the patterns of the two and three section implementations are, from a practical viewpoint, essentially perfect. The single section network does in fact increase the taper of the array distribution away from mid-band and this results in a better far out side lobe level and a reduced gain. Figure F20 shows however that the close in side lobes for this network are extensively degraded.

-167- It is clear that potentially infinite variety of feed network designs makes it impossible to present any exhaustive coverage of characteristics and while in studying feed techniques no network has surpassed the performance of the serial design, it is clearly not possible to claim that it is the optimum network of this type. In, any event, all networks of the type discussed in this section will be sensitive to mutual coupling effects and, having found a practical and effective feed network, it is perhaps of more interest to look at the magnitude'of these effects before expending further effort in search of feed improvements that may well in practice be swamped by the effects of coupling on the'array.

5. The Effects of Mutual Coupling on Feed Performance The work of chapter two showed thit the behaviour of an array of elements can be characterised by an array scattering matrix S that relates the distributions a and b incident on and ''eflected' from the array respectively:

b = S a1 S a ... 5.1 b2

b a N ; N

The diagonal elements of the matrix S represent array element reflection coefficients and the off diagonal elements represent the fields coupled from other array elements. In chapter two the effects of mutual coupling were studied on the assumption that the correct array distribution had been achieved. For many feed networks this does not give a complete picture of mutual coupling effects as the'array distribution can be altered by the fields 'reflected' back into the feed network. As derived earlier, in the presence of mutual coupling (represented by the matrix S) the distribution b obtained is related to the design distribution t through:

b = (I - TS)-1 t ... 5.2

where T as usual is that part of the network scattering matrix that relates to coupling between the ports connected to the array elements. Note that:

b = t + [(I - TS)- I] t

t + (I - TS) 1 TS t ... 5.3 giving the difference between b and t as E where: -168- .(I - TO-i TS t 5.4

It is clearly desirable if possible to obtain upper bounds on c and this

can be achieved using the concepts of vector and matrix norms. As is

common, the length or norm of a vector x will be defined as 1 x I where

I a 1 2 = ... 5.5

It is easy to prOve by use of Cauchy's inequality that

1 2s ± z1 lal 4-1z1 ... 5.6

Now the length of the vector Ax obtained when the vector x is multipled by the matrix A is 1 Ax I where

I A a 12 = (Aa)+ (Ax) = IF A+ Ax ... 5.7

Now the matrix A+ A is Hermetian and therefore has N positive or zero

eigenvalues and eigenvectors:

2 At Ax. = o. x. ... 5.8 -1

x. x. = 1 i = j —J 0 ... 5.9

Now, since these eigenvectors are complete, x can be expressed in the form:

a. x .

ti and from Eqn. 5.3, Eqn. 5.5, Eqn. 5.6 and Eqn. 5.7:

\ 2

1 a 1 2 = ai 5.11

and:

2 A x. 1 2 a. 1 0.1 2

cr2 1 a.1 2 max ... 5.12 -169-

Thus:

o- 1 A 1 2 2 ... 5.13 max1 - 1 2

2 where umax is the maximum eigenvalue of A+ A. Now it is natural to define the norm or length of a matrix as:

r IA xl 1 A 1 Maximum value of ... 5.14 7-77

for arbitrary x , giving the maximum length magnification that results from

its application to any vector. Thus from Eqn. 5.13: 0

1 A 1 2 Maximum eigenvalue of (At A) ... 5.15

and it is easy to show that:

A B I E ... 5.1 6

1A+BI E 1A1+1B1 ... 5.17

Returning to Eqn. 5.4 and using the equations just derivedit is straight-

forward to show that:

ITI IsI I EI 1-1Tlisl I t...5 .18

Now Eqn. 4.4 shows that:

Tt T . - t* tt 5.19

and

T+ T t* = it001 2 t* ... 5.20

+ T T u u if t+ u = o ... 5.21

Now as there are N-1 independent vectors that satisfy Eqn. 5.21 the matrix

T has N-1 unit eigenvalues and one of Ito°1 2 which is less than unity.

Thus 1 T 1 is unity which gives finally:

I.s t ICI 1 s ... 5.22

This gives a bound on the error 1E1 but in practice it is not very useful

as it generally yields a grossly exaggerated estimate of the error. When

the fields reflected by the elements are fairly small, the expansion:

-170-

T S t + (TS)2 t + ... 5.23

converges fairly rapidly and to first order c is equal to TS t. Generally the fields returning from the aperture can be visualised as the sum of two components, that due to element mismatch and that coupled from other elements. Often the former component is the larger of the two and it is therefore useful

0 to consider the effect of this component alone (ie ignoring mutual coupling). If the elements are all identical and have a reflection coefficient p , then S = p I and to first order:

pTt ... 5.24

e E Ip1 2 t+ T+ T t ... 5.25

Substituting for T+ T from Equation 5.17 gives:

• C 1P1 2 [(I+ - (tt t) (t+ t*)1 ... 5.26

Differentiating with respect to the components of t shows that this expansion is minimised if all components of t have the same phase, with:

I el — IP I I tool I tl ... 5.27

Thus, to first order, in phase distributions minimise the effects of element mismatch. In fact with S = p I and t real Eqn. 4.3 shows that Tt = Tt* , = -too -6 giving from Eqn. 5.2:

-1 b = (I - p T) t

= ( 1 + t*o o) t ... 5.28

showing that identical element reflection coefficients change the magnitude but not the form of in phase distributions.

The above derivations give some idea of the effects of array element reflections in the absence of mutual coupling. When such coupling is present the problem becomes more complex and while some effort has been, expended in attempting to derive realistic bounds on performance in this situation, no satisfactory results have been obtained. In order to gain some idea of array and network interaction effects in this general situation, the array analysis programs of

-171-

chapter two and the network analysis programs of this chapter have been used in conjunction. The former have been used to generate the array scattering matrix S that incorporates mutual coupling effects and the latter the feed network scattering matrix T. These are then substituted into Eqn.5.1 to yield the:distribution b which includes the effects of array and network interaction. Finally, using this excitation, the array element patterns allow the array far field pattern to be computed. These effects have been studied on a twelve element array with a mid-band element spacing of 0'75 wavelengths and an element width of 0.4 wavelengths (this array operates between parallel plates as described in chapter two. It operates in the TE mode and the term wavelength used with reference to the array means that applicable to 'free' propagation between the parallel plates). The feed network used is of the serial type using 3 section quarter wave transformers and designed to operate between 0'8 fo and 1'2 fo, 1'0 being the mid-band frequency. It has the sort of performance indicated in section 4, the worst phase error being less than 5 degrees and of quadratic form, giving an essentially perfect pattern in isolation. The design distribution is the usual -20 dB Chebyshev.

A typical set of array far field patterns for this array are shown in P21 for upper and low band edge and mid-band. The dotted curve is the pattern that would be obtained if the reflections from the array elements were absorbed in isolators while the full lines illustrate the effects of allowing these mutually coupled fields returning from the aperture to distort the array distribution. While there is little degradation at mid-band, the effects of interaction between the array and the feed are in evidence away from mid- band. The effects are quite marked at the upper edge of the band in particular where the redistribution of the reflected energy by the feed has produced side lobes at -32 dB, 8 dB above the design level and 5 dB worse than those predicted without interaction effects.

Adding lines of identical length into each input to output path of the feed network only changes the overall phase of the network design distribution and does not change its form. However, since fields 'reflected' by the elements pass along these paths more than once, it is clear that array- network interaction effects will be influenced by these line lengths. The patterns of F22 are for an array and network identical to those of P21 except that all input port to output port lines have been extended by a half wavelength. The no interaction patterns (dashed) are clearly unchanged but the patterns with interaction are quite different in form in that the worst side lobe degradation now occurs in a quite different position although of about the same magnitude. There may be 'lucky' feed arm lengths for which no gross -175- side lobe degradations result. This is illustrated in F23 which again shows the same feed and network with a third extension of all output port line lengths. In this case the interaction effects actually improve performance at the upper band edge and do not produce unacceptable errors at the low end of the band. However, a number of array and network combinations have been studied and the patterns of F21 and F23 are more typical of the results obtained with isolated side lobes between 3 and 10 decibels above the design level being common, mostly, but not exclusively, towards the edges of the design band. The array patterns at mid-band where the distribution is in phase are generally better than than those elsewhere bearing out earlier predicitions to this effect. However some caution is needed here as these predictions are based on element mismatch in isolation, the effects of inter- element coupling being specifically excluded by the use of a diagonal array scattering matrix S p I . Thus the agreement noted above could be taken to infer that the effects of mismatch predominate over those of inter-element coupling on the array currently being studied. In actual fact, although element standing wave ratios are appreciable, being typically 1.5 , the phase errors produced by the serial feed network are so small that the combined effect results in side lobe levels well below those in evidence in F21 to F23 which must therefore result from mutual coupling between array elements and the effect this has on the network distribution. This being so, it is all the more interesting to note that the earlier predictions of good performance at mid-band appear to hold true even when mutual coupling effects predominate over those of simple element reflection.

6. Summary When one element of an array is driven it excites the other array elements with two distinct results. First, the other elements radiate parasitically and modify the isolated element radiation pattern and secondly energy returns

into the feed lines to these elements. It is clear that the first of these • effects, often referred to as 'forward' coupling, will always influence the radiating characteristics of the antenna no matter what kind of feed network is employed. The second effect, that of 'backward' or 'reverse' coupling may or may not influence the radiating properties of the array depending upon the type of feed network.

Feed networks constructed using directional couplers or those containing isolators in their element ports are ideal in that they completely eliminate the effect of 'backward' coupling on array side lobe level which is therefore only sensitive to 'forward' coupling and distributional errors in the feed -176- • network. The work presented in chapter two suggests that the effects of forward coupling on side lobe level are not severe. They can to some extent be allowed for over narrow bands using the synthesis results of chapter three. The 'serial' directional coupler array in which some power is sacrificed in a load has been shown to produce quite low phase errors over large bandwidths provided that a suitable number of coupling sections are used in the couplers. Other coupler networks have been shown to yield quite large phase errors and are in general less suitable as array feed networks. A disadvantage of the parallel design is that, in its simplest form, it can only feed an array in which the number of elements is a power of two.

• The shunt junction coaxial network described earlier in this chapter is typical of a large class of networks for which the side lobe levels are modified by the 'backward' coupling effects on the array. In the absence of coupling effects, the 'serial' shunt junction network is capable of producing an extremely accurate distribution over large bandwidths, being better than the equivalent coupler network at the same level of complexity (number of coupler or transformer sections). It is unfortunate that the results of section 6 indicate that the degrading effects of 'backward' coupling are generally quite severe for this class of networks making them less than ideal for very low side lobe applications. They are however very much easier to construct than the directional coupler networks and this tends to offset their disadvantage of being sensitive to 'backward' coupling effects especially where antenna cost is an important factor. If this type of network is used it is clearly advisable to reduce the mismatch of array elements and the 'backward' coupling between them as far as is practical.

•

• -177- CHAPTER FIVE - A PRACTICAL ARRAY

-178- 1. Introduction The antenna array described in this chapter was built for low power operation, the principal requirements being low side lobe levels in the horizontal plane over a relatively wide band, coupled with light weight and low windage. The design was evolved by the author in conjunction with J Wyatt and the practical implementation was undertaken by Messrs Wyatt, Gregory and McMorland of ASWE Extension Funtington to whom the author is greatly indebted. The antenna (figure Fl) is designed to operate between 1215 and 1485 MHz and consists of an 18 element array feeding a parabolic cylinder reflector. The array has an overall length of 3.4 metres with an element spacing of 19 cm and a -40 decibel Chebyshev excitation. The reflector has a horizontal length of 3.66 metres, some 26 cm longer than the array, its vertical dimension being 2.4 metres. The antenna is horizontally polarised and advantage is taken of this in the construction of the reflector which is shown in cross-section in F2. It consists of hollow, thin walled aluminium slats slotted into templates that form the parabolic shape in the vertical plane. The gaps between slats act as cut-off waveguide sections and are designed to produce an attenuation of better than 40 decibles in the field transmitted through the reflector. This construction provides a low windage and also gives very good control of tolerances in the horizontal plane which are critical in achieving low side lobes. To prevent any serious degradation of the array side lobe levels at -40 decibels, reflector errors have to be reduced to about 0.5 mm r.m.s. As the levels achieved in practice have been less than half this value, antenna side lobe degradation can be attri- buted almost entirely to errors in the array that feeds the reflector. This array and its power dividing network will now be described.

2. The Array Element If, as in this case, the power dividing network is influenced by array element coupling and mismatch, it is clearly desirable to reduce these effects as far as is practical. Work with conventional flared waveguide horn elements was not encouraging and as a result the exponentially flared horn illustrated in F3 was adopted. This element consists of a coaxial line to ridged waveguide transition in a uniformly flared waveguide section. The ridges then taper onto the waveguide walls as illustrated, the final transition into free space being accomplished via an exponentially flared section. A Smith chart plot of element reflection coefficient over the band 1200 to 1500 MHz is shown in F4, this being taken in the presence of adjacent loaded elements (ie in the array environment). As can be seen, the element has a low reflection coefficient over the operating band, the worst voltage

0 -179- I

F1 The Antenna on the Measuring Site

-180- ) a

F2 Reflector Construction

-181- A

. 13.5

14.58

THE EXPONENTIAL HORN

ELEMENT

P3 The Exponential Horn Element

-182- F4 Smith Chart for the Exjonential Horn Element (1200-1500MHz)

TABLE ONE - RELATIVE ARRAY ELEMENT AMPLITUDES

elements 1 18 2 17 3 16 4 15 5 14 6 13 7 12 8 11 9 10 amplitudes 1127 1747 2861 4194 5644 7074 8329 9263 9761

TABLE TWO - FEED NETWORK LINE IMPEDANCES elements main line section impedances between junctions 1 to 2 50.00 55.92 77.50 107.40 120.12 2 to 3 35.30 40.05 57.84 83.52 94.75 3 to 4 32.73 36.08 47.98 63.79 70.33 4 to 5 29.22 31.52 39.33 49.06 52.92 5 to 6 25.71 27.23 32.22 38.12 40.38 6 to 7 22.34 23.29 26.30 29.71 30.97 7 to 8 19.12 19.65 21.27 23.02 23.66 8 to 9 16.06 16.2? 16.92 17.60 17.83

-183- Thr • 7

F5 The Array Construction Showing Chokes (hatched)

-184-

• •

F6 A Front View of the Array standing wave ratio being 1.06-. Further development of the element has produced a radiator with a worst V.S.W.R. of 1.1 over a greater than 2:1 operating band. There is a patent pending on the design.

As might be anticipated from the shape of the element, the coupling between adjacent array elements is fairly high, varying from -30 to -22 decibels from the low to high end of the band. The quarter wavelength choke illustrated in F5 was designed empirically and reduces the adjacent element coupling level to -33 to -28 decibels over the operating band at the expense of a small degradation in the element reflection coefficient. The final form of the array elements and chokes are shown in F6 which is a photograph of the array.

3. The Power Division Network The feed network is constructed in coaxial line using shunt junctions as described in the previous chapter. As the array excitation is symmetric about the array centreline, the feed actually consists of two identical half networks each feeding half the array. As both sum and difference patterns are required, these in turn are fed from a hybrid junction. Each 9 element feed is constructed in coaxial line with an outer conductor of 7/16" inside and 1/2" outside diameter. The feed configuration is shown in F7 where the output arms to, the array elements are labelled 1 to 9. These lines, which have a constant impedance of 50 ohms, join the 'main line' of the feed at a number of shunt junctions. Referring to F7, power enters each junction on the right-hand line and splits between the two output lines, one feeding an array element and the other the remainder of the network. Now, assuming matched elements, the former presents an impedance of 50 ohms at the junction and, knowing the power division required, this allows us to calculate the impedance that the 'main line' output arm must present at the junction. The impedance seen by the 'main line' input arm at the junction is simply the parallel combination of the output arm impedances. Thus for each main line section between adjacent junctions the impedance seen at its left-hand end and that desired at its right-hand end is known and a multiple quarter wavelength section impedance transformer can be inserted into the line to achieve the desired transformation. In the design described here three section transformers are used, the coaxial inner for each main line section being stepped in diameter five times to yield the desired characteristic impedances (three sections for the transformer and one each for the lines entering the junctions at each end of the line). Table 1 gives the array amplitude distribution and

-186- • •

F? The Coaxial Feed Network for Half of the Array F8 A Typical Network Junction

-188- a •

F9 A Rear View of the Array Showing the Feed Network and H brid Junction Table 2 gives the characteristic impedances of the sections in the half feeds.

The construction of a typical junction is illustrated in F8 where it is seen that three dielectric washers support the coaxial inner conductors which are undercut to maintain a constant characteristic impedance through these. washers. The three lines entering the junction are attached to a short cylindrical metal block running at right angles to the lines themselves. In addition to providing a convenient means of construction, this block provides a mechanism for matching out any lumped discontinuity impedances at the junction. Its inductive and capacitative impedances are controlled in the main by its diameter and length respectively and by adjusting these dimensions the total lumped impedance at the junction can be tuned out. Early junctions were made in a °T' configuration with 'main' line sections in line and the output ports at right angles but it was found that the symmetric 120° arrangement of the present feed junctions improved both the match and phase performance of the junctions. Using these techniques it has proved possible to match all junctions to 1.01 over the operating band from 1200 to 1500 MHz.

The output ports of the feed are connected to the array elements using semi- rigid coaxial line sections whose lengths are tailored to equalise the total path length between each array element and the network input port. The feed network can be seen in F9 which also shows the hybrid rat-race which is used to obtain both sum and difference patterns. Unfortunately, as the time available for the design and development of the antenna was very limited, a decision was taken that detailed bench measurements of feed network performance would only be undertaken if the array far field patterns proved to be unsatisfactory. Thus, as the latter were acceptable, no data is presently available on the distributional tolerances achieved in this 4 feed design.

The choice of a shunt junction coaxial feed which, in contrast to a directional coupler network, is sensitive to 'reverse' coupling effects on the array, is based upon its constructional simplicity. In the feed used nearly all precision components are produced by turning on a lathe, a process that can provide high accuracy at relatively low cost. This would not be true for a directional coupler feed which would in consequence be more costly to construct for the same degree of precision.

-190- 0 4.. Antenna Patterns The E-plane patterns of the 18 element feed are shown in figures F10 to F16 over the band 1215 to 1L1-85 MHz. The mid-band pattern (F13) achieves a side lobe level of -36 decibels, some 1+ decibels higher than the design target. Between 1250 and 1450 MHz there are no side lobes above -32 dB and except for the upper end of the band where grating lobes are in evidence, the side lobes are mostly below -35 decibels. These patterns were taken on a ground reflection range and as the vertical beam width of the feed is very wide it is likely that there are some site errors in these patterns.

Before fitting the feed to the reflector a near field phase and amplitude plotter was used to measure the phase front produced by the feed with a view to trimming the feed line lengths to improve feed performance. Predicting the amount by which each element feed line should be altered proved to be extremely difficult as it was found that changing the line length to a particular element often changed the phase of other elements in the array, almost certainly as a result of array-feed interactions as discussed in chapter four. Nevertheless some improvement was achieved. The E-plane patterns of the complete antenna (ie feed plus reflector) at 1300, 1350 and 1400 MHz are shown in F17, F18 and F19. Unfortunately patterns are not available outside of this reduced band. At 1350 MHz an excellent pattern has been obtained with good decay of side lobe levels away from the main beam and only two isolated side lobes above -40 decibels. At 1300 MHz there is an isolated side lobe at -32 decibels and only four other side lobes above -40 decibels and at 1400 MHz a good proportion of the side lobe structure lies below the design level.

At first sight patterns F17-F19 for the feed with reflector are distinctly different from the corresponding patterns F12-F14 of the feed alone. However, mounting the feed on a reflector reverses its pattern and as a result positive angles on F17-F19 correspond to negative angles on F12-F14 and vice versa. When this effect is taken into account the patterns are in fact fairly similar except for a better decay in side lobe levels for the complete antenna. The latter effect results from the fact that the reflector is only slightly longer than the feed. A greatly extended reflector would behave as an almost perfect reflector in the horizontal plane, reproducing the feed pattern almost exactly except in the immediate vicinity of endfire at t 90 degrees. When the reflector length is reduced, wide angle feed radiation, in effect, partially 'misses' the reflector, leading to a reduced antenna gain for the wide angle side lobes. Of course, -191- 1215 MHz -10

0 w -J -20

-30

-50

60 1 -90 -30 0 30 60 90 ANGLE (DEGREES)

E-plane Feed Pattern at 1215 MHz

-192- 1250MHz 1C

-4

11\1 f\(\111 1\11 1

) -90 -60 -30 0 30 60 9 ANGLE (DEGREES )

Fll Eplane Feed Pattern at 1250MHz

a -193- 1300MHz e (

-2

-4

6 .90 —60 30 0 30 60 0 ANGLE (DEGREES)

F12 E-plane Feed Pattern at 1300MHz

-194- •

1350 MHz 10 L't; 0

w w -20

4IP

-30

-40 p

-50

-60 I -90 -60 -30 0 30 90 ANGLE (DEGREES)

F13 E-plane Feed Pattern at 1330MHz

-195- 1400MHz 1

-6 -90 -60 -30 0 30 60 9 ANGLE (DEGREES)

F14_ E-plane Feed Pattern at 11400 MHz

-196- LEVEL -4 (DB) 6 5 2 3 -90 E15 E-planePeedPattern at1450Mils

-60

if -30 \

ANGLE (DEGREES) -197- 0

30 1450MHz

9 0

•

-10 in- Ca

-J -20

-40

-50

60 -90 -60 -30 0 30 60 90 ANGLE (DEGREES)

F16 E-plane Feed Pattern at 1485 MHz

-198- -66T-

ismooCT ve =owed aoweijell euTd peed euvd-g ad

(933W1340) 310S4V 6 09 OE 09- 06- g 05

•

DZ 31 A 13 CI) M 01• z}-10100El.

n e 1350MHz

e ( fY

-6 3 -90 -60 -30 . 0 30 60 9 0 ANGLE (DEGREES)

F18 E-plane Feed plus Reflector Pattern at 13,50 MHz

-200- • •

1400 MHz

-30

-40

60 -90 -60 -30 0 30 60 90 ANGLE (DEGREES)

F19 E-plane Feed plus Reflector Pattern at 1400)1He

-20i- • as reflector length is reduced, 'spillover' increases and the length is a compromise between these two effects. The side lobe at -32 decibels on F17 not in evidence to the same extent on F12 is almost certainly a result of the small changes in feed path lengths that were implemented between taking the two sets of patterns. Its presence provides evidence of the effects of the interactions of the array and the feed network as discussed in the previous chapter.

•

-202- CHAPTER SIX - CONCLUSIONS

-203- • The coupling between the elements of an antenna array can modify its behaviour in two distinct ways. Firstly there is the effect that will be called 'forward' coupling in which a portion of the fields coupled onto elements that are the neighbours of the driven element is reradiated. This parasitic excitation of passive elements modifies the intrinsic pattern of the driven element giving the so called 'active element pattern', the radiation pattern of an element in the array environment. Second there is coupling between the element feed lines. The excitation of an element produces 'reflected' fields not only in its own feed line but also in those of neighbouring elements. This effect will be referred to as 'reverse' coupling.

Considering the effects of forward coupling first and using the TE array of chapter two as an example, the array pattern is given by Eqn. 9.3 (Ch. 2) as:

G(o) = Po(o) op+B op ) exp (ikx cost)

+ P1(0) ) B._ exp (slap cos0) " p

+ Ps(6)) ‘ B exp (ikx 004) + 2p p

where A is the incident excitation of element p at position x and B is op mp the resultant excitation of mode m on element p. The 'model element patterns' P m(0) are defined in chapter two (section 9). The first term in this equation represents the pattern contribution that arises from the excitation of the lowest waveguide mode, other terms resulting from the excitation of the higher order evanescent modes in the aperture plane. Each individual • term is in the form of an array factor multiplied by an element pattern and it is immediately evident that the complete array pattern cannot generally be expressed in this form. However, when higher order mode excitation is small enough to be neglected, the array pattern can be approximated as:

G(0) = P0(0) ) '(A0_+Bop) exp (ikxp coml.)) -L" p

which is in the correct form. This is often called a 'single mode approximation' as only the pattern contributions arising from the lowest (propagating) waveguide mode have been included. -204- It is often said that the variation of the active element patterns with position in an array prevent the application of the principle of pattern multiplication except as an approximation. While this is correct, it is in a sense misleading as it suggests that the principle cannot apply when the active element patterns vary. In practice it continues to apply in the presence of wide variations in active element patterns provided that the radiation from higher order mode excitation is negligable. When an element is excited on its own, its pattern is the sum of its intrinsic radiated field with the field radiated as a result of the parasitic excitation of other array elements, it being the latter component that results in pattern variation with position. However, rather than accounting for the parasitic excitation of other elements in the pattern of the driven element, we can attribute an identical passive element pattern to all array elements and compute a modified array distribution which accounts for the direct and parasitic excitation of each array element. Equation 1.2 is in this form, each element havinga passive element pattern P0(0), and an 'effective' excitation A op +Bop that includes both direct and parasitic excitation of the element. Thus when the radiation from higher order modes can be neglected, the principle of pattern multiplication continues to apply with an array factor calculated from a modified distribution.

As nearly all array theory is based on the pattern multiplication principle, it is evident that there are distinct advantages in using arrays for which the principle applies. It is clear from the above discussion that it is not active element pattern variation as such that prevents its application, but rather the excitation of higher order waveguide modes in the aperture plane. One way in which the practical attainment of low side lobes could be simplified therefore is by use of 'single mode radiators', elements that can only support one mode. Clearly practical elements can only approximate this ideal but it should be possible to devise an element for which higher modes are so heavily damped that they are effectively non-existent. On the TE array • discussed in chapter two for example, one possibility is the use of very narrow waveguide elements for which the higher modes are well below cut-off. The main difficulty with this approach is that, for normal element spacings, such elements exhibit high reflection coefficients giving a low radiation efficiency for the array. If the element mismatch is tuned out, the array bandwidth is generally greatly reduced. If however narrow elements are used in conjunction with small element spacings, while element reflection coefficients in isolation are still high, the mutual coupling effects for distributions that vary only slowly from element to element are such that the reflected fields are low in actual use. The problem in this instance is clearly one of economy in that this course of action requires many -205- •

F1 HIGHER MODE REDUCTION

...

IN -206- more elements than would normally be used on a given aperture. An interesting combination of these two approaches is illustrated in F1 which shows both plan and front views of a possible TE array implementation. This can be considered in two distinct ways. First it can be viewed as a large number of narrow elements with a specialised feed network that feeds them in groups or secondly as an array of wide elements in which baffles have been inserted to suppress the higher order modes. While this should reduce higher order mode effects, in doing so it is likely that other effects such as the interactions between reflections from either end of the baffle structure have been introduced. However the geometry lends itself to a fairly precise analysis and it would certainly be interesting to determine its properties.

When higher mode effects can be neglected the array pattern is given by Eqn. 1.2 which can be written in the form:

G(o) Po(o) e (o) ( b) 1.3 by defining column vector array excitations a and b whose p'th components are Aop and Bop respectively and a column vector e(0) with exp ( -ikx cosp) as its p'th component (the superscript + denotes conjugate transposition). Using conventional array theory the pattern would be evaluated as:

G(0) = P0(0) 2t (0) a which shows that to this level of approximation the effect of mutual coupling is to modify the 'effective' distribution as far as radiation is concerned from a in isolation to a + b in practice. Thus it is a + b rather than a alone that has to be equated to a desired distribution to obtain the corresponding pattern. Remembering that b is related to a through the array scattering matrix S as discussed in chapters two and three, Eqn. 1.3 can be written:

G(95) = P0(0) e (p) (I + s) a 1.5

If t is a distribution giving some desired pattern, then using an incident distribution a given by:

-1 a = (I + S) t ... 1.6

is seen by substitution into Eqn. 1.5 to give the desired far field pattern corresponding to the array distribution t. This demonstrates that it is -207- OP possible to compensate exactly for the effects of mutual coupling on arrays consisting of single mode radiators. However it should, be noted that the array scattering matrix S and therefore the distribution a are functions of frequency and as a result compensation over an extended band requires the design of a power division network that provides a distribution that varies in a prescribed manner with frequency. This is a complex problem that, to my knowledge, has not been solved and we are forced in practice to adopt an approach that falls short of this ideal. Fortunately when the incident distribution a varies only slowly from element to element, the reflected distribution b is very similar in form. When this applies the effective distribution a b is not very different from the incident distribution except for an overall change in magnitude and as a result the array pattern ■ does not suffer any significant degradation. This behaviour is illustrated in the results presented in chapter two. An alternative approach to over- coming the effects of mutual coupling that is briefly discussed in chapter two relies on the addition of passive elements on each end of an array to provide an essentially identical environment for each of the driven elements. Clearly, given the addition of enough passive elements, the active patterns s' of the driven elements can be made identical to some prescribed tolerance. If this is achieved the principle of pattern multiplication will apply and the active element patterns can be factored out leaving an array factor for the incident distribution. This approach has a major advantage in that the 'effective' distribution is simply the incident distribution a which can therefore be a direct implementation of any low side lobe distribution. Most significantly this distribution will be independent of frequency, allowing the use of well established methods for the design of wide-band constant power division networks. The results presented in chapter two show the technique to be viable but suggest, as might be expected, that it makes inefficient use of the available aperture. When the total aperture is limited, the need for passive elements at each end decreases the length of the active part of the aperture thus reducing the gain and increasing the beam width in comparison to that available from the complete aperture. Nevertheless the technique may find application where aperture length is not limited. It is also worth noting that it takes account of the effects of higher mode radiation.

The synthesis technique presented in chapter three provides a method for allowing for the effects of mutual coupling even when higher mode radiation cannot be neglected. However it provides distributions that depend upon the eigenvectors of the array scattering matrix S and as a result they exhibit a variation with frequency that will make wide-band implementation difficult or impossible. -208- In summary then the above discussion suggests there are a number of techniques that can compensate for the effects of 'forward' coupling over narrow bandwidths. While not proven, it seems likely that compensation for such effects over extended bandwidths, using such techniques as added passive elements for example, are accompanied by a reduction in radiating efficiency or aperture utilisation. However, perhaps most significantly, the results presented in chapter two suggest that 'forward' coupling effects, even without compensation, do not significantly limit the attainment of low side lobes in practice. p

Turning now to 'reverse' coupling, it is clear that its influence on side lobe level depends upon the array feed network. First of all its effects can be completely eliminated by using feeds composed of directional couplers or any devices that prevent aperture 'reflected' fields returning through the feed network to other array elements. The results of chapter four show that when such fields can return to the aperture for a second time, the side lobe degradation can be quite significant. Rather than producing a general increase in side lobe level characteristic of random errors, it generally gives rise to isolated side lobes well above the design level. In addition these error side lobes exhibit a sensitivity to frequency variation, rising and falling over relatively narrow bands. As array feed networks exhibit a high degree of constructional regularity, it is hardly surprising that 'reverse' coupling effects result in errors that are more correlated than random in form. Equally, rapid frequency variation is to be expected as fairly long path lengths are involved and will result in rapid phase variations with frequency. Thus, in summary, for feeds in which fields can travel between the network ports feeding array elements, reverse coupling effects will generally lead to isolated side lobes at fairly high levels. If such effects cannot be tolerated, it is essential that the feed network is of a type that does not contain direct pathways between its ports that

feed the array elements. r

The design of wide bandwidth feed networks to yield an ideally constant array distribution has been covered in chapter four. While from a practical viewpoint the results presented are quite satisfactory, from a theoretical viewpoint they are perhaps less so in that they have been achieved via analysis of preconceived configurations rather than synthesis. Synthesis is generally a preferable technique as it often yields solutions that are known to be optimum in some defined sense. Analysis contains no such guarantees, it being highly probable that better performance can be obtained with networks that are simpler in form.

-209- REFERENCES

-210- R1 Woodward P M A Method of Calculating the Field Over a Plane Aperture to Produce a Given Polar Diagram. Proc. IEE, Volume 93, Part 3, Pages 1554-1558, March-May 1946. R2 Dolph C L A Current Distribution for Boadside Arrays which Optimises the Relationship Between Beam Width and Side Lobe Level. Proc. IRE, Pages 335-348, June 1946. R3 Taylor T T Design of Line Source Antennas for Narrow Beam Width and Low Side Lobes. IRE Trans., Volume AP-3, Pages 16-28, January 1955. R4 Rhodes D R Synthesis of Linear Aperture Distributions by Pattern Sampling. Proc INt, Volume 119, No 7, Pages 827-831, July 1972. R5 Rhodes D R On a New Condition for Physical Realisability of Planar Apertures. IRE Trans., Volume AP-19,No 2, Pages 162-166, March 1971.

R6 Rhodes D R On an Optimum Line Source for Maximum Directivity. IEEE Trans., Volume AP-19, No 4, PageS 485-492, July 1971.

R7 Ruze J Antenna Tolerance Theory - A Review. Proc. IEEE, Volume 54, Pages 633-640, April 1966. R8 Schanda E The Effects of Random Amplitude and Phase Errors on Continuous Apertures. IEEE Trans., Volume AP-15 Pages 471-473, May 1967. R9 Brantley E N Some Aspects of the Rapid Directional Fluctuations of Short Radio Waves Reflected from the Ionosphere. Proc. IEE, Volume 102B, Pages 533-540, 1955. R10 Special Issue on Electronic Scanning. Proc IEEE, Volume 56, November 1968. R11 Hansen R C Significant Phased Array Papers. (Editor) Artech House Inc., 1973. R12 Special Issue on Conformal Arrays. IEEE Trans., Volume AP-22, No 1, January 1974. R13 Watson G N A Treatise on the Theory of Bessel Functions. Cambridge University Press, 1966. R14 Handbook of Mathematical Functions. Dover Publications, Inc., New York. R15 Hissink A J Rapid Generation of Bessel Functions Required for and Solutions to Engineering Problems. Gladman B R Electronics Letters, Vblume 6, No 8, April 1970.

R16 Clenshaw C W Chebyshev Series for Mathematical Functions. Mathematical Tables, Volume 5. National Physical Laboratory. -211- R17 Ralston A A First Course in Numerical Analysis. McGraw-Hill, New York,1965.

R18 Watson G A An Algorithm for the Inversion of Block Matrices of Tceplitz Form. Department of Mathematics, University of Dundee.

R19 Wilkinson J H The Algebraic Eigenvalue Problem. Oxford University Press, 1965.

R20 Cheng D X Optimisation Techniques for Antenna Arrays. t Proc. IEEE, Volume 59, No 12, December 1971.

R21 Stein S On Cross Coupling in Multiple Beam Antennas. IRE Transactions on Antennas and Propagation, September 1962.

R22 Shaw E The Maxon Multi-Beam Antenna: Theory and Design for Non-Interacting Beams. The Radio and Electronic Engineer, February 1969, Volume 37, Page 117.

R23 Rhodes Synthesis of Linear Aperture Distributions by Pattern Sampling. Proc. _LEE, Volume 119, No 7, Pages 827-831, July 1972.

R24 Salt H An Investigation into Producing Superdirectivity from a Two Wavelength Aperture. Ph. D. Thesis, University of London, June 1973. R25 Montgomery, Principles of Microwave Circuits. Dicke and McGraw-Hill 1948. Purcell

R26 Levy R General Synthesis of Asymmetric Multi-element Coupled Transmission Line Directional Couplers. IEEE Transactions, MTT July 1963.

R27 Toulious P P Synthesis of Symmetrical TEM Mode Directional and Couplers. Todd A C IEEE Trans., MTT13, September 1965.

R28 Cristal E G Theory and Tables of Optimum Symmetrical TEM Mode • and Coupled Transmission Line Directional Couplers. Young L IEEE Transactions, Volume MTT13, No 5 September 1965.

229 Riblet H J General Synthesis of Quarter Wave Impedance Transformers. IRE Transactions, MTT January 1957.

-212- 4 -213- APPENDICES

• -214-

APPENDIX ONE - CHEBYSHEV ARRAY DESIGN

This Appendix contains further information on the design of Chebyshev arrays. Two methods of generating the element amplitudes are derived and implemented in computer programs. Tables are given for normalised beam widths and aperture efficiencies as functions of the number of elements and the design side lobe level.

Element Amplitudes by Summation

Let an be the amplitude of the n'th element of an N element array (1:nAN). In the notation of chapter one these amplitudes are such that: N

[s cos -(2/2)]4R = exp (N+1 ) 4/21 aN exp (inu) TN-1 n=1

where the phase reference is at the centre of the array rather than at the end. Now it is fairly easy to show that, if 1

Applying Eqn.2 to Eqn.1 with u = 2irq/N yields:

N-1 N 1 = T [s cos (v/N)] an NR N-1 q=0

x exp [i(N-2n+1)110] • • 3

and after further reduction:

N 1 a = Eq (-)1 TN-1 Es cos (qir/N)] n q q=0

x cos [(2n.1)(17r/N] ••• 4.

.215-

where the notation [ ] on the upper limit of summation demotes the integer part of the enclosed expression.

Element Amplitudes by Recurrence

Starting with Eqn. 1 a recurrence scheme for the a1\1 can be derived using the identity:

T ( 1 + TN(x) = 2x T (x) TN-2(x) N-1 .• • 5

Substituting from Eqn. 1 into 5 gives: N-1 N-1 N exp [-iNu/2] an exp (inu) + exp [-i(N+2)u/2]a exp (inu)

n=1 n=1

N = 2s cos (u/2) exp [-i(N+1)u/2] aN n exp (inu) ••• 6 n=1

Expanding the cosine term on the right-hand side and simplifying gives: N-1 N+1 aN-1 exp (inu) + aN+1 exp [i(n-1)u] 1 n=1 11=

N N = sXaNan (inu) + s) \ aN exp [i(n-l)u] ... 7 / / n n=1 n=1

Collecting coefficients of like exponentials: N aN+1 [an - s an - s aN n n+1 n+1 ] exp (inu) = 0 n=0

Where aN is zero if n is outside the range 1:n:N. Since the exponentials in Eqn. 8 are independent, all coefficients must be zero. Thus:

N+1 N N. a = s (a - an ) - an n+1 n+1 4 ... 9

a recurrence relationship is both n and N for the aN . Starting values

-216- for the scheme are easily obtained as:

ai = 1/R ; a2 = a2 = OR ... 10 2

Computer Programs Description Both of the methods given for element amplitude evaluation have been implemented in computer programs. The subroutine CHVDSTR uses the recurrence scheme, the coding being fairly self explanatory with the variable names corresponding with the notation of this chapter whenever possible. The inner loop recurrence is in the variable N (NU in the program, 'upper case' N) with the array T holding the a value produced for a particular n value n (NL in the program). Reference to the recurrence formula Eqn. 9 shows that, with the loops arranged thus, the computation of the set of aN requires a knowledge of the set being generated and also' the previous set. With the N recurrence order reversed, storing the set of a for a fixed N value, computation of the current set would require knowledge of the two preceding sets. The former method is thus slightly easier to implement. The subroutine CHVDSTS uses the summation formula derived above Eqn. and is again fairly self explanatory. When the number of elements N is small CHVDSTR is more efficient than CHVDSTS, the opposite being true when N is large. They are about equal in efficiency when N is 200 (on the computer used by the Author). Both routines operate in double precision arithmetic.

Design Tables Description The first four tables give the normalised beam width values for Chebyshev arrays (ie DsinG/X, D being the centre-to-centre spacing of the end elements) at -3dB and the first zero as functions of the number of elements and the design side lobe level. The last, pair tables give the aperture efficiency of the Chebyshev array distribution as function of the same parameters.

10 -217- SUBROUTINE CHVDSTR

* * SUBROUTINE FOR CHEBYSHEV AMPLITUDES BY RECURRENCE * SL= SIDELOBE LEVEL IN DECIBELS * N= NUMBER OF ELEMENTS * P OUTPUT ARRAY FOR ELEMENT AMPLITUDES * SUBROUTINE CHVDSTR(SL,N,A) DOUBLE PRECISION SL,A,R,S,T,U,SA,SB DIMENSION A(N)T(400)

* COMPUTE R, S AND STARTING VALUES FOR RECURRENCE * R=10.0D0**(DABS(SL)/20.0D0) S=DEXP(DLOG(R+DSORT(R*R-1.0D0))/(N-1)) S=0.5D0*(S+1.0DO/S) T(1)=1.0DO/R T(2)=0,5DO*S/R

* RECUR NU (UPPER CASE N) FOR NL (LOWER CASE N) = 1

DO 10 NU=3,N T(NU)=S*T(NtJ- 1) 10 CONTINUE A(1)=T(N) A(N)=T(N)

* OUTER LOOP RECURRENCE IN NL * NLX=(N+1)/2 DO 40 NL=2,NLX NUN=NL+NL SA=T(NUN-2) SP=T(NUN-1) U=2.0D0*S*SA-T(NUN-3) T(NTIN-1)=U IF(NUN.GT.N) GO TO 30 * * INNER LOOP RECURRENCE IN NU * DO 20 NU=NUN,N U=S*(T(NT1-1)+SP)-SA SA=SB SB=T(NU) T(NU)=U 20 CONTINUE * * SET AMPLITUDE VALUE T(N) INTO OUTPUT ARRAY (NOTE: U=T(N)) * 30 A(N+1-NL)=U A(NL)=U 40 CONTINUE RETURN * * DEAL WITH SMALL N VALUES

-218- SUBROUTINE CHVDSTR

50 IF(N.LE• 0) RETURN AC 1)=1 • ODO I F(N•FO• 1) P.ETIMN AC 1)=0 • 5D0 AC 2.)=0.5D0 RETURN END SUBROUTINE CRVDSTS

* SUBROUTINE FOP CHERYSHEV AMPL I TUDES BY SUMMATION

* 51.= SI DFLOBE LEVEL IN DECIBELS * N.= NUMBER OF ELEMENTS * A OUTPUT ARRAY FOR ELEMENT AMPLITUDES

SUBROUTINE CHVDSTS( SL, Ns A) DOUBLE PRECISION SL, As Co Br Sr T, Us TCH DIMENSION A(N),C( 400),TE 100) * * COMPUTE R.* S AND TABLES OF COSINE AND CHEBYSHEV VALUES

IF(N.LE• 2) GO TO 50 PI =3.14159265358981)0 EN=N R=10.0D0**(DABS(SL)/20.0D0) S=DEXP( DLOG( R+DSORT( R*R-1• ()DO) ) /( EN-1. °DO) ) S=0.5D0*CS+1.ODO/S) U=PI /EN NL X ta 2*N DO 10 NL= 1 NLX C(NL)=DCOS( (NL-1)*U) 10 CONTINUE IQX=CN-1)/2 U=-1.0 DO DO 20 10=1..10X TC I0)=U*TCH( S*C( I 0+1),N-1) U=—U 20 CONTINUE

* LOOP FOR ELEMENT NUMBER * NLX= I GX+ 1 DO 40 NL=1.•NLX I=2*NL— 1 U=0. ODO * * PERFORM SUMMATION EXCEPT FOR FIRST TERM * DO 30 I0=1.•IQX J=MOD( I O* I, 2*N ) 1 U=U+T( I 0 )*C( J) 30 CONTINUE * * ADD IN FIRST TERM AND EVALUATE AMPLITUDE

Uzz( 1 • OD0+2. 0 DO*U/R) /EN * * SET AMPLITUDE INTO OUTPUT ARRAY

A( N+ 1.-NL )=1) A(NL)=U 40 CONTINUE RETURN

—220— SUBROUTINE CHVDSTS

* DEAL WITH SMALL N VALUES

50 IF(N.LE.0) RETURN A(1)=1.0D0 IF(N.E0.1) RETURN A(1)=0.5D0 A(2)=0.5D0 RETURN END

* FUNCTION TO EVALUATE THE CHEBYSHEV POLYNOMIAL * FUNCTION TCH(X,N) DOUBLE PRECISION TCH,X,Y IF(X.GT01.0D0) GO TO 10 TCH=DCOS(N*DACOS(X)) RETURN 10 Y=DEXP(N*DLOG(X+DSORT(X*X-, 1.0D0))) TCH=0.5D0*(Y+1.0DO/Y) RETURN END

-221- CHEBYCHEV DISTRIBUTION HALF POWER BEAMWIDTH AS A FUNCTION OF SIDELODE LEVEL (SL) AND NUMBER OF ELEMENTS (N).

N SL= -20 -30 -40 -50 -60 -70 -80 1 3 0.3452 0.3577 0.3620 0.3634 0.3639 0.3640 0.3640 5 0.4108 0.4568 0.4845 0.5008 0.5102 0.5155 0.5185 7 0.4293 0.4917 0.5369 0.5691 0.5918 0.6077 0.6187 9 0.4365 0.5066 0.5613 0.6039 0.6370 0.6626 0.6822 11 0.4401 0.5141 0.5741 0.6231 0.6632 0.6959 0.7226 13 0.4420 0.5183 0.5816 0.6346 0.6793 0.7171 0.7490 15 0.4432 0.5209 0.5862 0.6419 0.6897 0.7311 0.7669 17 0.4440 0.5226 0.5894 0.6468 0.6969 0.7408 0.7795 19 0.4445 0.5238 0.5915 0.6503 0.7019 0.7477 0.7885 21 0.4449 0.5247 0.5931 0.6528 0.7056 0.7528 0.7953 23 0.4452 0.5253 0.5943 0.6547 0.7084 0.7567 0.8004 25 0.4454 0.5258 0.5952 0.6561 0.7105 0.7597 0.8044 27 0.4456 0.5262 0.5959 0.6572 0.7122 0.7620 0.8075 29 0.4457 0.5265 0.5964 0.6581 0.7136 0.7639 0.8101 31 0.4458 0.5268 0.5969 0.6589 0.7147 0.7655 0.8121 33 0.4459 0.5270 0.5972 0.6595 0.7156 0.7668 0.8138 35 0.4460 0.5271 0.5976 0.6600 0.7163 0.7678 0.8153 37 0.4460 0.5273 0.5978 0.6604 0.7170 0.7687 0.8165 39 0.4461 0.5274 0.5980 0.6608 0.7175 0.7695 0.8175 41 0.4461 0.5275 0.5982 0.6611 0.7180 0.7701 0.8184 43 0.4462. 0.5276 0.5984 0.6613 0.7183 0.7707 0.8191 45 0.4462 0.5277 0.5985 0.6616 0.7187 0.7712 0.8198 47 0.4462 0.5277 0.5986 0.6618 0.7190 0.7716 0.8204 49 0.4463 0.5278 0.5988 0.6619 0.7193 0.7720 0.8209 51 0.4463 0.5278 0.5988 0.6621 0.7195 0.7723 0.8213 53 0.4463 0.5279 0.5989 0.6622 0.7197 0.7726 0.8217 55 0.4463 0.5279 0.5990 0.6624 0.7199 0.7728 0.8221 57 0.4463 0.5280 0.5991 0.6625 0.7200 0.7731 0.8224 59 0.4463 0.5280 0.5991 0.6626 0.7202 0.7733 0.8227 61 0.4464 0.5280 0.5992 0.6626 0.7203 0.7735 0.8229 63 0.4464 0.5281 0.5992 0.6627 0.7204 0.7736 0.8232 65 0.4464 0.5281 0.5993 0.6628 0.7206 0.7738 0.8234 67 0.4464 0.5281 0.5993 0.6629. 0.7207 0.7739 0.8236 69 0.4464 0.5281 0.5994 0.6629 0.7206 0.7741 0.8238 71 0.4464 0.5281 0.5994 0.6630 0.7208 0.7742 0.8239 73 0.4464 0.5282 0.5994 0.6630 0.7209 0.7743 0.8241 75 0.4464 0.5282 0.5995 0.6631 0.7210 0.7744 0.8242 77 0.4464 0.5282 0.5995 0.6631 0.7210 0.7745 0.8243 79 0.4464 0.5282 0.5995 0.6632 0.7211 0.7746 0.8244

422.. CHFPYCHFV DISTRIBPTION HALF POWER BFAMWIDTH AS A FUNCTION OF SIDFLODE LEVEL (SL) AND NUMBER OF ELEMENTS (N). N S1..= -20 -30 -40 -50 -60 7() -80 2 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 4 0.3893 0.4206 0.4362 0.4437 0.4472 0.4488 0.4496 0.4225 0.4782 0.5160 0.5409 0.5571 0.5675 0.5742 8 0.4336 0.5005 0.5512 0.5892 0.6176 0.6386 0.6541 10 0.4386 0.5109 0.5686 0.6148 0.6517 0.6812 0.7046 12 0.4412 0.5164 0.5783 0.6295 0.6721 0.7076 0.7372 14 0.4427 0.5198 0.5842 0.6386 0.6850 0.7248 0.7588 16 0.4436 0.5219 0.5879 0.6446 0.6936 0.7364 0.7737 18 0.4443 0.5233 0.5905 0.6487 0.6996 0.7445 0.7843 20 0.4447 0.5243 0.5924 0.6516 0.7039 0.7505 0.7921 22 0.4450 0.5250 0.5937 0.6538 0.7071 0.7549 0.7980 24 0.4453 0.5256 0.5947 0.6554 0.7095 0.7583 0.8025 26 0.4455 0.5260 0.5955 0.6567 0.7114 0.7609 0.8060 29 0.4456 0.5264 0.5962 0.6577 0.7129 0.7630 0.8089 30 0.4458 0.5267 0.5967 0.6585 0.7142 0.7648 0.8111 32 0.4459 0.5269 0.5971 0.6592 0.7151 0.7661 0.8130 14 0.4459 0.5271 0.5974 0.6597 0.7160 0.7673 0.8146 36 0.4460 0.5272 0.5977 0.6602 0.7167 0.7683 0.8159 38 0.4461 0.5273 0.5979 0.6606 0.7172 0.7691 0.8170 h0 0.4461 0.5275 0.5981 0.6609 0.7177 0.7698 0.8179 42 0.4461 0.5275 0.5983 0.6612 0.7182 0.7704 0.8188 44 0.4462 0.5276 0.5985 0.6614 0.7185 0.7709 0.8195 46 0.4462 0.5277 0.5986 0.6617 0.7188 0.7714 0.8201 48 0.4462 0.5278 0.5987 0.6619 0.7191 0.7718 0.8206 50 0.4463 0.5278 0.5988 0.6620 0.7194 0.7721 0.8211 52 0.4463 0.5279 0.5989 0.6622 0.7196 0.7724 0.8215 54 0.4463 0.5279 0.5990 0.6623 0.7198 0.7727 0.8219 56 0.4463 0.5279 0.5990 0.6624 0.7200 0.7730 0.8222 58 0.4463 0.5280 0.5991 0.6625 0.7201 0.7732 0.8225 60 0.4464 0.5280 0.5992 0.6626 0.7203 0.7734 0.8228 62 0.4464 0.5280 0.5992 0.6627 0.7204 0.7736 0.8231 64 0.4464 0.5281 0.5993 0.6628 0.7205 0.7737 0.8233 66 0.4464 0.5281 0.5993 0.6628 0.7206 0.7739 0.8235 68 0.4464 0.5281 0.5993 0.6629 0.7207 0.7740 0.8237 70 0.4464 0.5281 0.5994 0.6630 0.7208 0.7741 0.8238 72 0.4464 0.5281 0.5994 0.6630 0.7209 0.7743 0.8240 74 0.4464 0.5282 0.5994 0.6631 0.7209 0.7744 0.8241 76 0.4464 0.5282 0.5995 0.6631 0.7210 0.7745 0.8243 78 0.4464 0.5282 0.5995 0.6631 0.7211 0.7745 0.8244 80 0.4464 0.5282 0.5995 0.6632 0.7211 0.7746 0.8245

423.. 0 CHET3YCHFV DISTRIBUTION ZERO POINT BEAMWIDTH AS A FUNCTION OF SIDELOPE LEVEL (SL) AND NUMPR OF FIX4ENTS (N). N SL= -20 -30 -40 -50 -60 -80

1 3 0.8050 0.8880 0.9365 0.9642 0.9799 0.9887 0.9936 5 0.9869 1.2090 1.3930 1.5388 1.6516 1.7376 1.8028 7 1.0338 1.3105 1.5661 1.7934 1.9907 2.1591 2.3015 9 1.0517 1.3521 1.6427 1.9150 2.1649 2.3910 2.5934 11 1.0603 1.3726 1.6820 1.9799 2.2618 2.5254 2.7695 13 1.0650 1.3342 1.7045 2.0179 2.3201 2.6082 2.8808 15 1.0679 1.3913 1.7185 2.0419 2.3573 2.6621 2.9545 17 1.0698 1.3959 1.7277 2.0579 2.3825 2.6989 3.0054 19 1.0711 1.3992 1.7342 2.0691 2.4002 2.7250 3.0419 21 1.0720 1.4015 1.7388 2.0773 2.4131 2.7441 3.0688 23 1.0727 1.4032 1.7423 2.0833 2.4228 2.7586 3.0891 25 1.0732 1.4045 1.7449 2.0880 2.4303 2.7697 3.1049 27 1.0736 1.4055 1.7470 2.0916 2.4361 2.7784 3.1173 29 1.0740 1.4064 1.7486 2.0945 2.4408 2.7854 3.1272 31 1.0742 1.4070 1.7500 2.0969 2.4445 2.7911 3.1353 33 1.0744 1.4076 1.7511 2.0988 2.4477 2.7958 3.1420 35 1.0746 1.4080 1.7520 2.1004 2.4502 2.7997 3.1476 37 1.0748 1.4084 1.7527 2.1018 2.4524 2.8029 3.1523 39 1.0749 1.4087 1.7534 2.1029 2.4543 2.8057 3.1563 41 1.0750 1.4090 1.7539 2.1039 2.4558 2.8081 3.1597 43 1.0751 1.4092 1.7544 2.1047 2.4572 2.8102 3.1626 45 1.0752 1.4094 1.7548 2.1055 2.4584 2.8119 3.1652 47 1.0752 1.4096 1.7552 2.1061 2.4594 2.8135 3.1674 49 1.0753 1.4097 1.7555 2.1067 2.4603 2.8149 3.1694 51 1.0754 1.4099 1.7558 2.1071 2.4611 2.8161 3.1711 53 1.0754 1.4100 1.7560 2.1076 2.4618 2.8172 3.1727 55 1.0754 1.4101 1.7562 2.1080 2.4624 2.8181 3.1740 57 1.0755 1.4102 1.7564 2.1083 2.4630 2.8190 3.1753 59 1.0755 1.4103 1.7566 2.1086 2.4635 2.8197 3.1764 61 1.0756 1.4104 1.7568 2.1089 2.4640 2.8204 3.1774 63 1.0756 1.4104 1.7569 2.1092 2.4644 2.8211 3.1783 65 1.0756 1.4105 1.7571 2.1094 2.4648 2.8216 3.1791 67 1.0756 1.4106 1.7572 2.1096 2.4651 2.8222 3.1799 69 1.0757 1.4106 1.7573 2.1098 2.4654 2.8226 3.1806 71 1.0757 1.4107 1.7574 2.1100 2.4657 2.8231 3.1812 73 1.0757 1.4107 1.7575 2.1102 2.4660 2.8235 3.1818 75 1.0757 1.4107 1.7576 2.1103 2.4662 2.8238 3.1823 77 1.0757 1.4108 1.7576 2.1105 2.4665 2.8242 3.1828 79 1.0757 1.4108 1.7577 2.1106 2.4667 2.8245 3.1832

-224- CH FPYCH FV DI SIR' MITI ON ZERO POINT Ff.:A:41i PTIf AS A FUNCTION OF SI DFL OPE I, FA/Ft. (St.) AND NUMBER OF FL FAT TS ( N -30 -40 -50 -60 -70 -80 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.9299 1.0976 1.2213 1. 3086 1.3691 1.4107 1.4391 6 1.0166 1.2721 1.4985 1.6909 1.8503 1.9804 2.0854 1.0445 1.3353 1.6113 1.8644 2.0913 2.2916 2.4665 10 1.0567 1.3639 1.6653 1.9521 2.2199 2.4668 2.6920 12 1 • 0629 1.3792 1.6947 2.0012 2.2944 2.5715 2.3311 14 1 • 0666 1.3881 1.7122 2.0312 2.3406 2.6379 2.9212 16 1.0689 1.3938 1.7236 2.0507 2.3711 2.6822 2.9322 18 1.0705 1.3977 1.7312 2.0640 2.3921 2.7130 3.0251 20 1 • 0716 1.4004 1.7367 2.0735 2.4072 2.7353 360563 22 1.0724 1.4024 1.7407 2.0805 2.4183 2.7518 3.0796 24 1.0730 1 • 4039 1.7437 2.0858 2.4268 2.7645 3.0975 26 1.0734 1.4051 1.7460 2.0899 2.433/4 2.7743 3.1114 28 1.0738 1.4060 1.7479 2.0932 2.4386 2.7821 3.1225 30 1 . 0741 1.4067 1. 7493 2.0958 2.4428 2.7884 3.1315 32 1 • 0743 1.4073 1.7506 2.0979 2.4462 2.7935 3.1388 34 1.0745 1.4078 1.7515 2.0997 2.4490 2.7978 3.1449 36 1.0747 1.4082 1.7524 2.1011 2.4514 2.8014 3.1500 38 1 • 0748 1.4085 1.7531 2.1024 2.4534 2.8044 3.1543 40 1 • 0749 1 • 4088 1.7537 2.1034 2.4551 2.8070 3.1580 42 1.0750 1.4091 1.7542 2.10143 2.4565 2.8092 3.1612 1 • 0751 1.4093 1.7546 2.1051 2.4578 2.8111 3.1639 1.0752 1.4095 1.7550 2.1058 2.4589 2.8127 3.1663 48 1 • 0753 1.4097 1.7553 2.1064 2.4599 2.8142 3.1684 50 1.0753 1.4098 1.7556 2.1069 2.4607 2.8155 3.1703 52 1.0754 1.4099 1.7559 2.1074 2.4615 2.8166 3.1719 54 1.0754 1 • 4100 1.7561 2.1078 2.4621 2.8176 3.1734 56 1 • 0755 1.4102 1.7563 2.1082 2.4627 2.8186 3.1747 58 1. 0755 1.4102 1.7565 2.1085 2.4633 2.8194 3.1758 60 1 • 0755 1.4103 1.7567 2.1088 2.4638 2.8201 3.1769 62 1 • 0756 1.4104 1.7568 2.1091 2.4642 2.8208 3.1779 64 1.0756 1.4105 1.7570 2.1093 2.4646 2.8214 3.1787 6.6 1.0756 1.4105 1.7571 2.1095 2.4650 2.8219 3.1795 68 1.0756 1.4106 1.7572 2.1097 2.4653 2.8224 3.1802 70 1.0757 1 • 4106 1.7573 2.1099 2.4656 2.8229 3,1809 72 1 • 0757 1.4107 1.7574 2.1101 2.4659 2.8233 3.1815 74 1.0757 1.4107 1.7575 2.1102 2.4661 2.8237 3.1820 76 1 • 0757 1.4108 1.7576 2.1104 2.4663 2.8240 3.1825 78 1.0757 1.4108 1.7577 2.1105 2.4666 2.8243 3.1830 80 1.0757 1.4108 1.7577 2.1106 2.4668 2.8247 3.1835

-225- CHFFYrHFV PISTRMTION APERTURE EFFICIENCY AS A FUNCTION OF SIIFLOPF LEVEL (SL) AND,,NTYMBER OF ELEMENTS (N). N SL= -20 -30 -40 -50 -60 -70 -80 1 1.0000 1.0000 1.000"0 1.0000 1.0000 1.0000 1.0000 3 0.9423 0.9071 0.8948 0.8908 0.8895 0.8891 0.8889 5 0.9372 0.9451 0.7933 0.7655 0.7504 0.7420 0.7374 7 0.9508 0.8397 0.7659 0.7201 0.6909 0.6719 0.6593 9 0.9597 0.8443 0.7593 0.7033 0.6651 0.6383 0.6192 0.9637 0.8501 0.7591 0.6969 0.6532 0.6213 0.5975 13 0.9642 0.8554 0.7608 0.6948 0.6474 0.6121 0.5852 15 0.9625 0.8597 0.7630 0.6945 0.6446 0.6070 0.5778 17 0.9591 0.8633 0.765.3 0.6950 0.6434 0.6041 0.5733 19 0.9546 0.8662 0.7674 0.6958 0.6429 0.6024 0.5704 21 0.9493 0.6686 0.7694 0.6969 0.6429 0.6015 0.5685 23 0.9415 0.8705 0.7712 0.6978 0.6432 0.6010 0.5674 25 0.9373 0.8721 0.7723 0.6988 0.6436 0.6008 0.5666 27 0.9309 0.9734 0.7742 0.6997 0.6440 0.6008 0.5661 29 0.9241 0.8744 0.7754 0.7006 0.6445 0.6009 0.5658 31 0.9172 0.9752 0.7766 0.7014 0.6450 0.6010 0.5657 33 0.9103 0.8759 0.7776 0.7022 0.6455 0.6013 0.5656 35 0.9033 0.8764 0.7785 0.7029 0.6460 0.6015 0.5657 37 0.8963 0.8767 0.7793 0.7036 0.6464 0.6018 0.5657 39 0.8893 0.8770 0.7901 0.7042 0.6469 0.6020 0.5658 41 0.8823 0.8772 0.7808 0.7048 0.6473 0.6023 0.5659 43 0.9754 0.8773 0.7814 0.7053 0.6477 0.6026 0.5661 115 0.8685 0.8773 0.7820 0.7058 0.6481 0.6028 0.5662 47 0.8616 0.9772 0.7825 0.7063 0.6484 0.6031 0.5664 49 0.8548 0.8771 0.7830 0.7067 0.6487 0.6033 0.5665 51 0.8481 0.9770 0.7835 0.7071 0.6491 0.6035 0.5667 53 0.8414 0.9768 0.7839 0.7075 0.6494 0.6038 0.5668 55 0.8348 0.8765 0.7843 0.7078 0.6497 0.6040 0.5670 57 0,8283 0.9763 0.7847 0.7082 0.6499 0.6042 0.5671 59 0.9219 0.8760 0.7850 0.7085 0.6502 0.6044 0.5673 61 0.8155 0.8756 0.7853 0.7088 0.6504 0.6046 0.5674 63 0.8092 0.9753 0.7856 0.7091 0.6507 0.6048 0.5676 65 0.9030 0.8749 0.7859 0.7094 0.6509 0.6049 0.5677 67 0.7968 0.9745 0.7961 0.7096 0.6511 0.6051 0.5678 69 0.7908 0.8741 0.7863 0.7098 0.6513 0.6053 0.5679 71 0.7848 0.8736 0.7866 0.7101 0.6515 0.6054 0.5691 73 0.7789 0.8732 0.7868 0.7103 0.6517 0.6056 0.5682 75 0.7731 0.8727 0.7870 0.7105 0.6519 0.6057 0.5683 77 0.7673 0.8722 0.7871 0.7107 0.6520 0.6059 0.5684 79 0.7616 0.9717 0.7873 0.7109 0.6522 0.6060 0.5685

-226- CHFPYCHEV DISTRIBUTION APERTURE EFFICIENCY AS A FUNCTION OF SIPELOBE LEVEL (SL) AND NUMBER OF ELEMENTS (N). N SL= -20 -10 -40 -50 -60 -70 -80 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 4 0.9326 0.8623 0.8286 0.8131 0.8061 0.8028 0.8013 0.9443 0.8399 0.7751 0.7372 0.7143 0.7004 0.6917 8 0.9559 0.8416 0.7613 0.7097 0.6755 0.6522 0.6361 10 0.9622 0.8473 0.7588 0.6993 0.6580 0.6285 0.6068 12 0.9643 0.8529 0.7598 0.6955 0.6498 0.6160 0.5906 1/' 0.9636 0.8577 0.7619 0.6945 0.6458 0.6092 0.5811 16 0.9609 0.8616 0.7642 0.6947 0.6439 0.6053 0.5753. 18 0.9570 0.8649 0.7664 0.6954 0.6431 0.6031 0.5717 20 0.9521 0.8675 0.7685 0.6963 0.6429 0.6019 0.5694 22 0.9465 0.8696 0.7703 0.6973 0.6430 0.601.2 0.5679 24 0.9405 0.8714 0.7720 0.6983 0.6434 0.6009 0.5669 26 0.9341 0.8728 0.7735 0.6993 0.6438 0.6008 0.5663 28 0.9275 0.8739 0.7748 0.7002 0.6443 0.6008 0.5660 30 0.9207 0.8748 0.7760 0.7010 0.6448 0.6010 0.5658 32 0.9138 0.8756 0.77/1 0.7018 0.6453 0.6011 0.5657 34 0.9068 0.8761 0.7780 0.7026 0.6457 0.6014 0.5656 36 0.8998 0.8766 0.7789 0.7033 0.6462 0.6016 0.5657 38 0.8928 0.8769 0.7797 0.7039 0.6467 0.6019 0.5658 40 0.8858 0.8771 0.7805 0.7045 0.6471 0.6022 0.5659 42 0.8789 0.8772 0.7811 0.7050 0.6475 0.6024 0.5660 A4 0.8719 0.8773 0.7817 0.7056 0.6479 0.6027 0.5661 A6 0.8650 0.8773 0.7823 0.7060 0.6482 0.6029 0.5663 48 0.8582 0.8772 0.7828 0.7065 0.6486 0.6032 0.5664 50 0.8514 0.8770 0.7833 0.7069 0.6489 0.6034 0.5666 52 0.8447 0.8769 0.7837 0.7073 0.6492 0.6036 0.5668 54 0.8381 0.8767 0.7841 0.7077 0.6495 0.6039 0.5669 56 0.8315 0.8764 0.7845 0.7080 0.6498 0.6041 0.5671 58 0.8251 0.8761 0.7848 0.7083 0.6501 0.6043 0.5672 60 0.8186 0.8758 0.7851 0.7087 0.6503 0.6045 0.5673 62 0.8123 0.8754 0.7854 0.7089 0.6505 0.6047 0.5675 64 0.8061 0.8751 0.7857 0.7092 0.6508 0.6049 0.5676 66 0.7999 0.8747 0.7860 0.7095 0.6510 0.6050 0.5678 68 0.7938 0.8743 0.7862 0.7097 0.6512 0.6052 0.5679 70 0.7878 0.8738 0.7865 0.7100 0.6514 0.6054 0.5680 72 0.7818 0.8734 0.7867 0.7102 0.6516 0.6055 0.5681 74 0.7760 0.8729 0.7869 0.7104 0.6518 0.6057 0.5682 76 0.7702 0.8724 0.7870 0.7106 0.6519 0.6058 0.5684 78 0.7645 0.8719 0.7872 0.7108 0.6521 0.6059 0.5685 80 0.7588 0.8714 0.7874 0.7110 0.6523 0.6061 0.5686

-227-

APPENDIX TWO - TAYLOR LINE SOURCE DESIGN

This Appendix provides further details of the Taylor Line Source Distribution.

Product Expansions for the Taylor Pattern As already described, the Taylor Pattern is derived from the 'stretched' ideal Chebyshev pattern:

E0(U) = ch (ch-iR)2- (U/a)2 ]/R ; IUI:ach iR

= cos [ I (U/a)2 (ch-iR)2 1/R ; lUI:ach iR.

with zeros at the positions:

_ i U = t Iry (n4)2 + A2 PA = h R ... 2 n 11

Now, by choice of the dilation factor a, the n'th zero can be positioned at (17+a/2)v. To achieve this end a has the value + a/2 a cE_i)2 4. A2 ... 3

The subscript on a will be dropped whenever confusion is unlikely. A new pattern function Et(U) is now constructed by moving all zeros of Eo(U) above the r'th to the positions (n+a/2)7r to correspond with the asymptotic behaviour required for the specified a value. In doing this we must ensure that none of the separations between adjacent zeros are increased as this would lead to higher pattern values in Et(U) than in Eo(U), a situation we want to avoid. As all these spacings after the move have a value of IT , the above condition is met if all zero spacings above the lft'th zero in the original pattern Eo(U) are it or greater. In Eo(U) the spacing in between the n'th and (n+l)'th zero is:

1n = va [ (n+i)2 + A2 - ,\I (n-2)2 + A2] 11 ••• 4. and is easily shown by differentiation to be a monotonically increasing function of n when the latter is positive. Thus if the spacing of the r'th and (1141)1th zeros is it or greater all higher zero spacings will be also. Thus no increase in separation will occur provided that:

-228-

ira_ 2 4. A2 n ... 5 which can be written as:

[ CE+144/2)/am+1 (E44/2)/an] 3 1 ... 6 and finally;

(71+1+a/2) (a.1 - also ) since all the ar are positive. Since the first factor is always positive, Eqn. 7 reduces to:

a li+1 ... 8 showing that usable a values are such that on increasing 11 by unity there is a decrease in value. Values not meeting this criterion are omitted from the tables at the end of this appendix. As Et(U) is an entire function of U V with known zero disposition it has an infinite product expansion. Summarising, Et(U) has zeros at positions:

U = n ; n ... 9

(11-1-41/ 2)7T ; n

The Taylor pattern Et(U) in infinite product form is given by:

E t(U) = EI(U) E2(U) 10 • where: 11-1 U2 E1(U) = [ 1- 7.1.2 a2 4. A.2] n=1 [(n-2)2 ]

E2(U) 1 U2 1 ... 12 L r2 (n+2/2)2J

With m = [(a+1)/2] , the integer part of (a+1)/2 E2(U) can be -229-

written as: 74m-1 E2(U) = sin U ; a even .6. 13 2 2 n= n 71.

[ 1U2 = cos U ; a odd. (n4)2 77.2] 11=

Equations 10, 11, and 13 provide a convenient means of computipg the Taylor pattern Et(U).

Computer Programs Description A set of subroutines have been written to facilitate the design of Taylor line sources. The subroutine TAYINIT uses the side lobe level, the transition point and the edge exponent to initialise variables in the common block TAYCOM used by the subroutines TAY and TAYDIST. In the notation used here, the • variables in this common block are w , A2, a2, "E-1, EA-m-1 and lastly IOD which is the remainder when a is divided by 2. This initialisation routine must be called before either TAY or TAYDIST. The fumotion TAY(Z) computes the Taylor pattern value at the point Z (U=712) using Eqn. 10, Eqn. 11 and Eqn 13 above. The coding is fairly self explanatory. The subroutine TAYDIST evalu- ates the Taylor line source distribution using Eqn. 5.7 or Eqn. 5.8 of chapter 1 and the function TAY. Its operation is controlled by the variable L. If L is zero, the positions for amplitude evaluation are input in the array X. If L is negative, position in X and amplitude in A are both computed for N points:

X = (2n-N-1)/N ; n = 1 (1) N n ... 14

The same happens for L positive except that in this case the N values are at the points:

X = (n-1)/(N-1) ; N = 1 (1) N ... 15 n

All routines operate in double precision arithmetic.

-230- Design Tables Description In the tables dilation factor, normalised beam width and aperture efficiency are given as functions of the design side lobe level, the transition point and the edge exponent a. As explained earlier, positions in these tables that have not been listed are unusable because the dilation factor at these positions does not meet the criterion in Eqn. 8.

•

-231- TAYLOR PROGRAMS

* PROGRAMS FOR TAYLOR LINE SOURCE DESIGN

* INITIALISATION ROUTINE

* SL= SIDELOBE LEVEL IN DECIBELS * NT= TRANSITION POINT * NA= EDGE EXPONENT (ALPHA) * SUBROUTI NE TAYINI TC SLo NTo NA) DOUBLE PRECISION SL, PI o A2, SG ao T COMMON /TAYCOM/PI s A2, SGao NTX,NBX, I OD PI =3.1415926535898D0 T=10• ODO**C DABS( SL)/20.0D0) A2=( DLOGC T+ DSORT( T*T••1• °DO)) /PI )**2 T=NT SG2=( T+0*SDO*NA)**2/(A2+( T...0.5D0)**2) NTX=NT-• 1 NBX=NTX+(NA+ 1) /2 I ODatNA — 2*(NA/2) RETURN END * * FUNCTION TO EVALUATE THE TAYLOR PATTERN AT U=PI*Z * FUNCTION TAY( Z ) DOUBLE PRECISION TAY, Za PI Aao SG 2s T, B, Cs EN COMMON/TAYCOM/PI a A2s SG2oNTXPNBX, I OD * * EVALUATE E2CU), UPPER FACTOR IN T. LOWER FACTOR IN B * IF(IOD.NE.0) GO TO 10 T=1.0D0 IF(E.NE.0.0D0) T=DSIN(PI*Z)/(PI*Z) EN=0.0D0 GO TO 20 10 T=DCOS(PI*Z) EN=-0'.5D0 * * EVALUATE LOWER FACTOR OF E2(U) * 20 B=1.0D0 DO 40 N=1,NBX EN=EN+1.0D0 C=1.0D0—(Z/EN)**2 IF(DABS(C).LT.1.OD-6) GO TO 30 B=B*C GO TO 40 * * E2(U) NEARLY INDETERMINATE WITH UPPER AND LOWER FACTORS * NEARLY ZERO. USE LIMITING FORM 30 T=0.5D0 IF(N.EO.2*(N/2)) T=—T IF(IOD.EO.0) GO TO 40 T=T*PI*EN TAYLOR PROGRAMS

40 CONTINUE T=T/B * * MULTIPLY E2(U) AN T BY FACTORS OF E1(U) * C=Z*Z/SG2 EN=-0.5D0 DO 50 N=1,NTX EN=EN-0.1.0D0 T=T-T*C/(A2+EN*EN) 50 CONTINUE TAY=T RETURN END * * SUBROUTINE TO EVALUATE THE TAYLOR AMPLITUDE DISTRIBUTION * * X AN ARRAY FOR INPUT OR OUTPUT OF POSITION VALUES * A AN ARRAY FOR OUTPUT OF AMPLITUDE VALUES * N= NUMBER OF VALUES TO, BE OUTPUT * L= CONTROL VARIABLE: * L=-1 OUTPUT POSITIONS IN X AND AMPLITUDES IN A FOR N POINTS SPACED EVENLY ON THE APERTURE (-1.LE.X.LE.1) WITH HALF SPACES AT EACH END * L= 0 OUTPUT AMPLITUDES IN A FOR POSITIONS INPUT IN X * L= 1 OUTPUT POSITIONS IN X AND AMPLITUDES IN A FOR N EVENLY SPACED POINTS, THE FIRST AT THE CENTRE (X=0) AND THE LAST AT THE EDGE (X=1) * SUBROUTINE TAYDIST(X,A,N,L) DOUBLE PRECISION X,A,E,PI,A2,SG2,EN,EM,T,XX DIMENSION X(N),A(N),E( 100) COMMON/TAYCOM/PI,A2SSG2,NTX,NBX,IOD * * STORE PATTERN VALUES REOUI RED TO GENERATE AMPLITUDES * MX=NBX-I0D+1 EN=0.0D0 IF(IOD.NE.0) EN0 0 , 5D0 EM=EN DO 10 M=1, MX E(M)=TAY(EM) EM=EM-1-1.0D0 10 CONTINUE IF(IOD.EQ.0) E(1)=0.5D0*E(1) IF(L) 20,60,40 * * GENERATE POSITION VALUES FOR L=-1 * 20 EM=N f=1.0DO-EM DO 30 I=1,N

-233- TAYLOR PROGRAMS

X(I)=T/EM T=T+2.0D0 30 CONTINUE GO TO 60

* GENERATE POSITION VALUES FOR L=1

40 EM=1.0D0/(N-1) T=0.0D0 DO 50 I=1,N X(I)=T T=T+EM 50 CONTINUE

* GENERATE AMPLITUDE VALUES BY SERIES SUMMATION

60 D0'80 I=1,N XX=X(I) BM=EN T=0.0DO:' DO 70 M=I,MX T=T+E(M)*DCOS( EM*PI*XX) EM=EM+I.ODO 70 CONTINUE A(I)=T 80 CONTINUE RETURN END

-234- TAYLOR DILATION FACTOR AS A FUNCTION OF SIDELODE LEVEL (SL) AND TRANSITION POINT (N). ALPHA = .0

SL= -20 -30 •40 -60 -80 1 2 1.1255 - - --- 3 1.1213 4 1.1027 1.0693 5 1.0870 1.0662 r 6 1.0749 1.0608 1.0430 7 1.0655 1.0554 1.0424 8 1.0582 1.0505 1.0407 9 1.0522 1.0463 1.0386 1.0292 10 1.0474 1.0426 1.0364 1.0289

11 1.0433 1.0394 1.0344 1.0282 12 1.0399 1.0367 1.0324 1.0272 1.0211 13 1.0370 1.0342 1.0307 1.0263 1.0210 14 1.0345 1.0321 1.0290 1.0252 1.0208 15 1.0323 1.0302 1.0276 1.0243 1.0204

16 1.0303 1.0285 1.0262 1.0233 1.0199 1.0160 17 1.0286 1.0270 1.0250 1.0224 1.0194 1.0159 18 1.0271 1.0257 1.0238 1.0216 1.0189 1.0158 • 19 1.0257 1.0244 1.0228 1.0208 1.0184 1.0156 20 1.0244 1.0233 1.0218 1.0200 1.0178 1.0153 1.0125

21 1.0233 1.0223 1.0209 1.0193 1.0173 1.0151 1.0125 22 1.0223 1.0213 1.0201 1.0186 1.0168 1.0148 1.0124 23 1.0213 1.0205 1.0194 1.0180 1.0164 1.0145 1.0123 24 1.0204 1.0197 1.0187 1.0174 1.0159 1.0142 1.0122 25 1.0196 1.0189 1.0180 1.0168 1.0155 1.0139 1.0121

26 1.0189 1.0182 1.0174 1.0163 1.0150 1.0136 1.0119 27 1.0182 1.0176 1.0168 1.0158 1.0146 1.0133 1.0117 28 1.0176 1.0170 1.0163 1.0154 1.0143 1.0130 1.0116 29 1.0170 1.0165 1.0158 1.0149 1.0139 1.0127 1.0114 30 1.0164 1.0159 1.0153 1.0145 1.0135 1.0124 1.0112

31 1.0159 1.0154 1.0148 1.0141 1.0132 1.0122 1.0110 32 1.0154 1.0150 1.0144 1.0137 1.0129 1.0119 1.0108 • 33 1.0149 1.0145 1.0140 1.0134 1.0126 1.0117 1.0106 34, 1.0145 1.0141 1.0136 1.0130 1.0123 1.0114 1.0105 35 1.0141 1.0138 1.0133 1.0127 1.0120 1.0112 1.0103

36 1.0137 1.0134 1.0129 1.0124 1.0117 1.0110 1.0101 37 1.0134 1.0130 1.0126 1.0121 1.0115 1.0108 1.0099 38 1.0130 1.0127 1.0123 1.0118 1.0112 1.0105 1.0098 39 1.0127 1.0124 1.0120 1.0115 1.0110 1.0103 1.0096 40 1.0124 1.0121 1.0117 1.0113 1.0108 1.0101 1.0094

-235- TAYLOR DILATION FACTOR AS A FUNCTION OF SIDELODE LEVFL'(SL) AND TRANSITION POINT (N). ALPHA = 1.0

N SL= -20 -30 -40 -50 -70 -80 1 2 1.4069 1.2512 3 1.3082 1.2380 1.1606 4 1.2406 1.2030 1.1583 5 1.1957 1.1728 1.1445 1.1120

6 1.1645 1.1492 1.1299 1.1072 1.0818 7 1.1416 1.1308 1.1169 1.1003 1.0814 8 1.1243 1.1162 1.1057. 1.0931 1.0786 1.0624 9 1.1107 1.1044 1.0963 1.0864 1.0749 1.0621 10 1.0997 1.0947 1.0882 1.0803 1.0711 1.0606 1.0490

11 1.0908 1.0867 1.0814 1.0749 1.0673 1.0586 1.0490 12 1.0832 1.0799 1.0755 1.0700 1.0637 1.0564 1.0483 13 1.0769 1.0740 1.0703 1.0657 1.0603 1.0541 1.0472 14 1.0714 1.0690 1.0658 1.0619 1.0572 1.0519 1.0459 15 1.0667 1.0646 1.0618 1.0584 1.0544 1.0498 1.0446 16 1.0625 1.0607 1.0583 1.0553 1.0518 1.0477 1.0432 17 1.0588 1.0572 1.0551 1.0525 1.0494 1.0458 1.0418 18 1.0556 1.'0541 1.0523 1.0499 1.0472 1.0440 1.0404 * 19 1.0527 1.0514 1.0497 1.0476 1.0452 1.0423 1.0391 20 1.0500 1.0489 1.0474 1.0455 1.0433 1.0407 1.0378 21 1.0476 1.0466 1.0452 1.0436 1.0416 1.0392 1.0366 22 1.0455 1.0445 1.0433 1.0418 1.0399 1.0378 1.0354 23 1.0435 1.0427 1.0415 1.0401 1.0385 1.0365 1.0343 24 1.0417 1.0409 1.0399 1.0386 1.0371 1.0353 1.0333 25 1.0400 1.0393 1.0384 1.0372 1.0358 1.0342 1.0323 26 1.0385 1.0378 1.0370 1.0359 1.0346 1.0331 1.0314 27 1.0371 1.0365 1.0356 1.0346 1.0334 1.0320 1.0305 28 1.0357 1.0352 1.0344 1.0335 1.0324 1.0311 1.0296 29 1.0345 1.0340 1.0333 1.0324 1.0314 1.0302 1.0288 30 1.0334 1.0329 1.0322 1.0314 1.0304 1.0293 1.0280

31 1.0323 1.0318 1.0312 1.0305 1.0296 1.0285 1.0273 32 1.0313 1.0308 1.0303 1.0296 1.0287 1.0277 1.0266 33 1.0303 1.0299 1.0294 1.0287 1.0279 1.0270 1.0260 34 1.0294 1.0291 1.0285 1.0279 1.0272 1.0263 1.0253 35 1.0286 1.0282 1.0278 1.0272 1.0265 1.0256 1.0247 36 1.0278 1.0275 1.0270 1.0265 1.0258 1.0250 1.0241 37 1.0270 1.0267 1.0263 1.0258 1.0251 1.0244 1.0236 38 1.0263 1.0260 1.0256 1.0251 1.0245 1.0238 1.0231 39 1.0257 1.0254 1.0250 1.0245 1.0240 1.0233 1.0226 40 1.0250 1.0247 1.0244 1.0239 1.0234 1.0228 1.0221

-236- TAYLOR DILATION FACTOR AS A FUNCTION OF SIDELOBE LEVEL (SL) AND TRANSITION POINT (N). ALPHA = 2.0

N SL= -20 -30 -40 -60 -70 -80 1 1.8587 2 1.6882 1.5014 1.3292 3 1.4951 1.4149 1.3264 1.2365 4 1.3784 1.3367 1.2870 1.2322 1.1751 5 1.3044 1.2794 1.2485 1.2131 1.1744

6 1.2540 1.2376 1.2168 1.1924 1.1650 1.1354 7 1.2178 1.2062 1.1913 1.1736 1.1535 1.1312 1.1074 8 1.1904 1.1818 1.1708 1.1574 1.1420 1.1249 1.1063 9 1.1691 1.1625 1.1540 1.1436 1.1315 1.1180 1.1031 10 1.1521 1.1469 1.1401 1+1318 1.1221 1.1111 1.0990

11 1.1382 1.1339 1.1284 1.1216 1.1137 1.1046 1.0946 12 1.1266 1.1231 1.1185 1.1128 1.1062 1.0987 1.0902 13 1.1168 1.1138 1.1099 1.1052 1.0996 1.0932 1.0860 14 1.1084 1.1058 1.1025 1.0985 1.0937 1.0882 1.0820 15 1.1011 1.0989 1.0961 1.0926 1.0884 1.0836 1.0783

16 1.0947 1.0928 1.0903 1.0873 1.0837 1.0795 1.0748 17 1.0891 1.0874 1.0853 1.0826 1.0794 1.0757 1.0715 18 1.0841 1.0826 1.0807 1.0783 1.0755 1.0722 1.0685 19 1.0797 1.0783 1.0766 1.0745 1.0720 1.0690 1.0657 20 1.0756 1.0745 1.0729 1.0710 1.0687 1.0661 1.0631

21 1.0720 1.0710 1.0696 1+0678 1.0658 1.0634 1.0607 22 1.0687 1.0678 1.0665 1.0649 1.0631 1.0609 1.0585 23 1.0657 1.0648 1.0637 140623 1.0606 1.0586 1.0563 24 1.0630 1.0622 1.0611 1.0598 1.0582 1.0564 1.0544 25 1.0604 1.0597 1.0587 1.0575 1.0561 1.0544 1.0525

26 1.0581 1.0574 1.0565 1.0554 1.0541 1.0526 1.0508 27 1.0559 1.0553 1.0545 1.0534 1.0522 1.0508 1.0492 28 1.0539 1.0533 1.0526 1.0516 1.0505 1.0492 1.0477 29 1.0520 1.0515 1.0508 1.0499 1.0489 1.0476 1.0463 30 1.0503 1.0498 1.0491 1.0483 1.0473 1.0462 1.0449

31 1.0487 1.0482 1.0476 1.0468 1.0459 1.0448 1.0436 32 1.0471 1.0467 1.0461 1.0454 1.0445 1.0435 1.0424 33 1.0457 1.0453 1.0447 1.0441 1.0433 1.0423 1.0413 34 1.0444 1.0440 1.0435 1.0428 1.0421 1.0412 1.0402 35 1.0431 1.0427 1.0422 1.0416 1.0409 1.0401 1.0391

36 1.0419 1.0415 1.0411 1.0405 1.0398 1.0391 1.0382 37 1.0407 1.0404 1.0400 1.0395 1.0388 1.0381 1.0372 38 1.0397 1.0394 1.0389 1.0384 1.0378 1.0371 1.0363 39 1.0386 1.0384 1.0380 1.0375 1.0369 1.0363 1.0355 40 1.0377 1.0374 1.0370 1.0366 1.0360 1.0354 1.0347

.4237+. TAYL.07; DI S TR' PPTI IJ J HALF POWER 13t AN I LITH AS A FUNCTION UF nFt..opF, LFUFL, (SL) AND TRANSITION POINT (N)• ALPHA = .0 N SL= -30 -40 -60 -80 1 2 0 ,11972, 3 0•4959 0•6991 0.5623 5 0.4831 0.5610 6 0•4784 0.5586 0.6242 7 0•4746 0.5561 0.6239 6 0•4716 0.5539 0.6231 9 0.4691 0.5519 0.6220 0.6824 10 0.4671 0.5501 0.6208 0.6822

11 0.4654 0.5485 0.6197 0.6817 12 0.4639 0.5472 0.6187 0.6812 0.7369 13 0•4627. 0.5460 0.6177 0.6806 0.7368 1/4 0•4616 0.5449 0.6168 0.6800 0•7366 15 0.4607 0.5440 0.6160 0.6794 0.7364 16 0•4598 0.5432 0.6152 0.6789 0.7361 0.7882 17 0.4591 0.5424 0.6145 0.6763 0.7357 0.7881 16 0.4584 0.5417 0.6139 0.6778 0.7354 0.7880 19 0.4578 0.5411 0.6133 0.6773 0.7351 0.7879 20 0.4573 0.5405 0.6127 0.6768 0.7347 0.7877 0.8366 21 0.4568 0.5400 0.6122 0.6764 0.7344 0.7875 0.8366 22 0.4564 0.5395 0.6118 0.6760 0.7340 0.7873 0.8366 23 0.4559 0.5391 0.6113 0.6756 0.7337 0.7871 0.8365 24 0.4556 0.5387 0.6109 0.6752 0.7334 0.7869 0.8364 25 0.4552 0.5383 0.6105 0.6748 0.7331 0.7867 0.8363 26 0.4549 0.5379 0.6102 0.6745 0.7328 0.7864 0.8362 27 0.4546 0.5376 0.6099 0.6742 0.7326 0.7862 0.8361 28 0.4543 0.5373 0.6096 0.6739 0.7323 00860 0.8359 29 0•4541 0.5370 0.6093 0.6736 0.7320 0.7858 0.8358 30 0.4538 0.5368 0.6090 0.6734 0.7318 0•7856 0.8357 31 0.4536 0.5365 0.6087 0.6731 0.7316 0.7854 0.8355 32 0.4534 0.5363 0.6085 0.6729 0.7314 0.7852 0.8354 33 0.4532 0.5360 0.6082 0.6726 0.7311 0.7851 0.8352 0•4530 0.5358 0.6080 0.6724 0.7309 0•7849 0.8351 35 0.4528 0.5356 0.6078 0.6722 0.7307 0.7847 0.8349 36 0.4526 0.5354 0.6076 0.6720 0.7306 0•7845 0.8348 37 0.4525 0.5353 0.6074 0.6718 0.7304 0.7844 0.8347 38 0.4523 0.5351 0.6073 0.6716 0.7302 04.7842 0.8345 39 0.4522 0•5349 0.6071 0.6715 0.7300 0.7841 0•8344 40 0•4520 0.5348 0.6069 0.6713 0.7299 0.7839 0•8343

-238- TAYLOP DISTRIBUTION HALF POWER BFAMWIDTH AS A FUNCTION OF SIDELOPF LEVEL (SL) AND TRANSITION POINT (N). ALPHA = 1.0 N SL= -20 -30 -40 -50 -60 -70 -80 1 2 0.5973 0.6415 3 0.5681 0.6373 0.6850 4 0.5448 0.6242 0.6841 5 0.5281 0.6118 0.6780 0.7309 4 6 0.5160 0.6016 0.6712 0.7286 0.7763 7 0.5069 0.5933 0.6648 0.7250 0.7761 8 0.4999 0.5866 0.6592 0.7211 0.7745 0.8211 9 0.4943 0.5810 0.6543 0.7174 0.7724 0.8209 10 0.4897 0.5764 0.6501 0.7140 0.7701 0.8199 0.8645 11 0.4860 0.5725 0.6464 0.7108 0.7678 0.8187 0.8645 12 0.4828 0.5692 0.6432 0.7080 0.7655 0.8172 0.8640 13 0.4801 0.5663 0.6404 0.7054 0.7634 0.8157 0.8633 14 0.4778 0.5638 0.6380 0.7031 0.7615 0.8142 '0.8624 15 0.4757 0.5616 0.6357 0.7011 0.7596 0.8128 0.8614 16 0.4739 0.5597 0.6338 0.6992 0.7580 0.8114 0.8604 17 0.4724 0.5580 0.6320 0.6975 0.7564 0.8101 0.8594 18 0.4710 0.5564 0.6304 0.6959 0.7549 0.8088 0.8584 0 19 0.4697 0.5550 0.6290 0.6945 0.7536 0.8076 0.8574 20 0.4686 0.5537 0.6276 0.6932 0.7524 0.8065 0.8565 21 0.4675 0.5526 0.6264 0.6920 0.7512 0.8055 0.8556 22 0.4666 0.5515 0.6253 0.6909 0.7501 0.8045 0.8547 23 0.4657 0.5506 0.6243 0.6898 0.7491 0.8035 0.8539 24 0.4649 0.5497 0.6234 0.6889 0.7482 0.8027 0.8531 25 0.4642 0.5489 0.6225 0.6880 0.7473 0.8018 0.8524 26 0.4635 0.5481 0.6217 0.6872 0.7465 0.8011 0.8517 27 0.4629 0.5474 0.6209 0.6864 0.7457 0.8003 0.8510 28 0.4623 0.5468 0.6202 0.6857 0.7450 0.7996 0.8503 29 0.4618 0.5462 0.6196 0.6850 0.7443 0.7990 0.8497 30 0.4613 0.5456 0.6190 0.6843 0.7437 0.7983 0.8491 31 0.4608 0.5450 0.6184 0.6837 0.7431 0.7977 0.8485 32 0.4604 0.5445 0.6178 0.6832 0.7425 0.7972 0.8480 33 0.4600 0.5441 0.6173 0.6826 0.7420 0.7966 0.8475 34 0.4596 0.5436 0.6168 0.6821 0.7415 0.7961 0.8470 35 0.4592 0.5432 0.6164 0.6816 0.7410 0.7956 0.8465 36 0.4589 0.5428 0.6159 0.6812 0.7405 0.7952 0.8461 37 0.4585 0.5424 0.6155 0.6807 0.7401 0.7947 0.8457 38 0.4582 0.5421 0.6151 0.6803 0.7396 0.7943 0.8452 39 0.4579 0.5417 0.6148 0.6799 0.7392 0.7939 0.8448 40 0.4576 0.5414 0.6144 0.6796 0.7388 0.7935 0.8445

-239-

04, TAYLOP rISTRIBUTION HALF PMEP BEAMWIDTH AS A FPNCTION 1 _11 SIB LOPE LEVEL (SL) AND TPANSITION PUIN1 (N). ALPHA = 2.0 N SL= -20 -30 -40 -60 -80 1 0.6125 2 0.-6904 0.7371 0.7636 3 0.6379 0.7122 0.7628 0.7977 4 0.5994 0.6845 0.7481 0.7961 0.8328 ■ ■ .41,■■ 5 0.5726 0.6618 0.7319 0.7877 0.8324 6 0.5534 0.6441 0.7175 0.7778 0.8278 0.8696 7 0.5391 0.6301 0.7053 0.7683 0.8217 0.8673 0.9066 P 0.5280 0.6190 0.6951 0.7597 0.8153 0.8637 0.9060 9 0.5193 0.6100 0.6865 0.7522 0.8093 0.8595 0.9040 10 0.5123 0.6026 0.6793 0.7455 0.8037 0.8553 0.9014 11 0.5065 0.5964 0.6731 0.7398 0.7986 0.8512 0.8985 12 0.5016 0.5911 0.6677 0.7347 0.7941 0.8474 0.8956 13 0.4974 0.5866 0.6631 0.7302 0.7899 0.8438 - 0.8927 14 0.4939 0.5827 0.6591 0.7262 0.7862 0.84-04 0.8899 15 0.4905 0.5792 0.6555 0.7227 0.7828 0.8374 0.8873 16 0.4880' 0.5762 0.6523 0.7195 0.7798 0.8346 0.8848 17 0.4856 0.5735 0.6495 0.7166 0.7770 0.8320 0.8825 18 0.4835 0.5711 0.6469 0.7140 0.7744 0.8296 0.8803 19 0.4816 0.5689 0.6446 0.7117' 0.7721 0.8273 0.8782 20 0.4798 0.5670 0.6425 0.7095 0.7700 0.8253 0.8763 21 0.4782 0.5652 0.6406 0.7076 0.7680 0.8234 0.8745 92 0.4768 0.5636 0.6389 0.7057 0.7662 0.8216 0.8728 23 0.4755 0.5621 0.6373 0.7041 0.7645 0.8200 0.8713 24 0.4743 0.5607 0.6358 0.7025 0.7630 0.8184 0.8698 25 0.4732 0.5595 0.6344 0.7011 0.7615 0.8170 0.8684 26 0.4722 0.5583 0.6332 0.6998 0.7602 0.8156 0.8671 27 0.4712 0.5572 0.6320 0.6986 0.7589 0.8144 0.8659 28 0.4704 0.5562 0.6309 0.6974 0.7577 0.8132 0.8647 29 0.4696 0.5553 0.6299 0.6963 0.7566 0.8121 0.8636 30 0.4688 0.5544 0.6289 0.6953 0.7556 0.8110 0.8626 31 0.4681 0.5536 0.6280 0.6944 0.7546 0.8100 0.8616 32 0.4674- 0.5528 0.6272 0.6935 0.7537 0.8091 0.8606 33 0.4668 0.5521 0.6264 0.6926 0.7528 0.8082 0.8598 34 0.4662 '0.5514 0.6256 0.6918 0.7520 0.8074 0.8589 35 0.4656 0.5508 0.6249 0.6911 0.7512 0.8066 0.8581 36 0.4651 0.5501 0.6243 0.6903 0.7504 0.8058 0.8573 37 0.4646 0.5496 0.6236 0.6897 0.7497 0.8051 0.8566 38 0.4641 0.5490 0.6230 0.6890 0.7490 0.8044 0.8559 39 0.4637 0.5485 0.6224 0.6884 0.7484 0.8037 0.8553 40 0.4632 0.5480 0.6219 0.6878 0.7478 0.8031 0.8546 TAYLOR DI STRI BUTI ON ZERO POINT BEAMWI DTH AS A FUNCTION OF SI MOPE LEVEL ( SL) AND TRANSI TI ON POINT ( N ) • ALPHA = • 0

N SL= -20 -30 -50 -60 -80 1 2 1.2110 3 1.2065 4 1.1865 1.5094 5 1.1696 1.5049 6 1 • 1566 1.4973 1.8347 7 1.1465 1./4897 1.8337 8 1 • 1386 1.4828 1.8306 9 1 • 1322 1.4768 1.8269 2.1747 ,10 1.1270 1.4716 1.8231 2.1740 11 1.1226 1.4672 1•8195 2.1725 12 1.1189 1.4632 1.8161 2.1705 2.5228 13 1 • 1158 1.4598 1.8130 2.1684 2.5226 14 1.1131 1.4568 1.8101 2.1663 2.5219 15 1 • 1107 1.4541 1+8075 2.1643 2.5209 16 1.1086 1.4518 1.8051 2.1623 2.5197 2.8756 17 1.1068 1.4496 1.8030 2.1604 2.5185 2.8755 18 1.1051 1.4477 1.8010 2.1586 2.5172 2.8751 19 1 • 1036 1.4460 1.7991 2.1568 2.5159 2.8745 20 1 • 1023 1.4444 1.7974 2.1553 2.5146 2.8738 3.2317 21 1.1011 1.4429 1.7959 2.1537 2.5134 2.8730 3.2316 22 1.0999 1.4416 1.7944 2.1523 2.5122 2.8722 3.2315 23 1.0989 1.4404 1.7931 2.1510 2.5110 2.8714 3.2312 24 1.0980 1.4392 1.7919 2.1498 2.5099 2.8705 3.2308 25 1.0971 1.4382 1.7907 2.1486 2.5088 2.8697 3.2303 26 1.0963 1.4372 1.7896 2.1475 2.5077 2.8688 3.2298 27 1.0956 14.4363 1.7886 2.1464 2.5068 2.8680 3.2292 28 1+0949 1.4355 1.7877 2.1454 2.5058 2.8672 3.2287 29 1 • 0943 1.4347 107868 2.1445 2.5049 2.8664 3.2281 30 1 • 0937 1.4340 1.7860 2.1436 2.5040 2.8656 3.2275

31 1 • 0931 1.4333 1.7852 2.1428 2.5032 2.8649 3.2269 32 1.0926 1.4326 1.7844 2.1420 2.5024 2.8642 3.2263 33 1.0921 1.4320 1.7837 2.1412 2.5017 2.8635 3.2257 34 1.0916 1.4314 1.7831 2.1405 2.5009 2.8628 3.2252 35 1 • 0912 1.4309 1.7824 2.1398 2.5002 2.8621 3.2246 36 1.0908 1.4304 1.7818 2.1392 2.14996 2.8615 3.2241 37 1 • 0904 1• 4299 1.7813 2.1385 2.4989 2.8609 3.2235 38 1.0900 1.4294 1.7807 2.1380 2.4983 2.8603 3.2230 39 1.0896 1.4290 1.7802 2.1374 2.4977 2.8597 3.2225 40 1 • 0893 1• 4286 1.7797 2.1368 2.14972 2.8591 3.2219

TAYLOP DISTRISHTION :?FRO POINT BEAlliabTH A8 p FI1NCTION OF SIDFLOPF LFVFL,(8L) 4Nr TPANSITION POINT (N) • ALPHA = 1.0 N SL= -20 -30 -50 *.60 -80 1 2 1.5138 1.7661 3 1.4076 1.7475 2.0416 1.3349 1.6980 2.0374 5 1.2866 1.6554 2.0132 2.3496 6 1.2530 1.6221 1.9875 2.3395 2.6726 7 1.2284 1.5961 1.9646 2.3249 2.6716 5 1.2097 1.5755 1.9450 2.3097 2.6648 3.0071 9 1.1951 1.5589 1.9284 2.2956 2.6557 3.0061 10 1.1833 1.5452 1.9143 2.2827 2.6462 3.0019 3.3482 11 1.1737 1.5338 1.9022 2.2712 2.6368 2.9963 3.3481 12 1.1656 1.5242 1.8918 2.2610 2.6279 2.9901 3.3459 13 1.1557 1-+5160 1.8827 2.2519 2.6196 2.9837 3.3425 14 1.1528 1.5088 1.8748 2.2437 2.6120 2.9773 3.3384 15 1.1477 1.5026 1.8678 2.2364 2.6049 2.9713 3.3340 16 1.1433 1.4971 1.8616 2.2298 2.5985 2.9655 3.3295 17 1.1393 1.4923 1.8560 2.2239 2.5926 2.9601 3.3251 18 1.1358 1.4879 1.8510 2.2185 2.5871 2.9550 3.3207 b 19 1.1327 1.4840 1.8465 2.2136 2.5821 2.9502 3.3165 PO 1.1298 1.4805 1.8424 2.2091 2.5775 2.9457 3.3125 21 1.1273 1.4773 1.8386 2.2050 2.5732 2.9415 3.3086 22 1.1249 1.4744 1.8352 2.2012 2.5693 2.9375 3.3049 23 1.1225 1.4717 1.8321 2.1978 2.5656 2.9338 3.3014 24 1.1209 1.4692 1.8292 2.1945 2.5622 2.9303 3.2981 25 1.4191 1.4670 1.8265 2.1915 2.5590 2.9271 3.2949 26 1.1174 1.4649 1.8241 2.1888 2.5560 2.9240 3.2919, 27 1.1159 1.4629 1.8217 2.1862 2.5532 2.9211 3.2890 2% 1.1145 1.4611 1.8196 2.1837 2.5506 2.9104 3.2863 29 1.1131 1.4594 1.8176 2.1815 2.5481 2.9158 3'.2837 30 1.1119 1.4579 1.8157 2.1793 2.5458 2'.9134 3.2813 31 1.1107 1.4564 1.8140 2.1773 2.5436 2.9111 3.2790 32 1.1096 1.4550 1.8123 2.1754 2.5415 2.9089 3.2767. 33 1.1086 1.4537 1.8107 2.1737 2.5396 2.9069 3.2746 34 1.1077 1.4525 1.8093 2.1720 2.5377 2.9049 3.2726 35 1.1068 1.4513 1.8079 2.1704 2.5359 2.9030 3.2707 36 1.1059 1.4502 1.8066 2.1689 2.5343 2.9012 3.2688 37 1.1051 1.4492 1.8053 2.1674 2.5327 2.8995 3.2671 38 1.1043 1.4482 1.8041 2.1661 2.5312 2.8979 3.2654 39 1.1036 1.4473 1.8030 2.1645 2.5297 2.8964 3.2638 40 1.1029 1.4464 1.8019 2.1636 2.5284 2.8949 3.2622

-242- 0 TAM% DISTRIPOTION ZERO POIN1 FIFAMWIDTH AS A FTINCTION OF SI ITLOPF LEVEL- ( SL) ANT TRANSITION POINT (N). ALPHA = 2.0 N SL= -20 -30 ••40 -50 -60 -70 1 2.0000 2 1.8165 2.1193 2.3381 3 1.6087 1.9971 2.3332 2.6127 4 1.4832 1.8867 2.2638 2.6037 2.9032 5 1.4036 1.8059 2.1962 2.5632 2.9013 4 6 , 1.3494 1.7468 2.1404 2.5195 2.8782 3.2136 7 1.3103 1.7025 2.0956 2.4798 2.8497 3.2019 3.5346 8 1.2809 1.6681 2.0594 2.4456 2.8215 3.1839 3.5309 9 1.2580 1.6409 2.0299 2.4164 2.7955 3.1643 3.5207 10 1.2397 1.6188 2.0054 2.3914 2.7722 3.1449 3.5077 11 1.2247 1.6005 1.9849 2.3700 2.7514 3.1266 3.4937 12 1.2122 1.5852 1.9675 2.3514 2.7330 3.1097 3.4797 13 1.2016 1.5721 1.9525 2.3353 2.7166 3.0942 3.4663 14 1.1926 1.5609 1.9394 2.3211 2.7020 3.0800 3.4535 15 1.1848 1.5511 1.9280 2.3085 2.6890 3.0671 3.4416 16 1.1779 1.5425 1.9180 2.2974 2.6772 3.0554 3.4304 17 1.1719 1.5349 1.9090 2.2874 2.6666 3.0446 3.4201 18 1.1665 1.5281 1.9010 2.2785 2.6571 3.0348 3.4105 19 1.1617 1.5221 1.8938 2.2704 2.6483 3.0258 3.4015 • 20 1.1574 1.5166 1.8873 2.2630 2.6404 3.0175 3.3933 21 1.1535 1.5116 1.8814 2.2563 2.6331 3.0099 3.3855 22 1.1499 1.5071 1.8760 2.2502 2.6264 3.0028 3.3783 23 1.1467 1.5030 1.8711 2.2445 2.6202 2.9962 3.3716 24 1.1437 1.4992 1.8665 2.2393 2.6144 2.9901 3.3654 25 1.1410 1.4957 1.8623 2.2345 2.6091 2.9845 3.3595 26 1.1385 1.4925 1.8585 2.2301 2.6042 2.9792 3.3540 27 1.1362 1.489,5 1.8549 2.2259 2.5996 2.9742 3.3488 28 1.1340 1.4868 1.8515 2.2221 2.5953 2.9696 3.3440 29 1.1320 1.4842 1.8484 2.2184 2.5913 2.9653 3.3394 30 1.1301 1.4818 1.8455 2.2151 2.5875 2.9612 3.3351 31 1.1284 1.4795 1.8428 2.2119 2.5840 2.9573 3.3310 32 1.1267 1.4774 1.8402 2.2089 2.5806 2.9537 3.3272 (it 33 1.1252 1.4754 1.8378 2.2061 2.5775 2.9502 3.3235 34 1.1237 1.4735 1.8355 2.2035 2.5745 2.9470 3.3200 35 1.1224 1.4718 1.8333 2.2010 2.5717 2.9439 3.3167 36 1-.1211 1.4701 1.8313 2.1986 2.5690 2.9410 3.3136 37 1.1198 1.4685 1.8294 2.1963 2.5665 2.9382 3.3106 38 1.1187 1.4670 1.8276 2.1942 2.5641 2.9355 3.3078 39 1.1176 1.4656 1.8258 2.1922 2.5618 2.9330 3.3051 40 1.1165 1.4643 1.8242 2.1903 2.5596 2.9306 3.3025

-243- 41( TAYLIIIc DISTRIPUTION APERIT EFFICIENCY AS A PINClION SIDELOPF LEVEL (SL) AND TRANSITIUN POINT (N). ALPHA= •0 N SL= -20 -40 -50 -60 -70 -80 1 2 0.9518 3 0.9535 4 0.9605 0.8534 5 0.9650 0.8553 6 0.9667 0.8586 0.7674 7 0.9661 0.8619 0.7678 8 0.9640 008649 0.7689 9 0.9607 0.8675 0.7702 0.6998 10 0.9566 0.8696 0.7716 0.7000 11 0.9519 0.8715 0.7729 0.7004 12 0.9467 0.8730 0.7742 0.7010 0.6464 13 0.9412 0.8742 D.7754 0.7016 0.6464 14 0.9355 0.8753 0.7765 0.7022 0.6466 15 0.9295 0.8761 0.7775 0.7028 0.6468 16 0.9235 0.8768 0.7784 0.7034 0.6471 0.6032 17 0.9173 0.8773 0.7792 0.7040 0.6474 0.6032 18 0.9111 0.8778 0.7800 0.7045 0.6477 0.6033 19 0.9049 0.8781 0.7807 0.7051 0.6480 0.6034 20 0.8987 0.8784 0.7813 0.7055 0.6483 0.6035 0.5674 21 0.8924 0.8785 0.7819 0.7060 0.6486 0.6037 0.5674 22 0.8862 0.8786 0.7825 0.7064 0.6489 0.6038 0.5674 23 0.8800 0.8786 0.7830 0.7068 0.6492 0.6040 0.5675 24 0.8738 0.8786 0.7834 0.7072 0.6495 0.6042 0.5676 25 0.8677 0.8786 0.7839 0.7076 0.6497 0.6043 0.5676 26 0.8616 0.8784 0.7843 0.7079 0.6500 0.6045 0.5677 27 0.8556 0.8783 0.7846 0.7083 0.6502 0.6047 0.5678 28 0.8496 0.8781 0.7850 0.7086 0.6504 0.6048 0.5679 29 0.8437 0.8779 0.7853 0.7089 0.6507 0.6050 0.5680 30 0.8378 0.8776 0.7856 0.7091 0.6509 0.6051 0.5681 31 0.8320 0.8774 0.7859 0.7094 0.6511 0.6053 0.5682 32 0.8262 0.8771 0.7862 0.7097 0.6513 0.6054 0.5683 33 0.8205 0.8768 0.7864 0.7099 0.6515 0.6056 0.5684 34 0.8149 0.8764 0.7866 0.7101 0.6517 0.6057 0.5685 35 0.8094 0.8761 0.7868 0.7103 0.6518 0.6058 0.5686 36 0.8038 0.8757 0.7871 0.7105 0.6520 0.6060 0.5687 37 0.7984 0.8753 0.7872 0.7107 0.6522 0.6061 0.5687 38 0.7930 0.8749 0.7874 0.7109 0.6523 0.6062 0.5688 39 0.7877 0.8745 0.7876 0.7111 0.6525 0.6063 0.5689 40 0.7824 0.8741 0.7877 0.7113 0.6526 0.6065 0.5690 TAYLOR DISTRIBUTION APERTURF FFFICIFNCY AS A FUNCTION OF SIDFLOPE LFVFL (SL) AND TRANSITION POINT (N). ALPHA= 1.0 N SL= -20 -30 -40 -50 -60 1 2 0.8069 0.7493 3 0.8465 0.7544 0.6990 4 0.8790 0.7704 0.7000 5 0.9021 0.7861 0.7064 0.6530 6 0.9181 0.7993 0.7137 0.6552 0.6134 7 0.9292 0.8103 0.7206 0.6585 0.6136 8 0.9368 0.8193 0.7268 0.6620 0.6148 0.5789 9 0.9418 0.8268 0.7322 0.6655 0.6165 0.5791 10 0.9450 0.8330 0.7370 0.6687 0.6184 0.5797 0.5491 11 0.9467 0.8383 0.7412 0.6717 0.6203 0.5806 0.5491 12 0.9472 0.8428 0.7449 0.6744 0.6221 0.5817 0.5494 13 0.9470 0.8466 0.7481 0.6769 0.6238 0.5827 0.5498 14 0.9460 0.8500 0.7510 0.6791 0.6255 0.5836 0.5504 15 0.9445 0.8528 0.7536 0.6811 0.6270 0.5849 0.5510 16 0.9425 0.8554 0.7559 0.6830 0.6284 0.5859 0.5517 17 0.9402 0.8576 0.7580 0.6846 0.6297 0.5868 0.5523 18 0.9375 0.8595 0.7599 0.6862 0.6309 0.5877 0.5530 19 0+9346 0.8612 0.7616 0.6876 0.6320 0.5886 0.5536 20 0.9316 0.8628 0.7632 0.6889 0.6331 0.5894 0.5542 21 0.9283 0.8641 0.7647 0.6901 0.6340 0.5902 0.5548 22 0.9249 0.8653 0.7660 0.6912 0.6350 0.5909 0.5554 23 0.9214 0.8664 0.7672 0.6923 0.6358 0.5916 0.5559 24 0.9178 0.8673 0.7683 0.6932 0.6366 0.5923 0.5564 25 0.9142 0.8681 0.7694 0.6941 0.6373 0.5929 0.5569 26 0.9104 0.8689 0.7703 0.6949 0.6381 0.5935 0.5574 27 0.9066 0.8696 0.7712 0.6957 0.6387 0.5940 0.5578 28 0.9028 0.8701 0.7721 0.6965 0.6393 0.5945 0.5583 29 0.8989 0.8707 0.7729 0.6972 0.6399 0.5950 0.5587 30 0.8950 0.8711 0.7736 0.6978 0.6405 0.5955 0.5591 31 0.8911 0.8715 0.7743 0.6984 0.6410 0.5959 0.5594 32 0.8872 0.8719 0.7749 0.6990 0.6415 0.5964 0.5598 33 0.8833 0.8722 0.7756 0.6996 0.6420 0.5968 0.5601 34 0.8793 0.8724 0.7761 0.7001 0.6424 0.5972 0.5605 35 0.8754 0.8727 0.7767 0.7006 0.6428 0.5975 0.5608 36 0.8715 0.8728 0.7772 0.7010 0.6432 0.5979 0.5611 37 0.8675 0.8730 0.7777 0.7015 0.6436 0.5982 0.5614 38 0.8636 0.8731 0.7781 0.7019 0.6440 0.5985 0.5616 39 0.8597 0.8732 0.7786 0.7023 0.6444 0.5988 0.5619 40 0.8558 0.8733 0.7790 0.7027 0.6447 0.5991 0.5622 TAYLOR DISTRIBUTION APERTURE EFFICIENCY AS A FUNCTION OF SIDELOPE LFVFL (SL) AND TRANSITION POINT (N). ALPHA= 2.0 N SL= -20 -30 -40 -70 -80 1 0.7805 0.6972 0.6503 0.6260 3 0.7550 0.6741 0.6267 0.5977 0.8017 0.7021 0.6394 0.5990 0.5715 5 0.8361 0.7265 0.6539 0.6056 0.5717 6 0.8615 0.7467 0.6673 0.6134 0.5750 0.5464 7 0.8802 0.7632 0.6790 0.6211 0.5793 0.5478 0.5234 0.8943 0.7767 0.6891 0.6282 0.5839 0.5502 0.5238 9 0.9049 0.7880 0.6978 0.6346 0.5883 0.5529 0.5250 10 0.9128 0.7974 0.7053 0.6403 0.5924 0.5556 0.5265 11 0.9187 0.8055 0.7118 0.6454 0.5962 0+5583 0.5282 12 0.9231 0.8124 0.7175 0.6499 0.5997 0.5609 0.5300 13 0.9262 0.8183 0.7225 0.6539 0.6028 0.5633 0.5317 14 0.9283 0.8235 0.7270 0.6575 0.6057 0.5655 0.5333 15 0.9296 0.8281 0.7309 0.6607 0.6084 0.5676 0.5349 16 0.9302 0.8321 0.7345 0.6637 0.6108 0.5696 045365 17 0.9302 0.8356 0.7377 0.6663 0.6130 0.5714 0.5379 18 0.9298 0.8388 0.7406 0.6688 0.6150 0.5730 0.5392 19 0.9290 0.8416 0.7432 0.6710 0.6168 0.5746 0.5405 20 0.9278 0.8442 0.7456 0.6730 0.6186 0.5760 0.5417 21 0.9264 0.8465 0.7478 0.6749 0.6201 0.5773 0.5428 22 0.9247 0.8485 0.7498 0.6766 0.6216 0.5786 0.5438 23 0.9228 0.8504 0.7517 0.6782 0.6230 0.5798 0.5448 2/ 0.9207 0.8521 0.7534 0.6797 0.6243 0.5808 0.5458 25 0.9185 0.8537 0.7550 0.6811 0.6255 0.5819 0.5466 26 0.9161 0.8551 0.7565 0.6824 0.6266 0.5828 0.5475 27 0.9137 0.8564 0.7579 0.6836 0.6276 0.5837 0.5482 28 0.9111 0.8575 0.7592 0.6847 0.6286 0.5846 0.5490 29 0.9084 0.8586 0.7604 0.6858 0.6295 0.5854 0.5497 30 0.9057 0.8596 0.7615 0.6868 0.6304 0+5862 0.5503 31 0.9029 0.8605 0.7626 0.6877 0.6312 0.5869 0.5510 32 0.9000 0.8613 0.7636 0.6886 0.6320 0.5876 0.5516 33 0.8971 0.8621 0.7645 0.6895 0.6327 0.5882 0.5521 34 0.8942 0.8628 0.7654 0.6903 0.6334 0.5888 0.5527 35 0.8912 0.8634 0.7663 0.6910 0.6341 0.5894 0.5532 36 0.8882 0.8640 0.7671 0.6917 0.6347 0.5900 0.5537 37 0.8852 0.8645 0.7678 0.6924 0.6353 0.5905 0.5542 38 0.8821 0.8650 0.7685 0.6931 0.6359 0.5910 0.5546 39 0.8790 0.8654 0.7692 0.6937 0.6365 0.5915 0.5550 1.0 0.8759 0.8658 0.7699 0.6943 0.6370 0.5920 0.5555 APPENDIX THREE THE'EYALUATION OF TIDE FIELD INTEGRAL

The Integral: co exp [-ik(xt+v/T:J)] I(x,z,p) = dt •• • 7 (t2-p2) )77

plays an important role in the analysis of the fields in the parallel plate • region and will now be put in a suitable form for evaulation. Making the substitutions:

x = r coe z . r sing 0 : g.. 71- ... 2

t . cos° dt = - sine de e . 0 + iy ... 3

and noting that the square root in Eqn. 1 must be positive real or negative imaginary, the integral can be transformed to :

I(x,z,m) = exp [-ikr cos (0-R)] de .•. 4 0052(9- 02

where the contour C in the complex e plane is shown in Fl. Now the magnitude of the exponential function in the integrand is:

exp [- kr sin (0-g) sh y] ..• 5

where e = 0 + i y (sh and ch will be used as shorthand notations for sinh and cosh respectively). Thus as y becomes large in magnitude and

sin (0 - fi) sh y > 0 ••• 6

the integrand goes to zero and the endpoints of the contour C can be moved to any 0 values satisfying Eqn. 6 without altering the value of the integral. Thus the allowed value of 0 for the endpoints are:

y < 0 - < <

y > 0 # <0

as illustrated in Fl. Note also that the exponential in the integrand decays most rapidly with y along the lines:

y < 0 # - 7r/2 ... 8 y > 0 = w/2 -247- -Tr

rc 1f3

Fl CONTOURS IN THE COMPLEX THETA PLANE

-248- 4 •

F2 DIFFERENT CONTOUR AND POLE CONFIGURATIONS IN THE THETA PLANE F3 DIFFERENT CONTOUR AND POLE CONFIGURATIONS IN THE THETA PLANE

• • •

that are part of the contour C' in F1. In order to transform the contour from C to C', the contributions from any intervening poles must be determined. The cases p < 1 and p > 1 will be considered separately. When p is less than unity Eqn. 4 can be written:

1 f exp [ -ikr cos (0-g)] gx,z1cosaY = de ... 9 Tr cos2e-cos2a

Noting that # lies in the range 0, # IT and that the poles of the integrand lie at:

= ± a 4- nv. 0:a v/2 ...10

then there are three positions of C' relative to C that must be considered as shown in F2 (the horizontal segments of C and C' lie on the real axis and are only shown displaced to clarify the diagram). With I and I' being the value of the integral along C and C' respectively we obtain:

I - I' = 2i cos [kr cos (a+(Q)]. / sin 2 a ; 0 < # < 7T/2-a = 0 ; 7r/2-a < 0 < v/2-Fa = 2i cos [kr cos (a-,#)] / sin 2 a ; v/2-0 < fi' < IT

4.• 11 The integral along C' is now evaluated by substituting the 8 values along the three straight line segments yielding: fir I' = . exp [-kr sh y] dy i exp [-kr sh sin2(13-iy)-cos2a ff sin2(4iy)-cos2a 0 0

v/2 exp [-ikr cose ] de + 12 1 f coS2(9+Q) - cos2a -v/2

Using theexpansions:

00 1 _ ( -)nsin 2na exp [-2n(y-i0)] sin2(g+iy) - cos2a sin 2a L_J n=1 ... 13

00 1 4 ) sin 2na cos [2n(9+19)] ... 14 ccs2(0+#) - cos2a sin 2a

-251-

and the conjugate of the former, all of which can be derived using some fairly straightforward algebra, we obtain:

00 8i I' - v sin 2a ( -)n sin 2na cos 2n/3 f exp [-kr sh y 2ny] dy 0

u/2 8 sin 2a sin 2na cos 2n0 cos (kr cose) cos 2ne ae n=1

v/2 - i sin (kr cose) cos 2n8 de ... 15 f ] 0

The integrals in this expression are discussed by Watson (R13) and when his expressions are substituted, Eqn 15 yields:

OD (-)nsin 2na cos 2re S (1o') /u] sin 2a [J2n(kr) 1.2n(kr) 2n

... 16 where Jn(x), Yn(x) and Sn(x) are Bessel functions of the first and second kind and Schlafli's polynomial respectively. The imaginary part of Eqn. 16 is only slowly convergent. A more useful expansion can "be obtained by using alternative functions defined by Watson (R13 section 10.6) and given in Eqn. 5.12 and 5.13 of chapter two. Thus defining Wn(x) in terms of these functions as:

2 W (x) = i [inx -1112 + W ] J (x) + i n n — Un (x) ... 17 gives:

2n J2n( kr ) Y2n( kr ) S2n(kr) / n = W2n(kr) - i T (kr) /11- ... 18

Now using Eqn. 5.12 (chapter 2) which can be written:

T (x) = n+2m(x) ... 19 n 131=-4)0 o it is laborious but straightforward to prove that: -252-

(kr) = a cos (kr cosa) ; lal u 2 ... 20 -)n sin 2na T2n r( n=1

It should be noted that this summation represents a function that is discontinuous at a = t u/2 as it is easy to show that f(a+ir) = f(a). Thus using Eqn. 18 and Eqn. 20 in Eqn 16 gives:

CO I' = ( -)n sin 2na cos 2ng W2n(kr) sin 2a n=1 CO 2i 2n(kr) sin2a )---.1 (-)n sin [2n (a+g)] T n=1

(-)n sin [2n (a-(3)] T2n(kr) 21

Combining Eqn. 11 and Eqn. 21 yields finally:

CO 4 I(x,z,cosa) _ sin 2a (-)n sin 2na cos 2nd' W2n(icr n=1

\-1 2i (a-116-7r) cos [kr cos (a+g).1 ... 22 7T sin 2a (a43) cos [kr cos (a-p)]

which, perhaps surprisingly, does not have the discontinuities of Eqn.11. It turns out that these discontinuities cancel exactly with those of the T (x) summations in Eqn 21 referred to above. n

When p is greater than unity Eqn 4 can be written: •

f exp [-ikr cos (0-,0)] Z ch a) = de ... 23 0052 e - ch2 a

Considering the paths C and C' as before and noting the pole positions at:

0 = ± is ± nv ... 24

two positions of O'relative to C have to be studied as shown in F3 (where again for clarity segments of C and C' on the real axis have been shown -253-

displaced). Thus with notation I and I' as before:

I - I' = -2i exp(-krshasin/3) sin (krchacoe) /sh2a ; 0 < 0 < v/2

= 2i exp(-krshasie) sin (krchacoe) /sh2a ; ff/2 < < w

or equivalently:

I - I' = -2i exp(-krshasie) sin (krcha lcoel) /sh2a ... 25

Expanding the integral along C'into its values along the three straight line segments gives:

CO 00 1'1 exp (-krshy) I' - i dy + i f exp (-krshy) dy 7 sin2 ch2a Jo sine (e+iy) ch2a o_

7/./2 exp (-ikrcose) + -1; f de ... 26 ccs2(e+0) - ch2a -ir/2

Using the expansions (discussed in section 5, chapter 2):

1 shy chy sh2a ... 27 sin2(g+iy)-ch2a 2C ch2(a+ifi) + sh2y

en(-)ncos 2110 exp E-212 (a+y)J

11= 0

2 exp (-2na) cos [2n(0+4q)] ... 28 ccs2(04-#) -ch2a sh2a 11=0

Eqn. 26 yields: 00 -2 n E (-) cos 2n0 exp (-2na) j2n(kr) iY2n(kr) If = sh2a n S2n(kr) /ff 11= 0 ... 29

00 00 4. 2i t exp (-krt) t exp (-krt) irsh2a f ch2(a+.10) t2 CL-G ch2(a-iP) + t2

-254- where the substitution t = shy has been used in the latter integrals. Now using simple substitutions the integrals can be put in the form: 00 g(z) = f u exp (-zu) 2 ... 30 u A- 1 0 which defines one of the two auxiliary functions for the sine and cosine integrals (R14). Thus combining Eqn. 25, Eqn. 29, and Eqn. 30 gives:

(kr) Y2n(kr)-1 I(x,z,cha) = -=1- en( -)n cos2n0 exp (-2na) [j2n sh2a S2n(kr) /r 11=0

;7E72i [g [krch(a+0)] + g [kr ch(a-i]]

2i sh2a exp (-krshasinO)sin(krchalcaerl) ... 31 completing the derivation. The coupling integral has z equal to zero and # therefore either zero or 71. giving from Eqn. 22 and Eqn. 31:

I(x,o„cosa) = 2 (1-2a/v) cos (kxcosa) sin 2a o a 4 /1/2 00 + 2 ) (-)n sin 2na W2n(kx)] ... 32 n=1

i sin(lkxlcha) - g (kxcha) I(x,o,cha) =sh -2 2a o a 00 Co + ' En(-)n exp (-2na)[W2n(kx) iT2n(kx) br]

... 33

The following derivative function can be obtained from Eqn. 32 and Eqn. 33 by diferentiation:

I(x,o,p) + kx Hi(kx) • [pI(x,o,p)] = p2 - kx K (x,o,p) .32+ aµ _ m2 where K(x,o,p) is given by:

-255-

1 i K(x,o,cosa) = sina (1-2a/g) sin (kxcosa) o a < g/2 00 sird(221.420apv_(cx) ... 35 n= 0

K(x,o,cha) = - shaI - f(kxcha) - i cos (kxcha)sgn (kx) o a 00 n kx (-)n exp -(2'1+1)a][w2 +1( ) n=0

T2n+1(1°11K )/17]] ... 36 where sgn (x) is +1 if x is positive and -1 if x is negative.

-256- APPENDIX FOUR - COMPUTER PROGRAMS FOR ARRAY SOLUTION

The following set of subroutines have been developed for solving the finite parallel plate waveguide array discussed in chapter two. As they are reasonably well documented in themselves only a minimum of additional information will be given here. The term 'transverse free space wavelength' refers to A in the main text. There are two major routines, SOLVE for deriving a solution for the modal fields in the waveguide regions and PAT for calculating the corresponding 'free space' solution in terms of the element patterns of the array elements. Subroutine SOLVE must be called first with the desired input parameters as given in its listing. Note that the modal indices input in array MI will start from 0 for TE modes and 1 for TM modes. SOLVE computes the'reflected' modal excitations in the waveguide elements when each of the latter is excited in turn. These are output in the array CREF with array CR . aand im(Cmp +Dmp) for TE containing the solution vectors/(Amp +Bmp)kzm and TM excitation respectively. The latter are used by PAT for element pattern computation. The array CS contains the aperture scattering matrix when viewed as an NE port network at the waveguide terminals with reference planes in the aperture. The subroutine PAT uses the output of SOLVE (array CR) and computes the element patterns for angles from broadside to +ve endfire. There are NPAT equally spaced pattern points giving an increment 90/(NPAT-1) degrees. Note that the first point is at broadside and the last at endfire with the angle 6 (TH in the subroutine) being measured from broadside (in contrast to the pattern angle 0 used in section 4, chapter 2 of the main text which is measured from the +ve x axis). The routine PAT outputs the element patterns in the complex array CPAT with CPAT(M,N) being the M'th pattern point in the N'th element pattern. Patterns for 6 less than zero are not given since they are identical to the patterns for 0 positive with the element numbering reversed (ie element NE becomes element 1 etc.). The pattern for an arbitrary complex excitation of the array elements contained in the array A(N) is easily obtained by computing the sum: NE, A(N) x CPAT(M,N) N=1 for each pattern point M from 1 to NPAT. The element patterns output by the subroutine are correctly normalised in such a way that the integrated power pattern gives the radiated power with unit incident excitation power on the excited element.

-257- The remaining subroutines with the exception of MATMULT are used by SOLVE in generating the solution. Subroutine MATMULT is a general purpose matrix multiplying routine. These routines are fairly well documented and will not be covered further.

•

-258- to MATRIX GENERATION AND SOLUTION

001 * 002 * THIS SUBROUTINE GENERATES THE COUPLING MATRIX ( CM) AND THE RIGHT 003 * HAND SIDE VECTORS ( CR) FOR THE FINITE ARRAY AND THEN SOLVES THI S 004 * SET. WI TH W AS THE "TRANSVERSE FREE SPACE WAVELENGTH": THE 005 * INPUT/OUTPUT PARAMETERS ARE: 006 * 007 * INPUT: AKA = 2*PI*A/W ; A = GUI DE WI DTH 008 * AKD = 2*PI*D/W ; D = GUI DE PITCH 009 * NE = NUMBER OF ELEMENTS 010 * NMD = NUMBER OF MODES USED IN SOLUTION 011 * MI ARRAY CONTAINING THE MODAL INDICES 012 * LEH +VE FOR H ( TE) MODES/ —VE FOR E (TM) MODES 013 * OUTPUT: CR ARRAY FOR THE OUTPUT OF THE SOLUTION VECTORS• 014 * VIEWED AS A THREE DIMENSIONAL COMPLEX ARRAY 015 * WITH DIMENSIONS CR( NMD, NE,NE) CRC I Ps NP•ND) 016 * I S THE SOLUTION FOR THE . I P' TH MODE ON THE 017 * NP' TH ELEMENT WITH THE LOWEST MODE ON ELEMENT 018 * ND DRIVEN WITH UNIT AMPLITUDE. 019 * CREF ARRAY CONTAINING THE REFLECTED MODAL AMPLI TUDES. 020 * ITS FORMAT IS THE SAME AS THAT OF ARRAY CR 021 * CS THE APERTURE SCATTERING MATRIX. AS A COMPLEX 022 TWO DIMENSIONAL ARRAY WITH DIMENSIONS CS(NE, NE), 023 * CS( IP, I D) I S THE OUTPUT AT ELEMENT I P WITH 024 * ELEMENT I D DRIVEN 025 * SCRATCH: CM WORKING STORAGE FOR THE COUPLING MATRIX. AS 026 * THE COUPLING MATRIX I S SYMMETRIC ONLY ELEMENTS 027 * ON OR ABOVE THE DIAGONAL ARE STORED IN CM. IN 028 * COMPLEX ARRAY FORMAT CM( J*( J... 1 ) /2+I ) CONTAINS 029 * C( J) WITH J GREATER THAN OR EQUAL TO I • 030 * 031 SUBROUTINE SOLVE( AKAs AKD, N E, NM MI LEH• CR, CM, CREF, CS) 032 DIMENSION MI ( 1), CR( 1), CM( 1), CREF( 1), CS( I), CSM( 800) 033 DATA PI /3. 14159 26 5/ 034 * 035 * LOOP TO SELECT ALL ELEMENT SPACINGS ON THE ARRAY 036 * 037 DO 100 IX=1,NE 038 AKX= ( 1— I X) *AKD 039 * 040 * OBTAIN MODAL COUPLING MATRIX FOR THIS SPACING 041 * 042 CALL SUBMAT( AKX, AKA, MI , NMI)/ L CSM ) 043 * 044 * INTER MODE COUPLING ON SINGLE ELEMENTS ( X = 0) 04 5 * 046 IF( IX•NE• 1) GO TO 40 047 DO 30 IP=1,NE 048 J= ( I P— 1) *NMID 049 KZ=J+J 050 DO 20 MP= Is NMD 051 J= J+ 1 052 K=KZ+J*( J— 1) 053 L= 2*NMD* (MP— 1 ) 054 DO 10 MD= 11MP 055 K=K+ 2 MATRIX GFNERATI ON AND SOLUTION

056 L=L+ 2 057 CMCK-1)=CSM(L-1) 058 CMCK)=CSM(L) 059 10 CONTINUE 060 20 CONTINUE 061 30 CONTINUE 062 GO TO 8 0 063 * 064 * LOOP TO SCAN ALL ELEMENT PAIRS WITH GIVEN SPACING 065 * 066 40 DO 70 IP=IXPNE • 067 t.J= ( I P• 1 ) *NM D 068 KZ=2*NMD*( I P*I X) 069 * 070 * LOOPS FOR SCANNING MODES 071 * 072 DO 60 MP= 1, NMD 073 Jr: J+ 1 074 H=KZ+J*( J— 1) 075 1.=2*NMD*( MP-. 1) 076 DO 50 MD= 1,NMD 077 K=K+2 078 L=L+ 2 079 CMCK-1)=CSMCL.• 1) 080 CM(K)=CSM(L) 081 50 CONTINUE 082 60 CONTINUE 083 70 CONTINUE 084 * 085 * GENERATE RIGHT HAND SI DES 086 * 087 80 MX= 40 0*AKA**2 088 I F(L EH•L T• 0) AMC= 2. 0*AKA*SORT( 1. 0—C PI /AKA)**2) 089 R= 2*NE*NMD 090 J=CIX*.1)*K+1 091 K=IX*K 092 DO 90 L=UP K 093 CR(L)=0. 0 094 90 CONTINUE 095 L=2*( 1)*NMD*(NE+1)+1 096 CRCL )=ARX 097 1 0 0 CONTINUE

098 * -411- 099 * SOLVE EQUATIONS USING SUBROUTINE CSGEL 100 * 101 CALL C SG EL ( CM, CRP N E*NM DP NE, 1 • 0 5) I C SM ) 102 I F( I ER• EO• 0) GO TO 1 1 0 103 STOP 104 1 1 0 NES=NE*NE 105 NM 2= 2*NM D 106 * 107 * COMPUTE MODAL REFLECTED FIELDS 108 * 109 I FCLFH•L T. 0) GO TO 160 110 DO 150 IP= loNMD

—260— MATRIX GENERATI ON AND SOLTJTI ON

111 T= PI *MI ( I P) /AKA 112 U= 1.0— T*T 113 T= 1 . 0 /( AKA*SORT( ASS( U) ) ) 114 K= I P+ I P 115 I F( U•LE. 0. 0) GO TO 130 116 DO 120 J=1,NES 117 CREF( K— 1) = T* CRC 1) 118 CREF(K)=T*CR(K) 119 K=K+NM 2 120 120 CONTINUE 121 GO TO 150 122 130 DO 140 J= 1, NES 123 CREF( K )= T*CR( K.- 1 ) 124 CREF( 1 ) • T* C R(K ) 125 K=K+ NM 2 126 140 CONTINUE 127 150 CONTINUE 128 GO TO 180 129 160 J= 2*NE*NE*NM D 130 DO 170 K=1, J 131 CREF( K)=CR(K) 132 17 0 CONTINUE 133 180 K= 1 134 J= 2*NMD*( NE+ I ) 135 DO 190 IP=1,NE 136 CREF(K)=CREE(K) —1. 0 137 K=K+J 138 19 0 CONTINUE 139 * 140 * LOCATE AND STORE SCATTERING MATRIX ELEMENTS 141 * 142 J=2 143 K=2 144 DO 200 L= lo NES 145 CS( J-1 ) = CREF( K-1) 146 CS( J)=CREF(K) 147 J=J+R 148 K=K+NM2 149 200 CONTINUE 150 * 151 * MODI FY ARRAY CR TO SIMPLIFY PATTERN CAL CUL ATI ON 152 * 153 DO 270 I P= 1,NMD 154 J=MOD( MI ( I P) 4)+ 1 155 K=IP+IP 156 GO TO ( 270,210s 2301250),J 157 210 DO 220 J=1, NES 158 T=CR(K) 159 CR(K )=CR( K— 1 ) 160 CR(K— 1 )=— T 161 K=K+NM2 162 220 CONTINUE 163 GO TO 270 164 230 DO 240 J= la NES 165 CR(K)=—CR(K)

-261- MATRIX GENERATION AND SOLUTION

166 CRC H•.1 = - CRC K- 1) 167 H=K+NM2 168 240 CONTINUE 169 GO TO 270 170 250 DO 260 J=1,NES 171 T= CRC 10 172 CR(f )=. -.CRC K- 1) 173 CR( K...1 )=T 174 K=H+NM2 175 260 CONTINUE 176 270 CONTINUE • 177 RETURN 178 END

•

-262- MATRIX GENERATION AND SOLUTION

179 * 150 * THIS SUBROUTINE GENERATES THE MODE COUPLING SIM—MATRIX FOR 181 * ELEMENTS AT POSITIONS XD ( THE DRIVEN ELEMENT) AND XP ( THE 182 * PASSIVE EL EMENT) • WI TH A AS THE TRANSVERSE GUI DE WI DTH AND 183 * W AS THE "TRANSVERSE FREE SPACE WAVELENGTH", THE INPUT 134 * PARAMETERS AKX AND AKA ARE 2*PI *( X D—XP) /W AND 2*PI*A/W 185 * RESPECT' VELY• THE ARRAY MI CONTAINS THE MODAL INDICES OF 186 * THE MODES TO BE CONSIDERED AND NMD CONTAINS THE NUMBER OF 187 * SUCH MODES• THE LAST INPUT PARAMETER LEH I S POSI TIVE FOR H 188 * (TE) MODES AND NEGATIVE FOR E ( TM) MODES• ON RETURN THE 189 * ARRAY CSM CONTAINS THE SUB—MATRIX IN THE FORMAT OF A TWO 190 * DIMENSIONAL COMPLEX ARRAY WITH ITS FIRST AND SECOND INDICES 191 * CORRESPONDING TO MODES OF THE DRIVEN AND PASSIVE ELEMENTS 192 * RESPECTIVELY AND MODE RANKING AS IN ARRAY MI • 193 * 194 SUBROUTINE SUBMATC AKX, AKA, MI NMD, L EH, CSM) 195 DOUBLE PRECISION AMU/ MR, DZI s Dn., DPI MR" EMI , 196 & EZR, EPR, EPI P, Oa Rs SR, SI, TR, TI 197 DIMENSION MI( 1),CSM( 1),AMU( 20), DMR( 20), DMI ( 20), DZPJ 20)1 198 & DZIC 20), DPR( 20), DPI( 20), ENRC 20), EMI( 20), EZERC 20), 199 & EZIC 20),EPR( 20), EPIC 20) 200 * 201 * COMPUTE MU VALUES AND CALL "DINT" FOR COUPLING INTEGRAL VALUES 202 * 203 R=AKA 204 P=3. 1415926535898DO/R 205 DO 10 IP=1,NMD 206 AMU( I P)=P*MI C I P) 207 10 CONTINUE 208 0=AKX 209 P= DABS( 0-.P.) 210 R= DABS( 0+R) 211 0= DABS((` ) 212 CALL DI N T( P, AMU, NM D, DMR, DM I EMR, FM I ) 213 CALL DINT( Oa AMU, NM Da DZR, DZ I EZ Fia EZ I ) 214 CALL DINT( R, AMU, NM D, DPP., DPI EPR, EN ) 215 * 216 * COMPUTE AND SAVE INTERMEDIATE VALUES FOR THE GENERATION OF OFF 217 * DIAGONAL SUB—MATRIX ELEMENTS AND GENERATE DIAGONAL ELEMENTS 218 * 219 DO 60 IP=1,NMD 220 P= 1 • ODO 221 IFCMOD(MI C IP), 2) •NE• 0) P=—P 222 SR=DZE( I P)—P*DMR( I P) 223 SI=DZI ( I P)—P*DMI ( IP) 224 TR=DZRC I P)—P*DPR( I ID) 225 TI =DM(IP)—P*DPI IP) 226 0=AMU( I P) 227 R= 1 • ODO— 0* 0 228 IF(LEH.LT• 0) 0=0*R 229 * 230 * STORE INTERMEDIATE VALUES 231 * 232 DMR( I P)= 0* SR 233 WI( IP)=0*SI

-263- MATRIX GENERATION AND SOLUTION

234 DPR( I P) = 0.*TR 235 DPI ( I P)=0*TI 236 * 237 * DIAGONAL ELEMENT COMPUTATI ON 238 * 239 P= 0 • 5DO*P 240 SR= 0 • 5D0*( SR+TR) 241 SI=0•5D0*( SI+TI) 242 TR=EZRC I P)-P*C EMR( I P)+EPR( IP)) 243 TI=EZI (IP)-P*C EMI( IP)+EPI( IP)) • 244 IF(LE1-i.GT• 0) GO TO 20 245 sR=( n-n• oDo)*sn 246 SI=t ODO)*SI 247 TR=R*TR 248 TI=R*TI 249 20 SR= S R+ TR 250 SI =SI+TI 251 IFCAKX•NE• 0. 0) GO TO 50 252 * 253 * ADD IN ADDITIONAL MAIN DIAGONAL TERMS 254 * 255 P=AKA*DSORT( DA13S(R)) 256 I F(LEN•GT* 0) P=P/R 257 IF(MI(IP)•NE• 0) GO TO 30 258 P=P+P 259 IF(LEH.LT• 0) P=0•0D0 260 30 IF( R•LT• 0. ODO) GO TO 40 261 SR=SR+P 262 GO TO 50 263 40 SI =SI -P 26/4 50 I=2*( ( I P- 1 )*NMD+I P) 265 * 266 * STORE DI AGONAL ELEMENTS IN CSM 267 * 268 CSM(I-1)=SR 269 CSMC I )=SI 270 60 CONTINUE 271 * 272 * GENERATE AND STORE OFF DIAGONAL ELEMENTS 273 * 274 DO 90 I= a. NM D 275 IP=I-1 276 MP=MI ( IP) 277 P=AMUC IP) 278 J=2*((IP-1)*NMD+I) 279 K=2*IP*(NMD+1) 280 DO 80 ID=IsNMD 2.81 R= 1 • ODO 282 I F(MODEMP+MI ( I D)s 2) •NE• 0) R=-R 283 rl= AMU( I D) 034 SR=P*P-O*0 285 SI =0/SR 286 SR=P/SR 287 IF(LEH.GT• 0) GO TO 70 288 TR= SR MATRIX GENERATION AND SOLUTION

289 SR= S I 2913 SI =TP 291 70 TR= SR*C DMR( I P)+R*DP11( IP) )—SI*( DPRC I D)+R*DMR( I D) ) 292 TI=SR*( DMI I P)+R*DPI ( IP) )—SI*C DPI( I D)+R*DMIC ID)) 293 CSMC J— 1 )= TR 294 CSIV J)=TI 295 CSM( K— 1 )=R*TR 296 CSMC K)=R*TI 297 J=J+2 298 K=K+ 2*Ntol D 299 80 CONTINUE 300 90 CONTINUE 301 RETURN 302 END

—265— PATTERN CALCULATION

001 * 002 * THIS SUBROUTINE GENERATES THE ELEMENT PATTERNS IN THE ARRAY 003 * ENVIRONMENT. INPUT PARAMETERS AKA, AKD, NE, NMD, MI,LEH 004 * AND CR ARE THE SAME AS DESCRI BED FOR SUBROUTINE SOLVE. THE 005 * INTEGER 'NPAT I S INPUT AND CONTAINS THE NUMBER OF PATTERN 006 * POINTS • THESE ARE EQUALLY SPACED WITH THE FIRST ON THE 007 * ARRAY NORMAL ( X= ) AND THE LAST AT "EN DFI RE" CX +VE) • THE 008 * FIELDS RADIATED BY THE ELEMENTS ARE OUTPUT IN THE ARRAY CPAT 009 * WHICH CAN BE VIEWED AS A COMPLEX TWO DIMENSIONAL ARRAY 010 * CPATC NPAT, NE) WI TH THE FIRST INDEX CORRESPONDING TO THE 011 * POSITION IN THE PATTERN AND THE SECOND TO THE EXCITED ELEMENT• 012 * 013 SUBROUTINE PAT( AKA, AKDD NE* NM D, MI L EH* CR* CPAT, NPAT) 01'4 DIMENSION MI ( 1), CR( 1), CPAT( 1), AU( 20), UM( 20),V( 21) 015 DATA PI 2, V/ 1 • 5707963'3s 0.0,1.0,0.0,—I. 0, 0.0,1.0,0.0, 016 -1. 0, 0. 0, 1. 0, 0. 0, 1. 0, O. Os 1.0,0.0,•1.Os 017 a O. 0, 1• 0, 0. 0,•• 1• Os 0. 0/ 018 DO 10 I=1,NMD 019 AU( 1 )*PI 2*MI ( I ) 020 10 CONTINUE 021 THI*PI 2/(NPAT— 1) 022 TH=0. 0 023 XZ*...0.5*(NE...1) 024 * 025 * GENERATE PATTERN NORMALISATION VALUES SO THAT THE INTEGRAL 026 * OF THE POWER PATTERN OVER ANGLE C RADIANS) GIVES DI RECTLY 027 * THE TOTAL RADI ATED POWER 028 * 029 Tas 16. 0*PI 2*AKA 030 I F(LEH.LT• 0) VI T*SCIRTC 1 • 0—C 2. 0*PI 2/AKA)**2) /2. 0 031 T= 1 • 0/SQRT( T) 032 * 033 * LOOP FOR SCANNING PATTERN POINTS 034 * 035 DO 9 0 Los 1, NPAT 036 IF(LEH.LT. 0) CT=AKA*COS( TH) 037 X=AKA*SIN( TH) 038 Yos O. 5*X 039 TS=SIN(Y) 040 TC=COSCY) 041 * 042 * GENERATE ARRAY OF V( KXA,M) VALUES FOR REQUIRED M VALUES 043 * 044 DO 40 I=1,NMD 045 JitsMI ( I ) 046 Z=AU( I ) 047 B1=Y+Z 048 B2=Y-Z 049 Bss131*B2 050 IF(B•EQ• 0. 0) GO TO 20 051 IF(LEH.GT• 0) Zir-Y 052 B=2. 0*Z*( TS*VC J+ 2) +TC*VC .3+ 1) ) /B 053 GO TO 30 054 20 IF(B1•EQ• 0.0) B=1. 055 IF(LEH•LT. 0) Thx....B

—266— PATTERN CALCULATION

056 IFCB2.EQ•0.0) B=B+1.0 057 IF(B2•EA• 0. 0•AND*J•NE.2*(J/2)) B*—B 058 30 UM( I )308 059 40 CONTINUE 060 X=X*AKD/AKA 061 Y=X*XZ 062 * 063 * LOOP FOR SCANNING EXCITED ELEMENTS 064 * 065 DO 8 0 J=1*NE 066 RE= 0.0 067 AI= O. 0 068 I=2*(J•1)*NE*NMD 069 Z=Y 070 * 071 * LOOP FOR SUMMING OVER ELEMENTS 072 * 073 DO 60 X= 1* NE 074 CCC=COS( t) 075 SSS=SIN( Z) 076 * 077 * LOOP FOR SUMMING OVER MODES 078 * 079 DO 5 0 M= 1 NM D 080 1=1+2 081 RE=RE+UM(M)*( CR( I• 1 )*CCC•Clt( I )*SSS) 082 AI :=AI+UM(M)*( CR( I.. 1)*SSS+CR( I )*CCC) 083 50 CONTINUE 084 Z=Z+X 085 60 CONTINUE 086 * 087 * STORE PATTERN VALUES 088 * 089 I = 2*( ( J• 1 )*NPAT+L) 090 RE=T*RE 091 AI=T*AI 092 IF(LEHoGT• 0) GO TO 70 093 Et= C T* RE 094 RE=CT*AI 095 AI =B 096 70 CPAT( I • 1 )=RE 097 CPAT( I )=AI 098 80 CONTINUE 099 TH= TH+ TH I 100 90 CONTINUE 101 RETURN 102 END

-267- PATTERN CALCULATION

103 * THIS SUBROUTINE COMPUTES THE PRODUCT OF TWO COMPLEX MATRICES. 104 * THE MATRICES A AND B ARE IN COMPLEX FORMAT WITH THE DIMENSIONS 105 *,A(IAsJA) AND B(IB,JB) RESPECTIVELY. THE RESULT IS RETURNED IN 106 * ARRAY Co THE INTEGER INPUT IN LL CONTROLLING THE WAY IN 107 * WHICH A AND B ARE USED IN FORMING THE PRODUCT. LL LIES IN THE 108 * RANGE 11 TO 44, THE TENS DIGIT REFERRING TO MATRIX A AND THE 109 * UNITS DIGIT TO B WITH THE MEANING: 110 * 111 * 1 FORM THE PRODUCT WITH A/B 112 * 2 FORM THE PRODUCT WITH THE CONJUGATE OF A/B 113 * 3 FORM THE PRODUCT WITH THE TRANSPOSE OF A/B 114 * 4 FORM THE PRODUCT WITH THE CONJUGATE TRANSPOSE OF A/B 115 * 116 * ON RETURN IER IS 0 IF NO ERRORS HAVE OCCURRED AND 1 IF THE 117 * MATRIX DIMENSIONS ARE INCOMPATIBLE FOR THE PRODUCT SPECIFIED. 118 * 119 SUBROUTINE MATMULTC A. I A, JA, B, I B, dB* C, LL, I ER) 120 DIMENSION A(1),BC1),C(1) 121 IER=0 122 I=LL/10 123 JaiLL-10*I 124 AA=1.0 125 BB-1.0 126 IF(I.EQ.2*(I/2)) AAP-1.0 127 IF(J.EQ.2*CJ/2)) BB=°1.0 128 IF(I.GT.2) GO TO 10 129 II=IA 130 JJ=JA 131 NA=2*IA 132 MA=2 133 GO TO 20 134 10 II=JA 135 JJ=IA 136 NA=2 137 MA=2*IA 138 20 IF(J.GT.2) GO TO 30 139 KK=JB 140 K=IB 141 NB=2 142 MB=2*IB 143 GO TO 40 144 30 KK=IB 145 K=JB 146 NB=2*IB 147 MB=2 148 40 IFCK.EQ.JJ) GO TO 50 149 IER=1 150 RETURN 151 50 ASSIGN 60 TO JUMP 152 IF(AA*BB.LT.0.0) ASSIGN 80 TO JUMP 153 KC=2 154 KLB=2 155 DO 120 K=loKK 156 ,KLA=2 157 DO 110 I=1,II

-268- PATTERN CALCULATION

158 LA=KLA 159 LB=KLB 160 SR=0.0 161 SI=0. 0 162 GO TO JUMP 163 60 DO 70 Ju 1 JJ 164 S=AC ) 165 T=A( LA) s 166 U=130.13...1) 167 V=BC L13) 168 SR= SR+ S*U.•T*V 169 SI =SI +S*V+T*U 170 LA=LA+NA 171 LB=LB+N/3 172 70 CONTINUE 173 GO TO 100 174 80 DO 90 J=1. JJ 175 S=AC LA•• 1 ) 176 T=-ACLA) 177 U=13C LB-.1) 178 V=Bt LB) 179 SR= SR+ S*13...T*V 180 SI=SI+S*V+T*U 181 LA=LA+NA 182 LB=L B+NB 183 90 CONTINUE 184 100 CCKC-1)=SR 185 1{C)=BB*SI 186 KC=K C+ 2 187 MLA= KL A+ MA 188 110 CONTINUE 189 KLB=KLB+MB 190 120 CONTINUE 191 RETURN 192 END

-269- COUPLING INTEGRAL EVALUATION

001 * 002 * THI S SUBROUTINE COMPUTES I C X, Os MU) AND THE DERIVATIVE WITH 003 * RESPECT TO MIT OF Mt.T*I(X, 0, MU) FOR AN ARGUMENT X AND A RANGE 004 * OF MIT VALUES INPUT IN THE ARRAY AMU• THE NUMBER OF VALUES IS 005 * INPUT IN NMU AND THE REAL AND IMAGINARY PARTS OF THE ABOVE 006 * FTJNCTI ONS ARE OUTPUT IN DOUBLE PRECI SI ON IN THE ARRAYS MR/ 007 * DI I, DDR AND DDI RESPECTIVELY • 008 * 009 SUBROUTINE DI NT( X, AMU, NMU., DI Rs DI I DDR, DDI ) 010 DOUBLE PRECI SI ON Xs AMU, DI R, DI I DDR, DDI, DUI DU, DT, PI 2, 01 1 DJK, DUI MK, CA, SA/ G, R, S, T 01 2 DIMENSION AMU( 99), DIN 99 ), DI I C 99), DDRC99), DDI C99) 013 COMMON/FUN S/ DJ( 20 0 ) • DU( 20 0 ), DT( 30 0 ) 014 DATA PI 2/ 1 • 57079632679489662D0/ 015 IF(X.LT• 1. OD-6) GO TO 130 016 * 017 * FIND LOWEST MU VALUE GREATER THAN UNITY TO DETERMINE LARGEST 018 * NUMBER OF 'FOC, N) NEEDED TO COMPUTE THE SUM OF THE TERMS 019 * X, 2N) I(MU+SQRT(MU*Mtl— 1) )**2N ETC. CALL DFUNS FOR REQUI RED 020 * FUNCTION VALUES 021 * 022 DO 1 0 M= 1, NMU 023 CA=AMUC M) 024 I F( CA•GE• 1 • 0 DO) GO TO 20 025 10 CONTINUE 026 NT= 0 027 GO TO 30 028 20 NT= 35. ODO /DLOG( CA+ DSORT( CA*CA 1 • (WO) ) 029 I F(NT.GT• 295) NT=295 030 30 CALL DFUNS(X,Ntis NT) 031 G=DLOGCX/20 ODO)+0•577215664901532861D0 032 * 033 * LOOP FOR MU VALUES 034 * 035 DO 120 M= 1, NMU 036 CA=AMUCM) 037 SA= DSORTC DABS( CA*CA— 1 • 0 DO) ) 038 I Ft CA• NE. 0 • 0 DO ) GO TO 50 039 * 040 * COMPUTE LIMI TING FORMS FOR MU = 0 041 * 042 DJK= 0 • 5DO*DJC 2) 043 DUK= 0 • 5D0* DUC 2) 044 DO 40 N=4, NJ, 2 045 DJK= DJK+ DJ( N ) 046 DUK= DUK+ DUC N ) 047 40 CONTINUE 048 R.=— 2 • 0 DO*X*DJK 049 DIRCM)=R 050 DDRC M)=R 051 R=— 2. 0 DO*C X*(DUK— G*DJK ) O. 5DO*Dtg 1) ) /PI 2 052 DI I CM)=R 053 DDICM)=R 054 GO TO 120 055 *

—270— COUPLING INTEGRAL EVALUATION

056 * SERI ES SUMMATIONS FOR MU < 1 057 * 058 50 IF( CA• GT• 1 • 0 DO) GO TO 70 059 R= 0 • 5D0*( PI 2- DATAN2C SA, CA) ) 060 DJI=O• ODO 061 DJK= 0. 0 DO 062 =R*DCOSC X*CA) 063 DUK=R* DSINC X*CA) 064 R=CA 065 S=SA 066 DO 6 0 N= 2, NJ, 2 067 DJK= DJK+ S* DJC N) 068 DUK= DUK+ S* DU( N) 069 T= S*SA-R*CA 070 S= -C S* CA+ R*SA) 071 DJI=DJI +S*DJC N+ 1 ) 072 DUI = DUI +S*DU( N+ 1) 073 R=T*CA-S*SA 074 S= T* SA+ S* CA 075 60 CONTINUE 076 GO TO 1 1 0 077 * 078 * SERI ES SUMMATI ONS FOR MU > 1 079 * 08 0 7 0 CALL DEG( X*CA, R, S) 081 DJI=-0•5D0*DJC 1) 082 DJK= 0. 0 DO 083 DUI=0•5D0*(S-DUC 1)) 084 DUK=- 0 • 5DO*R 085 R= 1 • 0 DO /C CA+ SA) 086 S= 0 • 5D0*R 087 DO 80 N= 2,NT, 2 088 DUK= DUK+ S* DT ( N ) 089 S=-S*R 090 DUI=DUI+S*DTCN+1) 091 S= S*R 092 IF( DABSC S)•LE• l• OD-16) GO TO 90 093 80 CONTINUE 094 90 S= -R 095 DO 1 00 N= 2,14J, 2 096 DJK= DJK+ S* DJC N ) 097 DUK= DUK+ S* DU( N ) 098 S=- S*R 099 DJI=DJI+S*DJ(N+1) 100 DUI= DUI + S* DU( N+ 1 ) 101 S= S*R 102 I FC DARSC S ) •LE• 1 • OD- 1 6) GO TO 1 1 0 103 1 0 0 CONTINUE 104 * 105 * COMPUTE FUNCTION VALUES FROM SUMS 106 * 107 1 1 0 DUI= 2. 0 DO*C DUI G*DJI ) /CPI 2*CA*SA) 108 DJI= 2. 0 DO*DJI CA*SA) 109 DUK= 2. 0 DO* ( DUK-G* DJK) 1. ( PI 2*SA) 11 0 DJK= 2. 0 DO* DJK/SA

-271- COUPLING INTEGRAL EVALUATION

111 P.= CA*CA 112 S= 1 . ODOR 1. ODOR) 113 R=R*S 114 T=C X*( DUC 2) -•G*DJC 2) )+1),J( 1) ) /PI 2 115 DIR(M)=DJI 116 DI I CND= DUI 117 DDIU M) =R*DJI DJK+ S*X* DJ( 2) 118 DDI CM)=R*DUI -X*DUK+S*T 119 120 CONTINUE 120 RETURN 121 * 122 * COMPUTE LIMITING FORMS FOR X = 0 123 * 124 130 DO 160 M=1,NMU 125 CA= AMUC M) 126 SA= DSORT( DAF3SC CA*CA- 1 • ODO ) ) 127 IF( CA• GT• 1 • ODO) GO TO 140 128 R=0.0D0 129 S= 1 . 0 DO /PI 2 130 IFCCA.E0.0.0D0) GO TO 150 131 S= C 1 • 0 DO-- DATANN SA, CA) /P12) CA*SA) 132 GO TO 150 133 140 R=-1 • ODOR CA*SA) 134 S=R* !LOG CA+ SA) /PI 2 135 150 DIR(M)=R 136 DI I (M)=S 137 CA*CA 138 DDR(M)=T*RR 1 • ODO-T) 139 DDI CM) = T* S+ 1 • 0 DO /PI 2) IC l• °DO-. T) 140 160 CONTINUE 141 RETURN 142 END

-272- COUPLING INTEGRAL EVALUATION

143 * 14'I * THIS SUBROUTINE COMPUTES J(X,N), U(X,N) AND T(X0N) IN DIMPLE. 145 * PRECI SION FOP AN ARGUMENT X AND ORDERS N FROM ZERO UPWARDS. 146 * FuNCTI ON VALUES ARE OUTPUT VIA THE COMMON BLOCK /FUNS/ WITH 147 * THE (N+1) 'TH ARRAY ELEMENT'S CONTAINING THE ORDER N VALUES. 148 * THE INTEGER NJU I S OUTPUT AND I S THE HIGHEST ORDER FOR 149 * WHICH J(X,N) AND U(X,N) APE OUTPUT ACCURATELY. THE HIGHEST 150 * ORDER FOR WHICH T( X.,N) IS REQUIRED IS INPUT IN INTEGER NT. 151 * 152 StJT3ROUTI NE DEWS( X, NJUiN T) 153• DOUBLE PRECISION X, Ws DUs DT, X I a Rs So T, ti 154, COMMON/FUNS/ DJC 200)s MC 200), DT( 300) 155 * 156 * INITIALISE AND PREPARE TO GENERATE JC N ) BY BACKWARD 157 * RECURRENCE. ALSO COMPUTE SUMS OF RECIPROCAL INTEGERS 158 * NEEDED FOR THE GENERATION OF U(X,N) 159 * 160 R=X**0• 3333333333D0 161 mx--=X+ 1 5. ODO*R+ 10. ODO 162 NL=X+ 1 1. 0 DO* R+6. ODO 163 NL=2*( (NL+ 1) /2) 164 NJU=NL 165 NX=X 166 NX=2*( (NX+ 1) /2) 167 XI =14X 168 DJ(MX+2)=0• ODO 169 DJ(MX+1)=1.0D-50 170 R= 0 • ODO 171 S=O• ODO 172 T=2• ODO/X 173 U= MX* T 174 * 175 * JC X.. N) BY BACKWARD RECURRENCE AND RECIPROCAL SUMMATION 176 * 177 DO 10 I=1,MX 178 N=MX+ 1—I 179 DJ(N)=U*DJ(N+1)—DJ(N+2) 180 U= 0— T 181 DUC I )=S 182 R=R+I• ODO 183 S=S+ 1 • ODO/R 184 10 CONTINUE 185 * 186 * COMPUTE NORMALI SATI ON SUM FOR JCX,N) AND NORMALISE 187 * 188 P= 0. 5DO*DJC 1) 189 DO 20 N=3..MX, 2 190 R=R+DJ(N) 19 1 20 CONTINUE 192 O• 5DO/R 193 DO 30 N=1,MX 194 DJCIN)=R*DJ(N) 195 30 CONTINUE 196 * 197 * COMPUTE EVEN ORDER U(X,N) FOR N>X BY SUMMATION OF BESSFL

—273- COUPLING INTEGRAL EVALUATION

198 * FUNCTION SERIES REPRESENTATION 199 * 200 R=XI 201 DO 50 N=NXANL,2 202 S=DU(N+1)*DJ(N+1) 203 T=1.0D0 204 U=-1.0D0 205 I=N+3 206 DO 40 M=IpMX,2 207 S=S+U*(1.0DO/T+1.0D0/(T+R))*DJ(M) 208 T=T+1.0D0 209 U=-U 210 40 CONTINUE 211 DO(N+1)=S 212 R=R+2.0D0 213 50 CONTINUE 214 * 215 * COMPUTE ODD ORDER U(X,N) FOR N>X USING THE RECURRENCE RELATION 216 * 217 I=NX+1 218 R=XI+1.0D0 219 DO 60 N=I,NL,2 220 DUCN+1)=(0.5D0*X*(DU(N)+DUCN+2))+DJ(N+1))/R 221 R=R+2.0D0 222 60 CONTINUE 223 * 224 * COMPUTE TI(X,N) FOR NX IF REQUIRED BY SUMMATION OF 236 * BESSEL FUNCTION SERIES REPRESENTATION. COMPUTE STARTING VALUES 237 * FOR BACKWARD RECURRENCE FOR N

-274- COUPLING I NTEGPAL EVALTIATI ON

251 DO 100 M=3,11Y., 2 2511 T= T+ 1 . 01)0 255 IF(M.EO.I) GO TO 100 256 S=S+C 1 • 0 DO /T- 1 • ODO/C T+U) )*DJ(M) 257 1 0 0 CONT. / NUE 258 DT01+1)=S 259 R= R+ 1 • 0 DO 260 1 1 0 CONTINUE 261 * 262 * COMPUTE ODD ORDER T( X,N) FOR N>X USING THE RECURRENCE RELATION 263 * 264 I =NX+ 1 265 R=2. ODO*XI+2. ODO 266 DO 120 N=L•NL, 2 267 DTCN+1)=CX“ DT(N)+DTCN+2))+4.0DO*DJCN+1))/}1 268 R=R+ 4* 0 DO 269 120 CONTINUE 270 * 271 * COMPUTE TC X.,N) FOR N

—276— COUP!, I NG INTEGRAL EV AI, TYA TI 339 * 340 * EXPANSION COEFFI CI ENTS FOR GC 1 OY) FOR Y < 341 * 342 DATA AG / 9• 6 07 1 40896328 18717 11)- 03) 4.742176365839144373D- 03, 343 & -5.917077872512348303D-05, 2. 0814677519 03784934D- 06, 344 & -1.288042971729056144D-07, 1.148300322022694445D-08, 345 & -1.• 32547 0 1751 1 1856950D-09, 1•857289258 1 0265860 OD-1 0) 346 & -3. 028953824139 09 29 00D-1 1, 5.5840892845363583861J-12, 347 & - 1 • 1394835298300 10097D-12, 2.533437664265603311D-13, 348 & -.6* 0629642309757 14837D-14.. 1.546948 073544872,548D-• 14, 349 & -4.175815125592602601D-15, 1. 185088037779694496D-15, 350 & -3. 5175407 0 5676882629 D- 16, 1. 087 1892220558 08 002D- 16/ 351 3• 4860 53785408 351339D• 17, 1 • 155957286820813016D- 17, 352 & -3.953040337592667803D-18/ 353 C2=0.000 354 S2= 0 • 0 DO 355 IFCX.GT• 1 O• ODO) GO TO 20 356 * 357 * COMPUTE CI C X) AND SI C X) FOR SMALL ARGUMENTS 358 * 359 T=X*X1 25. ODO•2. ODO 360 C1=ACC 17) 361 S1=AS( 17) 362 I=16 363 10 C3=C2 364 C2=C1 365 S3=S2 366 52=51 367 C1=T*C2-C3+ACC I ) 368 S1=T*S2-S3+ASC I ) 369 1=1-1 370 I Ft I • NE• 0 ) GO TO 10 371 S2=X*C S1-S3) /20. ODO 372 C2=0•5D0*C Gl-C3)+DLOG(X)+0• 5772156649 0 1532861D0 373 * 374 * COMPUTE. FCC X) AND GSCX) FOR X < 10 375 * 376 S1=DSINC X) 377 C1=DCOSC X) 378 FC=C2*S1-S2*C1 379 G5=-CC2*C1+S2*S1) 380 RETURN 381 * 382 * COMPUTE Ft X) AND GC X) FOR LARGE ARGUMENTS 383 * 384 20 T=C20• ODO/X)**2-2. ODO 385 C1=AF( 21) 386 S1=AGC 21) 387 1=20 388 30 C3=C2 389 CP=C 1 390 53=52 391 " S2=S1 392 CI=T*C2-C3+AFC I ) 393 SI=T*S2-S3+AG( I )

-277- COI JP1-1 NG I NT EG PAL EV AL LIAT I ON

3914 I = I - 1 395 IF( I .NE• 0) GO Ti] 30 396 * 397 * COMPUTE: FCC X) AND GSCX ) FOR X > 10 398 * 399 FC=5. 0 Do*c C 1- C3) /X-PI 2*VCOS( X) 400 GS= 0 • 5D0*( S1- S3) -PI 2*DSINC X) 401 RETURN 402 END

-278— COMPLEX SYMMETRIC CA1TSSIAN ELIMINATION

ont * 002 * THIS SUPPOTITINF SOLVES THE SET 01. EnUATIONS C*X = Y f.OR THY 003 * VECTORS X (AVM THE COMPLEX SYMMETRIC MATRIX C(NN,NN) AND 004 * THE MATRIX Y(NN,MM) CONTAINING MM RIGHT HAND SIDE VECTORS 005 * FOR WHICH THE SOLUTION IS REQUIRED. INPUT/OUTPUT PARAMETERS 006 * ARE: 007 * 008 * A A COMPLEX ONE DIMENSIONAL ARRAY CONTAINING THE 11, 009 * ELEMENTS OF C THAT ARE ON OR ABOVE THE DIAGONAL. 010 * THAT IS A(J*(J-1)+I) CONTAINS C(I,J) WITH J 011 * GREATER THAN OR EQUAL TO I. C HAS DIMENSIONS 012 * C(NN,NN) AND A HAS THE LENGTH A(NN*(NN+1)/2). 013 * THE ARRAY A IS DESTROYED DURING THE ELIMINATION 014 * PROCESS. 015 * Y TWO DIMENSIONAL COMPLEX ARRAY Y(NNAMM) WHICH 016 * CONTAINS MM RIGHT HAND SIDE VECTORS ON INPUT 017 * AND THE MM SOLUTION VECTORS ON RETURN. 018 * NN ORDER OF THE EQUATIONS 019 * MM NUMBER OF RIGHT HAND SIDE VECTORS 020 * EPS SMALL NUMBER INPUT TO DETERMINE WHEN A LOSS 021 * OF SIGNIFICANCE HAS TAKEN PLACE. THAT'IS IF 022 * AT ANY STAGE OF THE ELIMINATION ALL REMAINING 023 * MATRIX ELEMENTS HAVE MAGNITUDES LESS THAN 024 * EPS*(L.ARGEST MAGNITUDE ELEMENT IN MATRIX C) 025 * THEN THEY ARE TO BE TAKEN AS ZERO (TYPICALLY 026 * FPS IS 1.0E-5 IN SINGLE PRECISION). 027 * I ER ERROR CODE, ZERO FOR NO ERROR, —1 FOR INPUT 028 *' PARAMETER ERROR (NN OR MM) AND- EQUAL TO THE 029 * ORDER FOR WHICH LOSS OF SIGNIFICANCE OCCURRED 030 * OTHERWISE. 031 * W SCRATCH STORAGE OF 2*NN WORDS (I.E. NN COMPLEX) 032 * 033 SUPROUTINE CSGEL(A,Y,NN,MM,EPS,IER,W) 034 DIMENSION A(1),Y(1)sW(1) 035, IF(NN.F0.0) GO TO 16 036 IF(MM.E0.0) GO TO 16 037 IER=0 038 NNT=NN+NN 039 NNL=NNT-2 040' DO 11 N=1,NN 041 NT=N+N 042 ND=N*(N+1) 043 J=ND—NT 044 PA=0.0 045 DO I I=NT,NNTs2 046 J=J+I 047 S=A(J-1)*A(J-1)+A(J)*A(J) 048 IF(S.LF.PA) GO TO 1 049 PA=S 050 K=I 051 1 CONTINUE 052 IF(N.E0.1) TOL=PA*EPS*EPS 053 IF(PA.E0.0.0) GO TO 15 054 IF(IER.NF.0) GO TO 2 055 IF(PA.GT.TOL) GO TO 2

—279— COMPLEX SYMMETRIC GAUSSI AN ELIMINATION

056 I ER=N- 1 057 2 I DF=K-NT 058 K=K*(11+2) / 4 059 PR=ACK- 1 ) /PA 060 PI=-A(K) /PA 061 I=NT 062 DO 3 M= 1, MM 063 J=I+IDF 064 S=Y( J-1) 065 T=Y( J) 066 Y( J- 1)=Y( I -1) 067 Y( J)=YC I ) 068 Yt I - 1)=PR*S.- PI*T 069 YC I )=ER*T+PI*S 070 I=I+NNT 071 3 CONTINUE 072 IF(N•EQ•NN) GO TO 12 073 I=ND 074 J= I +I DF*(NT+NT+I DF+2) /4 075 AC J-1)=A( I -1) 076 AC J)=At ) 077 A(I)=IDF 078 J=J- I DF 079 L=J 080 DO 8 K=NT,NIVL, 2 081 I=I+K 082 J=J+ 2 083 I F( I -L ) 6,4,5 084 4 S=A( L- 1) 085 T=A(L) 086 GO TO 7 087 5 J=I+IDF 088 6 S=AC J- 1) 089 T=AC J) 090 At J- 1)=AC I - 1) 091 A( J)=AC I ) 092 7 W(K- 1)=S 093 W(K)=T 094 A( I - 1 )=PR*S- PI*T 095 AC I )=PR*T+PI*S 096 8 CONTINUE 097 I DE= 0 098 L=ND 099 DO 1 1 I=NT,NNL, 2 100 I DE= I DF+ 2 101 K=L 102 P11=- WC I - 1) 103 PI =-WC I ) 104 DO 9 J= 1 NNL , 2 105 K=K+J 106 M=K+ I DF 107 ACM- 1)=AC M- 1 )+PR*A(K- 1).- PI *ACK) 108 ACM)=A(M)+PR*A(K)+PI*ACK- 1) 109 9 CONTINUE 110 J=NT

-280- COMPLEX SYMMETRIC GAUSSIAN ELIMINATION

111 DO 10 M=1,MM 112 K=J+IDE 113 Y(K-1)=Y(K-1)+PR*Y(J-1)-PI*Y(J) 114 Y(K)=Y(K)+PR*Y(J)+PI*Y(J-1) 115 J=J+NNT 116 10 CONTINUE 117 L=L+I 118 11 CONTINUE 119 12 IF(NN.E0.1) RETURN 120 I=NNT 121 J=NN*(NN+1) 122 DO 14 N=2,NN 123 J=J-I 124 1=1-2 125 IDF=A(J)+0.5 126 X=I 127 DO 14 M=1,MM 128 S=Y(K-1) 129 T=Y(K) 130 KK=K 131 JJ=J 132 DO 13 L=I,NNL,2 133 KK=KK+2 134 JJ=JJ+L 135 S=S-A(JJ-1)*Y(KK-1)+A(JJ)*Y(KK) 136 T=T-A(JJ.'1)*Y(KK)-A(JJ)*Y(KX-1) 137 13 CONTINUE 138 L=K+IDE 139 Y(K-1)=Y(L-1) 140 Y(X)=Y(L) 141 Y(L-1)=S 142 Y(L)=T 143 K=K+NNT 144 14 CONTINUE 145 RETURN 146 15 IER=N-1 147 RETURN 148 16 IER=-1 149 RETURN 150 END

-281- APPENDIX FIVE - PROGRAMS FOR MATRIX EIGENVALUES AND EIGENVECTORS

Program for obtaining the eigenvalues and eigenvectors of a complex symmetric or Hermitian matrix.

-282- EIGENVALUE AND VECTOR CALCULATION

001 * 002 * THIS SUBROUTINE COMPUTES THE EIGENVALUES AND EIGENVECTORS OF A 003 * COMPLEX SYMMETRIC OR HERMITIAN MATRIX INPUT IN THE ARRAY CS WITH 004 * COMPLEX FORMAT AND DIMENSIONS CS(NMNE).. THE INPUT PARAMETER 005 * MT IS POSITIVE FOR SYMMETRIC AND NEGATIVE FOR HERMITIAN 006 * MATRICES. THE EIGENVALUES AND VECTORS ARE OUTPUT IN THE ARRAYS 007 * CE AND CV RESPECTIVELY. IN COMPLEX FORMAT CE(J) CONTAINS THE 008 * J'TH EIGENVALUE AND CV(IsJ) CONTAINS THE I•TH COMPONENT OF THE 009 * CORRESPONDING VECTOR. THE PROGRAM GENERATES A SEQUENCE OF 010 * SIMPLE ORTHOGONAL OR UNITARY TRANSFORMATIONS THAT PROGRESSIVELY 011 * REDUCE THE MAGNITUDE OF ALL OFF DIAGONAL ELEMENTS TO YEILD A 012 * DIAGONAL MATRIX WITH THE SAME EIGENVALUES. THE INPUT PARAMETERS 013 * NIT AND TOL DETERMINE THE MAXIMUM NUMBER OF TRANSFORMATIONS 014 * GENERATED AND THE TOLERANCE FOR OFF DIAGONAL ELEMENTS 015 * RESPECTIVELY. THIS REDUCTION PROCESS TERMINATES IF EITHER NIT 016 * TRANSFORMATI ONS ARE USED OR THE MAGNITUDE OF THE LARGEST OFF 017 * DIAGONAL ELEMENT AT ANY STAGE IS LESS THAN TOL*C THE MAGNITUDE 018 * OF THE LARGEST ELEMENT IN THE MATRIX CS). THE OUTPUT PARAMETER 019 * ERR IS ZERO IF TERMINATION RESULTS FROM THE LATTER CONDITION 020 * AND EQUAL TO THE 'TOL' VALUE ACHIEVED /F NIT TRANSFORMATIONS 021 * ARE USED. 022 * 023 SUBROUTINE EIG(CSANEaMT,CEPCV,NIT,TOL,ERR) 024 DIMENSION CS(1),CE(1)..CV(1)..DS(300)oEU(300) 025 DOUBLE PRECISION DS,EN,ARDAI,BR,BLvDR.DI,ERAEIPBC..EPS 026 * 027 * TRANSFER MATRIX TO DOUBLE PRECISION WORKING STORAGE AND 028 * DETERMINE THE LARGEST ELEMENT MAGNITUDE. INITIALISE THE 029 * ARRAY DV FOR EIGENVECTOR CALCULATION 030 * 031 ERR=0.0 032 J=2*NE*NE 033 DR=0.0D0 034 DO 10 I=1,JP2 0I5 DV(I)=0+0D0 036 DV(I+1)=0.0D0 037 ER=CS(I) 038 E/22CS(1+1) 039 DS(I)=ER 040 DS(I+1)=EI 041 DI=ER*ER+EI*EI 042 IF(DI.GT.DR) DR= DI 043 10 CONTINUE 044 J=2*(NE+1) 045 Kael 046 DO 20 I=1,NE 047 DV(K)=1.0D0 048 K=K+J 049 20 CONTINUE 050 EPS=DR*TOL*TOL 051 NN=0 052 * 053 * TRANSFORMATION LOOP. START BY LOCATING THE LARGEST OFF DIAGONAL 054 * ELEMENT 055 *

-283- El GENVALUE AND VECTOR CALCULATI ON

056 30 DR= 0.O DO 057 L-NE• 1 058 DO 50 JnloL 059 K= 2*NE*J 060 DO 40 In1PJ 061 KnK+ 2 062 DI =DSC )**2+ DSCK)**2 063 I Ft DI .LE. DR) GO TO 40 064 DR= DI 065 IInI 066 JJ0J 067 40 CONTINUE 068 50 CONTINUE 069 * 070 * TEST FOR LOOP TERMINATION 071 * 072 I FINN. EQ• NI T. OR• DR•LT. EPS) GO TO 130 073 NN=NN+ 1 074 * 075 * DETERMINE SINGLE AXIS ORTHOGONAL OR UNITARY TRANSFORMATION 076 * THAT REDUCES THE LARGEST OFF DIAGONAL ELEMENT TO ZERO 077 * 078 100 2*( JJ*NE+I I ) 079 AR*DSC K• 1 ) 080 AI LIDS( 10 081 I922*(t I I*1)*NE+II) 082 Jgo 2*( JJ*NE+ JJ+ 1 ) 083 EiReDSC I •• 1 ) DSC (.1 1 ) 084 BI :IDS( I ) DSC J) 085 ERnIEIR*BR+13I*B1 086 IF( ER.NE. O. 0 DO ) GO TO 60 087 ARn0.707 1 0678 1 186548D0 088 AI=0.0D0 089 BRIBAR 090 BI-0.0D0 091 I MT. GT. 0) GO TO 10 0 092 ER=AR/DSGIRT( DSC K.• 1 )**2+ DSC K)**2) 093 BR= ER* DSC K•• 1 ) 094 BI nER*DSC K ) 095 GO TO 100 096 60 DR=( AR*BR+AI *SI )1ER 097 DI 1st AI*BR■AR*BI ) /ER 098 AR=4. 0 DO*( DR*DR+DI*DI )+1. ODO 099 AI00.0D0 100 IF(MT.LT• 0) GO TO 70 101 AR*IAIP.8. 0 DO*DI*DI 102 AI so...8 0 DO*DR*DI 103 70 BC*IDSORTC AR*AR+Al *AI ) 104 BR= DSART( 0. 5D0*( BC+ DABS( AR) ) ) 105 IF( AR•LT• 0. ODO) GO TO 80 106 ER= BR/BC 107 EI 113 0 • 5DO*AI /C EIR*BC) 108 GO TO 9 0 109 8 0 EI =BR/BC 110 ERn 0 5DO*A1 /C DR*BC) EIGEHVALUE AND VECTOR CALCULATION

111 I FC ER.GE• O• ODO) GO TO 90 112 ER= — ER 113 EI =— EI 114 90 BR= 0. 5D0*( 1. ODO+ER) 115 BC= DSORTC BR*BR+ 0 • 25DO*EI *EI ) 116 AR= DSQRT( 0. 5 DO*( BC+BR) ) 117 AI =O. 2 5DO*EI /AR 118 BI ot ( DR* El + DI *ER) /BC 119 DR= (DR* ER— DI *EI ) /BC 120 BR= DR*AR+ BI * AI 121 BI =BI *AR• DR*AI 122 100 BC=BI 123 I FCMT.LT. 0) BColt•BI 124 * 125 * PERFORM REQUI RED COLUMN OPERATIONS 126 * 127 K=2*(II-1)*NE 128 L=2* JJ*N E 129 DO 110 I=1PNE 130 K=K+ 2 131 L=L+ 2 132 DR= DS( K• 1 ) 133 DI=DS(K) 134 ER= DSC L• 1 ) 135 EI=DS(L) 136 DSC K- 1 ) =AR* DR•AI *DI + BR* ER•BC* EI 137 DSC =AR*DI + AI *DR+ BR*EI +BC*ER 138 DS (1...• 1 )=AR*ER•AI*EI • BR* DR+ HI * DI 139 DSC L )0AR*EI +AI *ER•BR*DI • BI*DR 140 DR= DV C K— 1 ) 141 DI=DV(K) 142 ER= DV( L• 1 ) 143 EI=DV(L) 144 DV( K• 1 )=AR* DR•AI * DI + BR*ER• BC* EI 145 DV( K ) = AR* DI +AI * DR+ BR*EI +BC*ER 146 DV( L • 1 )=AR*ER•AI*EI •BR* DR+ BI * DI 147 ) = AR* EI +AI *ER• BR* DI — BI *DR 148 110 CONTINUE 149 * 150 * PERFORM REQUIRED ROW OPERATIONS 151 * 152 153 LI2 JJ+JJ+ 2 154 DO 120 J=IPNE 155 DR= DSC 14... 1) 156 DI =DS(K) 157 ER=DSC 1.• 1 ) 158 EI=DS(L) 159 DS( K•1 )=AR*DR•AI*DI+BR*ER•BI*EI 160 DSCK)=AR*DI+AI*DR+BR*EI+BI*ER 161 DS (1. • 1 ) AR* ER• AI *EI • BR* DR+ BC* DI 162 DS L ) = AR* EI +AI *ER• BR* DI • BC* DR 163 K=K+NE+NE 164 L=L+NE+NE 165 120 CONTINUE

-285- El GENVALUE AND VECTOR CALCUL ATI ON

166 GO TO 30 167 * 168 * TRANSFER EI GENVALUES AND VECTORS TO ARRAYS CE AND CV 169 * 170 130 Jl 0 171 K=-2*NE 172 L=2*(NE+ 1) 173 DO 140 I=1PNE 174 J•J+ 2 175 K+L 176 CE(+3.1.1)111)SCK.1) 177 CECJ)221)5(11) 178 140 CONTINUE 179 Jot 2*NE*N 180 DO 150 I=1.041 181 CV( I )=1)VC I ) 182 150 CONTINUE 183 I FC NN•NE.N I T) RETURN 184 ERR= TOL* DSORTC DR/EPS) 185 RETURN 186 END

-286- APPENDIX SIX - IMPEDANCE TRANSFORMER AND DIRECTIONAL COUPLER SYNTHESIS

As there is an extensive literature on these topics, only an outline of the synthesis techniques will be given. Cbebyshev impedance transformers are derived by expressing the transfer function of a cascade of n quarter wave transmission line sections in the form:

2 E 1 2 E 2 Tn (a COS #/) where:

2-rf Tn(a) / (R-1) 2 and R is the desired transformer ratio. The quantity a is related to the operating band and, for coaxial line transformers, is given by:

a cos br/(1+fm/fL)] ••• 3

where f and f are the maximum and minimum operating frequencies respec- H L tively. The maximum reflection coefficient in the operating band is related to e by:

= 141 + e2 ... 4- Irl max

For a given number,of sections n, equation 2 relates E and a so that either the bandwidth fH/f or the maximum reflection coefficient irl can be L max specified but not both. Alternatively, given both fli/fL and Iri max, the number of sections n can be chosen such that:

i r n ch-la ch le(R-1)/247] ••• 5

which gives the minimum number of sections for the specified performance.

With this n value either or can be respecified as before. fH/fL Inlmax These adjustments are performed in the first part of the transformer synthesis

program. From equation 1, 11%6012 = 1 - IT(g01 2 is given by:

T2 (a cos #1) Ir(ft01 2 = ••• 6 2 2 / E + T (a cos go n

and transforming to the variable t = i tan gi gives:

-287-

T2 2 n [a/, 1-t ] r(t)r(-t) = ... 7 E2 + T2 [a/ 1.7.e)

As r(t) represents a passive reflection coefficient, all its poles must be in the left half plane. Choosing its zeros also in the left half plane gives a minimum phase network. The numerator and denominator of equation 7 have factors which can be obtained by solving:

Tn(a/ VrTIT = ± ie ... 8 with the solutions t where: k

2 2a2i t = 1 - 1+ch [2sh le/n1 cos [(2k+1)./Vn] - ish[2sh ic/n] in [(2k4-1)111/1]1 k fch[2sh 1t/n] + cos [(2k+1)71/n]i2

These factors allow r(t) to be determined and from this Z(t) using:

z(t) - r ••• 9

The impedance of the first line in the cascade is given by Z(1) and the impedance function Zred(t) of the rest of the network is computed using:

Z(t) - Z(1)t - ... 10 Zred(t) 1-tZ(t)/Z(1)

This is of reduced degree since both the numerator and the denominator have the factor (1-t2) which can therefore be removed. This process can be iteratively repeated to yield characteristic impedances for all cascade line sections. A program implementing these steps is given at the end of this appendix. 4p,

The design of TEM mode coupled line directional couplers is almost parallel to transformer synthesis as it can be shown that if the even and odd mode impedances of all coupled sections satisfy:

Z Z = 1 oe oo then the coupling and transfer functions C(t) and T(t) of the coupler are identical to the reflection and transfer functions R(t) and T(t) respectively

-288-

of a cascade of quarter wave line sections whose characteristic impedances are the even mode impedances of the coupler sections and whose input and output lines have unit impedance (it is assumed that all input/output lines of the coupler have unit characteristic impedance). For the Ghebyshev coupler the transfer function is:

2 2 — 12 IT(al)12 2 2 / E T (a cos #1) 2 n

and 2 2 / E •-• T (a n cos #1) IcV012 ... 13 2 2 •-• E2 T (a cos 131)

With the maximum and minimum coupling within the operating band given by Cox and G respectively, the mean coupling value C and the ripple R are min defined as:

C = \IC C R = \ max min I C max / min 14

These can be obtained from equation 13 as:

e2 (e2 1 ) e2 e2 1 ) 1 1 (k 2 C4 R.4 .41. 1 5 2 / 2 E ke — I) 2 2 E: 1) or equivalently:

E 2 = R2 (R2 — C2) 2 R2 - C2 1 E2 = ... 16 (R4 1) c2 (R4 - 1) with a related to the bandwidth by equation 3 as before. For a given number of sections n and mean coupling C, either the bandwidth or the ripple can be specified but not both as el and a are related by:

T (a) n ... 17

since C(o) must be zero. As for impedance transformers, given both bandwidth and ripple the number of sections can be chosen to obtain the performance specified. Setting t = i tan 181 gives:

-289- E2 T2 (a / I1-7) c(t) c(-t) n ... 18 - Ir2 (a / 11-7) 2 n and the subsequent analysis is parallel to that for transformers except that the factors of the numerator and denominator are in this case solutions of:

T [a / I1TC2] = n ... 19 and given by:

2a2[11-ch(2ch-le/n) cos (2k7r/n) - ish(2ch 1c/n) sin (21o/n)i tk2 - Ich(2ch-1c/n) 4. cos (2kn/n)]2

This proCedure has been implemented in the program at the end of this appendix.

-290- TRANSFORMER DESIGN

001 * 002 * THIS PROGRAM CALCULATES THE SECTION IMPEDANCES/ADMITTANCES 003 * FOR A MULTI-SECTION CHEBYSHEV TRANSFORMER. INPUT 004 * PARAMETERS ARE: 005 * 006 ZYS SOURCE IMPEDANCE/ADMITTANCE 007 * ZYL LOAD IMPEDANCE/ADMITTANCE 008 * NS NUMBER OF TRANSFORMER SECTIONS 009 * BWR BANDWIDTH RATIO (LONGEST/SHORTEST 010 * WAVELENGTH ON THE TRANSMISSION LINE) e 011 * SWR MAXIMUM STANDING WAVE RATIO WITHIN 012 * OPERATING BAND IN UK OR US UNITS 013 * IND GIVES WHICH OF THE THREE PREVIOUS 014 * PARAMETERS IS NOT BEING SPECIFIED 015 * (ONLY 2 OF THE 3 CAN BE CHOSEN): 016 * =-1 NS UNSPECIFIED* INCREASE 017 * BANDWIDTH IF POSSIBLE 018 * = 1 NS UNSPECIFIED, IMPROVE VSWR 019 * IF POSSIBLE 020 * = 2 BWR UNSPECIFIED 021 * = 3 SWR UNSPECIFIED 022 * 023 * OUTPUT PARAMETERS ARE: 024 * 025 * ZYT AN ARRAY CONTAINING THE SECTION 026 * IMPEDANCES/ADMITTANCES 027 * IER NON-ZERO ON RETURN IF AN ERROR HAS 028 * OCCURRED* WITH A VALUE INDICATING 029 * THE TYPE OF ERRORt 030 * = 1 SOURCE/LOAD IMPEDANCE/ADMITTANCE 031 * LESS THAN OR EQUAL TO ZERO 032 * = 2 IND OUT OF RANGE (SEE ABOVE) 033 * = 3 BWR LESS THAN OR EQUAL TO ONE 034 * = 4 SWR LESS THAN OR EQUAL TO ZERO 035 * = 5 NS LESS THAN OR EQUAL TO ZERO 036 * = 6 TOO MANY SECTIONS (.12) 037 * 038 * IMPEDANCE/ADMITTANCE DATA CANNOT BE MIXED IN INPUT 039 * TO THE PROGRAM. 040 * 041 SUBROUTINE QblTD(ZYS,ZYL*NS,BWR,SWR*IND,ZYTaIER) 042 IMPLICIT DOUBLE PRECISION (D-F) 043 DIMENSION ZYT(1)*FNT(15),FNC(15),FDT(15),FDC(15)* 044 & DCN(15),DCD(15),DZN(15),DZD(15) 045 EQUIVALENCE UNT(1)*DZN(1))*(FDT(1)*DID(1)) 046 DATA PI*DP1/3.1415927,3.141592653589800/ 047 DARCH(DX)=DLOG(DX+tSQRTCDX*DX-1.0Dfl)) 048 DFN(DX)=0.5D0*DABS(DX-16,0D0)/DSQRT(DX) 049 * 050 * CHECK INPUT DATA* CALCULATE UNSPECIFIED PARAMETER 051 * AND GENERATE IMP/ADM DATA IF NS IS UNITY 052 * 053 IER=0 054 IF(ZYS.LE.0.0.OR.ZYL.LE.0.0) GO TO 70 055 IF(IND.LT.-1.OR.IND.GT.3) GO TO 80 056 DR=ZYL/ZYS 057 DS=SWR 058 I=IND+2 -291.. • TRANSFORMER DESIGN 059 GO TO (10.80,10,20,40),I 060 10 IF(BWR.LE.1.0) GO TO 90 061 DAL=1.0/COS(P1/(100+BWR)) 062 IF(SWR.LE.0.0) GO TO 100 063 EPS=DFN(DS) 064 DT=DFN(DR)/EPS 065 NS=DARCH(DT)/DARCH(DAL)+0.99999D0 066 ENEWS 06? IF(IND.EQ.1) GO TO 50 068 GO TO 30 069 20 IF(NS.LE.0) GO TO 110 070 EN =NS 071 IFCSWR.LE.0.0) GO TO 100 072 EPS=DFN(DS) 073 DT=DFN(DR)/EPS 074 30 DAL=DCOSH(DARCH(DT)/EN) 075 BWR=DPI/DACOS(1.0D0/DAL)-1.0D0 076 GO TO 60 077 40 IF(NS.LE.0) GO TO 110 078 EN=NS 079 IF(BWR.LE.1.0) GO TO 90 080 DAL=1.0/COSCPI/(1.0+BWR)) 081 50 DT=DCOSH(EN*DARCH(DAL)) 082 EPS=DFN(DR)/DT 083 SWR=CEPS+DSQRT(EPS*EPS+1.0D0))**2 084 60 IF(NS.NE.1) GO TO 120 085 ZYT(1)=SQRT(ZYS*ZYL) 086 RETURN 087 70 IER=1 088 RETURN 089 80 IER=2 090 RETURN 091 90 IER=3 092 RETURN 093 100 IER=4 094 RETURN 095 110 IER=5 096 RETURN 097 120 IF(NS.LE.12) GO TO 150 098 IER=6 099 RETURN 100 * 101 * THE NUMERATOR AND DENOMINATOR EACH HAVE INTEGER(NS/2) 102 * QUADRATIC FACTORS OF THE FORM: 103 * NUM: T**2 + FNT*T + FNC 104 * DEN: T**2 + FDT*T + FDC 105 * STORE THE COEFFICIENTS OF THESE FACTORS IN THE ARRAYS 106 * FNT(I), FNC(I), FDC(I) AND FDC(I). 107 * 108 150 M=NS/2 109 DS=DPI/EN 110 DT=0.5D0*DS 111 DU=DEXP(DLOG(0.0DO+DSQRT(1.0D0+EPS*EPS))/EPS)/EN) 112 DV=0.5D0*(DU-1.0DO/DU)/DAL 113 DU=DU/DALDV 114 DO 160 I=1..M 115 DW=DCOS(DT) 116 FNCC.I)=(DAL/DW)**2-1.OD0 -292- TRANSFORMER DESIGN

117 FNTCI)=0.0D0 118 DXE(DU*DW)**2 119 DY=CDV*DSINCDT))**2 120 DW=CDX+DY)**2 121 DZ=CDW-DX+DY)/DW 122 DW=DSCIRTCDZ*DZ+4.0DO*DX*DY/CDW*DW)) 123 FDC(I)=DW 124 FDT(I)=DSCIRT(2.0D0*(DZ+DW)) 125 DT=DT+DS 126 160 CONTINUE 127 * 128 * MULTIPLY OUT THE QUADRATIC FACTORS TO OBTAIN THE 129 * COEFFICIENTS OF THE NUMERATOR AND DENOMINATOR POLY- 130 * NOMIALS. STORE THE LATTER IN THE ARRAYS DCN(I) AND 131 * DCD(I) RESPECTIVELY WITH THE I•TH ELEMENTS CONTAINING 132 * THE COEFFICIENTS OF T**CI-1). IF NS IS ODD INSERT 133 * THE EXTRA LINEAR FACTOR OF THE DENOMINATOR. 134 * 135 DO 170 I=1.15 136 DCNCI)=0.0D0 137 DCD(I)=0.0D0 138 170 CONTINUE 139 DCN(I)=1.0D0 140 DCD(1)=1.0D0 141 N=NS+I-M-M 142 IFCN.EQ.1) GO TO 180 143 DCD(2)=1.0D0 144 DCD(1)=DSQRT(1.0D0+1.0D0/(DV*DV)) 145 * 146 * SCAN THROUGH THE QUADRATIC FACTORS 147 * 148 180 DO 200 I=1.M 149 DV=FNC(I) 150 DV=FNT(I) 151 DX=FDCC/) 152 DY=FDT(I) 153 * 154 * SCAN POLYNOMIAL COEFFICIENTS GENERATED SO FAR FROM THE 155 * TOP DOWN AND GENERATE NEW COEFFICIENTS FOR THE POLY- 156 * NOMIAL THAT INCLUDES THE PRESENT QUADRATIC FACTOR. 157 * 158 DO 190 J=1.N 159 K=N+1-J 160 DCNCK+2)=DCN(K)+DW*DCNCK+1)+DV*DCN(K+2) 161 DCD(K+2)=DCD(11)+DY*DCDCK+1)+DX*DCD(11+2) 162 190 CONTINUE 163 DCN(2)=DW*DCNC1)+DV*DCN(2) 164 DCD(2)=DY*DCD(1)+DX*DCD(2) 165 DCNC1)=DV*DCN(1) 166 DCDC1)=DX*DCDC1) 167 N=N+2 168 200 CONTINUE 169 * 170 * DERIVE NUMERATOR AND DENOMINATOR COEFFICIENTS FOR THE 171 * /MP/ADM FUNCTION FROM THOSE GENERATED FOR THE 172 * REFLECTION COEFFICIENT FUNCTION. - 173 * 174 DS=DCD(1)*(DR-1.0D0)/(DCNC1)*(DR+1.0D0)) -293- TRANSFORMER DESIGN 175 DO 210 I*1,N 176 DZNCI)*DS*DCNCI)+DCD(I) 177 DZDCI)*DCDCI)-DS*DCN(I) 178 210 CONTINUE 179 * 180 * REMOVE CASCADE LINE SECTIONS FROM THE IMP/ADM FUNCTION 181 * 182 DO 250 I=1,14 183 * 184 * EVALUATE IMP/ADM OF NEXT LINE TO BE REMOVED 185 * 186 DU=0.0D0 187 DV=0.01)0 188 DO 220 J=1,N 189 DU*DU+DEN(J) 190 DV=DV+DZD(J) 191 220 CONTINUE 192 DW=DU/DV 193 ZYT(I)=ZYS*DW 194 ZYTCNS+1-I)=ZYL/DW 195 * 196 * REMOVE LINE FROM IMP/ADM FUNCTION 197 * 198 DO 230 J=1,141 199 DCN(J+1)=DZNCJ+1)-DW*DZDCJ) 200 DCDCJ+1)=DZDCJ+1)-DZNCJ)/DW 201 230 CONTINUE 202 * 203 * REMOVE C1-T**2) FACTOR FROM THE NUMERATOR AND THE 204 * DENOMINATOR. 205 * 206 DZN(2)=DCNC2) 207 DZD(2)=DCD(2) 208 DO 240 J=2,14 209 DZNCJ+1)=DZNCJ-1)+DCNCJ+1) 210 DZDCJ+1)=DZDCJ-1)+DCDCJ+1) 211 240 CONTINUE 212 N*N-1 213 250 CONTINUE 214 IF(NS.EQ.M+M) RETURN 215 ZYTCM+1)=SCIRTCZYS*ZYL) 216 RETURN 217 END

-294- COUPLER DESIGN

001 002 * 003 *THIS PROGRAM CALCULATES THE NORMALISED EVEN MODE IMPEDANCES 004 *FOR A MULTI—SECTION ASYMMETRIC COUPLED LINE DIRECTIONAL 005 *COUPLER. INPUT PARAMETERS ARE: 006 * 007 * CDB COUPLED POWER IN DECIBELS WITHIN BAND 008 * RDB RIPPLE IN COUPLING VALUE WITHIN BAND 009 * BWR BANDWIDTH RATIO (LONGEST/SHORTEST 010 * WAVELENGTH ON TRANSMISSION LINE) 011 * NS NUMBER OF COUPLING SECTIONS 012 * IND GIVES WHICH OF THE THREE PREVIOUS 013 * PARAMETERS IS NOT BEING SPECIFIED 014 * (ONLY 2 OF THE 3 CAN BE CHOSEN) 015 * =-1 NS UNSPECIFIED, INCREASE BWR 016 * IF POSSIBLE 017 * = 1 NS UNSPECIFIED, DECREASE RDB 018 * IF POSSIBLE 019 * = 2 BWR UNSPECIFIED 020 * = 3 RDB UNSPECIFIED 021 * 022 *CDB IS NEGATIVE AND RDB IS POSITIVE, THE COUPLING VARYING 023 *BETWEEN CDB—RDB AND CDB+RDB WITHIN THE BAND. OUTPUT 024 *PARAMETERS ARE: 025 * 026 * ZNE AN ARRAY CONTAINING THE SECTION EVEN MODE 027 * IMPEDANCES 028 * IER NON—ZERO ON RETURN IF AN ERROR HAS OCCURRED, 029 * WITH A VALUE INDICATING THE TYPE OF ERROR: 030 * = 1 CDB OUT OF RANGE (CDB POSITIVE OR 031 * CDB+RDB POSITIVE IF RDB SPECIFIED) 032 * = 2 IND OUT OF RANGE (SEE ABOVE) 033 * = 3 BWR LESS THAN OR EQUAL TO ONE 034 * = 4 RDB LESS THAN OR EQUAL TO ZERO 035 * = 5 NS LESS THAN UNITY 036 * = 6 TOO MANY SECTIONS (>12) 037 * 038 SUBROUTINE CPLRD(CDB,NS,BWR,RDB,IND,ZNEsIER) 039 IMPLICIT DOUBLE PRECISION CD—F) 040 DIMENSION ZNE(15),DCNC15)sDCD(15),FCN(15),FCD(15) 041 DATA PI,DPI/3.1415927,3.1415926535898D0/ 042 DARCH(DX)=DLOG(DX+DSART(DX*DX-1.0D0)) 043 * 044 *CHECK INPUT DATA AND CALCULATE UNSPECIFIED PARAMETER. 045 *PERFORM COUPLER DESIGN IF NS IS UNITY 046 * 047 1E2=0 048 IFCCDB.GE.0.0) GO TO 70 049 IF(IND.NE.3.AND.CDB+RDB.GE.0.0) GO TO 70 050 IF(IND.LT.-1.OR.IND.GT.3) GO TO 80 051 DCP=10.0**(CDB/10.0) 052 I=IND+2 053 GO T0(10,80,10320..40),I 054 10 IF(BWR.LE.1.0) GO TO 90 055 DAL=100/COS(PI/(1.0+BWR)) 056 IF(RDB.LE.0.0) GO TO 100 057 DRP=10.0**(RDB/10.0) 058 DT=(DRP—DCP)/(DRP*DRP-1.0D0) —295— COUPLER DESIGN

059 EP1=DSQRT(DT*DRP) 060 EP2=DSQRT(DT/DCP) 061 NS=DARCH(EP1)/DARCH(DAL)+0.99999D0 062 EN=NS 063 IF(IND.EQ.1) GO TO 50 064 GO TO 30 065 20 IF(NS.LE.0) GO TO 110 066 EN=NS 067 IF(RDB.LE.0.0) GO TO 100 068 DRP=10.0**(RDB/10.0) 069 DT=(DAP-DCP)/(DRP*DRP-1.0D0) 070 EP1=DSQRT(DT*DRP) r 071 EP2=DSQRTCDT/DCP) 072 30 DAL=DCOSH(DARCH(EP1)/EN) 073 BWR=DPI/DACOS(1.0DO/DAL)••100D0 074 GO TO 60 075 40 IF(NS.LE.0) GO TO 110 076 EN=NS 077 IF(BWR.LE.1.0) GO TO 90 078 DAL=1.0/COS(PI/(1.0+BWR)) 079 50 EP1=DCOSH(EN*DARCH(DAL)) 080 DT=4.0DO*EP1*EP1*(EP1*EP1...1.0D0)/DCP**2 081 EP2=DSCIRT(005D0*(1.0DO+DSCIRT(1.0DO+DT))) 082 DT=CEP2*EP2•1.0D0)/(EP1*EP11.0D0) 083 RDB=5.0DO*DLOG10(DT*(EP1/EP2)**2) 084 60 IF(NS.NE.1) GO TO 120 085 ZNE(1)=DSQRT(CEP1+EP2)/(EP2EP1)) 086 RETURN 087 70 IER=1 088 RETURN 089 80 IER=2 090 RETURN 091 90 IER=3 092 RETURN 093 100 1E11=4 094 RETURN 095 110 IER=5 096 RETURN 097 120 IF(NS.LE.12) GO TO 150 098 IER=6 099 RETURN 100 * 101 *PREPARE ARRAYS DCN(I) AND DCD(I) TO CONTAIN COEFFICIENTS

102 *OF THE NUMERATOR AND DENOMINATOR POLYNOMIALS RESPECTIVELY rr 103 *WITH THE I'TH ELEMENTS CONTAINING THE COEFFICIENTS OF 104 *T**(I-1). INSERT FACTORS ARISING FROM PURELY REAL ROOTS. 105 * 106 150 DO 160 11=1,15 107 DCN(I)=0.0D0 108 DCD(I)=0.0D0 109 160 CONTINUE 110 DT=2.0D0*DAL*DAL 111 DCH1=DCOSH(2.0DO*DARCH(EP1)/EN) 112 DCH2=DCOSH(2.ODO*DARCH(EP2)/EN) 113 IF(NS.EQ.2*CNS/2)) GO TO 170 114 DCN(2)=EP1/EP2 115 DCD(2)=1.0D0 116 DCD(1)=DSQRT(1.0D0-DT/(1.0DO+DCH2))

• COUPLER DESIGN

117 GO TO 180 118 170 DCN(3)=DSART(CEP1*EP1-1.0D0)/(EP2*EP2-1.OD0)) 119 DCN(2)=DCN(3)*DSART(1.0D0...DT/(1.01)0...DCH1)) 120 DCN(1)=0.0D0 121 DU=DSART(1.0D0—DT/(1.0D0+DCH2)) 122 DV=DSART(1.0DO—DT/(1.0D0-DCH2)) 123 DCD(3)=1.0D0 124 DCD(2)=DU+DV 125 DCD(1)=DU*DV 126 * 127 *ADD FACTORS ARISING FROM THE INTEGER(CNS-1)/2) COMPLEX 128 *CONJUGATE PAIRS OF ROOTS TO THE NUMERATOR AND DENOMINATOR 129 *POLYNOMIALS, COMPUTING THE NEW COEFFICIENTS AS EACH ROOT 130 *IS ADDED. 131 * 132 180 Noc(NS....1)/2 133 DANG=24.0DO*DPI/EN 134 IF(N.LT.1) GO TO 190 135 CALL FACTOR(DT,DCH1,DANG0N,DCN) 136 CALL FACTORCDT*DCH2,DANG,NsDCD) 137 138 *CONVERT TO NUMERATOR AND DENOMINATOR POLYNOMIALS OF 139 *THE IMPEDANCE FUNCTION 140 * 141 190 N=NS+1 142 DO 200 I=1,N 143 DU=DCNCI) 144 DV=DCD(I) 145 DCN(I)=DV+DU 146 DCD(I)=DV...DU 147 200 CONTINUE 148 * 149 *REMOVE COUPLED LINE SECTIONS FROM THE IMPEDANCE FUNCTION 150 * 151 DO 240 I=1,NS 152 * 153 *EVALUATE IMPEDANCE OF NEXT LINE TO BE REMOVED 154 * 155 DU=0.0D0 156 DV00.0D0 157 DO 210 J=l,N 158 DU=DU+DCN(J) 159 DV=DV+DCD(J) 160 210 CONTINUE 161 DU=DU/DV 162 ZNECI)=DU 163 * 164 *REMOVE THE LINE FROM THE IMPEDANCE FUNCTION 165 * 166 DO 220 J=14N 167 FCN(J+1)=DCNCJ+1)...DU*DCD(J) 168 FCD(J+1)=DCDCJ+1)—DCNCJ)/DU 169 220 CONTINUE 170 * 171 *REMOVE (1...T**2) FACTOR FROM THE NUMERATOR AND DENOMINATOR 172 * 173 DCN(2)=FCNC2) 174 DCD(2)=FCD(2) —297— COUPLER DESIGN

175 DO 230 J=2,W 176 DCN(J+1)=DCNCJ-1)+FCNCJ+1 ) 177 DCD(J+1)=DCD(J-1)+FCD(J+1 ) 178 230 CONTINUE 179 N=N-1 180 240 CONTINUE 181 RETURN 182 END

-298- COUPLER DESIGN

183 * 184 *THIS SUBROUTINE CALCULATES THE COEFFICIENTS OF A POLYNOMIAL 185 *BY INTRODUCING QUADRATIC FACTORS ONE AT A TIME 186 * 187 SUBROUTINE FACTOR(DTsDCH,DANG,NO,DPY) 188 IMPLICIT DOUBLE PRECISION CD—F) 189 DIMENSION DPY(15) 190 DSH=DSQRT(DCH*DCH-.1.0D0) 191 J=3 192 DO 20 I=1,NO 193 DTH=I*DANG 194 DCS*DCOSCDTH) 195 DW=DT/CDCH+DCS)**2 196 DR=1.ODO—DW*C1.0D0+DCH*DCS) 197 DI=DW*DSH*DSIN(DTH) 198 DW=DSQRT(DR*DR+DI*DI) 199 DR=DSQRT(2.0D0*(DR+DW)) 200 DO 10 K=IsJ 201 L=J+1—K 202 DPY(L+2)*DPY(L)+DR*DPYCL-1.1)+DW*DPYCL+2) 203 10 CONTINUE 204 DPYC2)*DR*DPY(1)+DW*DPY(2) 205 DPY(1)=DW*DPY(1) 206 J=J+2 207 20 CONTINUE 208 RETURN 209 END

—299— APPENDIX SEVEN - PROGRAK( FOR SHUNT JUNCTION NETWORK ANALYSIS

This group of subroutines facilitates the specification and analysis of shunt junction networks. Before describing the purpose of each routine, some constraints on usage will be given. First of all the programs as written are limited to networks with less than 100 lines and 100 nodes since line and node numbers are in the range 1 to 99. Secondly all network input/output ports can only be connected to one line and cannot have any shunt components specified at the port. In order to prevent the open circuit end of an open circuit stub being recognised as a network input/output port, an open circuit shunt admittance must be specified at the corresponding node number. Third, direct connection between two input/output ports is not allowed; there must be at least one intervening node even if it is only a dummy. Each line is specified as connecting two nodes, the line variables being defined at the first node specified. In specifying a line which is joined to an external node the latter must be the first node specified in the input sequence for the line. If it is not, the input data is altered to correct this and a non fatal warning is given. All line and node numbers are always two digits, line 1 being specified as 01 etc. Each line specified can have up to nine subsections specified, each of constant characteristic impedance. The number of lines connected to any node cannot exceed five. Reactive shunt admittances can be specified at internal nodes and are specified as capacitive and inductive admittance values at centre band (FREQ = 1.0) and are scaled accordingly by the program at other frequencies (both admittances are input as positive numbers). The purpose of the various subroutines will now be briefly described.

a. SUBROUTINE INPUT

This routine inputs data specifying the network or alters data already present, adding or deleting nodes and lines as directed. The input sequences are given in the program header and are fairly self explanatory. For example the sequence:

DL26

removes line 26 while:

L13: 13, 17, 0.2, 1.0

enters line 13 connecting node 13 and node 17 with a length of 0.2 wavelengths at centre band (FREQ = 1.0) and unit characteristic impedance. Lines containing multiple sub-lines are input using the sequence:

L31: 27, 29, 3 LTH: 0.25, 0.30, 0.30 IMP: 1.45, 2.46, 3.19 -300- where the last number on the first line gives the number of sub-lines with their lengths on the second line and characteristic impedances on the third. All line lengths are specified in wavelengths at mid-band. Illegal input sequences are ignored. b. SUBROUTINE INCHECK

This program compiles arrays as indicated in its header and sets the variable IER if any input errors are detected. The value of IER gives the type of error as follows:

1: number of sub-lines out of range 1 to 9.

2: node number out of range 1 to 99.

3: line has same node at both ends.

4: illegal line specification. Length less than zero or characteristic impedance less than or equal to zero.

5: more than 5 lines join at a node.

6: illegal shunt element specification (array NS element outside range -1 to 1).

7: shunt element specified for a node that has no connection to remainder of network.

8: short circuit specified at node with more than one connected line.

9: negative admittance data specified at a node.

10: program error.

11: line has an external node at both ends.

-1: external node hasnot been the first specified on a line. Input data has been rearranged to correct this.

c. SUBROUTINE MATGEN

This program generates the coefficient matrix as discussed in chapter four, section three. The frequency is specified by the variable FREQ which is unity at mid-band. d. SUBROUTINE MATRED

This program reduces the coefficient matrix to the form described in chapter four, section three allowing the scattering matrix and the internal matrix to be extracted.

e. SUBROUTINE OUTHDR

This program outputs data that describes the network lines and node connections.

f. SUBROUTINE OUTMTXS

This program outputs the network scattering matrix and the internal matrix if required. -301- /NETWORK ANALYSIS

001 * 002 * THIS PROGRAM INPUTS NETWORK DATA INTO THE ARRAYS IN COMMON 003 * BLOCK "INDATA", DESCRIBED IN THE HEADER OF SUBROUTINE 004 * INCHECK. LEGAL INPUT SEQUENCES ARE IN THE FORM OF A FOUR 005 * CHARACTER "COMMAND" WHICH IS IN ,SOME CASES FOLLOWED BY 006 * DATA. THE SEQUENCES IMPLEMENTED AND THE ACTIONS INITIATED 007 * ARE AS FOLLOWS WHERE ** IS A NON—ZERO TWO DIGIT NUMBER 008 * WITH BOTH DIGITS PRESENT: 009 * 010 * DALL DELETES ALL LINE AND NODE DATA 011 * DNDS DELETES ALL NODE SHUNT DATA 012 * DLNS ---- DELETES ALL LINE DATA 013 * DL** •••• DELETES DATA ON LINE ** 014 * DN** DELETES SHUNT DATA ON NODE ** 015 * 016 * N**S ---- ENTERS SHORT CIRCUIT ON NODE ** 017 * N**O ---- ENTERS OPEN CIRCUIT ON NODE ** 018 * N**: CAP ADMs IND ADM \ 019 * ---- ENTERS REACTIVE TERMINATION ON NODE ** 020 * 021 * L**: NODE!, NODE2s LENGTH* CH IMPEDANCE 022 * ---- ENTERS DATA ON LINE ** (1 SUB—LINE) 023 * L**: NODE!, NODE2s NUMBER OF SUB—LINES 024 * LTH: SUB-LINE LENGTHS 025 * IMP: SUB—LINE CH IMPS 026 * ---- ENTERS DATA ON LINE ** (>1 SUB—LINES) 027 * 028 * END END DATA INPUT 029 * 030 * ILLEGAL INPUT SEQUENCES ARE IGNORED. LUI GIVES THE LOGICAL 031 * UNIT NUMBER FOR INPUT, PROMPTS BEING GIVEN IF THIS IS THE 032 * TERMINAL. 033 * 034 SUBROUTINE INPUT(LUI) 035 COMMON/INDATA/LT(99),N1(99),N2(99)sD(9,99),Z(9,99), 036 & NSC99),BC(99)sBL(99) 037 DIMENSION ICH(17),B(9),C(9) 038 EQUIVALENCE (B(1)sICH(1)),(C(1)sICH(10)) 039 DATA IDsIEsILsINsIOsIS,NDsLNsIALPLTHsIMPsICL/'D l s'E's 040 & 11.'s'N's'O's'S'siND's'LN's'AL's'LTHOs'IMPOs s :'/ 041 GO TO 30 042 10 IFCLUI.EQ.0) WRITECOp20) 043 20 FORMATC'LAST INPUT SEQUENCE IGNORED') 044 30 IFCLUI•EQ.0) WRITEC0,40) 045 40 FORMAT('?'s$) 046 READ(LUIs50) IAPIB,ICsICH 047 50 FORMATCA1sA2sA1s17A4) 048 IFCIA.EQ.IL.OR.IA.EQ.IN) GO TO 160 049 IFCIA.EQ.ID) GO TO 60 050 IF(IA.NE.IE.OR.IB.NE.ND) GO TO 10 051 RETURN 052 * 053 * DELETION SEQUENCE 054 * 055 60 IFCIB.NE.IAL) GO TO 80 056 IFCIC.NE.IL) GO TO 10 057 DO 70 1=1,99 058 LTCI)=0 -302- NETWORK ANALYSIS 059 MS(1)=0 060 70 CONTINUE 061 GO TO 30 062 80 IF(IB.NE.ND) GO TO 100 063 IF(IC.NE.IS) GO TO 10 064 DO 90 I=1,99 065 NS(I)=0 066 90 CONTINUE 067 GO TO 30 068 100 IF(IB.NE.LN) GO TO 120 069 IF(IC.NE.IS) GO TO 10 070 DO 110 I=1,99 071 LT(I)=0 072 110 CONTINUE 073 GO TO 30 074 120 DECODE(4,130,IB) IAJPI 075 130 FORMAT(A1,R1) 076 DECODE(4.140,IC) J 077 140 FORMAT(R1) 078 I=I•256*(I/256)•240 079 J=J-256*CJ/256)-240 080 IF(I.LT.O.OR.I.GT.9) GO TO 10 081 IF(J.LT.O.OR.J.QT.9) GO TO 10 082 I=10*I+J 083 IF(I.EQ.0) GO TO 10 084 IF(IA.NE.IL) GO TO 150 085 LT(I)=0 086 GO TO 30 087 150 IF(IA.NE.IN) GO TO 10 088 NS(I)=0 089 GO TO 30 090 * 091 * NODE OR LINE INPUT SEQUENCE 092 * 093 160 DECODE(49170s/B) IsJ 094 170 FORMAT(2R1) 095 I=I-256*(1/256)-240 096 J=J-256*CJ/256)-240 097 IFCI.LT.O.OR.I.GT.9) GO TO 10 098 IFCJ.LT.O.OR.J.GT.9) GO TO 10 099 I=10*I+J 100 IF(I.EQ.0) GO TO 10 101 IFCIA.EQ.IL) GO TO 210 102 IF(IA.NE.IN) GO TO 10 103 * 104 * NODE INPUT SEQUENCE 105 * 406 IF(IC.NE.IS) GO TO 180 107 NSCI)=-1 188 GO TO 30 109 180 IF(IC.NE.I0) GO TO 190 110 NS(I)=1 111 BC(I)=0.0 112 BL(I)=0.0 113 GO TO 30 114 190 IF(IC.NE.ICL) GO TO 10 115 DECODE(68p200,/CH) BC(I).BL(I) 116 200 FORMAT(2F13.0) -303-

• NETWORK ANALYSIS

117 NSCI)=1 118 GO TO 30 119 * 120 * LINE INPUT SEQUENCE 121 * 122 210 IFCIC.NE+ICL) GO TO 10 123 DECODEC68,220,ICH) JoKsAoBB 124 220 FORMATC2I8o2F13,0) 125 IFCBB0EQ.0.0) GO TO 230 126 111(I)=J 127 N2CI)=K 128 LT(I)=1 129 D(1,I)=A 130 Z(IsI)=BB 131 GO TO 30 132 230 L=A 133 /F(LeLT.1.0R.L.GT.9) GO TO 10 134 IPCLUI.NE.0) GO TO 270 135 WRITE(0,240) 136 240 FORMATCILTHS0,$) 137 READ(0,250) CDCMJI),M=1,L) 138 250 FORMAT(5F1344/T5s5F13.0) 139 WR/TE(0*260) 140 260 FORMATC'IMPSO0S) 141 READ(0s250) (Z(111,I)sM=1,L) 142 N1(I)0J 143 N2CI)=K 144 LT(I)=L 145 GO TO 30 146 270 READCLU/s280) IAP(BCM),M=1J,L) 147 280 FORMAT(A4,(T5,5F13.0)) 148 READCLUI,280) IBs(CCM),M=1,L) 149 IFCIA.NE.LTHoOR.IB.NE.IMP) GO TO 300 150 DO 290 M=1,1. 151 DCM.I)=B(M) 152 Z(MAI)=C(M) 153 290 CONTINUE 154 N1CI)=J 155 N2C/)=K 156 LT(I)=L 157 GO TO 30 158 300 WRITE(0,010) LoIA,IB 159 310 PORMAT(I8,2A4) 160 GO TO 10 161 END

-304- NETWORK ANALYSIS

162 * 163 * THIS PROGRAM CHECKS THE INPUT DATA AND GENERATES 164 * TABULAR ARRAYS THAT FACILITATE SCATTERING MATRIX 165 * COMPUTATION. NETWORK LINE AND NODE NUMBERS ARE IN 166 * THE RANGE 1 TO 99, THE NETWORK BEING SPECIFIED BY 167 * THE ARRAYS: 168 * 169 * LT(L) 0, FOR UNUSED LINE NUMBERS 170 * J, IF J SUB—LINES ON LINE L 171 * N1CL) FIRST NODE ON LINE L 172 * N2CL) SECOND NODE ON LINE L 173 * DCJ,L) MID-BAND LENGTH IN WAVELENGTHS OF J'TH 174 * SUB—LINE ON LINE L 175 * Z(J,L) CHARACTERISTIC IMPEDANCE OF J'TH SUB—LINE 176 * ON LINE L 177 * NS(N) —1, IF NODE IS SHORT CIRCUITED 178 * 0, IF NO SHUNT ELEMENT SPECIFIED 179 * 1, IF NODE IS TERMINATED IN AN OPEN CIRCUIT 180 * OR AN INDUCTIVE/CAPACITATIVE ELEMENT 181 * BLCN) MID—BAND SHUNT INDUCTIVE ADMITTANCE AT NODE N 182 * BCCN) MID—BAND SHUNT CAPACITATIVE ADMITTANCE AT 183 * NODE N 184 * 185 * THE FOLLOWING VARIABLES ARE GENERATED: 186 * 187 * NT(N) 0, FOR UNUSED NODE NUMBERS J LINES ENTER NODE N 188 * J., IF 189 * -J, IF NODE N IS THE J'TH EXTERNAL NODE 190 * LAN(J,N) LINE NUMBER OF J'TH LINE ENTERING NODE N 191 * LVNCL) VARIABLE NUMBER ON LINE L 192 * NMX MAXIMUM NODE NUMBER IN USE 193 * NIL NUMBER OF INTERNAL LINES 194 * NEL NUMBER OF EXTERNAL LINES 195 * 196 * IF IER IS NON ZERO ON RETURN AN ERROR HAS BEEN DETECTED 197 * IF IER IS NEGATIVE THE ERROR HAS BEEN CORRECTED BY 198 * MODIFYING THE INPUT DATA WHILE IF IER IS POSITIVE THE 199 * ERROR IS FATAL. THE VALUE OF IER INDICATES THE TYPE OF 200 * ERROR THAT HAS OCCURRED. NH AND NC ARE THE NUMBER OF 201 * ROWS AND COLUMNS RESPECTIVELY IN THE COEFFICIENT MATRIX 202 * OF THE SYSTEM OF EQUATIONS DESCRIBING THE NETWORK. 203 * 204 SUBROUTINE INCHECKCIER,NR,NC) 205 COMMON/INDATA/LT(99),N1(99),N2C99),D(9,99),Z(9,99), 206 & NS(99),BCC99)sBLC99) 207 COMMON/NTABS/NT(99),LAN(5,99),LVNC 99),NMX,NIL,NEL 208 * 209 * CHECK NODE NUMBER RANGE AND VALI DITY AND COMPILE NTCN) 210 * AND LAN(J,N) ARRAYS 211 * 212 IER=0 213 DO 10 N=1,99 214 NT(N)=0 215 10 CONTINUE 216 NMX=0 217 NIL=0 218 DO 30 L=1,99 219 LVN(L)=0 -305- NETWORK ANALYSIS 220 I=LT(L) 221 IF(I.LT.0.011.1.GT.9) GO TO 200 222 IF(I.EQ.0) GO TO 30 223 J=N1(L) 224 K=N2(L) 225 IF(J.LT.1.0R.J.GT.99) GO TO 220 226 IF(K.LT.1.0R.K.GT.99) GO TO 220 227 IF(J.EQ.K) GO TO 240 228 DO 20 N=1,I 229 IF(D(NoL).LT.0.0.0RoZ(NoL).LE.0.0) GO TO 260 230 20 CONTINUE 231 LMX=L 232 NIL=NIL+1 233 IF(J.GT.NMX) NMX=J '234 /FCK.GT.NMX) NMX=K 235 N=NT(J)+1 236 IFCN.GT.5) GO TO 280 237 NTCJ)=N 238 LAN(N,J)=L 239 N=NT(K)+1 240 IF(N.GT.5) GO TO 280 241 NT(K)=N 242 LANCNAK)=L 243 30 CONTINUE 244 * 245 * CHECK NODAL SHUNT ADMITTANCES AND FIND EXTERNAL NODES 246 * 247 NEL=0 248 DO 60 N=1.,99 249 I=NTCN) 250 J=NS(N) 251 IF(J.LT.1.0R.J.GT.1) GO TO 300 252 IF(I.EQ.O.AND.J.NE.0) GO TO 320 253 IF(I.EQ.0) GO TO 60 254 IF(I.GT.1) GO TO 40 255 IFCJ.NE.0) GO TO 50 256 NEL=NEL+1 257 NT(N)=NEL 258 LVN(LAN(laN))=-..NEL 259 GO TO 60 260 40 IF(J.EQ.-1) GO TO 340 261 50 IFCJ.NE.1) GO TO 60 262 IFCBC(N).LT.0.0.OR.BL(N).LT.0.0) GO TO 360 263 60 CONTINUE 264 NIL=NIL—NEL 265 NC=2*CNIL+NEL) 266 NR=NC—NEL 267 * 268 * SET INTERNAL VARIABLE NUMBERS, CHECK FOR LINES DIRECTLY 269 * BETWEEN EXTERNAL NODES AND ENSURE THAT FIRST NODE OF AN 270 * EXTERNAL LINE IS THE EXTERNAL NODE 271 * 272 N=2*NEL+1 273 DO 100 L=1,LMX 274 I=LT(L) 275 IFCI•EQ.0) GO TO 100 276 IF(LVN(L).NE.0) GO TO 70 277 LVN(L)=N —306— NETWORK ANALYSIS

278 N=N+2 279 GO TO 100 280 70 J=N1(L) 281 K=N2(L) 282 IF(NT(J).LT.0) GO TO 90 283 IFCNTCK).GT.0) GO TO 380 284 N1CL)=K 285 N2CL)=J 286 IER=-1 287 J=I/2 288 DO 80 K=1,J 289 M=I+1-K 290 T=DCK,L) 291 D(K,L)=D(M,L) 292 D(M,L)=T 293 T=ZCK,L) 294 Z(K,L)=ZCM,L) 295 Z(M,L)=T 296 80 CONTINUE 297 GO TO 100 298 90 IFCNT(K),LT4.0) GO TO 400 299 100 CONTINUE 300 RETURN 301 200 IER=1 302 RETURN 303 220 IER=2 304 RETURN 305 240 IER=3 306 RETURN 307 260 IER=4 308 RETURN 309 280 IER=5 310 RETURN 311 300 IER=6 312 RETURN 313 320 IER=7 314 RETURN 315 340 IER=8 316 RETURN 317 360 IER=9 318 RETURN 319 380 IER=10 320 RETURN 321 400 IER=11 322 RETURN 323 END

-307- NETWORK ANALYSIS

001 * 002 * THIS PROGRAM GENERATES THE ARRAY CCCNR,NC) CONTAINING 003 * THE COEFFICIENTS OF THE SYSTEM OF EQUATIONS FOR THE 004 * NETWORK WAVE AMPLITUDES. THE FREQUENCY RELATIVE TO 005 * MID—BAND IS INPUT IN VARIABLE FREQ. 006 * 007 SUBROUTINE MATGEN(FREQ,CCsNRaNC) 008 COMMON/INDATAYLTC99),N1C99),N2(99),D(9,99)sZ(9,99), 009 & NSC99)sBC(99)..BL(99) 010 COMMON/NTABS/NT(99),LAN(5s99),LVWC99)..NMX,NIL,NEL 011 COMPLEX CCCNR,NC)sCVAsCVBsCIAsCIBsCRPCYsCPHsCIP 012 DO 10 I=IsNC 013 DO 10 J=IsNR 014 CCCJ,I)=(0.0,0.0) 015 10 CONTINUE 016 PH=6.283185308*FREQ 017 * 018 * SCAN ALL INTERNAL NETWORK NODES 019 * 020 I=0 021 DO 110 N=lsNMX 022 J=NTCN) 023 IF(J.LE.0)60 TO 110 024 * 025 * DETERMINE COEFFICIENTS OF "CURRENT" EQUATION FROM NODAL 026 * SHUNT ADMITTANCES 027 * 028 K=NSCN) 029 CY=C0.0s0.0) 030 CR=(1.04.0.0) 031 IFCK.EQ.0) GO TO 30 032 IFCK.EQ.1) GO TO 20 033 CY=(1.0.0.0) 034 CR=(0.0,0.0) 035 GO TO 30 036 20 CY=C0.0s1.0)*CFREQ*BC(N)-.BLCN)/FREQ) 037 * 038 * SCAN EACH LINE AT THE NODE AND DETERMINE THE COEFFICIENTS 039 * IN THE EQUATIONS: 040 * 041 * NODE VOLTAGE = CVA*A + CVB*B 042 * CURRENT INTO NODE = CIA*A + CIB*B 043 * 044 * WHERE A AND B ARE THE NORMALISED WAVE AMPLITUDES ON THE 045 * LINE, DEFINED AT ITS FIRST NODE 046 * 047 30 DO 100 K=1,J 048 L=LAN(K,N) 049 ZZ=SCIRTCZCI,L)) 050 IFCN.NE.N1CL)) GO TO 40 051 * 052 * A AND B ARE DEFINED AT THIS NODE 053 * 054 CVA=ZZ 055 CVB=ZZ 056 CIA=-1.0/ZZ 057 C/B=—CIA 058 GO TO 60 -308- NETWORK ANALYSIS

059 * 060 * A AND B ARE DEFINED AT OTHER END OF THE LINE 061 * 062 40 DD=PH*1)(1,L) 063 CPH=CMPLX(COS(DD),SIN(DD)) 064 CVA=ZZ/CPH 065 CVB=ZZ*CPH 066 CIA=1.0/CVB 067 CIB=-1.0/CVA 068 M=LT(L) 069 IF(M.EQ.1) GO TO 60 070 * 071 * PERFORM TRANSFORMATIONS ALONG MULTIPLE SUB—LINES 072 * 073 DO 50 JJ=20M 074 ZZ=ZCJJ,L) 075 DD=PH*1)(JJA,L) 076 CPH=(0.021.0)*SIN(DD) 077 DD=COS(DD) 078 CTP=CVA*DD—CIA*CPH*ZZ 079 CIA=CIA*DD—CVA*CPH/ZZ 080 CVA=CTP 081 CTP=CVB*DDCIB*CPH*ZZ 082 CIB=CIB*DD—CVB*CPH/ZZ 083 CVB=CTP 084 50 CONTINUE 085 * 086 * DETERMINE LINE VARIABLE ADDRESSES 087 * 088 60 IA=LVNCL) 089 IB=IA+1 090 IFCIA.GT.0) GO TO 70 091 IA=-.IA 092 IB=IA+NEL 093 70 IF(K.NE.1) GO TO 90 094 * 095 * COMPILE COEFFICIENTS FOR "CURRENT" EQUATION AND INSERT 096 * FIRST LINE VOLTAGE COEFFICIENTS INTO SUBSEQUENT "VOLTAGE" 097 * EQUATIONS IF NEEDED 098 * 099 CC(I+1*IA)=CR*CIA—CY*CVA 100 CC(I+1,IB)=CR*CIBCY*CVB 101 IFCJ.EQ.1) GO TO 100 102 DO 80 JJ=2,J 103 CC(I+JJ,IA)=CVA 104 CC(I+JJ,IB)=-.CVB 105 80 CONTINUE 106 GO TO 100 107 * 108 * COMPILE "VOLTAGE" AND "CURRENT" COEFFICIENTS FOR SECOND 109 * AND SUBSEQUENT LINES AT THE NODE 110 * 111 90 CC(I+K,IA)=CVA 112 CC(I+KsIB)=CVB 113 CC(I+1,IA)=CR*CIA 114 CC(I+1,IB)=CR*CIB 115 100 CONTINUE 116 I=I+J -309- NETWORK ANALYSIS

117 110 CONTINUE 118 RETURN 119 END

-310. NETWORK ANALYSIS

120 * 121 * THIS PROGRAM REDUCES THE COEFFICIENT MATRIX TO A MINIMUM 122 * FORM USING STANDARD GAUSSIAN ELIMINATION TECHNIQUES. IF 123 * IND IS NON ZERO A COMPLETE REDUCTION IS PERFORMED GIVING 124 * BOTH THE SCATTERING MATRIX AND THE INTERNAL MATRIX WHILE 125 * ZERO IND ONLY YEILDS THE FORMER (THE INTERNAL MATRIX 126 * GIVES THE DEPENDENCE OF THE INTERNAL WAVE AMPLITUDES ON 127 * THE WAVE AMPLITUDES ENTERING THE NETWORK). ON RETURN A 128 * NON ZERO VALUE OF IER INDICATES A SINGULAR MATRIX. 129 * 130 * CC(NRsNC) COEFFICIENT MATRIX (INPUT) 131 * CSN(NN.NN) SCATTERING MATRIX (OUTPUT) 132 * CIV(NIVoNN) INTERNAL MATRIX (OUTPUT) 133 * 134 SUBROUTINE MATRED(CCINRANC,IND,IER,CSN,CIV) 135 COMPLEX CC(NILINC),CSN(1),CIV(1)sCX,CY 136 DIMENSION EC2) 137 EQUIVALENCE (CX,E(1)),(E(1),,S)*(E(2),T) 138 IER=0 139 NN=NC—NR 140 DO 50 II=1ANR 141 I=NR+1-II 142 M=I+NN 143 * 144 * FIND LARGEST ELEMENT IN COLUMN AND INTERCHANGE ROWS 145 * TO BRING IT ONTO DIAGONAL NORMALISED TO —1 146 * 147 S=0.0 148 DO 10 J=IsI 149 T=CABS(CC(J,M)) 150 IF(T.LE.S) GO TO 10 151 S=T 152 K=J 153 10 CONTINUE 154 IF(S.EQ.0.0) GO TO 120 155 CX=—CC(KsM) 156 DO 20 J=1,M 157 CY=CC(K,J)/CX 158 CC(KsJ)=CC(I,J) 159 CC(IsJ)=CY 160 20 CONTINUE 161 * 162 * ELIMINATE ABOVE DIAGONAL ELEMENTS IN COLUMN BY ADDING 163 * PROPORTION OF "DIAGONAL" ROW TO ROWS ABOVE 164 * 165 IFCI.EQ.1) GO TO 50 166 L=I-1 167 DO 40 J=1.0. 168 CX=CC(J,M) 169 IF(S.EQ.0.0.AND.T.EQ.0.0) GO TO 40 170 DO 30 K=IPM 171 CC(J,K)=CC(JoK)+CX*CC(IsK) 172 30 CONTINUE 173 40 CONTINUE 174 50 CONTINUE 175 * 176 * ELIMINATE ELEMENTS BELOW THE DIAGONAL 177 * —311— NETWORK ANALYSIS

178 L=NR 179 IF(IND.EO.0) L=NN 180 II=L-1 181 DO 80 I=1,II 182 M=I+NN 183 N=I+1 184 DO 70 J=14,1, 185 CX=CC(JsM) 186 IFCS.EQ.0.0.AND.T.EQ.0.0) GO TO 70 187 DO 60 K=1,NN 188 CC(JsK)=CC(JsK)+CX*CC(IsK) 189 60 CONTINUE 190 70 CONTINUE 191 80 CONTINUE 192 * 193 * TRANSFER MATRICES TO OUTPUT ARRAYS 194 * 195 K=0 196 L=0 197 M=NN+1 198 DO 110 J=1,NN 199 DO 90 I=1sNN 200 K=K+1 201 CSN(K)=CC(IsJ) 202 90 CONTINUE 203 IF(IND.EQ.0.OR.NN.EQ.NR) GO TO 110 204 DO 100 I=MsNR 205 L=L+1 206 C/V(L)=CC(IsJ) 207 100 CONTINUE 208 110 CONTINUE 209 RETURN 210 120 IER=1 211 RETURN 212 END

—312— NETWORK ANALYSIS

101 * 002 * THIS PROGRAM WRITES THE NETWORK DATA TO OUTPUT UNIT LUO. 003 * 004 SUBROUTINE OUTHDRCLUO) 005 COMMON/INDATAILT(99),N1C99),N2C99),DC9,99),Z(9,99), 006 & NSC99),BCC99),BLC99) 007 COMMONINTABSINT(99)PLANC5,99),LVNC99),NMX,NIL,NEL 008 DIMENSION ICHC12) 009 DATA II,IEsISP,I0sIS/"I ','E 'a' ',"0 ','S 'I 010 WRITECLU0,10) 011 10 FORMATC///'LINE N1 N2 SL LENGTH CH IMP', 012 & ' LINE VARIABLES'/) 013 DO 70 1=1,99 014 J=LT(I) 015 IFCJ.EQ.0) GO TO 70 016 K=1 017 A=DC1s1) 018 B=Z(1,I) 019 L=LVNCI) 020 IFCL.LT.0) GO TO 20 021 L=L-2*NEL 022 M=L+1 023 IA=II 024 GO TO 30 025 20 L=—L 026 M=L 027 IA=IE 028 30 WRITECLU0,40) 029 40 FORMATC4I4s2F10.4," A=',I30A1,' B=',13,A1) 030 IFCJ.EQ.1) GO TO 70 031 DO 60 K=2PJ 032 WRITECLU0,50) KsD(KPI),ECK,I) 033 50 FORMATC12X,I4,2F10.4) 034 60 CONTINUE 035 70 CONTINUE 036 WRITECLU0s80) 037 80 FORMATC///'NODE INT/EXT ----LINES SHUNT pp 038 & 'CAP ADM IND ADM'/) 039 DO 170 I=1,NMX 040 J=NTCI) 041 IF(J.EQ.0) GO TO 170 042 IA=ISP 043 IB=II 044 IFOJ•GT•O) GO TO 100 045 J=—J 046 ENCODEC4,90.IA) J 047 90 FORMATCI2) 048 IB=IE 049 J=1 050 100 ENCODEC44,110,ICH) CLANCK,I),K=1,J) 051 110 FORMATCI3o4I4) 052 J=J+1 053 DO 120 11=Ja12 054 ICHCK)=ISP 055 120 CONTINUE 056 K=NSCI) 057 IFCK.EQ.0) GO TO 150 058 IF(K.EQ.-1) GO TO 140 -313- NETWORK ANALYSIS

059 ICH(7)=IO 060 A=BC(I) 061 B=BL(I) 062 IF(A.EG1.0.0.AND.B•E61.0.0) GO TO 150 063 ENCODE(18a130sICH(8)) AaB 064 130 FORMAT(2F8.3) 065 GO TO 150 066 140 ICH(7)=IS 067 150 WRITECLUO,160) IaIAPIBa(ICH(K)*K=1,11) 068 160 FORMAT(I4s6KJA2,A1s2Ka5A3a2X,6A4) 069 170 CONTINUE 070 WRITECLU00180) 071 180 FORMAT(///) 072 RETURN 073 END NETWORK ANALYSIS

074 * 075 * THIS PROGRAM WRITES THE SOLUTION MATRICES TO UNIT LUO. 076 * 077 SUBROUTINE OUTMTXSCLUOPCSN,CIVPIND,FREQ) 078 COMMON/NTABS/NT(99),LANC5,99),LVNC99)..NMX,NIL,NEL 079 COMPLEX CSNC1),CIVC1),CTPL 080 WRITECLU0s10) NELoNEL,FREQ 081 10 FORMATC//'SCATTERING MATRIX BY isI2s 1 )", 082 & FREQ= '.F7.4) 083 DO 50 I=1*NELs4 084 J=I+3 085 IFCJ.GT.NEL) J=NEL 086 WR/TECLU0s20) CK,K=IrJ) 087 20 FORMATE/'ROW COL=',I14,3I16/) 088 DO 40 K=1,NEL 089 L=K-.-NEL 090 WRITECLU0,30) Ko(CTPLCCSNCM*NEL+L)),M=1,J) 091 30 FORMATCI3s3X,4CF84,5,'C'sF6.2,')")) 092 40 CONTINUE 093 50 CONTINUE 094 IFCIND.EQ.0,DOR.NIL.EQ.0) GO TO 110 095 NIV=2*NIL 096 WRITE(LU0,60) NIV,NEL,FREQ 097 60 FORMATC/I'INTERNAL MATRIX (',130' BY ',I2,")', 098 & ' FREQ= ',F7.4) 099 DO 100 I=1,NELs4 100 J=I+3 101 IFCJ.GT.NEL) J=NEL 102 WRITECLUO$70) CK,K=I,J) 103 70 FORMATC/"ROW COL=',114,3I16/) 104 DO 90 K=1,,, NIV 105 L=11.q4IV 106 WRITECLU0,80) KoCCTPLCCIVCM*NIV+L)).,M=I,J) 107 80 FORMATCI3s3Xs4CF8.4s'C',F6.2;')')) 108 90 CONTINUE 109 100 CONTINUE 110 110 WRITE(LU0s120) 111 120 FORMATC///) 112 RETURN 113 END

-315-

• NETWORK ANALYSIS

114 FUNCTION CTPL(C) 115 COMPLEX CTPL(C 116 RE=REAL(C) 117 AI=AIMAG(C) 118 D=SQRT(RE*RE+AI*AI) 119 E=0.0 120 IF(D.NE.0.0) E=57.29578*ATAN2(AI,RE) 121 IF(E.LT.0.0) E=E+360.0 122 CTPL=CMPLX(DoE) 123 RETURN 124 END

-316-