Research Collection
Doctoral Thesis
Synchronisation of reflex-oscillators
Author(s): AbdelDayem, Aly Hassan
Publication Date: 1953
Permanent Link: https://doi.org/10.3929/ethz-a-000099179
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ETH Library Prom. No. 2165
Synchronisation of Reflex-Oscillators
THESIS
PRESENTED TO
THE SWISS FEDERAL INSTITUTE OF TECHNOLOGY
ZURICH
FOR THE DEGREE OF
DOCTOR OF TECHNICAL SCIENCE
BY
Aly Hassan Abdel Dayem
of Egypt
Accepted on the recommendation of
Prof. Dr. F. Tank and Prof. Dr. M. Strutt
Zurich 1953 Dissertationsdruckerei Leemann AG. Leer - Vide - Empty Table of Contents
Preface 5
Chap. 1. Synchronisation of Oscillators 7
1.1. Introduction 7
1.2. Summaries of Some Simplified Theories on Synchronisation ... 8 1.3. Calculation of the Steady State Amplitude and Phase by Using the Energy Equation 14
Chap. 2. Synchronisation of Reflex-Klystron Oscillators 26
2.1. Introduction 26 2.2. Choice of the Equivalent Circuit 31 2.3. The Energy Equation and Steady State Solution 34 2.4. Amplitude and Phase Behaviour of the Synchronised Reflex- Oscillator 37
2.5. Calculation of an Example 43
Chap. 3. Mutual Synchronisation of two Klystrons 44
3.1. The Equivalent Circuit 44 3.2. Steady State Equation Under Mutual Synchronisation 49 3.3. Synchronisation of Two Identical Klystrons 52 a) Effect of the Coupling Phase Angle 53
b) Case of Small Coupling . 59 3.4. The Long Line Effect 63
Chap. 4. Synchronous Parallel Operation of Reflex Klystrons 64
4.1. General Requirements 64 4.2. Magic-T; Scattering Matrix 67 4.3. Alternative Combining Networks for Synchronous Parallel Ope¬ ration 72
1. Parallel Operation with an External Synchronising Signal . . 73 2. Symmetrical Combining Networks; Coupling through Reflec¬ tion 76
3. Combining Network Composed of a Single Magic-T with Compli¬ mentary Bethe-Hole Coupler 78
3 Chap. 5. Experimental Results 80
5.1. Introduction 80 5.2. Cold Test to Estimate the of the Q-Factors Klystron Cavity . . 87 5.3. Mutual Synchronisation 91 5.4. Parallel of Two . Synchronous Operation Reflex-Oscillators . . 101
5.5. a from a Harmonic Generator Synchronisation by Signal .... 106
Literature 110
4 Preface
The early experiments and theoretical treatments of the problem of syncohinisation have led to a considerable interest in the possible practical applications of the synchronised oscillator. There is a continually growing literature on the possible applications, especi¬ ally on the subject of using an oscillator as a synchronous-amplifier limiter for f. m. reception. Also in the microwave region, synchroni¬ sation has already found application in the "linear accelerator". Here a chain of synchronous high power magnetron-oscillators are used to drive the linear accelerator for the production of high- energy atomic particle. In the present work a theory is presented which predicts the behaviour of any self-limiting oscillator, when synchronised by an external signal of any magnitude and any waveform. The theory is based on the principle of conservation of energy and enables the calculation of the steady state amplitude and phase, when the nonlinear characteristic of the oscillator is representable by a simple mathematical function. The theory is then extended to include the mutual synchronisation of two reflex oscillators of arbi¬ trary properties and with any degree of coupling between them. This part of the work is included in the first three chapters. In the fourth chapter some bridge-circuits are suggested, which enable the synchronous parallel operation of 2 or 2n reflex oscilla¬ tors. The experimental work is then described in chapter 5. Com¬ plete verification of the predictions of the theory presented has been established by the experimental results.
5 Leer - Vide - Empty Chapter 1. Synchronisation of Oscillators
1.1. Introduction
The nonlinear theory of electrical and mechanical oscillations has been extensively studied by a great number of authors. It is a well known fact that in any self-exciting, self-limiting source of steady harmonic oscillations there must exist some form of a non¬ linear relation between the acting forces and the resulting harmonic motion. This nonlinear relation is responsible for the self-excitation as well as for limiting the steady state amplitude to a finite value. Although the formulation of the differential equation for the general case of an oscillatory system disturbed by some external signal is a rather simple matter, it is very difficult and often impos¬ sible to find its exact solution. It is often advantageous to trans¬ form the obtained equation, if possible, into the form of the Hill's differential equation, for which many useful approximate solutions have been developed and in some very special cases its exact solution is known. In this way it is sometimes possible to gain a clear insight into the behaviour of the system or at least to be able to discuss its general features. In treating the problem of synchronisation it was, therefore, found more appropriate to discuss the behaviour of the oscillator from a physical point of view without primarily laying stress on mathematical rigor. Such discussions have led to simplifying assumptions which enabled to get a solution for the special case where the amplitude of the external signal is small compared with that of the undisturbed oscillation. In what follows it is intended to give a short summary of some of the theories developed to study the synchronisation problem. A simple method based on the prin¬ ciple of conservation of energy is then described. This method enables to calculate the steady state amplitude and phase, if the nonlinear characteristic can be represented by a simple function.
7 1.2. Summaries of some Simplified Theories on Synchronisation
Van der Pol [1], in his well known paper on "The Nonlinear Theory of Electrical Oscillations" considered the case of a triode oscillator disturbed by an external signal. He assumed a solution of the form
v = b1 sin co11 + b2 cos w11,
substituted it in the differential equation, and was thus able to calculate the steady state amplitude as a function of the detuning
w1 }' An interesting work on the synchronisation of oscillators by modulated signals was published by F. Diemer [3]. He treated this problem by transforming the differential equation into the Hill's form and derived a solution in an integral form. This enabled him to discuss many general features of the problem and to show the range of validity of other theories based on the assumption of a linear characteristic. He could further derive the condition of stability of the synchronised oscillation in terms of the charac¬
teristic exponent and the auxiliary phase — these are parameters generally used in the solution of the Hill's equation. The exact solution, however, was impossible, except for the case of a very small signal having a small degree of modulation. Therefore he had to resort at last to a more physical consideration. Like Van der Pol he plotted curves for the steady state amplitude and phase as functions of the detuning for the case of an unmodulated external signal. He then based the rest of his discussion on these curves by considering the synchronised oscillator as a filter having the above curves as gain and phase characteristics. Representing the filter curves in the synchronisation region by a simple approximate equation he could also calculate the amount of distortion produced in an FM signal while transmitted through the filter. In the above two papers the transient state has not been con¬ sidered. The study of the transient state is especially important if the applied signal is modulated in the frequency. If the oscillator is capable of following the frequency deviations of the impressed signal, we have a state of a continually disturbed oscillation. Let us
8 assume that the amplitude of the impressed signal is big enough so that synchronisation can be obtained with an unmodulated signal within a band of frequencies greater than twice the maximum frequency deviation. The knowledge of the time constant of the pull-in process is still required to determine the highest modulation frequency allowable in order to attain steady locking. It is evident, however, that if the changes in frequency of the external signal occur in a time long compared with the pull-in time constant, the oscillator can be assumed in equilibrium at each instant of time and a succession of such steady states are a good enough approxi¬ mation to the actual situation.
Turning now to the theories on synchronisation by a small exter¬ nal signal, we begin with that published by H. Samulon [2]. He studied the synchronisation of a triode oscillator by a small signal of frequency o)x= — (a>0 + A a>0) where - is any rational ratio.
According to his treatment the process of synchronisation in this case can be described briefly as follows: "Cross-modulation, occur - ing between the impressed signal and the oscillator voltage, pro¬ duces components of frequency given by cofc= i^tuj ±q0cj0. Those
which have a components frequency approximately equal to oj0, can be considered as a synchronising signal for locking with a one- to-one frequency ratio". This is valid if the frequency of the oscillator is controlled by some tuned circuit which presents a very low impedance to all harmonics other than those having a>k ~ coQ.
Superposing the anode current component 70 of frequency cd0 with those components Ik of frequency u)k^.io0 a beating current is obtained which, under the assumption Ik-^I0, gives a current of constant amplitude and varying phase. The oscillator tuned circuit, when driven by this current, oscillates at a variable frequency. From the phase characteristic of the tuned circuit together with the phase relationship between anode current, anode voltage, and grid voltage the instantaneous frequency deviation from co0 can be derived. The solution of the differential equation thus obtained is either periodic or aperiodic depending on the circuit parameters, the initial frequency deviation and the relative magnitude of the cross modulation product f-pl . The aperiodic solution leads to a steady
9 state where the oscillator frequency is equal to —co1; i. e. synchroni¬ sation, and is attainable within certain limits of the initial frequency deviation A a>0 given by
** A ^ A « \ fr ^ft d«^A" = \f0\-2%'
Q being the figure of merit of the tuned circuit. The periodic solution, occuring if A o)^>Aojm, shows that the oscillator should vary its frequency periodically between the limits
— the instantaneous value. The tu0 A o)m^ojt^oj0 + A com, a), being period Tf of this cyclic variation of the frequency is a function of
-j-4^- and tends to infinity as -r-^5- approaches unity. Samulon A 0)m A U)m plotted curves showing the relative instantaneous frequency devia¬ ~~^- ~ a)° °i" — *".' tion for different values of . . These curves have A com A (om A utm
of so that the a nearly sinusoidal form for large values -—-, average
to value of (at taken over a complete period Tf is very nearly equal oscillator cu0. If -r-^-^l, the curves show that the frequency remains nearly constant over the greatest part of T, at a value equal
— — = A to
n
= — the value of or (x>x (o>0— A oj0) respectively. In this case average ro to A io( over Tf is approximately equal (oj0 ± com).
Returning to the synchronising component Ik of cok ^ w0 it is obvious that its amplitude is dependent on the form of the nonlinear characteristic present. Representing this characteristic by a power series it can be easily proved that for the case m = 1, the lowest order term which enables the required Ik to be produced is the ra-th order term. All terms of lower order will contribute nothing to the amplitude of Ik. The coefficient of the w-th order term is therefore the decisive factor in the synchronisation process. Remembering that the coefficients of such a power series diminish rapidly with increasing order in most practical cases, it becomes directly evident that synchronisation becomes more difficult for larger values of n. However, it has been experimentally observed that such oscillators like multivibrators and relaxation oscillators synchronise easily
10 even for larger values of n. Samulon explained this behaviour as follows: Such oscillators contain higher harmonics of amplitude comparable with that of the fundamental and may thus have a considerable contribution to the amplitude of the synchronising component Ik. Remembering that the contribution to Ik originating from cross-modulation between the external signal and one of the higher harmonics requires a term of the power series of an order much lower than n, we notice that this contribution may even be bigger than that due to the fundamental component itself, depending on the rapidity with which the coefficients of the power series decrease with increasing order. As an example consider the case n = 25. If the oscillation is purely sinusoidal the lowest order term necessary is the 25-th, while if it contains e. g. the 4-th harmonic then the 7-th order term already contributes to the required com¬ ponent. This may lead to an increase in the amplitude of Ik and consequently to an improved ability to synchronise.
A treatment, quite similar to the previous one, has been pub¬ lished by Adler [4]. Here only the special case of o^^coq has been treated. The differential equation derived as well as the results obtained are essentially the same. However, Adler studied the requirements which an oscillator must meet so that the above analysis may be applicable. These requirements are fulfilled if the different elements of the oscillator circuit are dimensioned in such a manner that there are no aftereffects from different conditions which may have existed in the past. To explain this we quote a part of his discussion. "If an oscillator is disturbed but not locked by an exter¬ nal signal, we observe a beat note — periodic variations of fre¬ quency and amplitude. If these variations are rapid, a sharply tuned circuit in the oscillator may not be able to respond instantaneously, or a capacitor may delay the automatic readjustment of a bias voltage. In either case the above assumption would be invalid. To validate it, we shall have to specify a minimum bandwidth for the tuned circuit and a maximum time constant for the biasing system.'' Thus, if a tuned circuit is to reproduce variations of phase (i. e. frequency) and amplitude without noticeable delay, its decay time constant must be short compared to a beat cycle l-j—1, or stated
11 in other words, its pass band should be wide compared to the "undisturbed" beat frequency (Acu0), i.e. the frequency of the external signal should be near the center of the pass band. Also, any amplitude control mechanism present in the oscillator circuit should have a time constant short compared to one beat cycle, in order that we may be able to assume that amplitude variations are reproduced instantaneously everywhere in the oscillator circuit. But when the amplitude control mechanism acts too slow to acco¬ modate the beat frequency, phenomena of entirely different charac¬ ter appear. Such an oscillator would fall outside the scope of our mathematical analysis. We consider again the phenomena occuring outside the limits of
synchronisation (A w0 > A cum) where the oscillator is disturbed but
not locked by a small external signal of frequency t^. The above
theories have shown that the oscillator frequency u>t varies periodi¬
with a between the — A cally period Tf limits co0 cvm S cot S coQ + A u>m,
and that the value taken over one average wt period Tf has the values:
oi,~ a>n for —. ;=- 1
A con -> A -> 1 and o)( a>„ ± wm as -.—-
The instantaneous beat frequency A
same between the limits A — period Tf io„ A cum g A u> ^ w0 + A wm. The average beat frequency Aco, taken over T^, has the values
A ^ 0 A~oj A for -r^- 1 oj( , ^ cu0 ^ A wn
and A -> A Jcu^O as -—- -> 1 cot wm ,
Thus, as the value of o1 approaches one of the synchronisation limits (w0 + A com), the average frequency of the oscillator approaches the same limit and it can be said, that the oscillator frequency is being "pulled" towards that of the external signal. Such a "pulling" phenomenon would not exist, if the oscillator were linear. In a
12 linear system of natural frequency cd0 the application of an external signal of frequency iox does not disturb the frequency oj0 of the
"free" oscillation; large amplitude changes may occur if ojx is near
enough to w0 but the output will contain no other frequencies than
o>x and a>0. In the above theories the nonlinearity of the oscillator characteristic has been accounted for by assuming that the appli¬ cation of the external signal results in negligible change .of the amplitude of the output voltage. Such an assumption implies the existence of an oscillator which has developped its "undisturbed" amplitude up to the saturation value determined by its nonlinear characteristic. This means that the internal source of energy which maintains the oscillation has a limited power-carrying capacity and that the amplitude of the self-excited oscillation has been developed up to this limit. Hence, the application of the external signal cannot affect any further increase of the amplitude, yet it can change the instantaneous rate at which energy is supplied to the oscillatory circuit. This results in disturbing the phase balance of the oscillator associated with a subsequent change in the frequency. As long as the frequency of the oscillation is not equal to that of the impressed signal, the instantaneous rate of energy supplied is continually disturbed i. e. this rate does not correspond to that necessary to
maintain an oscillation of a frequency either equal to cj1 or to o>,.
Thus a>( should vary. But as the frequency of the external signal is fixed, its contribution to the rate of energy supplied may be expec¬ ted to favour any oscillation of a frequency nearer to its own and thence the observed "pulling" of the average frequency of the
oscillator away from w0 towards cuj. This pulling increases (i.e.
w° -^- becomes smaller, tending to zero by = 1) as a>1 appro-
aches io0 ± A a>m, till at the limit locking occurs and the oscillator
attains a fixed frequency a>1. Thus the synchronisation phenomenon
may not be described as that state where the "free" oscillation is being suppressed by the external signal and only the "forced" oscillation exists. It is rather that state in which the phase balance
of the oscillation can only be attained at the frequency o>1. Such a state, in which the oscillation does not take place at the natural frequency of the oscillatory circuit occurs, if the feed back circuit
13 introduces a phase shift other than 180° between the anode voltage and the voltage fed back to the grid. Here the self-excited oscilla¬ tion adjusts its frequency to a value w#oj0 at which the phase balance is fulfilled. Thus the small synchronising signal may be replaced for example by a ficticious impedance in series with the feed back circuit which causes the oscillation to take place at a>x. A similar concept is used by Huntoon [5] in his general treatment of synchronisation. All the above theoretical treatments of oscillator synchronisation have been concerned with the internal mechanism within a triode oscillator which accounts for synchronisation. The theory presented by Huntoon discusses certain features of synchronisation without reference to the internal mechanism which accounts for it. His theory is therefore generally applicable to all types of oscillators. He defines a set of compliance coefficient which show how the amplitude and the frequency of the oscillation depend upon the load impedance. The values of these coefficients may be derived theoretically or measured for the particular oscillator. He considered the external signal voltage as equivalent to the IZ drop on a ficticious increment in the load impedance. The oscillator's frequency and amplitude shift in accordance with its compliance coefficients and the magni¬ tude and phase of the incremental load impedance. He obtained a differential equation similar to that developped by Adler but more general. In addition he was able to discuss the amplitude behaviour of the- oscillator. An important value of this theory lies in the fact that it can be easily extended to include the mutual synchronisation of two oscillators of arbitrary properties, if the coupling between the oscillators is weak.
1.3. Calculation of the Steady State Amplitude and Phase by Using the Energy Equation
The method of calculation that we have developed, is explained by applying it to the pentode oscillator shown in Fig. 1.1. The non¬ linear characteristic is assumed to be representable by the third degree parabola
i = -avg + bv(l3, (1)
14 where i is the variable part of the anode current, and vg the variable part of the grid potential. vg is the sum of the voltage — kv fed back from the output plus the voltage vx of the impressed signal, so that
vg = -kv + v1 (2) and
i = oc(v + u) — y (v + u)3, (3) with
a k, y = bk and u = (4) k
Fig. 1.1. Simplified circuit of a pentode oscillator with an external signal applied in series with the grid coil.
The differential equation for the oscillatory circuit driven by the current i is
v 1 f „dv jA (5)
The principle of conservation of energy states that the amount of energy supplied to a system during any time interval (t —10) should be equal to the sum of the energy consumed in the system during the same time interval plus the increase of the energy stored in the system. Calling Pt the instantaneous power supplied to the system, Pc the power consumed and WSo, Ws the initial and final energy stored, the energy equation will be given by:
\ptdt = {Ws-W,o) + \pcdt. (6)
Differentiation gives
rt~ +r° (7) dt
15 Eq. (7) could have been used to derive the differential equation for the oscillatory circuit. For the simple case under consideration (5) can be easily transformed to (7) by multiplying both sides by the voltage v, thus yielding
+ + (8) L)vdt Cvdt\ K' with Pt = iv, P<- = R' (9) dW, v C dv , , ^ and —r-1 = -=- \vdt + (v~. dt L J dt
If in the steady state the system oscillates with a fixed period T, the functions Pt, Pc, Ws and -~ are all periodic functions of the time having generally a period — \T\ whereas the functions $ Ptdt t u and J Pcdt are monotonically increasing functions of time indicating u the continuous energy supply and energy consumption. Thus, inte¬ grating (7) over a complete period T yields
U + T J" Ptdt = J Pedt (10) U U since, because of the periodicity of^s
wsito+T) - WS%) = Q.
Equation (10) says that the energy supplied to the system over a complete period is equal to the energy consumed; the energy storage assumes again its initial value. Thus if v in Eq. (8) has a period T, the integration of the different terms over a complete period, using the notation
Sf(t)dt = f(t).T, T gives J ivdt = iv • T, T
16 V J ($vdt)dt = ^T[($vdt)*]=0.. T and -=dt = -^--T.
T
This yields -% iv=^. (11)
It is obvious that Eqs. (10) and (11) are equivalent, and that they will give us the steady state amplitude of v. It is still required to derive, in a similar manner, another equation which gives the frequency. It may be argued, that differention of (7) gives an equation containing the frequency as a multiplying factor and then integrating over a complete period gives the required expression. If this is applied on (7) the result obtained will be 0 = 0 because of the periodicity of all the functions contained. But if we observe that the energy equation is always associated with a "force" equa¬ tion, the differentiation of the energy equation and the cancel¬ lations of those terms which satisfy the force equation will lead to an expression which, if integrated over a complete period will not lead to a result 0 = 0. Thus differentiating (8) gives
v v2 „ dv dv v di dv [ If, ^,dv .) .,_, VTt=RTt+L+Cvdi + -dt\B+L)vdt + Cdt-l\- (12) Because of (5) the expression between the brackets {} vanishes and we are left with
di v dv v2 dv ^ ,,„. = v-r -^ -T- + T- + Cv~. (13)K ' dt B dt L dt
Since
\V%dt = \vd{ty[Vt]T-lttdV -'-I®'*--®'-*- the integration of (13) over a complete period yields
^i^V-Cv'2, (14) Li
17 and -=- is different where i' and v' mean -3- respectively. Eq. (14) from Eq. (13) by the fact that it contains terms differentiated with respect to the time and thus having the frequency as a multiplying factor. = can be in the form Calling w02 y^, (14) put
Ti' = C(u>Q2v2-v'2) (14a)
This gives the frequency of oscillation in terms of the natural fre¬ the nature of the quency of the system and in accordance with driving function i. Although we can easily recognize that the terms contained in (11) familiar give average power, it is difficult to assign any meaning to the terms contained in (14). To apply (11) and (14) to the case under consideration we assume for the voltage v a solution of the form
v = V sin w11, (15)
Such a is where u>1 is the frequency of the external signal. solution of the only valid within those limits of o>1 where synchronisation oscillation by the external signal can take place. Moreover, we have assumed that the oscillatory circuit is slightly damped, so that the oscillation developing in it is very nearly sinusoidal and hence higher harmonics are negligible. Further, let the external signal be given by u = U sin ( so that e = v + u = £ sin (t^t+ ), (17) with E cos if) = V + U cos From (3) the current is given by = — % ote ye3 . (19) Substituting (19) in (11) and (14) yields »2 v2 xve — y ve3 = (20) 3 = — G ave' yve'3 n2v2-{ 18 As the frequency of the oscillation is assumed to be given Eqs. (20) are expected to give the steady state phase cp and amplitude V as a function of the detuning, as well as the limits within which the solution (15) is valid. Using (15), (17) and (20) and performing the integration over 2-7T the period T = - we get = -= t COS i K .*(«-3^) (21) L (^-l)=^sin^(a-^). Separating V and i/r yields -J— v i/- + l^~wA2 = e(*-^bA (22) tan^^/^-^V w0L Putting 7i = = factor of the circuit Q —r Q oscillatory co0L (23) § = (j^-^L detuning \o)0 wx/ in Eq. (22) we get ^ / 3 (22a) tan i/j = § which give the steady state phase and amplitude as functions of of U E, a, y, 8. It is, however, required to find V and V = {V+Ucos 19 From Eqs. (24) we can discuss the phase and amplitude behaviour of the synchronised oscillator. The undisturbed amplitude V may be obtained by putting U = 0 giving F°2 = <25> 37 (*-]!>)' which yields also the familiar condition for self-excitation (with a parallel resonant circuit), namely: 1 a>iT The effect of the nonlinearity of the oscillator characteristic in limiting the steady state amplitude to a finite value is also indicated in (25); putting y = 0 (i.e. linear characteristic) yields F0 = oo. the relative v = and Normalising (24) by using amplitudes -^- v„ u = and jj- putting O.K -A=X (26) 4X and V 2 ' 0 3yR we get v = (v+ ucosy) [1 -x{(v + ucosip)2 + u2sin29> -1}] (27) USmy = 8 . (28) V + UCOS Eliminating v between (27) and (28) and denoting § by tan tfi, the phase sin3© sin 1 qs cos Putting of = signal frequency w1 w0 and having any initial phase angle against the oscillator voltage results also in a transient state where the oscillator phase rotates towards that of the external signal until both voltages are in phase. It is also obvious from (28) that values of 20 that in the v and u are denoting steady state not in phase if w1 4= oj0 . This is the natural result to be expected for, as the oscillation occurs at a frequency w1 =#o)0, the current and voltage in the reso¬ nant circuit must have a phase difference & which, for a single tuned circuit is given by tan 0 = 8, (30) with the current leading the voltage for cox > a>0 or 8 > 0 and lagging for o)x < w0 or 8 < 0. Remembering that the fundamental component of the anode current i is in phase with the total grid voltage e, the vector representation in Fig. 2 shows that @ must be equal to t/t, Fig. 1.2. Vector diagram showing the phase relations between disturbing signal u, anode voltage v, and current i and total grid voltage e for io^Wq. as is also obvious from (22a) and (30). This vector representation shows the new phase balance to be established under the influence of the external signal, if synchronisation takes place. But synchroni¬ sation can only take place for those values of 8 which satisfy (28). The limits are determined by the maximum value of S given by (28) and occurs at \K\ = ~ (31) m where vm is the value of v corresponding to 8 ~2QA so that Result (32) is the same as that obtained by the theories discussed above, if Vm is replaced by V0. Thus if Vm is less than V0, the 21 result (32) shows that synchronisation will take place over a band of frequencies broader than that given by the mentioned theories; the deviation between the two results being larger for larger values of U, where the external signal can produce a considerable change in the amplitude of the output voltage. To calculate the steady state phase we use (28). Putting ?"Z x = ^? and y = (33) sm ijj cosi/r (28) gives u2 x3 — x = — y (34) which is the equation of the third degree parabola illustrated in and have Fig. 1.3. The above discussion has shown that Fig. 1.3. the same sign with ft, and that for 8 = 0, 99 = 0 and S = Sm, AB of the is y=\, 9 = 0, ft = 0 and'8 = 0, and at A y = Q, (p = 90°, sini/fm = u and 8 = 8m. As to the amplitude behaviour of the oscillator, the assumption of a third order parabola for the nonlinear characteristic yields amplitude variations which cannot hold for an actual oscillator. The assumed parabola is a good approximation of the actual oscil- 22 iator characteristic between the points P and P' (see Fig. 1.4), beyond which the parabola gives a decrease in the current for increasing v while in the actual characteristic the current further increases up to some saturation value. Thus the results obtained below will describe the amplitude behaviour of an actual oscillator only for values of u small compared with unity. flcl-ual characherishc Fig. 1.4. Deviation of the assumed third degree parabola from the nonlinear characteristic of an actual oscillator. Calling v0 and vm the values of v for any u at 95 = 0 and = (v0 + u)3-(v0 + u) —, v^ + u2 = 1, (35) u and 8„, = Vl-u2' These relations are plotted in curves in Fig. 1.5 and Fig. 1.6. Here we notice the effect of the assumed characteristic on the amplitude behaviour by the fact that the amplitude of the output voltage decreases if the amplitude of the external signal increases beyond a certain limit. For x < 0,5, v0 increases with increasing u up to a maximum value given by = (36) 3(1+*> 3* 23 at a value of u given by l-2Xl/l+v (37) 3X and then decreases for larger values of u. We also find that v0max is attained at u > 0 for x < 0,5 and at u = 0 for x = 0>5 and would be attainable at u<0 for x>0,5. Thus it may be concluded that the form of the assumed characteristic does not allow the development -U 1.5. Relative of the 1.6. Relative Fig. amplitude Fig. amplitude F„. oscillation v as function of rel. amp. and limiting value of the detuning of the ext. signal u for 8 = 0 and 8m as functions of rel amp. of ext. different values of x = «R — 1. signal u. of an output voltage of amplitude greater than ~v0max; this maximum value occurs when the driving voltage applied to the grid has a certain amplitude, say e0. The value of e0 can be easily determined by using Eq. (19). The current maximum occurs at a voltage E* = ^ ' (38) 3y Relating now all amplitudes to Es we get from (26), (36), (37) and (38) the values: * 0 2 _ (40) Es2 1 + X' '2 4(l+x) vL(a= (41) v,2 "27v 24 and (42; V,2 27 y Remembering that the driving voltage applied to the grid is pro¬ portional to (v + u), the value of e0 may be taken to be equal to (vom«* + uc')- This gives 4 2 _ c. 2 _ (Vo, • + u« (43) From (40) and (43) it is obvious that vr2 = e for x = 0,5. Thus for X > 0,5 the undisturbed amplitude develops to a value which drives the grid voltage in excess to e0, so that the application of an exter¬ nal signal can only result in a decrease of the output voltage. This decrease is due to the falling part of the assumed characteristic which is shown dotted in Fig. 1.4. We may therefore conclude that the amplitude behaviour of an actual oscillator will be similar to that of the oscillator under consideration for x < 0,5 with u ^ u0; for u > uc the amplitude does not fall but rises slowly to its saturation value. In Fig. 1.6 the dropping of vm to zero by u = l is due to the infinite value of the detuning accompanying it. 0 S 0J 0,2 0,3 0A 0.S 0,1 0,2 0,3 0A 0,5 6 Fig. 1.7. Kg, 1.8. Calculated Curves showing phase (Fig. 1.7) and (Fig. 1.8) behaviours of a synchronised oscillator, whose nonlinear characteristic is assumed ta be a third order parabala. v0 = undisturbed aplitude. u = relative amplitude of extenal signal. S = detuning =Q(---^. \m0 w! 25 Figs. 1.7 and 1.8 show the behaviour of our hypothetical oscil¬ lator when synchronised by an external signal. The data used in calculating the curves shown are a = 1,5 X 10-3 A/V b = 0,6 X 10"3 A/V k = 2,5 X 10-2 a = 3,75 X 10~5 A/V y = 9,4 X 10-9 A/V3 Q = 120 R = 34 kQ x = °>275 < °>5 Bearing in mind the range of validity of the various assumptions used above in representing the properties of an actual self-exciting, self-limiting oscillator, the above discussion together with the cal¬ culated curves enable us to have a clear idea of the behaviour of such an oscillator when synchronised by an external signal. Chapter 2. Synchronisation of Reflex-Klystron Oscillators 2.1. Introduction The theory of velocity-modulated tubes, operating as amplifiers or oscillators, has been given by various authors [6—9]. Although the theory differs somewhat among the various presentations, all of them give essentially the same results. One useful form of the theory has been developed by using a number of simplifying assumptions, some of which are justified by the special design and simple geometry of the tubes used in practice. In such a tube the electron beam passes down the axis through a succession of regions separated by plane grids. Some of these are regions of acceleration, drift and reflection, which are relatively free from r-f fields. Others are gaps forming the capacitive portions of the resonator circuits, where interaction between the r-f gap fields and beam current takes place. These gaps have depthes that are usually small compared with the diameters of the gap areas. Moreover, the excitation of the resonators is generally such that the electric fields in the gaps are directed to the axis parallel and are nearly uniform over the gap areas. in If, addition, the beams are nearly uniform and fill the gaps, phenomena in the gaps are approximately one-dimensional. This 26 idealisation of the gap phenomena to uniform fields and a uniform beam composed of electrons moving parallel to the axis of the tube is a tremendous simplification that makes possible the analysis and discussion of tube behaviour. There are, of course, many limitations to a treatment of gap phenomena based on the assumption of uniformity. Since all gaps have finite areas and all beams have limited cross sections, there are edge effects. Uneven cathodes, fluctuation in emission, nonparallel grids, grid structure, and uneven reflector fields make the beams nonuniform. In addition, the conduction current is carried by the electrons, which are finite charges with local fields and hence contribute to the unevenness in the gap currents and fields. Electrons have transverse velocities, and the electron velocities must be well below the velocity of light if magnetic forces are to be neglected. Yet, the simplifying assump¬ tions have considerable validity in most practical tubes and have CaHiode Resonator Grids Reflector Region oFd acceleraKo gap Fig. 2.1. Schematic drawing of reflex oscillator with the d.c. voltages applied to the different electrodes. 27 made possible much of the theoretical treatment of these tubes. Moreover, they enable the further development of the theory to include the effect of one or more of the different factors, which have been neglected in the simplified form of the treatment. In the following chapters the simplified theory of velocity modu¬ lated tubes is going to be applied. As mentioned above this theory is valid under the assumption of uniformity together with the assumptions of linear reflecting field, negligible space charge effects and negligible thermal velocity spread. A short summary of the main relations is given in the following paragraphs. Fig. 2.1 shows a schematical representation of a reflex tube together with the potential distribution in the different regions, neglecting space charge effects. Thus, if all electrons are emitted from the cathode with zero initial velocity, then under the action of the d. c. accelerating voltage V0 they arrive at the r-f gap all having the same velocity u given by \ m u02 = e V0, with e and m the mass and charge of an electron respectively. If the cathode emission is uniform, the input current I0 from the accelerating region into the r-f gap is constant in time. Between the grids of the r-f gap the injected current will be velocity modulated under the action of the gap fields. Thus if the r-f voltage in the gap is given at any instant by v = V sin cot, (1) electrons arriving at the gap at any instant t' gain (or lose) a' kinetic energy eVsinait' during the transit time T1 through the gap, if this time were negligibly small compared to the period of the r-f voltage i.e. w T1<2tt. With a small but finite transit time T±, the energy gained will be eM V sin cot', where the factor M = -§f-, (0! = ^) (2) T is termed the "beam coupling coefficient" and is included to account for the reduction in the energy gained by the electron stream due to the change in the r-f field during the time of passage through the gap. Thus the velocity at exit is given by 28 M V . , , = - u un { 1 + =— sin to t (3) MV . , 1 + sin w t —.y- " for *< i-e- if the of modulation is small. -„y- 1> "depth ly- In the reflector region the electrons will be decelerated by the reflecting field, stopped and then returned back to the r-f gap to make a second transit. As a result of the initial velocity modulation, the stream is density modulated on returning to the gap. The pro¬ cess, that an initial velocity modulation of an electron stream results by drift action in a density modulated stream, is known as "bunching". It can be easily shown that those electrons, which have made their first transit at the instant t' return back to the gap at a time t, given in terms of t' by the relation / MY \ cot = wt' + @0l 1- -—=-sintat'\ (4) which holds for a small of modulation i. e. again depth WyF The quantity <90 is the d. c. transit time through the reflector region, measured in radians of the input frequency co: 0o = ^, (5) where d is the depth in the reflector region (measured from the center of the r-f gap) attained by the center-of-the-bunch electron, i.e. by that electron which has made its first transit through the gap at an instant where the r-f voltage is zero and changing from one which decelerates to one which accelerates the electrons. It is also convenient to define x_mv&0 r =cot; (7) X is a dimensionless quantity known as the "bunching parameter". The transit time relation is thus given by 29 = — t t' + @ X sin t' . (4a) Now let the instantaneous density modulated beam current return¬ ing to the gap at the instant t be denoted by i (t). The charge car¬ ried by the electrons arriving at the instant t during an interval of time A t will thus be (i (t) -At). But the same electrons have departed from the gap at an instant t' during an interval A t' and the current was constant and given by the d. c. current I0. Thus i(t)-At = -I0-At', or dividing both sides by A t and using t instead of t we get ,-(t) = -/0|J, (8) Eqs. (4a) and (8) give the instantaneous beam current i (t) and show its nonlinear dependence on the voltage. During the second transit this modulated current interacts with the gap fields. If the relative phase of this current and the r-f voltage lies in the proper range, power can be delivered from the stream to the resonator. If this power is sufficient for the losses and the load, steady oscillations can be sustained. It is clear, that for the phase to be optimum the center of the bunch should arrive at the gap when the field exerts the maximum retarding effect on the electrons at the center. Thus at optimum phase the d. c. transit angle © should have the value ' @w = 277(7* + !) (9) = n 1,2, 3, . . . As has been mentioned above, a finite gap transit angle @1 causes the gap voltage to be not fully effective in producing bunching and similarly the bunched current not fully effective in driving the resonator. The latter effect arises due to the partial cancellation in phase of the current in the gap. This introduces again the factor M defined by (2). Thus if the beam current is given at any instant by i (r), the driving current i. e. the current in the external circuit induced by i(r), will be Mi(t). Now, if an external signal is injected into the klystron resonator, 30 the new voltage appearing at the gap will disturb the phase and amplitude balances between the bunched current and gap voltage. If the disturbing signal frequency is near enough to the free-running frequency of the klystron oscillator, a new state of balance may be achieved at the disturbing' frequency associated with a change in the output power. In the following sections we are going to study the behaviour of the reflex klystron oscillator when synchronised by an external signal. Again the energy equation is going to be used to derive the steady state phase and amplitude. 2.2. Choice of the Equivalent Circuit The equivalent circuit to be chosen should consist of elements which can be easily measured for a given tube under test. Moreover, these elements should be so arranged that the applied voltages and currents may be easily related to the power of the external signal injected into the klystron cavity as well as to the power output from the klystron. Receiver 1 3 Calib. Aften. — Det. Direct. or CRT - Coupler Term. 2 4 Calib. Atten. _(~)s Fig. 2.2. Block diagram of a possible circuit arrangement to study the behaviour of a reflex-oscillator synchronised by an external signal. Let us, therefore, first consider a circuit arrangement which enables an experimental investigation of the klystron behaviour when synchronised by an external signal. In Fig. 2.2 let 8 be the signal source and assume that its power carrying capacity is some 100 times greater than that of the reflex klystron K under test, so that a large line attenuation is allowable between K and 8, to inject a signal power to K of the order of magnitude of the output power from K. The coupling between 8 and K takes place over a calibrated attenuator and a directional coupler of known coupling coefficient. Power output from K is led over a second calibrated attenuator to a crystal detector which serves to measure the relative 31 output power level from K. A known part of the output power from K couples through the directional coupler and the first attenuator into the signal source 8. This power, being very small compared with the output power from S, is assumed to produce no effect on the signal source. The output from the crystal detector may either be a beat note or a d. c. voltage depending on whether beating or synchronisation takes place. The existing state can be detected by supplying the crystal output to a spectrum analyser, a cathode ray tube or hetrodyne receiver. When synchronisation occurs, a d. c. meter together with a calibrated attenuator serve to determine the relative output power level from K. Further, it is assumed that 8 is provided with some device which indicates accurately its frequency of oscillation. Assuming that 8 delivers a constant power output over the frequency range necessary for our investigation, the known coupling coefficient of the directional coupler together with the reading of the calibrated attenuator will enable to determine the power incident on the klystron K. Call this incident power Pi. Now, if at signal frequency the klystron is matched to the line, the whole of P( will be absorbed into the klystron cavity and thus contribute to synchronisation. Such a match can only exist if the klystron cavity is in tune with the impressed frequency. Thus if P=yeja denotes the reflection coefficient at the signal frequency as seen in the line looking towards K, that part of P{ which couples into the cavity is given by Pt=(l~Y*)Pt, (10) the rest is reflected and absorbed in the different attenuators. Thus, Pi is a function of the frequency for a given Pt. Although only the fraction P, will contribute to the synchronisation, it is obvious that we should take Pt as a measure of the magnitude of the disturbing signal, especially because Pt is independent of the frequency deviation and can be easily measured. It has been tacitly assumed that all circuit elements are matched to the line over a wide frequency band. From Fig. 2.2 it can be easily seen that the klystron supplies its power to a very nearly matched line. 32 Thus our equivalent circuit should contain some current or voltage source (independent of the frequency) to represent P{, and its elements should be so arranged that the power absorbed into the elements representing the cavity is given in terms of Pt by (10). Moreover the circuit must allow the output from the klystron to be supplied to a matched line. This requirement can be easily fulfilled by shunting the output terminals of the equivalent circuit by an admittance equal to the characteristic admittance of the line. It is usual to consider the reflex klystron as a parallel resonant circuit (GB, C and L) driven by a negative electronic transadmit- tance. For our case a similar equivalent circuit can be used. lL M~) V I f" fatfi} Pig. 2.3. Equivalent circuit of a reflex oscillator disturbed by an external signal represented by the current source 2 i2 so that incident power is given The simple circuit in Fig. 2.3 is found to fulfil all the above require¬ ments. Here Mix is the driving current due to the electron beam and is given as a function of the gap voltage v by the nonlinear relation contained implicitly in Eqs. (1), (4a) and (8). C, L and GR are the equivalent capacitance, inductance and conductance respec¬ tively as seen at some reference plane in the waveguide. Y0 is the characteristic admittance (real) of the waveguide; thus the klystron supplies a matched line. A simple calculation will show that the incident power Pt is given by ~ and that Eq. (10) is fulfilled for -* 0 the above equivalent circuit. It is also obvious that the power output from the klystron is equal to the power absorbed in Y0 due to Mix and v. The elements GE, C and L can be easily measured by the cold test procedure. With the help of this equivalent circuit our problem reduces to a simple parallel resonant circuit driven by 2 current sources. 33 One current source is independent of the terminal voltage v and represents the disturbing signal. The other current source is depen¬ dent on v as given by velocity modulation and bunching. It is quite obvious that the voltage v is the result of the simultaneous action of both dependent and independent currents; it must be considered as a whole and not as the sum of two voltages, each being the con¬ tribution of one current source. (Due to the nonlinear relation Mi1 = Mi1(v) the law of superposition does not hold.) 2.3. The Energy Equation and Steady State Solution Reference to Fig. 2.3 shows that the differential equation and energy equation for the system are given by = ~ [vdt Mi1 + 2i2 CC^ + v(GR + Y0) + (11) and ft fj f) j + = + Cv~ + \vdt (12) (Mi1 2i2)v v*(GR+Y0) j- respectively. Putting @ = GB+Y0 and using (7) we get Mi1 + 2i2 = toCv' + vG-\ f\vdr (Ha) a>L J and (Mi1 + 2i2)v = v2G + coCvv' + ~\ \vdr. (12a) a>LJ with v' = with to t and j-. Differentiating (11a) respect using (12a) yields v' C v' = Gvv' (Mi,+2i2) + cx)Cv'2-\ = vd-> (13) uLJ If in the steady state v is a periodic function of t and has the period T, then integrating (11a) and (13) over a complete period yields Mi^v + 2t2~v= GV2 (14) 34 and = a>Cv'2 Mi^v'-Vli^v' + —T v'ivdr, (15) where the horizontal bar denotes again the average of the quantity in question taken over a complete period. In Fig. 2.3 the klystron cavity has been represented by the parallel resonant circuit (GR, G and L), which has one natural fre¬ quency of oscillation given by &>02 LC = 1. But, in a lossless resonant cavity free oscillations can take place at any of an infinite number of resonant frequencies which correspond to the infinite number of normal modes of the cavity. Thus in Fig. 2.3 it has been assumed that one of the resonant frequencies oj0 refers to the mode of particular interest and that all other modes are widely separated from it. This assumption is safe for ordinary reflex oscillator cavities, provided that the transmission line out to the chosen reference plane is not too long. For a long line, coupling between cavity and line may result in modes that are close together. For our case we suppose that the modes are widely separated. Further, due to the high Q of the cavity the effect of all higher harmonics may be neglected. Under these assumptions the steady state terminal voltage v is very nearly sinusoidal. Thus, if synchronisation takes place by the impressed current »8 = /asin(T + j8), (16) the steady state terminal voltage is v = V$mr, which gives rise to the steady state bunched current given by (4a) and (8). Let us consider first the energy supplied by the bunched beam current to the oscillatory circuit over one cycle. This is given by the first term of (14) Mi1v2rr = M \ixvdr. o Using (1), (4a) and (8) this expression yields 2w 27r-Mi1v= - MI0V J sin(r' - X sinr' + 6) dr' o = - MI0V { cos © /"sin (T'- X sin t') dr' + sin s[cos (t'~ Zsin t') dA = - JfJ0Ffo + sin©-27r JX(X)), 36 where Jx is the first order Bessel function. Thus the average power is given by P1=-i-7-2Jf/0J1(X)-sin© (17) Here V is the amplitude of the gap voltage, 2 MI0 J1(X) is the fundamental current component induced by the bunched beam current and (? + ©) is the phase angle between them. Similarly, the first term in Eq. (15) gives Miy = -^-V-2MI0J1(X)-cos@ (18) Here again, only the fundamental component of the bunched beam current appears. Using (1) and (16) to perform the integration over a complete cycle, (14) and (15) yield respectively - 2MI0 JX(X) -V sin© + 2/2Fcos£ = V2G (19) - = V2(coC 2MI0J1(X)-Vcos© + 212V sinp ^ (20) We notice that (19) and (20) could have been directly ob¬ tained by applying the steady state circuit theories to the circuit of Fig. 2.2, with Mi1 replaced by its fundamental component. Eqs. (19) and (20) are then nothing else than stating that the vector sum of the currents flowing into any node of the network is zero. However, when applying the methods of linear-circuit analysis to our nonlinear problem, we must bear in mind that the law of super¬ position does not hold. Thus all current sources should be applied simultaneously to the circuit and the resulting voltages are due to this simultaneous action and cannot be considered as the sum of the voltages resulting due to successive application of the current sources — each alone — to the circuit. That only the fundamental component of the bunched beam current appeared in our equations, arises from our assumption that the gap voltage is purely sinusoidal. Under this assumption the higher harmonics contained in the bunched beam current cannot contribute to the average power supplied by the beam to the cavity, although they affect the instantaneous power supply. 36 2.4. Amplitude and Phase Behaviour of the Synchronised Reflex Klystron In discussing the information contained in (19) and (20) we use the following abbreviations: M*6I0 Ge ... ——° g = -~ with Ge = (21) where Ge is known as the small signal electronic transconductance, = = loaded — of the Q -79— Q cavity, 8=e(^-^U2Q("^), (22) \ © = ®n + = — = sin ® cos cos 0 sin . 9 , 9 Further, as a measure of the external signal we define a parameter — similar to the bunching parameter — by M&0 i2 ^-^Tv'W (23) Substituting these quantities in (19) and (20) we get ^7^gcos«p + ^-2cosi3 = l (24) = 8 . and ^—-g sin 9 + -^sinjS (25) These equation give the amplitude of the oscillation voltage (X) and its phase (j8) relative to the impressed current in terms of the impressed signal amplitude (X2) and the frequency deviation of the impressed frequency from the resonance frequency of the cavity (8). The angle cp is the deviation of the reflection transit angle from the value @n = 2n(n + %). The power output from the klystron is Y obtained by multiplying (17) by -^ ; this gives P = ^-^-°-XJ1(X)cos9, (26) 37 which can be considered proportional to X J1(X), if we neglect the variation of © (and The Undisturbed Behaviour In the absence of an external signal (X2 = 0) the above relations reduce to: ^^gcos =1, (14a) X — = 8 tan93 , (15a) 2)-XJ1{X)cos Y 2 T V V ' ( G ® ' Inspection of these expressions show the effect of the reflector voltage and the conductance parameter g on the output power and on the frequency of oscillation. If the reflector voltage is adjusted at the center of some mode i. e. such that 93 = 0, the oscillation takes place at the resonance frequency of the cavity and maximum power is delivered to the load Y0. Changing the reflector voltage to either side of the center results in a change in the frequency and in a reduction of the power output. The effect of the conductance para¬ meter is quite important. At the center of any mode (95 = 0) the power output is proportional to the product X Jx (X), X being now solely determined by g as given by (14a). The product XJX (X) has a maximum = 1,252 at Xop = 2,40, which is the value of the bunching parameter for optimum power conversion. The necessary value of as is As sr given by (16a) g =2,31. g =7r~^~, it is seen that the power conversion to the load can be varied by changing the load conductance. Again, if the conductance parameter is 38 changed by increasing Y0 so that gcos Beflector Voltage Initially Adjusted to y = 0 The reflection transit angle as a function of the frequency is given by (5). Calling @0 = —— and using (5) and (22) we get = °_0J=^S + 0. If the reflector voltage is adjusted to the center of the mode in the absence of = an external signal, then (p0 0. @ At any frequency other than co0 we have 95=5-^8. This gives 5° for = — (p p& the following typical values: ®0 50 and Q 200 when the frequency deviation is 15 Mc/s from a central frequency of 8830 Thus as a = Mc/s. first-order approximation we may put cos - Ge a^80 vary with the fre¬ quency, but it is readily seen that its variation is very small \Ge = Geo 11 + yo) an^ mav ^e neglected. Substitution in Eqs. (24)—(26) gives 2^£>g+2^cos^l (24b) S{l + l^l^j = ^sin^ • (25b) p-XJ1(X) = P (26b) These expressions are quite simple and enable to deduce easily some important results. Consider first the case where 8 = 0, i.e. case of synchronisation by an external signal in tune with the undisturbed frequency. Eq. (25a) gives j8 = 0 or the gap voltage is in phase with the injected current. The external signal, being in tune with the oscillator fre¬ quency, results simply in a transient state where the oscillator 39 phase rotates towards that of the external signal until they coincide. The transient state then disappears and the oscillator phase remains "tied" to that of the external signal. This simple fact may have some important applications. For instance, let it be required that a number of klystrons run in synchronism so that their outputs should have prescribed phases at some definite reference planes. This can be easily accomplished in the following manner. Let the synchronising power be supplied by a klystron through a length of waveguide with a reflectionless termination at its other end. In the guide we have thus a single travelling wave. If at the appropriate planes some sort of coupling device (a hole in the common wall with directive properties or a probe etc.) is provided to couple power from this travelling wave into the different klystrons — all klystrons being pre-tuned to the same frequency with x{l-2^-g) = 2X2. (29) 2 J (X) Remembering that —^—- = 1 at X = 0 and decreases with in¬ creasing X, Eq. (29) shows that the amplitude of the gap voltage increases continuously with increasing X2. At -3l=3,83, Jt(X) = 0 and = (29) gives 2X2 3,83. At this value of X2 the power output is zero. Thus if a strong external signal of 2X2 = 3,83 is injected into the klystron cavity no oscillation can take place. For optimum power output (X = Xop =2,4) the amplitude of the external signal should be Thus, if for some tube g 40 cation of a synchronised reflex klystron. It may be generally stated that the amplitude of the synchronising signal should be as small as possible in order to obtain an output power which is not much smaller than the optimum. It is obvious from (24b) and (25b) that the amplitude and phase curves with X2 as a parameter are symmetrical about the 8 = 0 axis. It is also obvious that the maximum frequency deviation which satisfies the above equation is given by that value of 8 = ±8m which makes j3= ± 90°. At this value of |3 Eq. (24b) gives 2Jj^-9=l, (31) where Xm is the amplitude at the boundaries of the synchronisation region. We notice that Xm is independent of X2 and has a value equal to its undisturbed amplitude (compare with (24a)). Thus if an external signal is applied having some X2, the amplitude of oscillation X is greater than the undisturbed amplitude over the whole synchronisation region, it decreases with increasing 8 to attain its undisturbed amplitude at S = 8m. The value of 8m is given from (25b) and (31) by \*-\ =—r-^rrTTr^ (32) Xm(l+eg^) As Xm is independent of X2, |8m| is proportional to X2. The above discussion enables us to predict the shape of the curves describing the dependence of the amplitude and the output power on the frequency deviation for different values of X2 and g. These are shown in Figs. 2.4a and 2.4b. In Pig. 2.4a it is assumed that g is smaller than gop and thus the undisturbed amplitude is smaller than Xop. Curves 1 and 2 are drawn for 2Jf2<3,83, curve 3 for 2X2 = 3,83 and curve 4 for 2X2>3,83. It is obvious that the best operation is obtained by the state represented by curve 2, where the output power remains nearly constant over the greatest part of the synchronisation region. It is interesting to notice that for 2X2>3,83 the oscillation stops in the middle of the synchroni¬ sation region up to a value of the frequency deviation at which 41 xmm >x.' Aop b) 9 >9 a) 9<9op op Fig. 2.4. Amplitude of oscillation and output power for a reflex-oscillator when oscillator synchronised by an external signal. PQ = Power output is undisturbed, P = Optimum power output. JT = 3,83 where oscillation starts with zero output power. Fig. 2.4b shows the amplitude and power output for g > gop; the curves are numbered in accordance with the values of X2 taken in Fig. 2.4a. Here the power output is always smaller than the undisturbed value, which is less than the optimum power output. Operation under such conditions are therefore less advantageous than those represented by Fig. 2.4a. Reflector Voltage Initially adjusted to some 0 Substituting the value of ^—g(cos gsaup0 + = 8 1 + —~ (25c) X --g-Bmp egco$cp0\ P = p X Jx (X) • (cos 42 These expressions are comphcated and their discussion is rather laborious. The effect of 2.5. Calculation of an Example The values used in the following calculation "may be taken to represent some average values typical to the 2 K 25 reflex klystron. Some of these values has been experimentally measured. (See the experimental part.) The values assumed are: oj0 = 2ttX 8,9 X 109 C = 1 pP Q == 250 Q0 = 700 0 = 220 ftv GR = 80 Ge = 440 p The assumed values of G and Ge give a conductance parameter (/ = 2<<7op = 2,31 in order to obtain such curves as those shown in Fig. 2.4a. The first family of curves Fig. 2.5 has been calculated from (24b)—-(26b). The second family Fig. 2.6 has been calculated from (24c)—(26c) for ^0 = 30. Xi'0.5 x2-1 x2*1,5 x2=1,9S 1,5 6 a) b) Fig. 2.5. Synchronisation of a reflex-oscillator by an external signal of nor¬ malised amplitude X2. a) relative phase, b) amplitude X and power output (proportional to 2XJ1(X)) as functions of the detuning S, with relative reflection angle adjusted to zero. 43 T 1.0 o 1.0 eo Fig. 2.6. Same as Fig. 2.5 with relative reflection angle Chapter 3. Mutual Synchronisation of two Klystrons 3.1. The Equivalent Circuit Fig. 3.1 shows a circuit arrangement which enables the experi¬ mental investigation of the mutual synchronisation of two klys¬ trons. The directional couplers are supposed to couple a small ^_^ ^^ War. rnase var. LalD. Kt 1 3 1 3 (~) Direct. Shifter Atten. Direct. -fc)«<, Coupler Coupler 2, 4, Receiver Det. Det. Receiver •Jr / To indicating Instruments Fig. 3.1. Circuit arrangement for the experimental investigation of mutual synchronisation of 2 klystron-oscillators. 44 portion of the power flowing in the main line 121' 2' to be used for measuring the frequency and the power output as well as to indicate whether synchronisation takes place or not. The coupling bracnh 21' contains a variable calibrated attenuator and a phase shifter and thus enables to adjust the magnitude and phase of the coupling between the two klystrons. The power output from one klystron is partly absorbed in the attenuator; the rest is coupled into the other klystron to affect synchronisation. The circuit arrangement does not provide for obtaining useful power output to be supplied to some external load. This is intentionally done so as to extend our investigation to include any degree of coupling between the two klystrons. A schematic diagram of a possible arrangement where useful power output can be supplied to some external load is shown in Fig. 3.2. Here the directional coupler provides a small fixed Useful output from Kj — 1 3 Direct. H0K> Var. Phase Coupler c,0- 2 4 —"• Useful output Shifter from Kt Fig. 3.2. Simple circuit to operate two klystron-oscillators in synchronism; useful power out is available for external loads. coupling between the lines 12 and 34, so that the greatest part of the power output may be obtained from arms 1 and 4. The variable phase shifter serves to adjust the phase of the coupling to its optimum value. In either of these circuits the coupling branch may be represented by a symmetrical 2-terminal-pair network having the following characteristics: 1. Its characteristic admittance is equal :to that of the line con¬ necting the klystrons. 2. It produces an attenuation equal to that of the attenuator (Fig. 3.1) or of the directional coupler (Fig. 3.2). 3. It produces a phase shift equal to that produced by the phase shifter and the line length between some arbitrarily chosen reference planes. 45 This 2-terminal-pair network is either connected in series between the klystron outputs to represent the circuit in Fig. 3.1, or in par¬ allel between the 2 output lines to represent the circuit in Fig. 3.2; this is shown in Fig. 3.3. Without making any reference to the © 14 < i Hi . , , i2_Jk ° Coupling Coupling ~^[^) o— Nefworh —g—\_y Nehvork 2i & Fig. 3.3. Circuits of Figs. 3.1 and 3.2 with coupling branch replaced by a two-terminal-pair network. circuit elements to be contained in this 2-terminal-pair box or to their arrangement which may fulfil the prescribed requirements, we may define the electrical properties of the network completely by any of the matrices used in the 2-terminal-pair network theory. I, r o *— -30,1 ' V2 v'L 6 Fig. 3.4. Thus with the directions of currents and voltages chosen in Fig. 3.4 we define an ||a||-matrix by the equations vi = anV2 + a12I2 (1) so that "11 *12 \CL\\ = (2) For a symmetrical 2-terminal-pair network with characteristic and impedance g0 propagation constant y the a-matrix is given by 46 cosh y — £0 sinh y ft = (3) — sinhy — coshy ' ' So Thus if j0 is taken equal to the characteristic impedance of the output line from the klystron, we have only to define y in terms of the attenuation and phase shift existing in the line between the chosen reference planes. Let the reading of the attenuator (usually in decibels) correspond to a power ratio a2 (ratio of input to output power), and let the equivalent line length between the 2 reference planes (including the phase shifter) be ifs electrical degrees. Thus closing the line at one end by a reflectionless termination and supplying power from the other end we get: ' in ' out ho ^ -* in out P = \V- in and P \V T \ = * out I ' out *- out I IF- /• I IF- I2 ' r \ iri m I in I so that P W T \ W I2: x out I r out * out\ I ' out I with j0 taken to be real. As the phase of the output voltage is retarded tp° behind that of the input, we have £*- = «e->* (4) ' out Now closing terminals 2 — 2 of Fig. 3.4 by j0, we get from (1) and (3) = fr C • (5) ^ out Eqs. (4) and (5) define y of the equivalent 2-terminal-pair network in terms of the attenuation taken as a power ratio and the equi¬ valent line length tfi by the relation = ey a.e-H . (6) It is to be noted that due to the symmetry of the 2-terminal- pair network together with the symmetry of the chosen directions 47 of the terminal voltages and currents, the a-matrix defined by (3) is directly applicable to the network with either terminal pair 2 of taken as the input terminals. Thus the pairs equations Fj = V2 cosh y — 72 g0 sinhy (7) h So = V2 sinh y-lzho cosh Y (7a) and F2 = V1 cosh y — Ix j0 sinh y (8) = h i o Visinh Y-hbo sinh7 (8a) from the other. are both valid; the one pair being directly derivable 4 describe Therefore we may choose any 2 of the above equation to the relations between the terminal voltages and currents. For our problem the currents are functions of the voltages, so that the most convenient pair are (7) and (8) as they give 2 similar expres¬ sions containing the 2 unknowns V1 and F2. The reference planes are chosen such that the distance between mul¬ one plane and the gap of the adjecent klystron is an integral the transfor¬ tiple of half a guide wavelength. This choice reduces mation of voltages and currents from any reference plane to the a real adjecent gap and vice versa to a simple multiplication by number representing the transformation ratio of the line coupling the klystron cavity to the waveguide. Ci 4 Q, Fig. 3.5. Simplified equivalent circuit for two reflex-oscillators coupled together by a two-terminal-pair network. (Coupling between each cavity and the guide is assumed lossless.) According to the above considerations our equivalent circuit is as shown in Fig. 3.5. Here we have a system containing 2 resonant circuits representing the cavities of the 2 klystrons; the resonant circuits are coupled by a 2-terminal-pair network of known charac¬ teristics. The system is driven simultaneously at both ends by the 48 bunched beam currents M1^1 and M2$2 which are given by the process of velocity modulation and bunching as nonlinear functions of the and gap voltages SSX 23 2- 3.2. Steady State Equations under Mutual Synchronisation The resonant circuits shown in Fig. 3.5 have large Q-factors (also when loaded by the attenuator) so that in the steady state the terminal voltages 23x and 932 are very nearly sinusoidal and the effect of all higher harmonics may be neglected. It has been shown in the preceding chapter that for such a system the relations obtained by integrating the energy equation and its first derivative over one period can be directly obtained by simple circuit theories under the restriction that the law of superposition does not hold. If such an analysis is applied the driving currents are replaced by their funda¬ mental components obtained by representing the former currents by a Fourier's series for the fundamental frequency of oscillation a> and its higher harmonics. The fundamental component of the bunched beam current is itself a nonlinear function of the corres¬ ponding gap voltage and is given — as was shown in the last chap¬ ter — by MQ1 = -jM1Im2J1(X)er'*. (9) The different terms in (9) have exactly the same meanings as defined in the preceding chapter. We substitute again for the reflec¬ tion transit angle <9X the value 01=2ir(n + i)+ »! = V1 (real), Eq. (9) gives (H) M1Ql = M1I01-2J1(X)e-^^. Referring all phases to V1 we may write 232 = F2e^ (12) Jf,3, = lfsJM-2J1(Z)e-' With reference to Fig. 3.5 and using Eqs. (3), (7) and (8) the steady state values of the voltages 931 and 932 are given by 49 »! = 85,00^7-(8,-«,D,)i0sinhy »! = *! cosh y - (St - »! $1) So sinh y Where ^ and ^)2 are the admittance of the shunt (C, L, 0) at the working frequency. If the cavities are tuned to the resonant fre¬ quencies = and = <**x TTc a'«a L~c~ (14) and if the working frequency is co, the admittances ^ and |)2 are given by Si^i + Jfoo + Wi. » = lor2, (15) with B^QJ^L-^). (16) QLi is the loaded-# of the cavity and is given by «*-£& <"> i/0 = — is the characteristic admittance of the line. So The bunching parameters are given by M-V-&- Y l I l ""* 2 V so that -ta v1 )1 2 ^2 -k^ with k = uyJ (18) X,' and the conductance parameter is 9t = with Gei = (19) 2^oi Using (6) we get coshy 1 \ a and (20) = -iA sinhy -(ae^- -e- 2 \ a 50 Using (11)—(20) we get finally 1 2kX1e~*P 1/ ., ,,\ and (21) e+# 1 21, e+JP / 1 2 ., .A &X a i yi 2/i\ / ,1.4l(Z1).e-M-^- + ,-81)}(«^-i^), 2J1(Xi) with A1 (Xf) (22) and «,• 2/o In terms of 82 and 82 given by (15) we define the relative detuning of the klystron cavities by «i.= /^i;(^-^r)- (23) For a narrow band of frequencies around one of the oj0's we may use the approximation 3. = l^(^_Wo.) (24) which gives 82 = q281-q§2 (25) with a Gi. q Qu We keep co01 constant and choose co02 as the independent variable. The complex Eqs. (15) contain four unknowns: the amplitudes of oscillations Xt and X2, their relative phase angle j8 and the fre¬ quency of oscillation 81; with the two parameters: the voltage attenuation ratio a and the line length i/j. The relative reflection transit angles 51 meters, which depend on the initial adjustment of the reflector- voltages. In addition all the three angles 3.3. Synchronisation of two Identical Klystrons As was indicated in the preceding chapter, the main effect of adjusting the relative reflection phase angle to destroy their symmetry about the 8 = 0 axis. It was also indi¬ cated that for q> =t=0 synchronisation takes place over a band of fre¬ quencies, the width of which is effectively the same as with (21) shows that the effect of 93 4=0 in the case of mutual synchroni¬ sation is approximately the same as in synchronisation by an "independent" external signal. The present discussion may thus be limited to the case where the reflector voltages are both preadjusted to give 9>1 = oj2 = 0; the preadjustment being carried out with the klystrons connected to matched loads. Further we may neglect the dependence of 52 the coupling branch may be assumed constant over the frequency band of interest; an assumption which holds good, if the line length lying between the two cavities is not excessively long. The long-line effect will be considered later on in this chapter. Under these assumptions (21) yields 2X (26) $T-iH+H-{h(i->-is)->MKH where, for two identical klystrons k = i, yx = y% = y, G1 = G2 = G, q = \ so that from (25) §2 = 3i-812 (27) Equations (26) are still complicated due to the presence of the coupling phase angle. Therefore, we may begin our discussion by studying the effect of tfi on the behaviour of the system 3.3a. Effect of the Coupling Phase Angle Using the abbreviation g' = and separating real and imagi¬ nary parts, Eqs. (26) yield =r^cos/3 = -cosift — b(g J1(X2)— g') costfj — a sin if/S2 V ^2 y 1 2 X a =r-^cos/S = -cos^ — b(gAl (XJ—g') cos t// — a sin tfi 8X y ^i y (28) =r^sin,6 = - simfj — a (gA1(X2) — g') smtp + b cos^S2 y ^2 y —=r-isin/J = - sin ^4 — a(gA1(X1) — g')sraifi + bcosifih1 y ^2 y — with a = 53 Eliminating 8X between equations (28) we get \ (J1- ~x) °OS^ = gb cos^^i(Z^ - Ji (Xa)} +a sin,/.812 (30) -l^ + ^Jsin/S = -£asin«/,{J1(X1)-J2(X2)} + &cos As we are dealing with two identical klystrons, we may expect some identity in their behaviour for different values of the indepen¬ dent variable 8la. A closer study of (28) and (30) shows that, if for some value of 812 = 8[2 the four unknowns, as determined from these equations, have the set of values: Z1 = X', X2 = X", S1 = 8', 32 = 8", /? = /?' then a second set of values, possessing some sort of "reflection symmetry" with respect to the first set, namely the set X2 = X", X2 = X', 81 = 8", 82 = 8', £ = —/?' will satisfy our equations for 812 = — Sj2. Stated in other words this behaviour can be described as follows: Taking the resonant fre¬ of 1 quency cavity of klystron as reference we adjust w02 of klystron 2 to two values each on successively either side of cu01 but equally i.e. = + = displaced from it, oj02 a>01 Aw0. With a>02 w01 + Aw0 the system will oscillate at the frequency oj = oj01 + Aui, whereas if 54 that the frequency of oscillation of the system when both cavities are tuned to the same resonant frequency i. e. when S12 = 0, is not equal to that resonant frequency but deviates from it by an amount which depends on the magnitude and sign of tp. Further, although the power output from each klystron is not a symmetrical function of 812, the sum of the 2 powers is symmetrical. However, remem¬ bering that the frequency of oscillation is not a symmetrical func¬ tion of 812, the total power output of the system is also not a sym¬ metrical function of the oscillation frequency. Summarising, we may write -^l(812) = ^2(~812) »i (8U) = 82 (-M (31) P (8M) =-j8(-8u) which shows that the function X1 is simply the reflection of the function Xa about the S12-axis. The same holds for 8X, and 82; but j3 has a skew symmetry. Now let us find the value of if/ by which the following will be satisfied = = x %i (8i2) ^2 (812) (812) . 2. 81 (812) = - 82 (812) = 8 (812) i. e. the amplitudes of oscillations are always equal for any value of 812 and the frequency of oscillation lies always middle way between the resonant frequencies of the two cavities. Since Eqs. (31) are valid for any value of if/, thus combining (31) and (32) gives X(S12) = X(-8lt) 8 (812) = -S(-S12) which means that, with the auxiliary condition (32) satisfied, the behaviours of the 2 klystrons are exactly identical, each being in addition symmetrical about the S12-axis and consequently a sym¬ metrical behaviour of the system as a whole may be obtained. Substituting the first of conditions (32) in the first of Eqs. (30), the condition for symmetrical operation is given by a-sin^-812= 0. - (33) 55 If this is to be satisfied for all a and S12, we get = nrr with n = 2 xfi 0, 1, , or the equivalent line length between the chosen reference planes should be an integral multiple of half a guide-wavelength. As the reference planes were chosen at distances equal to —^ from the respective control grids, the line length between the 2 grids should thus be an integral multiple of ~. Putting ^r = 0 in (28) and remembering (32) we get the two simple expressions gAx(X) = i + -^J-^(i- a cos^ and (34) The first gives the amplitude of oscillation and the second the fre¬ quency of oscillation and the relative phase angle as a function of the relative detuning. As described by (34) the behaviour of the system is quite clear and simple. Consider first the limits of the frequency band over which syn¬ chronisation can take place. This is directly given by putting j8= ±90° 2oc l812lm /ok\ _ ~^--y(«>-l)> (35) giving an infinite value of |812|m at ¥"-4' (36) 18 1' 1 i.e. LMm varies nearly linearly with - for «> 1. Consider next the amplitude parameter at the middle of the synchronisation region i. e. at S12=j8 = 0. This is given as a function of the coupling by 56 The = undisturbed amplitude is given by putting a oo (zero coupling) g-A^XJ-1. (38) It is to be noted that for (0^X^3,83) the function A1(X) is (1^A1(X)^0). Thus, as the coupling is increased (a decreases), AX(X) decreases and the amplitude of oscillation increases. If Xu < 2,4, the power output increases by increasing the coupling, reaching a maximum at X0 = 2,4 and then decreases to zero by X0 = 3,83 or A1(X0) = 0. Thus, the maximum coupling with which the system can still oscillate at 812 = 0 is given from (37) by 2 a, = 1 for y < 2 and by V (37 a) a, = 1 for y > 2 For values of gAl(XJ = l + ^~-T) (39) From we notice that is smaller than both and X.t (39) Xm X0 M> whereas X0 is always bigger than Xu. Thus if Xu < 2,4, the power output v s detuning will possess 2 maxima unless X0 < 2,4. Another important fact is included in Eq. (39). We notice that as 1 + -A-K (39a) i 2/(^-1) 57 X=S,8S Fig. 3.6. Amplitude and power output as functions of the detuning with a as parameter and 0 = 0, in a system of 2 identical reflex-oscillators. Fig. 3.7. Maximum magnitude of 812, de¬ fining the boundaries of the synchronisa¬ tion region, as function of — with At values of 58 where Xu 1. For large values of a the system oscillates in synchronism up to the boundaries |S12|m given by (35). Beyond these boundaries synchronisation stops but oscillation continues at more than one frequency. 2. For a = ac, oscillation can only occur at one frequency (syn¬ chronisation). Just at the boundaries oscillation stops. 3. For values of a smaller than both a, and 3.36. Case of Small Coupling In a system of 2 reflex-oscillators, where mutual synchronisation is affected by introducing a small coupling between them, a great percentage of the power output from each will be available for use in an external load. This case is, therefore, of special interest for practical applications and will be discussed in the present section in more detail. The simplifications introduced by assuming we may neglect —er^ with respect to ae^ in (26). This yields the following four expressions A|iCos(«A + /3) = l-!7J1(Z2), ccy X2 _l|«oo8(^-j3)=I-!741(X1), «y Zl (40) yisin(>£ + jS)=82, ay X2 X, ^sin(«/,-J8)=81. 59 Since the percentage change in the amplitudes over the frequency band of interest is very small, we may express Xx and X2 in terms of their value at the middle of the band by the approximations X^Xoil + x), X2 = X0(l + x). (41) For the functions A1(X1) and A2(X2) we use the Taylor expansion up to the first order; thus A1(X) = A1(X0) + (X-X0)A1'(X0), which yields A1(X)=A1(X0){l-xK(X0)}, where (^) JT /y \ 2 (Xq) *(*o) = A(x0y Substitution in the first equation of (40) we get -^-(l+x1-x2)cosy + p) = {l-gA1(X0)} + x2gJ2(X0). (43) Remembering that at B = 0, XX = X2 = X and x1 = x2 = 0, the value of X0 is given by the expression ACoS^ = l_<7Jl(X0) (44) Since by assumption 2 — (cos (iff+B) - cos iff) = g x2 J2 (X0), 2 — (cos (if,-B) - cos^r) = gx1J2(X0), (45) 2 2 — = sin (iff B) 8, . ay 60 Eliminating §x between the last pair we get 4 — sin /S cos i/j = S12. (46) For any value of ifi the maximum values of 813 that define the boundaries of the synchronisation region is obtained from (46) by putting j8 = +90, giving 8l2im = — |0OSi/r|. (47) a y Eq. (47) shows an important feature in the mutual synchronisation of two identical oscillators. By varying the phase of the coupling, the width of the band over which synchronisation is possible, varies between zero and a maximum = the maximum takes ocy at = mr n = place -90 *90° ft -90° *90' ft Fig. 3.8. Relative amplitudes developed by either reflex oscillator as func¬ tions of the relative phase angle /! for the case a5>l, (a) #=0, (b) ^ = 0. 61 for ^ by reflecting the former on the vertical axis. Also the maxima Xq of the curves are displaced to the right or to the left from the vertical axis instead of lying directly upon it as in the case where i^ = 0. The corresponding curves with $ = 0 are shown in Fig. 3.8b. ^ ^ Here both and coincide. In Fig. 3.9a and 3.9b the 8's are X, X, shown. Fig. 3.9. The different S's as functions of p for the case a* 1, (a) ^#=0, (b) * = 0. For the case of a small coupling but non-identical oscillators, the following relations can be derived in a manner similar to that used in deriving Eqs. (45): (cos]8-l) = x2-g2J2(X0 *«/2 ^02 Xn —- ^ (cos p - 1) = xx -9l J2 (X01) «2/i kX01 (48) sin/3 Si. *2/i &^oi 2 JfeX, X, fj_ 1 02 91+ • X, sin/3 Jia Lo2 Vx "'-^oi; In (48) ifi has been put equal to zero. No further discussion of (48) is to made. It is going only intended to use them in deriving a simple equation which enables us to compare quantitatively the with theory the results of the experiment. Since we are going to use similar tubes we may put k = 1, for it depends only on the beam voltages, the beam coupling coefficients and the d. c. transit angles, 62 which may be expected to be the nearly equal. Further, if we choose the case of very small coupling where changes in the amplitudes may be neglected, we may define X01 and X02 by the equations ?1J1(X01)=g2J1(Z02)=l, (49) which hold for relating |Sia|m to — will be given by an expression which contains values that can be easily measured. 3.4. The Long Line Effects If the equivalent length of the line lying between the two planes containing the klystron grids is excessive the phase angle ip cannot be assumed constant and its dependence on the frequency should be taken into consideration. Due to the implicit form of the general equations (26) we may limit our present consideration to the case of small coupling described by the simpler expressions (45). l /\ \ f \ \ Mm.// / \ \\l6'2' m- ' V / \ \ \ 1 1 / \ \ \ 1 / \ x \ 1 1 / \ * \ 11/ \\ \ 1 '/ \\\ *9~ Physical •> line-length Fig. 3.10. Detunings at the boundaries of synchronisation region |Slajm as functions of physical line length; showing the long line effect. Curve denoted |812|Bl+ occurs when frequency co01. 63 From Eq. (47) it is obvious that |§12|m reaches its maximum if the physical line length makes tfi = 0 or n at the corresponding frequency of oscillations. Since the 2 frequencies occuring at the boundaries of the synchronisation region are different, different physical lengths of the coupling line is necessary to obtain the maximum values of |S12|m. Thus if we try to measure experimentally |812|m as, a function of the physical line length, we get such curves as those shown in Pig. 3.10. Another long-line effect will be noticed, if the physical length of the coupling line is adjusted to give if/ = 0 at the center of the syn¬ chronisation region i.e. at a> = w01. At the boundaries of the syn¬ chronisation region (^8 = 90°) will be different from zero, resulting in a corresponding reduction in the limiting value of S12 as is directly seen from (46). If the value of ip at the boundaries is denoted by tfim, where i/jm is a small angle, the reduction factor is 1 — this cosi/im= -~~.... Thus, long-line effect will only be noticed for an extensively long-line, where Chapter 4. Synchronous Parallel Operation of Reflex Klystrons 4.1. General Requirements A combining network which allows two klystron oscillators to be operated simultaneously into a common load must fulfil the requirements that: 1. The two klystrons remain locked-in to the same frequency, in spite of a possible error in tuning them and in spite of the random variation of the oscillation frequency of each klystron. 2. The total power supplied to the common load is optimum. 3. Simultaneous modulation of one of the oscillators' parameters (e. g. of the reflector voltages) does not throw the system out of synchronism, provided that the frequency of the modulating signal 64 is small compared with the synchronisation time-constant. In other, words simultaneous modulation of both reflector voltages should result in a single instantaneous frequency of oscillation -of the system as a whole. In addition the resulting frequency deviation should be proportional to the amplitude of the a. f. signal. Both conditions should be fulfilled under the assumption that the static characteristics of the oscillators relevant to modulation are not exactly identical. Thus, the combining network, should allow a certain fraction of the power output from either klystron to couple into the other, in order that mutual synchronisation may take place. Further, the phase of this coupling should correspond to optimum width of the band over which synchronisation is possible. Under such circums^ tances both the first and third requirements mentioned above are fulfilled. Concerning the second requirement some features of the available tubes should be mentioned. Attention is concentrated on the 723 A/B family of reflex klystron, for all the experiments carried out in the present work were performed on the 2 K 25 tube. These tubes have integral resonators with built in output circuits that consist essentially of a coupling device and an output transmission line. The coupling device is an inductive pickup loop formed on the end of a coaxial line and inserted in a region of the resonator where the magnetic field is high. The output lines are small coaxial lines provided with beads, which are also vacuum seals, and carry an antenna that feeds a waveguide. The design of the coupling loop arid output line together with the position of the antenna in the waveguide are so chosen that the tube may be correctly loaded by a matched guide. Yet most of the tubes of the above family require individually adjusted transformers .of some sort dn order to deliver full power to a resistive load that is matched to the "waveguide. This is because of the relatively 'big tolerences, which must be- allowed in manufacturing the small coupling loops and the coa,xiaJ. line bead seals. However, the fact that the tube used will deliver its optimum power to a load which is not matched to the wave¬ guide, may be used with advantage in the case of synchronous parallel operation of two klystron oscillators. This may be explained 65 as follows. Consider the combining network with the two klystrons connected to two different arms. Suppose that, if one of the klys¬ trons is replaced by a reflectionless termination, the admittance seen by the other klystron is a perfect match i. e. no reflected wave exists in the guide connecting this klystron to the network. Now, with both klystrons connected to their respective arms and due to the intentionally introduced coupling between them, a wave travelling towards either of them will exist in the guide connecting it to the network. In a steady state of mutual synchronisation such a wave has a frequency which is exactly equal to the frequency of oscillation of the klystron. For the klystron itself it does not matter at all whether such a "reflected" wave does originate from a non- matched line or from another oscillator connected elsewhere in the network. The only thing that really matters is the fact that there exists a standing wave in the connecting guide. Now, if the "standing wave ratio" is adjusted to the correct value optimum power will be supplied from the klystron to the network. In this event the synchronising signal coupled from either klystron to the other serves to affect synchronisation as well as to optimise the power output. The required "SWR" may be achieved by proper choice of the strength of the coupling introduced between the two oscillators. Here the question arises whether the magnitude of the coupling required for optimum power output will be associated with the required width of the frequency band over which synchro¬ nisation is possible. Fortunately, by the 2K25 tubes used in the coarse of our experiments optimum power was obtained by a coupling whose magnitude is about 10 db. The bandwidth associ¬ ated with this .coupling is about 10Mc/s°or more; a width which may be considered adequate for most applications. For the experimental investigation of the synchronous parallel operation the most suitable network element which enables an easy fulfillment of the above requirements is the magic-T. If the two oscillators are connected to two adjacent arms of the magic-T with the two remaining arms connected to reflectionless terminations, the two oscillators are completely decoupled from one another. The required magnitude and phase of the coupling may then be adjusted at will by using, for example, reactive screws to introduce reflection. ' r - 66 Before going into the details of the combining circuit, the magic-T used in this work may be described and its scattering matrix may be derived. This is done in the next section. 4.2. Magic-T; Scattering Matrix The magic-T used here consists of two parallel waveguides having a common wall in the broad side. The common wall contains a number of small coupling slots, which are dimensioned and arranged in such a manner that power incident on the junction from one arm splits equally between the two opposite arms with no power coupled into the adjacent arm. Also, no power will be reflected back to the source if the two opposite arms are connected to matched termina¬ tions. The measured characteristics of these magic-T's gave the following results: 1. Voltage standing-wave ratio (VSWR) in the feed arm <1,1. 2. Power outputs from the two opposite arms are equal to within 1 db or less. 3. Isolation between the two adjecent arms is better than 40 db. (The values given up represent the limits up to which our measuring devices could give reliable indications rather than the actual values of matching, balance and isolation.) In our subsequent calculations we may, therefore, suppose that the magic-T used is perfectly matched with equal power division between the two opposite arms and perfect isolation from the adjacent arm. The scattering matrix is a useful tool which is often used in cal¬ culating microwave circuits containing waveguide junctions with several arms. This matrix is a simple extension of the wave forma- D 1 |E0, Eos ! U— —-i 1 »j E04 ' 1 D Fig. 4.1. 67 lism generally employed in transmission line theory. Consider for example the four terminal pair junction shown in Fig. 4.1. Let Ein be a complex number representing the amplitude and phase of the transverse electric field of the incident wave at the n-th ter¬ minal pair. Let Eon be the corresponding measure of the emergent wave. It is assumed that Ein is normalised in such a way that \EinE*in is the average incident power, and correspondingly for E^. (E*n is the complex conjugate of Ein.) It is obvious that the amplitude of any emergent wave Eon may be related to the ampli¬ tudes of all incident waves by a simple linear combination of these amplitudes, each being multiplied by an appropriate proportionality (complex) constant. For example Eon =snlEil + Sn2^o2+ +SnnEon+ (l) The meaning of the coefficients snk is rather important. Consider the case where power is supplied to the junction from the n-th terminal pair with all other terminal pairs connected to matched loads. In this case all Enk — 0 for all k #=n, and (1) reduces to E =s E- Thus, although all other arms are perfectly terminated, there may exists an emergent wave in the feed arm, which is reflected by the discontinuities at the junction. Thus, snn describes a property of the junction itself independent of the manner in which its terminals are terminated. If snn — 0 the junction is said to be matched looking into the n-th. arm, meaning that all power incident on the junction from this arm will couple into the other arms, no direct reflection at the junction back into the n-th arm takes place. If all snn = 0 the junction is then matched looking into all arms. Again, for the case considered — power fed through the n-th arm only — the wave coupled into the k-th arm is given by E0k ~ snk Ein Thus, while snn represents a reflection coefficient, snk indicates the coupling coefficient between the n-th and k-th arm. Further it is obvious that, due to the reciprocity theorem, snk ~ skn 68 These coefficients may be written in the form of a matrix that is known as the scattering matrix of the junction. Thus for the four- terminal pair junction shown is Fig. 1, the scattering matrix has the form *11 S1Z S13 *H S12 522 S23 S24 l-SII = (2) 513 S23 S33 *34 su 524 S34 S44 Further, it may be shown that in addition to its symmetry the scattering matrix is also unitary i.e. < Zj \snk\ ~ 1 anC* Zj snk8mk ~ 0 (3) Now, in the magic-T described at the beginning of this section, the junction is matched looking into all arms i. e. = " snn > the adjacent arms are decoupled i. e. = = " S12 534 ' thus the scattering matrix reduces to 0 0 S13 su 0 0 *23 S2i 1-81 (4) S13 S23 0 0 «14 S24 0 0 Remembering that power fed from one arm divides equally between the two opposite arms, the magnitudes of the four unknowns con¬ tained in (4) are equal. Using the first of (3) we get U 1 — 4- I2 — * + + *24 Since all magnitudes are equal, we have 1 ~ — — (5) 312 I '14 I \S?a\ lS24l 71 and 11$|| may be written 69 0 0 c f 1 0 0 9 h 151 (6) a c 9 0 0 / 9 h h with the new complex unknowns having all unity magnitudes. The determination of these unknowns will obviously give the phases of the coupling coefficients. This can be done by considering the sym¬ metry of the magic-T. Choosing the reference planes at CC and DD (see Pig. 4.1), the junction has a reflection symmetry about the plane AA. This reflection symmetry may be described in the form of the matrix (see [10], chap. 12). 0 1 0 0 1 0 0 0 \F\\ = (7) 0 0 0 1 0 0 1 0 which commutes with the scattering matrix \\8\\ so that ||£|j.||.F|j = plJ-H^II. (8) Performing the multiplication, (8) gives c — = h and / g . (9) Using the second of (3) and (9) gives hg* + gh* = 0. (10) Remembering that \g\ = \h\ = 1, we may write g = ei@l, h = ei0'i. (11) Substitute (11) in (10) gives i (Si-02) e + e->«9i-<92) = 2 cos (©i-©2) = 0 (12) or g and h are in quadrature. Putting @1-&2=^ and ®2 = * we get = = ioiQ h = g je eie, (13) 70 and the scattering matrix assumes the simple form 0 0 1 i e;0 0 0 i i 1511 = (14) 12 1 j 0 0 1 1 0 0 The incident and emergent waves shown in Pig. 4.1 are now related by the expressions: aE01 = Ei3 + jEti a^oi = i®iz+ En (15) aE03 = Etl +jEi2 aEn = jEtl + Ei2 with the abbreviation = ]/2 e-'e (16) It is obvious that with the proper choice of the reference planes the phase angle @ may be made zero with a subsequent simplification of the expressions (15). However, if for any practical consideration the reference planes are preferably chosen to coincide with the physical terminal planes of the magic-T, the angle © may be measured experimentally as follows: If arms 2 and 4 are connected to reflectionless termination and arm 3 is short circuited while power is supplied through arm 1, we have E* Eu = 0 and Ei3 En so that (15) gives g-j'2® ej(«r-»8> 01 (17) E< Using a standing wave indicator to measure the distance of the first standing wave minimum from the terminal plane, this distance expressed in electrical degrees will give directly (n — 2®), as indi¬ cated by (17). A special case of loading the magic-T is going to be often met in the subsequent sections and may, therefore, be derived here. Let a 71 matched generator be connected to arm 1 with arm 2 perfectly terminated and arms 3 and 4 connected to loads producing at the respective reference planes the reflection coefficients rz and /\. 'Jhis gives ^£3 ^u _ ^2 = 0, r, = -=**, pA = -=i* (is) ^03 "^04 Substitution in (15) yields Em 1 01 = f.(A--T a (19) E02 02 i _ = E Mrs+ri) H2 Placing the generator on arm 2 with arm 1 perfectly terminated and arms 3 and 4 loaded as before, similar expressions are obtained: 4> Ei2~ a2( 3 (20) | = -^(A+r4) — We notice that with rs ri = r the generator sees a match in either of the above cases. The relative voltage coupled into the arm adjacent to that containing the generator is then given by ^f (21) which is the coupling coefficient between the two adjacent arms 1 and 2 produced by two identical loads on arms 3 and 4. 4.3. Alternative Combining Networks for Synchronous Parallel Operation In this section some combining networks which allow the synchondtfs parallel -operation of 2 or 2n klystrons, are suggested and their main features calculated. These networks represent by no meafis all the possible circuit arrangements, which may be used 72 points of view which should be taken into consideration to attain the proper behaviour of the system. In the following discussion it is assumed that all generators are matched to the line, so that any signal incident on the generator is absorbed there. Each generator is mounted on a length of wave¬ guide, whose terminal plane is assumed to be at a distance 4.3.1. Parallel Operation with an External Synchronising Signal Here, a simple combining network is described which enables 2 or 2n klystrons to operate in parallel and supply their output power to a common load. No direct coupling exists between the different generators and synchronisation as well as the required phase relationships are affected by an external signal supplied from a matched generator that is assumed to be completely uninfluenced by any signal coupled into it from the system. bignai 1 3 Source MT 2 4 Matched Load Fig. 4.2. Parallel operation with an external synchronising signal. If Kx and if2 are identical no power couples from them to the signal source; all is to the load. ! power output supplied The circuit suggested is shown in Fig. 2. The four terminal pair network denoted MT is the magic-T. The planes denoted by 1, 2, 3 and 4 are the terminals of the MT and are taken to be our reference planes as mentioned above. K1 and K% are the two klystron oscil- latprs to be synchronised. 73 Consider first the case where Kx and K2 are replaced by matched loads and power is supplied from the external source through arm 3. Let the incident wave at plane 3 be denoted by Es — real i. e. we refer all phases to the phase of Es at plane 3. Using (15) the emergent waves at planes 1 and 2 are given respectively by E01 = ~ES and EM = ?-E8 (22) which are in quadrature and of equal magnitudes. Taking plane 1' on line 1 and 2' on line 2 such that the distances ll' = 22' = emergent waves at 1' and 2' lag by an angle EW = E,— and EW = E„!—— (23) a n Under the assumption that the external source is uninfluenced by any signal coupled to it from the system the emergent waves Eor and E02. will hold their phase and magnitude if Kx and K2 are reconnected to arms 1 and 2. Remembering that synchronisation by an external signal, which is in tune with the resonant frequency of the oscillator cavity, results in phase coincidence between exter¬ nal signal and self-exciting signal, the klystrons will deliver 2 waves given at planes 1 and 2 by Eix = £e-'(»-8) and E = jEe-J With E real; these waves are of equal amplitudes and have the phases of Eor and E02,, given by (23). At planes 1 and 2 these become Eil^Ee~J^v-&) and Ei2 = jEeri<* Now considering that both arms 3 and 4 are perfectly terminated and the two waves given by (25) are incident on planes 1 and 2, substitution in (15) gives #03 = 0 and EQi = ji~2Ee-J2 i. e. the whole power comes out of arm 4, the 2 waves coupled in arm 3 cancel. It is now obvious that in the system shown in Fig. 2 74 an external signal fed in arm 3 divides equally between arms 1 and 2 and compels the oscillators connected there to adjust their phases to be in quadrature so that the total power output from Kx and K2 comes out of arm 4. The common load can thus be connected to arm 4. From the above results it can be easily shown that, if the line connecting terminal 1 to Kt is longer by a quarter guide wavelength than that connecting terminal 2 to K2, the total power output comes out of arm 3. This fact can be used to design a simple net¬ work for the parallel operation of four klystrons. This network is shown in Fig. 4.3. The signal source is shown connected to arm 3 of M and are to the left a distance from the Tz. Kx Kt displaced -j- plane containing K2 and Kz, so that power output from K1 and K2 comes out of arm 4 of M Tl and that from K3 and Kt comes out of _ Matched K,Q J Termination MT, Signal K2QM2 Source MT3 Load ML Matched Termination Fig. 4.3. Parallel operation of 4 reflex-oscillators synchronised by an exter¬ nal signal. If all oscillators are identical all power output is supplied to the load; no power couples into signal source. arm 3 of M T2. These output waves are in quadrature, with the latter wave leading the former one, so that the total power output comes out of arm 4 of M T3 to the load. No power couples from the system into the signal source. The main features of this network are: 1. that the total power output is directly available from one arm without the necessity of using any phase shifter and 2. that no power originating from the synchronised oscillators couples into the arm containing the signal source. This latter feature is quite important, if the signal source is 76 a harmonic generator using some crystal rectifier as a multiplier element. In such a case the crystal rectifier is generally fully loaded from the source of the fundamental wave so as to obtain high har¬ monic power and thus no additional loading can be allowed fpr. 4.3.2. Symmetrical Combining Network; Coupling through Reflection This combining network is especially convenient for laboratory work to investigate the parallel operation of two klystron oscillators wh'ch mutually synchronise. As shown in Fig. 4.4 coupling between Movable Sifrew Phase 4 Matched Shifter Tunerss .miner t=p j Termination MT, MT2 K2gh Load Fig. 4.4. Symmetrical combining network appropriate for the experimental investigation of synchronous parallel operation of 2 reflex-oscillators. Kx and K2 is achieved by two movable screw-tuners connected to arms 3 and 4 of M Tx. The phase shifter is used to adjust the phase of the incident wave in line 3—V to a value such that the two waves incident on terminals 1 and 2 of M T2 are exactly in quadra¬ ture. If these two waves are of equal amplitudes all the power will couple either in 3 or 4 of M T2, where the common load may be connected. Thus the part of the network to right of the plane AA serves only to convey the total power output to the common load. To simplify the following discussion we may suppose that looking into either terminal pair to the right of the plane AA the admittance seen is a match. With this assumption the reflection coefficients at terminals 3 and 4 of M Tx are solely determined by the movable screw-tuners. The magnitude of the reflection from the screw depends only on the insertion and does not vary with changes in the position of the tuner. Thus by the help of the movable screw- 76 tuners used the required magnitude of the reflection coefficients at terminals 3 and 4 can be obtained by choosing the proper insertion of the screw, while the required phase can then be independently obtained by adjusting the position of the screw along the guide. Choosing equal insertions and adjusting the positions at equal distances from the terminals, the coupling coefficient between Kx and K2 is given by (21): jTej2@. (21) As derived in section 3.2 this coupling coefficient is valid for the reference planes at terminals 1 and 2. If the length of the line between either plane and the grids of the klystron oscillator con¬ nected to it is given by yjV2*®-**). (27) If we put r=yeJoi, (22) becomes * V i) ye (28)' giving a magnitude of the coupling coefficient between Kx and K2 equal to the magnitude of the reflection coefficient introduced by either screw and a phase given by ^ = 2©-2^. + a + ~ (29) It has been shown in the preceding chapter that for symmetrical operation of either oscillator and identical behaviour of both as well as for maximum width of the frequency band over which synchroni¬ sation is possible, the phase of the coupling coefficient should be zero. The necessary phase of the reflection coefficients is then given from (29) by a = 2 It is seen that this combining network allows in a simple manner the adjustment of each parameter completely independent from 77 the others and thus enables the fulfilment of all the requirements as well as the of for proper synchronous parallel operation study the effect of each parameter separately. An alternative of this circuit which allows the omission of that part to right of the plane AA is shown in Fig. 4.5. It can be easily shown that the requirements imposed on the coupling magnitude C0 -jfj-|w MT, <>0I- 2 Load Fig. 4.5. Alternative network of that in Fig. 4.4. and phase are also fulfilled by this combining network, if K1 as well = In as both screws are displaced to the right a distance ~. addition to its simplicity this network has also all the advantages of that shown in Fig. 4.4. 4.3.3. Combining Network Composed of a Single Magic-T with Complimentary Holes for Coupling The circuit described above suggests another simpler one, where neither screw tunes nor phase shifters are necessary. If the common wall of the magic-T is designed in the usual manner to fulfil the requirements of equal power division and directivity, a Bethe-hole 78 coupler may then be bored at one end of the common wall to intro¬ duce the required coupling between the oscillators. Thus as shown in Pig. 4.5 the Bethe-hole perovides for the coupling while the magic-T serves to convey the total power output to the load. For two parallel, equal waveguides coupled through a circular hole in the centre of the wide side, a wave travelling in one guide in a certain direction, say to the right, excites two waves in the other guide, one travelling to the right (termed forward wave) and the other to the left (backward wave). With unit amplitude of the wave in the exciting guide, the squares of the amplitudes of the excited waves are given by [12] 16 tt2 r6 ( • A 2\ for forward wave A* = — ^^ (2-^) (31) and for backward wave B = -2-^(2+V\ r — radius of the whole a, b = dimensions of guide cross section A0 = free space wavelength \g = guide wavelength. The amplitude of the forward wave is zero at a wavelength satisfying the relation A9=V2A0 = 2a. (32) For a standard 1-in by J-in waveguide (32) is satisfied at a free space wavelength of 3,2 cm. At this wavelength the directivity of a Bethe-hole coupler is infinite and a wave travelling in one guide in the forward direction excites only a backward wave in the other guide. Such a coupler is thus suitable to produce the required coupling between the two klystron oscillators. If planes AA and BB are taken to represent the planes con¬ taining the grids of the oscillators, the best phase angle of the coupling is obtained with the length of the dotted line equal to -~-; with the phase shift introduced by the Bethe-hole being taken into consideration in calculating this length. If, in addition, the distance between AA and terminal 1 is ~ shorter than that between 4 79 BB and 2, the incident waves at 1 and 2 are in quadrature and the total power couples into arm 3 where the common load is connected. If the system is to operate at a wavelength other than that where the directivity of the Bethe-coupler is infinite, forward waves are excited in both guides. If the main waves are in quadrature at the plane of the whole with the wave in arm 2 lagging, the excited waves are also in quadrature with the wave in arm 2 leading. Thus the excited waves will couple into arm 4 instead into the load, resulting in a reduction of the net power supphed to the load. But if the Bethe-hole is designed for coupling of about 20 db at A0 = 3,2, thus in a system working at A0 = 3,4 the power in the excited forward waves will be 4 3,1 to 3,5 mm, it becomes obvious that the reduction in the power delivered to the load is negligibly small all over that range. Thus the Bethe-hole coupler is quite suitable for such a purpose. Chapter 5. Experimental Results 5.1. Introduction A number of experiments were carried out to investigate the behaviour of the synchronised reflex-klystron oscillator and the possibility of synchronous parallel operation. Nearly all measure¬ ments were performed at a wavelength of 3,4 cm using the 2K25 reflex-tube. The choice of this working wavelength was primarily determined by the available cavity-wavemeters which have a range extending between 3,37 and 3,43 cms. The experiments are going to be described here in the sequence of their development rather than the sequence of the theoretical treatment of the preceding chapters. The present investigation necessitates the knowledge of the pro¬ perties of the tubes under test as seen from the waveguide, as well as their performance under different loading conditions. Prom the properties of the tube we have to measure the different (J-factors 80 (unloaded- and radiation-Q) of the cavity resonators and the appa¬ rent position of the cavity grids relative to the terminal plane of the waveguide, over which the tube is mounted. The knowledge of the Q-factors are necessary to calculate the expected width of the band over which synchronisation is possible, as well as to compare the experimental results with those predicted by the theory. The knowledge of the tube performance with different loads enables to determine the conditions most appropriate for synchronous parallel operation. Measurement of impedance involved in the experimental proce¬ dures to obtain the informations mentioned above is carried out by using a standing-wave indicator. This consists of a section of wave¬ guide into which a small, movable probe can be introduced through a slot. The probe extracts a small fraction of the power flowing in Square Wave a-f Calib. Selective Modulator Attenuat. Amplifier Power Preamplif. Supply Slotted Klystron — 15 db Atten. Test Piece Section Fig. 5.1. Arrangement of circuit for standing-wave measurements. the guide to deliver it to a crystal rectifier, the output of which is then supplied to some indicating device. In order to achieve the required sensitivity and in the same time to allow the signal source to be isolated from the line by attenuating pads, the system shown in Fig. 5.1 is used. A square-wave modulation voltage is applied in series with the reflector voltage of the klystron oscillator to produce an "on-off" modulation. Power from the klystron is fed through a 15 db attenuator to the measuring device. As the 2K25 reflex- klystron supplies about 20 mW at cw operation, the average power output when square-wave modulated will be about 10 mW. Power in the incident wave in the slotted line is therefore about 0,3 mW. 81 Suppose it is required to measure VSWR's up to about 35 which corresponds to a PSWR about 1000. At this value the power at the standing-wave maximum is about 4 times the power in the incident wave i.e. about 1,2 mW while that at the standing-wave minimum is 1,2 x 10-3 mW. To reduce the errors attributable to the presence of the probe in the line, we may assume a small probe- coupling of the order 0,5%. Thus the power supplied to the crystal detector by the probe is 6 /xW at the standing-wave maximum and 6x 10~3 /xW at the minimum. The sensitivity that can be achieved with standard microwave crystals is of the order of 1 mV of rectified open-circuit voltage for 1 /xW of r-f power absorbed in the crystal. The rectified voltage is in most cases very nearly proportional to the power absorbed for powers not greater than a few microwatt. Thus the 6 /xW power absorbed at the standing-wave maximum in the above example may be considered as the maximum allowable for the assumption of proportionality between the output voltage and the absorbed power to be valid. Now, at the standing-wave mini¬ mum the voltage output from the crystal is of the order of 6 /xV. Thus, if the full-scale meter deflection is obtained by a 6 V output from the amplifier, the total amplification required is about 106 or 120 db. To achieve the high sensitivity imposed on the indicating device we designed and constructed a preamplifier unit and an amplifier modulator unit shown in the block diagram of Fig. 5.2. The sensiti¬ vity of an amplifier, working in the audio range of frequencies, is usually limited by microphonics, nicker effect, and harmonics of power-line frequency rather than by thermal noise. These effects are considerably reduced if the recurrence frequency is increased to 15 to 20 kc/sec. Therefore, we chose our modulation frequency at 17 kc/sec. The first stage of the preamplifier uses a 6F5 triode, that has low microphonics, little hum modulation and low equi¬ valent-noise resistance, while the last stage is a cathode-follower with an output resistance of 75 Q equal to the characteristic resistance of the a.f. calibrated attenuator. The amplifier-modu¬ lator unit consists of an R-C oscillator which drives through a buffer stage a slicer circuit (used to produce the square wave modulating voltage), and supplies the reference signal through a 82 Preamplifier Jnit r From i Cath. V Amp. 2nd Amp. 4-» a-f Calib. Attenuator Crystal 1 Follower L r x Amp. Coherent Phase Selective Butter Signal - <5 Inverter Amp. Amp. Detector 1 Amp. H Amplifier- modulator Unit. Buffer Phase RC Buffer Slicer Shifler Oscillator Circuit Amp. Amp To Klystron Reflector Fig. 5.2. Block diagram showing the preamplifier unit and the amplifier- modulator unit. phase-shifter and a buffer to the coherent-signal detector. The coherent-signal detector is preferred to the conventional detector circuit because it reduces the pass band to about 1 cps and thus allows to measure the amplitude of the fundamental component of the signal with greater accuracy. The output from the a-f atte¬ nuator is fed to a matched buffer amplifier followed by a selective amplifier stage, using a twin-T feedback network. The output from this selective stage is fed to the coherent-signal detector through a phase invertor and two buffer amplifiers. In spite of the high selectivity of the coherent-signal detector the twin-T selective stage is not omitted since it removes a great deal of noise and inter¬ ference prior to the detector. Measurements on the constructed units at the working frequency gave the following results: 1. The preamplifier gives a maximum undistorted output of about 300 mV to a load resistance of 75 Q with a voltage gain of about 600. 2. The slicer circuit gives a square-wave of amplitude 40 V max. with a time of rise of about 1 % and a smaller time of fall. 83 3. The selective stage has a pass band of about 1,5 kc/sec cen¬ tered at the working frequency. 4. The coherent-signal detector has a linear characteristic up to full-scale deflection of the indicating meter (an ampermeter of range 1 mA). The detector has a pass band which is certainly less than 1 cps. 5. The voltage gain of the amplifier up to the detector input is about 7000. Thus total gain of both units is about 130 db. The system allows to measure input voltage ratios up to 60 to 70 db corresponding to VSWR's 30 to 35 db with an accuracy < 1 db. Small VSWR's (as low as 0,1 db) can be measured by using the output meter. Apart from standing-wave measurements we still require methods to enable the measurement of absolute and relative power, attenuation, wavelength and frequency differences of the order of Slotted Matched —to Atten. 1 m Atten. II Section Detector ' Inicating Device Fig. 5.3. Circuit arrangement used for calibrating the attenuators. magnitude of some Mc/sec. At the time these experiments were carried out no thermistor mount for 3-cm band was available, also no attenuation standards. The only attenuators available were two variable waveguide attenuators of the flap type using IRC resistive strips as dissipative elements. To use these in measuring attenuation and comparing power levels, they were calibrated in the following simple manner. A crystal detector, matched to a VSWR < 1 db receives power from a modulated source through the 2 attenuators under test as shown in Pig. 5.3. The indicating device is that shown in Fig. 5.2. With the dissipative strip of attenuator I fully inserted in the guide (maximum attenuation) and that of II fully withdrawn (zero attenuation) the reading of the output meter was adjusted by the a-f attenuator to 0,7 its full deflection. A magic-T, also 84 matched to a VSWR < 1 db is then inserted between the detector and the attenuator II. Power division between the output arms is then checked and found to be better than 0,2 db. With the crystal detector connected to one output arm and the 2 other arms con¬ nected to reflectionless terminations, the reading of the output meter is observed. In this manner we have obtained 2 values on the output meter with 3 db seperation between them. Now, using the circuit of Fig. 5.3 the method of calibration is quite obvious. Increasing the attenuation of II till the meter indicates the half- power point and then decreasing the attenuation of I to the full- power point enables to determine the attenuation curve against 40 -Q ~°30 c c o 1520 c tt 10 60 SO 40 30 20 10 Sceale Reading Fig. 5.4. Measured attenuation characteristics for the attenuators used. scale reading by a succession of points with any 2 successive points separated by 3 db. The slotted line was used to check the VSWR during the coarse of measurement. An example of the curves thus obtained is shown in Fig. 5.4. It is not worth while to discuss the sources of error in the method used nor to mention the disadvan¬ tages of such an attenuator, which is generally used as a buffer attenuator only. The important thing is that with the help of this attenuator we could, with a good enough approximation, estimate relative power levels and measure attenuation. 85 For the measurement of frequency differences of the order of some Mc/sec an obvious method is to use a mixer crystal connected to a superheterodyne receiver that produces a tone when tuned to the frequency of the signal received from the crystal. In some cases it was found more appropriate to use the cavity wavemeter for this aim in order to avoid undue complexity of the circuit. The circuit shown in Fig. 5.5 was used to measure the difference between any 2 resonant frequencies corresponding to 2 different settings of the wavemeter tuning plunger. Thence, the frequency difference pro scale division near a particular resonance was calculated. This was done by tuning the wavemeter to the frequency of Kz and mea¬ suring the frequency difference between the frequency of K2 and that of Kx by using a superheterodyne receiver connected to a crystal mixer (Det. (2) in Fig. 5.5). The frequency of K2 is then — 20 db 2 _. ,4 Term. *-&i Direct. Coupler — _ Superhet. Term H2 4H1 3 Det 2 Direct. Receiver Coupler (~)-20db- C.W. — Det.1 -/?) Fig. 5.5. Block diagram for the circuit used to measure the frequency difference in Mc/sec pro scale division for a cavity wavemeter. changed and the same steps are repeated. The 2 sets of readings thus obtained (2 readings on the wavemeter scale and the cor¬ responding 2 readings on the receiver scale) enable to calculate the frequency difference in Mc/s pro scale division. This was done for a number of resonant frequencies covering the whole tuning range of the wavemeter. The measurement gave a value of 10 Mc/s pro scale division at the resonant frequency lying in the middle of the tuning range. A slightly higher value was observed at higher resonant frequencies towards one end, and a slightly lower one towards the other end of the tuning range. It is to be noted that the scale of the wavemeter used has 18 divisions, each containing 10 subdivisions and that the resonance wavelength at the middle of the tuning range was estimated to be about 3,39 cm. 86 5.2. Cold Test to estimate the Q-Factors °f tne Klystron Cavity Near a particular resonant frequency, the impedance looking into the cavity across an appropriately chosen reference plane d0 can be written in the form (see [8], chap. 12) Zw = ±^tl^\ + " (1) with Z(do) and r expressed in terms of the characteristic impedance 30 of the measuring hne. Thus, at that reference plane d0 the cavity R vwwvvv o c it to Fig. 5.6. .Equivalent circuit of a klystron cavity at particular reference planes and near a particular resonance. can be represented by the equivalent circuit Fig. 5.6, if Q0 = —^j— = the unloaded-Q of the cavity, and QR = oj0Cj0 = the radiation or external Q of the cavity. Using the abbreviations a = ^ and S = QB(^-^)=2Q^, Eq. (1) becomes Z = ~^ + r. (la) From Eq. (1) it is obvious that the reference plane to be chosen should coincide with one of the standing-wave minima planes, when the cavity is detuned far enough from resonance to make 87 1 0 and Z It is to be noted that Q0 is a measure of the internal losses of the cavity whereas the series resistance R is included to account for the losses in the line coupling the cavity to the waveguide. To estimate the parameters contained in Eq. (1) it is customary to plot the VSWR as a function of the frequency throughout the resonance curve of the cavity. The circuit we used to perform this measurement is shown in Fig. 5.7. The procedure of measurement Square-wave Indicating modulated Device Klystron under Source I Test. (Cold.) Slotted 15 db Atten. ' 3 0- Direct. Line O I . Coupler :.w Det. D-0 Fig. 5.7. Block diagram of apparatus used for making the cold test. is as follows. The signal source is tuned to the frequency of interest and the klystron under test (not oscillating) is sufficiently detuned away from this frequency. In this case the position of the minimum as indicated by the slotted line defines the position of the reference plane d0 and the measured VSWR is equal to reciprocal of the series resistance r. With the probe of the slotted line at the plane d0 the cavity is then tuned to the frequency of the signal. This is indicated by the fact that either a standing-wave maximum or minimum will be found at the plane d0, if the cavity is in tune with the frequency of the incident signal. With the cavity thus tuned the resonance curve is plotted by changing the frequency of the signal source and measuring the corresponding VSWR and position of the minimum. The frequency of the signal is indicated by the wavemeter. The data obtained from the above measurement is shown in the curves Fig. 5.8 for the 4 reflex-klystrons to be used in the coarse of our experiments. In Fig. 5.8 the VSWR is given in db and the frequency axis indicates frequency differences from the resonance 88 i 1 r -60 -40 -20 20 40 60 SO -60 -40 -20 0 20 40 60 80 20 40 -60 -40 20 40 -60 -40 Fig. 5.8. Measured resonance curves of the four reflex-oscillators under test. r = VSWR in db AF = Frequency deviation off resonanbe. calibration frequency. These are estimated from the results of the of the wavemeter that was mentioned in the introduction. The determination of the cavity parameters from the measured data is carried out in the following manner. Consider first the input impedance Z(do)o when the cavity is in tune with signal frequency. This is given from Eq. (la) by 89 A*,),, = -~ + r (3) a Let the corresponding VSWR be r0. Since the quantity Z^ may be greater or less than unity, two cases must be distinguished. In Case 1, the standing-wave maximum at resonance is found at d0 and Z(do) is greater than unity; in Case 2, Z^ is less than unity and the standing-wave minimum occurs at d0. This gives r0=I + r, Casel, (4) = h r, Case 2. r0 a For the 4 tubes under test it was found that Case 2 holds. In what follows only Case 2 will be considered. Let roii denote the VSWR with the cavity sufficiently detuned. From Eq. (2) we have ~y=Z{di))-r. (2a) — — can De determined. To From Eqs. (2a) and (4) rr separate Q0 and QB it is customary to determine the width of the resonance curve at some properly chosen value of the VSWR. To find the relation between 8 and the measured VSWR, let us denote the value of the latter for a certain 8 by rg and let us introduce the complex reflection coefficient r defined by the equation Z(dn\ — 1 Wo) Ado) +l Since only the phase of r will vary by varying the position of the reference plane, we may calculate the magnitude y = \r\ for any 8 from (la) and (5), thus giving + + 2 Mr-l) l}2 82(r-l)2 7 = ( ' {ff(r+l) + l]2 + 82(r+l)2- Since a and r are known, the value of y for any 8 can be calculated from (6). Substituting this y in the relation 90 gives rs at the assumed value of 8. Let the width of the resonance curve at the calculated value of rg be A F Mc/s. Since from defi¬ nition QR can be obtained. If QR is thus calculated for a number values 8 of and the average is taken, this average will represent a better approximation to the value of QR. This calculation was carried out for the 4 tubes under test. Let us denote these by K1, K2, K3 and K±\ these symbols are going to be used to refer to any of these tubes when used later on in the following experiments. The results obtained are: Klystron r0 r Qr Qo *i 1,26 0,05 885 650 K2 1,42 0,10 885 535 K3 1,42 0,03 650 440 K, 2,82 0,03 950 310 = VSWR at resonance. r0 „ r = normalised series resistance — giving losses ho in the line coupling cavity to waveguide. QR= radiation Q. Q0 = unloaded Q. 5.3. Mutual Synchronisation An arrangement of a circuit appropriate for the experimental investigation of the mutual synchronisation of 2 reflex-oscillators has been described in section 3.1. The arrangement actually used for the following measurements is shown in Fig. 5.9. The directional coupler used here couples about 3 % of the power flowing in the main line 13 into the auxiliary line 24. Signals coupled into this line are used for indication and measurement. The cali¬ brated attenuator in series with the phase shifter enable to vary the magnitude and phase of the coupling between the 2 oscillators. In the phase shifter used the change in guide wavelength is brought 91 K, Phase Calib. 1 3 0- Shifter Atten. Direct. Oor£ Coupler 4 Det.2 C.W. Receiv. Det.1 CRT & Fig. 5.9. Circuit arrangement used to measure the width of the synchroni¬ sation region for mutual synchronisation of 2 reflex-oscillators. about by moving a long polystyrene slab laterally across the interior of the waveguide. The position of the slab is indicated by a micro¬ meter scale; with the slab near the side wall of the waveguide the reading on the scale is zero. Thus, this reading corresponds to the longest equivalent length introduced by the phase shifter. The aim of the measurement is to determine the width of the synchronisation region as a function of the phase shifter setting for different values of the attenuation (i.e. |812|OT as a function of ifi with a as parameter). The same measurement was carried out twice; once for the pair of klystrons K1 — K3 and the other for the pair Ki — Ki. The following description is going to be referred to the pair K1 — K3. Consider first the case where the attenuator is set to give maxi¬ mum attenuation (about 45 db for the attenuator used). Since the power coupled from one oscillator to the other is now very small, each can be assumed to be oscillating at its "free-running" fre¬ quency. Signals arriving at detector (2) will be mixed by the crystal mixer and an i-f signal, of frequency equal to the difference between the "free-running" frequencies of Kx and K3, is supplied to the receiver. The latter is adjusted to receive unmodulated signals. Thus an audio tone will be heard if the receiver is tuned to the frequency of the i-f signal. Now if the reflector voltages were pread- justed such that the relative reflection phase angles are both zero, the "free-running" frequency of each will be equal to the resonant 92 frequency of its cavity. Thus, the frequency measured by the receiver gives directly the difference between the resonant frequen¬ cies of the cavities. Let us now consider the case where the attenuation is adjusted the resonant to give a certain coupling of magnitude a. We keep constant and that of If is frequency oj01 of iC1 vary K3 (co03). oj03 takes we observe near enough to ioQ1 but no synchronisation place, beats — each oscillation undergoes periodic variations of frequency and amplitude. The period of these variations becomes longer, as one of the at the a)03 is brought nearer to limiting frequencies boundaries of the synchronisation region. At the same time the from the resonant average frequency of each beat is "pulled" away frequency of the corresponding cavity and is brought nearer to that limiting frequency. The signals reaching the detectors are com¬ from the i-f posed of a mixture of the 2 beats. Hence, the output terminals of the crystal rectifiers is again a beat of average fre¬ the of quency equal to the difference between average frequencies the r-f beats. Thus, if the receiver is tuned to this average inter¬ mediate frequency, a tone will be heard. Due to the complex character of the i-f beat received, it is obvious that the tone will be heard, if the receiver is detuned to either side of the average. However, the middle frequency of the band, over which the tone of the can be heard, may be taken equal to the average frequency the i-f beat. Also, if this average frequency is low enough, signal of received by the cathode ray tube (CRT) will produce a wave if is towards the some particular form. Now, co03 brought slowly the i-f beat will be boundary, a reduction in the frequency of indicated by both the CRT and the receiver. The disappearance of of the the wave seen on the CRT accompanied by the vanishing tone heard from the receiver will indicate that synchronisation has taken place. Due to the lack of a fine adjustment in the tuning mechanisms of the existing reflex-tubes great care should be taken the oscillates in at to adjust w03 so that system synchronism just full one of the boundaries. If this is done, then by introducing again attenuation the difference between the resonant frequencies of the cavities can be measured as explained in the preceding paragraph. and In this manner we obtain for each setting of the attenuator 93 6 8 2 4 6 8 Phase-shifter Scale reading Fig. 5.10a. Measured width of the synchronisation region as a function of the phase shifter setting for different values of the magnitude of the coupling a (expressed in db), for the Klystrons Kx and K3. The circles o give AFly and the crosses + AF2. 94 (24 db) (21 db) (18 db) (16 db) ^ 7 f\l r~—i 1 T I I I (12 db) (Wdb) "i 1 r—i ~i 1 r 16- 14- (8db) (6db) (4db) (2db) 12- 10 ^8^ %6A 4-\ 2 i 1 1 —I r 0 —I 1 1 —i 1 1 r 24632463246 2 4 6 Phase-shifter Scale reading and Fig. 5.10b. Same as Fig. 5.10a for the Klystrons Kt Kt. 95 phase shifter the 2 frequency differences defining the synchroni¬ sation region. Denoting by A F± the measured frequency difference at the upper boundary i.e. for «j03>aj01 and by A F2 that at the lower boundary i.e. for oj03 and 3.7 ted as functions of —. Comparisons between this figure Fig. shows the qualitative agreement between theory and experiment. To make a quantitative check we use the expression (3.50) giving the initial inclination of the curve relating |812|m to — (with -^~ from Eq. (3.49). This contains the conductance parameters which, as given by Eq. (3.19), are functions of the small signal transconductance Ge as well as G and Q; Ge and G have not been measured. Thus we may either assume values for Ge and C or pur y^- = 1. This latter assumption is rather justified since the first 96 1 11 i > i 111 i i i—i—i—i 1 1 1 i—r ~ 2t 16 12 10 8 6 « 2 Odb 24 16 12 10 8 6 0db'20logw a I I I I 1 I I I I I I I I I I I I I I I I I 0 1 0 01 0,2 0,3 0,k 0,5 0,6 0,7 O.H 0,9 11 0,1 0,2 0,3 0,6 0,5 0,6 0,7 0,8 0,9 ± a. « b) as funktion of Fig. 5.11. Maximum width of synchronisation region a) for the pair K1, Ka. b) for the pair K2, K4. term of (3.50) is multiplied whereas the second term is divided by this ratio. Thus, putting this ratio equal to 1 or 0,8 wjll give two results which differ only by a few per cent. Taking ^~ = 1 and substituting (3.22)—(3.25) in (3.50) yields for the initial inclination the simple expression: Mi+i)Mc'8'^ <8) with F0 = 8850 Mc/s = working frequency. For the first pair Kx and K3 the initial inclinations as determined from Fig. 5.11 are: for A Flmax 27 Mc/s and for A F2max 22,8 Mc/s while the value cal¬ culated from (8) is 23,8 Mc/s. For the second pair K2 and K4 the experimental values are: for A Flmax 12,7 Mc/s and for A F2max 11,3 Mc/s, while that calculated is 19,3 Mc/s. Reference to the table in section 5.2 shows that the klystron K2 has a series resistance r which is rather big. As shown in Fig. 5.6 this series resistance reduces the magnitude of the signal coupled into the cavity and hence will result in a reduction in the width of the synchronisation region. Taking this fact into consideration and remembering that Eq. (3.50) was deduced under the assumption of a lossless line coupling the cavity to the waveguide (i.e. R = r = 0), we notice that the theory also agrees quantitatively with the experiment. From Figs. 5.10 and 5.11 we notice that A Flmax is always bigger than A F2max. It can be shown that inequality between the 2 limiting frequencies can be the result of an error in the preadjust- ment of the reflector voltages to some other values than those making the relative reflection phase angles equal to zero. To show this we use Eqs. (3.21) under the assumptions: 1. identical klystron with a small coupling, 2. cPl = ^ = 0, and 3. cp2is small so that e~J'<*'2 = l—j = = X = Xx X2 , oosi8 l-j/J1(Z), xy (9) 2 = -\ sin 8 8, . y 98 Since 82 = 8t — 812 we have -*' S12 = amP + MA^X), (10) a y which shows that the maximum values of 812 obtained by putting (8= +90° are no more equal and are given by 4 S12 = S12, = — + 4 and S12 = 8U_= --tpa-gA^X). (3=-9o° Thus, if one of the relative reflection phase angles has a value other than the zero, the values of A Flmax and A F2max are not equal. Eqs. (11) show that A FlmaT > A F2mar ^T 0 and A Flmax < A F2max for Subtracting Eqs. (11) yields |§12+-Sl2_| = 2l Assuming that (12) remains valid for large values of the coupling as well, the value of gAx(X) in (12) is then given by Eq. (3.39), so that IV-^h^Wl + ^t!,-) d3) which shows that the difference increases by increasing the coupling (i.e. reducing a). Comparison between (13) and Fig. 5.11 shows again the agreement between theory and experiment. Further Eqs. (9) enables us to make some remarks on the behaviour of a system of 2 mutually synchronised reflex-oscillators, if either or both of the reflector voltages are modulated. Under the assumption that the period of the modulating signal is small com¬ pared with the synchronisation time constant, the system can be assumed in equilibrium at each instant of time and a succession of steady state solutions give a good enough approximation. Hence, Eqs. (9) give the solution when only the reflector voltage of K2 is 99 modulated; 8ia = 0 8i = 82 = - |?a^iffl which shows that the frequency at which the system oscillates is directly proportional to q>2 and hence also proportional to the ampli¬ tude of the modulating signal, if the latter is small. The factor \ arises naturally from the fact that only one reflector voltage is modulated. This enables us to conclude that if the static charac¬ teristics (relative reflection angle as a function of reflector voltage) of the klystrons considered are different from one another, the fre¬ quency of oscillation of the system remains proportional to the amplitude of the modulating signal, when both reflector voltages are simultaneously modulated. The equivalent static characteristic relevant to the modulating signal, for the system as whole, will be somewhere between the characteristics of the individual oscillators. The behaviour of a "disturbed" oscillator in the region of beats (a) (b) Fig. 5.12. Frequency "Pulling" in the region of beats. AF0= "undisturbed" frequncy difference. Af = "disturbed,, frequency difference. (a) for the'pair Ky, K3, (b) for the pair K2, K^. WO has been fully discussed in Chap. 1. A similar behaviour may be expected in the case of two oscillators coupled together and gene¬ rating frequencies that are not widely different. If the two frequen¬ cies differ by only a small percentage, they are both shifted from their normal values in such a way as to reduce the difference. This attraction of the two frequencies becomes more pronounced as the difference between the normal oscillating frequencies is reduced and finally becomes so great that the oscillators pull into synchro¬ nism. This behaviour is illustrated in the experimental curves Fig. 5.12a and b for the 2 pairs of klystrons under test. Here, the average frequency of the i-f beat is plotted against the difference between the undisturbed frequencies. 5.4. Synchronous Parallel Operation of two Reflex Oscillators This problem was theoretically discussed in chapter 4, and the condition for identity and symmetry of the operation was given in Eq. (4.30) which defines the phase of the coupling coefficient. It was also mentioned, that the magnitude of the coupling should be chosen in such a manner that optimum power may be supplied to the common load. The choice of the magnitude of the coupling necessitates the knowledge of the behaviour of the particular kly¬ stron under various loading conditions, i. e. the knowledge of its rlieke diagram (see [8], chap. 12). However, for the particular informations required for our present application it is not necessary to plot the whole of the Rieke diagram. It is only necessary to locate that region of the diagram where optimum power is supplied to the load. This is done as follows. A movable screw tuner is calibrated when connected to a matched termination and the com¬ bination is used as a standing-wave introducer. Here, the VSWR depends solely on the screw introduction, and the phase on its position. If this is used as a load for the klystron a quick deter¬ mination of the required data is possible. With a constant screw introduction and variable position the point on the Smith chart representing the load seen by the klystron traces a constant-y circle (y = magnitude of the reflection coefficient). On this circle we have to determine the position of the two points at which maxi- 101 from the If this mum and minimum power is supplied klystron. the results on is repeated for a number of values of y and plotted is determined. a Smith chart, the region of maximum power readily The circuit we used for this purpose is shown in Fig. 5.13. The reference klystron together with the mixer and receiver are used to of measure the change in the frequency of oscillation the klystron under test when the particular load of interest is presented to it from its frequency when acting into a matched load. The cavity- wavemeter (C. W.) is used to check the frequency of the reference klystron during the coarse of the experiment. The calibrated /•A 9 n A OutputMput fa De,\^_ Receiv. s~\ Slotted Screw Calib. Direct. \y~ Line HMov.Tuiner Alien. Coupler Term — Mixer Klystron Direct. under test Coupler 20 db Reference C.W. Klystron /M0- Det. Fig. 5.13. Arrangement of circuit used to determine the behaviour of the klystron oscillator under different loading conditions. attenuator together with the output detector and the micro-ampere¬ meter are used to measure the level of the power output from the klystron under test. The slotted line is used to determine the position of the standing-wave minimum of any particular load of interest. The data obtained for the 4 klystrons Kl, K2, K3 and Kt is plotted on the Smith chart Fig. 5.14; the reference plane used here is the terminal.plane of the waveguide on which the klystron is mounted. Curves in Fig. 5.15 give the level of the maximum and minimum power outputs as functions of the VSWR of the load. It is now obvious that with the help of Fig. 5.15 the location of the regions of maximum power output can be easily determined on Fig. 5.14. These are shown dotted in Fig. 5.14. 102 Fig. 5.14. Loci of maximum and minimum power output obteined by fixing the magnitude of the reflection coeff. of the load and varying its phase. For the four Klystrons under test the points determined are shawn: x for Ky, o for K2, • for K3 and + for Ki. The regions enclosed within the dotted curves denote those for optimum power outpus. 103 max. 5 rswR Ki 1.0- 0.9 - \ \^max. 0.8 - \° ^^ 0.7- 0.6 - \ min. 0.S- 0.4 i 0.3- • 0.2- \> 0.H 1 1 1 1 srswR 6 snwR min Fig. 5.15. Plat of ^E and ^ as functions of the VSWR. 104 Choosing K% and K3 to be operated in parallel by using the combining circuit Pig. 4.4, the "operating point" should lie within the common part between their respective dotted regions in Fig. 5.15. At this operating point the necessary magnitude of the cou¬ pling coefficient is about 0,262 (corresponding to a VSWR 1,75) and the necessary phase is read directly on the "Wavelength towards generator" scale. Adjusting the screw tuners in Fig. 4.4 to give this value of coupling, the adjustment of their positions to give the required phase at terminals 1 and 2 of M T1 (see Fig. 4.4) can be carried out by a simple standing-wave measurement. With everything properly adjusted the level of the maximum power supplied to the load as well as that coupled into arm 3 of M T2 were measured. The results of the measurement are: level of maxi¬ mum power supplied to the load is 3 db over the optimum power level from K3 alone and 2 db over that from K2 alone; the level of power coupled into arm 3 is 18 db beneath that supplied to the load. Reference to Fig. 5.15 shows that the power supplied to the load is at least equal to the sum of the powers supplied from each klystron separately to a matched load. That complete cancellation of the waves coupled into arm 3 does not take place, is obviously due to the unequal power outputs from the klystrons. To investi¬ gate the stability and reliability of the operation of the system, it was left running for a couple of hours, switched on and off; the system remained always in synchronism with practically constant power supplied to the load. With the appropriate adjustments performed the alternative combining circuit Fig. 4.5 was then examined. The same results were obtained here as for the original circuit Fig. 4.4. Further, it was found possible, by slight readjustment of the resonant fre¬ quency of one the klystrons, that complete cancellation of the waves coupled into arm 3 may be attained, accompanied by a slight increase of the power coupled into the load. It was also found that the adjustment of the positions of the screws is not critical: the power supplied to the load varies slowly with varying the positions of the screws, whereas their insertions produced marked effects. This enables the proper adjustments of this circuit for the synchronous parallel operation of 2 klystrons without having any 105 knowledge of their active performance. This is done by tuning each separately to the same frequency with its reflector voltage adjusted to the center of the mode. When connected to the circuit of Fig. 4.5, a simple trial enables to find the proper insertions of the screws; then maximisation of the power delivered to the load is performed by adjusting the positions. It is advisable in this case to take insertions of the screw which produce a VSWR 1,5 in order to insure stability of operation. A circuit adjusted in this manner may be useful for application as a signal source of a higher power output. 5.5. Synchronisation by a Signal from a Harmonic Generator Frequency standards available in the microwave region are either the frequencies of the absorption lines in the spectrum of an absorbing gas (mostly NH3); these being found in the 1 cm region, or a signal of microwave frequency produced by multiplying the frequency of a crystal oscillator. The multiplication into the vhf- region is made by vacuum tube multipliers. The vhf multiple of the crystal frequency is raised to about 900 Mc/s by using a light¬ house triode multiplier. The last step in multiplication into the microwave region is accomplished by harmonic generation in a silicon crystal or by using klystron multipliers. Although the klystron multiplier provides a larger power than does the crystal rectifier, the alignment of the 2 cavities of the former is critical, and the multiplier is difficult to use. However, the possibility to synchronise a reflex klystron oscil¬ lator gives another alternative which provides a large power without having the disadvantages of the klystron multiplier. To investigate this alternative we constructed the harmonic generator shown in Fig. 5.16. The signal of fundamental frequency is applied to the crystal through the i-f terminals and the 3-cm harmonic is propagated down the waveguide. The transmission characteristics of the waveguide provides adequate filtering of the fundamental and thus no power of fundamental frequency exists in the output. The input line is fitted with chokes which prevent any harmonic power from flowing into this line. For best harmonic generation one adjustable short circuit and 2 screw tuners are 106 Tuning Screws Auxiliary Rings Fig. 5.16. Harmonic Generator for the 3 em band. 3 included to enable matching; in addition the crystal can be moved across the guide by use of the auxiliary rings. Fundamental power at a wavelength of about 10,3 is supplied by a lighthouse triode oscillator. Proper matching was attainable by using a three stub tuner so that the VSWR seen by the 10,3 oscil¬ lator was about 1,22. The power supplied to the crystal was mea¬ sured and found to be about 45 mW. The harmonic power output measured by comparison, using our calibrated attenuator, was found to be at level 15 db beneath the power level of the 2K25 reflex-klystron; thus estimated to be about 0,8 mW. Thus the conversion loss of the harmonic generator is fundamental logi power input = 310 17,5 db harmonic power output in producing the third harmonic. = 1 -= 3,43 cm X 10,3 cm I r ^ i Synchronised Harmonic 3-Stub Slotted Light-house ^ 1 3 Klystron o> Generator Tuner Line Oscillator MT Term. 2 4 Atten. Det. ~l To CRT. Fig. 5.17. Synchronisation of one reflex oscillator by signal from harmonic generator. The signal thus obtained was used to synchronise a 2K25 reflex-klystron. The circuit used is shown in Fig. 5.17. The signal reaching K is thus \ that generated by harmonic generator, i. e. of about 0,4 mW. An external signal of this magnitude will affect synchronisation over a band of frequency, the total width of which is about 1,5 Mc, which is rather narrow. However, with proper adjustment, the system remained in stable synchronism for a couple of hours giving a power output of about 10 mW or more. Thus the total conversion loss of the system as a whole is given by fundamental power input log = 6db synchronised harmonic power output 108 with an increase of about 11,5 db over that obtained by the har¬ monic generator alone. When the system was switched off and left to cool, to be switched on afterwards, no synchronisation occured. This is due mainly to the fluctuation in frequency of the lighthouse triode oscillator, which, even when small, will result through multi¬ plication in a rather high frequency difference. Next, the signal from the harmonic generator was applied to synchronise simultaneously a system of 2 reflex oscillators by using the circuit shown in Fig. 4.2. The stability of the system was bad; but as it once happened that the system remained in synchronism for a few minutes the power output was double that obtained from a single klystron with a corresponding conversion gain. However, to attain a good stability of such a system the signal provided by the harmonic generator should be of about 3 mW to affect syn¬ chronisation over a band of 3 Mc/s. With the same conversion loss of the harmonic generator alone the fundamental signal power should be about 150 mW. However, if the fundamental signal source has a stable frequency, only about 100 mW fundamental power is necessary to obtain good stability and steady locking. Since the harmonic power output in such a system is about 50 mW the conversion loss of the system as whole will be about 3 to 5 db. I wish to express my deep gratitude to my Professor Dr. F. Tank for his valuable advice and superior guidance. It is also to acknow¬ ledge that some of the apparatus used in carrying out the above experiments has been paid through the fund of the Jubilee 1930 of the Swiss Federal Institute of Technology. I wish here to express my best thanks and appreciation. 109 Literature 1. Balth. van der Pol, "The nonlinear Theory of Electric Oscillations". Proc. I.R.E. vol. 22, pp. 1051—1086. Sept. 1934. 2. Heinz Sarnulon, ,,tjber die Synchronisierung von Rohrengeneratoren". Helvetica Physica Acta, vol. 14, pp. 281—306, 1941. 3. Fritz Diemer, „YJber Synchronisierung von Rohrengeneratoren durch modulierte Signale". Mitteilungen aus dem Institut fur Hochfrequenz- technik an der ETH in Zurich, Nr. 7, Verlag Leemann Zurich. 4. Robert Adler, "A Study of Locking Phenomena in Oscillators". Proc. I.R.E., vol. 34, pp. 351—357, June 1946. 5. Robert D. Huntoon and A. Weiss, "Synchronisation of Oscillators". Proc. I.R.E., vol. 35, pp. 1415—1423, December 1947. 6. J. R. Pierce and W. O. Shepherd, "Reflex Oscillators". Bell System Techn. Journal, vol. 26, pp. 97—113, March 1946. 7. E. L. Ginzton and A. E. Harrison, "Reflex-Klystron Oscillators". Proc. I.R.E., vol. 34, pp. 97—113, March 1946. 8. D. R. Hamilton, J. K. Knipp and J. B. H. Kuper, "Klystrons and Microwave Triodes". McGraw-Hill Book Co. Inc., New York, N. Y., 1948. 9. A. H. W. Beck, "Velocity-Modulated Thermionic Tubes". The Macmillan Co., New York, N. Y., 1948. 10. Montgomery, Dicke and Purcell, "Principles of Microwave Circuits". McGraw-Hill Book Co. Inc., New York, N. Y., 1948. 11. Montgomery, "Technique of Microwave Measurements". McGraw-Hill Book Co. Inc., New York, N. Y., 1948. 12. M. Surdin, "Directive Couplers in Waveguides". Journal I.E.E., vol. 34, pp. 725—745, Sept. 1946. 110 Course of Life I was born on 18th November 1921 in Alexandria (Egypt). After finishing the primary and secondary schools I obtained my certi¬ ficate of maturity in 1939. Then I joined the Military College in Cairo and studied there as a student-officer for 2 years. In 1941 I entered the Faculty of Engineering in the Farouk I University in Alexandria. After 5 years study I obtained in 1946 my B. Sc. degree in Electrical Engineering. I worked in the same faculty as assistant for one year and then came to Switzerland. Since the summer term 1948 I studied under the guidance of Prof. Dr. F. Tank in the Institute of High-Frequency Techniques at the Swiss Federal Institute of Technology.