Design of electrical filters in the audio frequency range
Item Type text; Thesis-Reproduction (electronic)
Authors Stucky, N. Paul, 1919-
Publisher The University of Arizona.
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Download date 26/09/2021 18:42:54
Link to Item http://hdl.handle.net/10150/553588 DESIGN OF ELSCtRICAL FILYBBS IS THE AUDIO FREQUENCY RATGE
by
K. Paul 3 tucky
A Thesis submitted to the faculty of the
Department of Physics
in partial fulfillment of the re mirenxmts for the degree of
i.'astor of Science
In the Graduate College University of Arizona
1941
Approved * ,# Director of Thesis . J./? 4 ^
« A.*.
»eW ;2 t*
###Ng# ;.,?r ^
5?>:.-'; $ ¥ ' ' - ; £ 9 7 9 / / 9 * 7 / & /
KaoroiisDoaBet
The author wishes to express his
appreciation to Dr. Vinton A. Brown and
Mr. Leroy R. Alldred^e for suggesting
this interesting xxroblem and aiding in
its coinrlotion.
i ' l i t i L U 3JA3IS OF OOliTEZTS
Chapter I'age I. Introduction 1 II. Effect of Frequency Range on Articulation 2 III. Distortion in Networks ...... 5 IV. 5 and Pi Networks ...... 7 Y. Resonant Circuits In Eetworto A. Reactance and Reactance Curves •.••.••••• 15 B. Susceptanoe . . • . . . • . . . . . • . • . • 16 VI. Infinite Lines A. Characteristic Impodanoe . . . • ...... • * • 18 B. Attenuation ...... 22 VII. Introduction To Filters ...... 28 VIII. Losses In Electrical Networks A. Insertion Losses ...... 31 B. Reflections • • ...... • 33 IX. F ilters A. Reactance Curves . . . • • . . . • • . . • • • . • 42 B. Low- and High-Pass Filters ...... 43 C. Band-Pass F ilters ...... 55 .o X. Inductance Construction .••••..••••••••• 64 XI. Band-Pass F ilter Design ...... 68 XII. Summary ...... 72 Bibliography 73 CHAFES X IKTfiODUCTIOT
The oommunloatlon engineer Is often confronted, with the problem of the transmission ef certain frequencies while others are to bo completely attenuated. Since at present the most economical and quickest ways of aoBEQunicatlon over any great distance are electric al, the communication engineer is most interested In the transmission of electrical energy. Electrical energy has few uses by Itself^ Ordinarily sound energy is transformed into mechanical energy which la turn Is transformed into electrical causing fluctuations In the current. This may bo transmitted by means of wires or may further bo changed Into electromagnetic radia tion and transmitted throu#i:sp@@e& In either ease electrical energy ' forms a very Important link in the conmnlcatlea process arri a t the receiving eM it Is again tramforaed tc the desired form of energy. CiLUrfBB II BFF3CI Of fKSatlBSCX KAMOS ON /SflODLAtlOK
Since Intelligenoe by -sire and radio can be transmitted by means of varying currents only, the problem of oozmmnication becomes one of alternating current. To transmit such a wide band of frequencies as to cover the entire audio range would be extremely expensive so far as equipment Is oonr cerned and then could be reached to an approximation only. 1 It was found, however, by Crandall and Mackenzie that most of a group of syllables could still be Interpreted even if certain freqnenelee were not reproduced. Figure 1 show the results of Crandall and MacKensle. •The abscissae represent the limit of the frequencies used and the ordinates the percentage of correct judgments as to the sounds received, for the "H’’-curve, all frequencies below the abscissa were eliminated; whereas for the MLM-ourve, all frequencies above wore eliminated. Take for example the MHrt-curvQ. For the frequency of 1000 cycles the articulation is 06^« The interpretation is that if all frequencies below 1000 were eliminated, 86# of the syllables could still be correctly understood. From the same curve it can be seen that 40# of the syllables were understood if a ll frequencies below 1960 cycles ware eliminated. The interpretation of the nLw-ourvo is similar except that elimination of
Crandall and MacKeasie, ihy. Bov., 19, 1928, p.221. the fre inoncles is above the limit rather than belo?. On this curve 40%
] cc
c RO
r - b fC
4 0 d) r* (h PC pL
1CC0 .2000 3000 4000 5000 freq u en cy 1 — Effect TJpon Interpretation of ITLiirinatinp V arious P o rtio n s of The frequency ^ n f e
of the syllables were understood if all of tlie frequencies above 1000 were eliminated# From these curves it is possible to conclude that at a frequency of 1550, or in the third octavo above middle C, the Importance of the frequencies higher than this value is as great as that of the frequencies lower than this value* This would indicate that if a filter would pass a band of frequencies around 1550 cycles, more syllables could be under stood than the same width band passed at any other frequency* It would be reasonable to conclude that a transmission from 400 to 3000 cycles >4»
mlght be acceptable for some purposes where Intelligibility only Is required. It appears also that the limit of consideration is at 5000 cycles. In many cases it is desirable not only to have the words of the speaker understood but also to reproduce speech or music with natural ness. Since the quality of any tone is determined by the combination of the overtones, it is often desirable to transmit a greater range of frequencies* This is the ease in broadcast transmission. The limits of a band of such high fidelity is somewhat a matter of opinion although it has been generally ascertained that it is desirable to have a frequency range from 30 or 60 cycles to 8000 cycles. Some sounds, however, are much higher still, but the Increased cost of equip ment for increased fidelity cannot be forgotten. So in each case the commonlcation engineer must strike a median between economy and fid elity . In each case then it is desired to receive at the other end the same complex wave, the form of vdxieh must not be altered to any great degree. If, however, the form of the wave is altered. It is said to be distorted. 6HAPSEB III DISTOItflOl IS NE’SWOKKS
If a change occurs in the relative magnitudes of the different frequency component#, frequency distortion Is saM to be present* If new frequencies are present in the output which were not present at the input or if some frequency consonants have been cut out there is non-linear distortion. If the tine of transmission or delay varies with frequency, there is said to be phase distortion present. Frequency and phase distortion are due to the linear elements of inductance and capacitance, because of the variation of impedance with frequency. However, it is often possible to correct for both of these typos by additional meshes or elecents which provide a counteracting distortion. For example, in a cable the higher frequencies travel faster and are attenuated more than the lower ones. How an equalizing network at the end of the line would attenuate the lower frequencies and delay the higher ones. This should then neutralise the effect of the lin e. Ordinarily a small phase distortion will not be perceptible to the ear. In long transmission lines the effect may become appreciable. If, however, physical instruments are used to interpret the signal as in telegraphy, phase distortion is of great importance. Also since the eye is very sensitive to small changes it is important to keep phase distor tion to a minimum in television transmission. nonlinear distortion is not so easily corrected. Once row fre quencies are produced, they can be eliminated only if they fall outside the band of those which are desired* If they fa ll outside the desired bands they can be disposed of by networks, called "filters", which effectively transmit certain bands of frequencies and greatly attenuate others. In the ease of total absence of phase distortion, nonlinear distortion may be corrected by another nonlinear impedance with a reverse curvature. In general nonlinear distortion should be prevented rather than corrected after its occurrence. nonlinear distortion, however, has extremely important uses. A complex wave, represented by a band of frequencies, can be translated to any other band of equal width, and later retranslated to its original form. Often this is desirable so that several messages may be sent in the same medium or because of a greater effectiveness of the trans mitting medium at the new band. This is used in radio and all carrier telephone transmission. ok nmd o te same reason* the named for work, is well to consider two types of structures. Figure 2 shows a T net- net- T a shows 2 Figure structures. of types two consider to well is nenl meac Z ad eeae vlae , n a od meac ZB impedance load a and E, voltage generated and Zs Impedance Internal ok s cle bcue f t cnijrtin ad iue , 7T a 3, Figure and ion, jurat ccnfi its of because called so •ork, (?)—\ZW\A- D uo to inherent inherent Duo to Both Both figures figures F ' t— t vFf'. i. Flfir. 3 2 ?iF. 2 ---- show ntok oncig gnrtr o load. a to generator a connecting T network t T tfr cnetn a re tr o load. a to tor prner a connecting fork et impedances impedances these networks connected between a generator of of generator a between connected networks these A) KS7W0BK3 H AS) £ CHAPTSR IV present in a ll electrical systems, it it systems, electrical ll a in present —8—
If it can now bo shown that when the proper relationships are esta blished between 2%, Zpt Z^, and ZA, Z0f Zq9 there is no difference in the two currents and Ig for the two networks, there would be no way
to differentiate between the two combinations if they were placed in a box and only the t.;rae terminals brought out. In such a case the equivalence of the two networks would then bo established.
The most convenient measurements are those made between any two
terminals with the third disconnected. As a result consider only the
Tir- 1 — T-se^ticn. \ . . --TT-f - '’ t i
T section and TT section shown in r i^ure 4 and , i'ure 5 respectively.
It will be noted that figures 4 and 5 are identical with ! igures 2 and 3 with the except ion that the generator and terminating load resis tance are omitted, leaving only tho 2 and TT configurations. If tlio im pedance between the terminals 1 and 2 are equated for tho two config urations, the following relation is obtainedt
2it22 ID ZA+23+2C
In a similar smnner measuring the impedances between terminals 2 and 3;
Zq(Z v+Zn) ■.-w&issS- . 121
Likewise between terminals 1 and 3; 2A(ZBt20) Z1TZ3 ' Z ^ Z o ' : 131
To solve for Zj* add Equations (1) and (3) and subtract Squation
(Zj+ZgJ'+lZjj+'Zg)—(Zg^Zg) = ZAtZB*2G
2a2b ZAtZBfZ0
By adding Equations (1) and (2) and subtracting Equation (3),
. ^ ZmZn ^2 ZA+ZB4Zc ,
In a similar way.
ZlZc 2A+2B+2 C -16-
So far it has been shown that a T and TT sootlon cro equivalent without any considwation of gemrator or load. Consider''next the generator and load oonmoted as shewn in Figures Z and 3. From Figure 2, the current Is;
B ( 7 ) 2s-+2q 4 Z2+Z3^2B
Consider aw that tho imncdanoes Z \t 2 2 * Zg, of Figure 2 have
. - 1 f magnitmes in terras of 2A, Zg, Z0 as determined by Equations (4), (5), and (6)• For sim plicity le t the sum of Z^+Zg+Zc be represented by Zj» Then
Z8 ^ Z2 t z Zm T fi
Xl - ,, t / ZB2c‘t'2B20'f2AZBZC‘,>ZBZC3!g42BZR2T:CtZB; j 3 Z"2 V \ ■ - ZB^G+2A2C+%2SZnZ^+Z .Z^+Z^Zm / and substituting for Z$ its value ZA+Zg+ZCt
s' . . I l 34__ r;ZA4Zb4Zg)(ZbZC»ZhZg4ZbZR) 1 Zot ST %l+zBf2G[ ZBZc4ZAZG+ZHliA+ZB4-Zc) J
r zb^ g ) 42b2q ZS4ZA L(2a*42b)(Zb4Zc)4Zr2q ] 'll-"
Tba value of Ig In Pigure 2 will be
( 9 ) 12 " 11 ZgfZgtZg
By substitution from Equations (4), (5), and (6),
z „ I12ASG ( 10) lZAtZB)(Zy*Zo)42g2o
Equations (8) and (10) give the currents flowing In Figure 2, while the Impedance arms have the relations to the arms of Figure 3 shown by Equations (4), (5), and (6)» The currents flowing in the network of Figure 3 will next be determined. The Impedance of Zn and Z* in parallel is This, in series 0 R 2Bf 20 with ZB, constitutes one of the branches between terminals 1 and 3, the other braitoh being ZA. Therefore the impedance between terminals 1 and 3 w ill be W q ) = W b ■'T&b+Hq . / i (ID ZRZC 2A*zb‘ 2E4Zq
. The current Is thereforo
: . l ' - — 5L- 1 2g 4Z15
Zg + za*2B 4 ZR4ZC ______B ___ (12) ZafZ, A (ZAfZBK % 42c )t% S c
Hew the current In Zg will bo, z I
and.
Therefore (13) (2 A^ZB ^ G ^i‘2H2C
I t w ill be noticed at once that Equations (8) and (12) are Identical, and a like statement can be made for Equations (10) and (1C). By the symastry of tho equations, it is apparent that the transmission networks are equivalent when the load and generator are each connected to any two of the throe terminals. Since 3 , Zg, and Zg are general values, the networks of Figures Z and 3 are equivalent in all respects. If the net works are enclosed in a box, with only the three terminals carried out, there would be no way of distinguishing between them at the frequency for which they are equalized. It should be noticed that while the equations for equivalence hold good at any one frequency to give relations specified in Equations (4) to (6), the induetances, capacitances, and resistances required to satisfy these equations may vary with frequency. It is frequently desirable to make the conversion in the opposite direction, !•©., from the T to the TT form. From Squat ions (4), (5), and (6) the following may he obtained:
221^2*2223*^123 (2A*ZB+Z0)2 ZiZ2tZ223+zl z3 2asbsc (14)
Divide Equation (14) by Equation (5) on both sides;
(14) Z122»22Z34.Zi 23 (5) 22 2A (15)
Similarly (Mi 'h&zs?3&izz (16) • 161 za
- — i M ± z ?*l £ 3 , = z0 (17) 21
By means of these equations any T section can be converted into a TT section—a means of simplification which will be used considerably la te r. CHAFi’EH 7 REsomir cimmiTS m m«ORKS
In networks used in cmsminioation system, icmodanoes are dtie chiefly to the induotames and capacitances of the network. Inductive reactances increase and capxcltivo reactances decrease with increased frequency. Therefore, at higli frepuemles, even the inductive reactance and the capacitive susceptanoe (l.e., negative reciprocal of reactance) of a length of wire connecting two elements cannot be neglected. Use, can then be made of "resonance" (i.e., neutralization of capacitive and inductive reactance for series resonance; or neutralisation of capaci tive and inductive susceptance for parallel resonance). Because of the fact that one type of reactance increases with fre- -queney while the other decreases, the total reactance or susceptanoe w ill be zero at one particular frequency, depending upon the values of induc tance and capacitance. This property of alternating current circuits would indicate that by the proper combination of inductance and capaci tance, a circuit can be designed which will freely transmit certain fre quencies and greatly impede others. This enables the use of a single medium, such as a telephone line or free space, for the transmission of several messages simultaneously, selective circuits at the receivers picking out those bands of frequencies associated with a given message for routing it to its proper destination. , In a complicated network, either type of resonance may occur at dif- -1 5 -
fereut frequencies one or more times, the nu-hor of such resonant points depending on the number and character of the impedance elements.
Reactance and Reactance Curves
The reactance of an Inductance is = tuL and that of a capacitance is Xq = - It can be seen that the inductive reactance is a linear function starting from zero for direct current; whereas the capacitive reactance has an equation which forms a hyperbola starting at -oo for zero frequency and approaching zero when the fre [uency becomes infinite.
Figure 6a shows the variation of reactance of a condenser with frequency.
Figure 6b is the corresponding curve for the reactance of an inductance -1 6 -
ri.^ure 60 is a combination of the other two curves and shows the sum total of the circuit reactance* The frequency at which Xp crosses the abscissa is the series resonant frequency of the circuit*
Susceptanoe
In parallel resonance, two equal and opposite susceptanoes oppose each other, so that the admittance. Instead of the Impedance, is a minimum at the resonant frequency*
In such a case It would be ranch more desirable to plot susceptanoes
Instead of reactances* The curves of Figure 7 show that the susceptanoes
Vi£. 7 .-- sketch of r* $cn;int ^ i i ^ -1 7 -
of parallel resonance are olnllsr to series resonance reactance curves* Figure 7a shows both sweep tame curves with the total susceptanoe. Figure 7b is obtailed by taking the negative reciprocal of the eusoep- tance, thereby forml% the total roactance curve. It may often be found useful to sketch reactance and sweep tame curves and carry on such transforaations so as to arrive at the total reactance of a more complex circuit. CHiPBR VI I1FI1I2S UTJS
It often happens that transmission lines are many miles in length. Under such conditions it is observed that neither the voltage nor the cur rent at the receiving end is equal to that at the transmitting end, even though there is no apparent intermediate connection. Here the admit tance between the wires is of such a magnitude that it 1ms an important bearing on the current and voltage delivered to the load.
Characteristic Impedance For purposes of analysis, consider such a lino of infinite length and measure the impedance between the two conductors of the lin e. If several miles of the line are cut off, the line will still be of infinite length and the impedance measured w ill be the sane. This is known as tho "characteristic impedance’* of the line, and is designated by the symbol Z0* Duo to the fact that the input impedance of a line terminated in Z0 would be equal to that of an infinite line, terminating any length in 20 w ill produce an input impedance equal to Z0 a t the other end. For practical purposes this could then be considered an infinite line. A transmission line, like any other network, can be represented at any single frequency by a T section. Due to symmetry, the impedance of both series arms would be equal, mad could be represented as shown In
Figure 8. As Zg, the load impedance, is increased, 2 ^ , the input -1 9 -
iraijedanco, will increase from the short-circuited value 21 ZiZo % = — + ------(16) JSC 2 Zx+2Z2 and w ill ap roach the op n-circuit value
(19)
How when terminated in Its characteristic impedance, tlie input impedance will also be Z0, and therefore this value where % » Zin will be 20«
AVW\A AAAAA/V
Fip. 9.--T network .vith t°ryi ~ 1 o' d i - ^ "e
I f in the circuit of Figure 8 , Z is made e%ual to Z0, then will also R be ejual to Z0 and under this condition 20
( ) Zln = *0 “ T v z2 *■ ~r ♦
Clear of fractions.
+ + z* , M i f _M + Mo t _M2 1 Z22c- "2 4 2 2 21 20 “ ~ 4
Zo - i Zjzot ^ ( 20)
In tlie case of a 7T sectL e 9,
A/WWWWVW\A
vi r. : - 3y n etri 7T T t •.n r1-. , t 1 ZZp+Zr, 1 2in * “ SSgZo 22 g^Z 1*^ 2ZZ+*0
2z2z0+z1z0+ 22^‘‘°. .. = azsZit 22gtZ0 2Zg*Z0
422Z02-#-Z1Z02 - 4Z22Z1
2^(42^^ ■ 4222Z1
s -i/iS 0 V z ^ f l 2 4
'222Z ^
2i V
21"2iZ i£a. (21I % V21Z2+ 4 where 20j applies to a T section with the same values of 2^ and Zg.
If this is written in terns of admittances zT2" Z1Z2* - f T ' « ....- V OT 2, Z1Z2
. V ^ F F 4
% « 1+Y2_ APS 2
$ (21a) iflilch is so far as form is conoerned. sim ilar to Equation (20) for 5 networks• If the open- and short-circuited inpedanees of Equations (16) and (19) are multiplied.
Zoc2sc ” * zlz2 * zoJ
Therefore
z0 = voc v sc ( 22)
Since any network can be reduced to a T configuration without altering the open- and short-circuited impedances, the characteristic impedance of any network can be determined by these two measurements.
Attenuation In general if a voltage is applied to a $ section with a terminating impedance, a current w ill flow* The current at the end w ill, however, be smaller than the initial current. If the terminating impedance is equal to the characteristic Impedance of the line, the ratio of the two currents will be a constant h —i - = a . (23) : ' ' ' X2 - . - - ■ ..-■■■•______* Except if standing waves are present. If a long lino ms considered as being connoaod of a series of similar T sections, the ratio of the input of each to its output would be constant.
fl (24) 12
The ratio of the input current to the output current at the end of the series would be the products of the individual ratios.
I1 I s
So for n sections terminated in Z0
I 1 n a ( 26) In+1
The numeric a11 is called the "attenuation factor" of the line, but must not be confused with the "attenuation constant" to be introduced la te r. Except in the case of pure resistances the ratio of 1%/lg w ill be a complex number indicating a change in both magnitude and phase. In the T section of Figure 8, the ratio of the currents would bo
Zt h _ -g+- Zg + Zp (26) z2 Z2 provided i t was terminated in its character is tio impedance. In terms of an exponent of e, the relation would be - 24-
n f i * A 2 (27) ' h oV = i-vz2 2Z2 where y is called the "propagation constant" or "cosplox attenuation constant". Its real and imaginary parts are defined by the relation
r - o ^ / s (28) and it can be calculated from Equation (27) y = log.:{l' % fit (29)
I f it is w ritten in the polar form as A//9 whei'o A is the absolute value and yd the angle, then ' .
y ” lo^ Ae^ ■ loge A ♦ log0
*= lto$e A ♦ J/8 . ’
Tli© attenuation factor Ii/l2 will actually consist of two factors, a factor e* which represents a change in magnitude, and a factor e^whioh represents a p’nase. shift. o( is called the "attenhation constant", and the "wave-length constant’*. The "attenuation factor" as previously defined equals o* for one section and ene< for n sections* • Consider a transmission line of length &£ • Let Z » impedance per unit le^th along the lino -2 § r Let Y » aclmlttanoo per unit length acres6..the lin e, The values of 2^ and 2^ wuld bo 2% * 2 & JL Z z' i h - . To find the characteristic impedance Equation (20) is applied. ( 2 4 i r ' Y 4 Since a line can be represented by an infinite nuu&er cf 5 sections each of length , let approach sero. Then (31) which is the equation for the characteristic impedance of a distributed lino. By tlie expansion of Equation (27) by the binomial theorem l/2 5/2 * , 2, e = 1 f- ——* ♦ © * 4 © * - If represents the propagation constant for a length by substitution of the values of Figure 10 in the last equation. e*A< = 1 ♦ (2Yl^V«. — — ' ^ # # # * (32) 2 By means of Taylor’s expansion the following would also be true. y _ 2 3 o « 1 -* Y 4 —— t ___ ■f • (33) —26— vi p . 10. - - f ifrtirn rer iPpent. i rr So by dropping hitler order te rra s and equatin’ the first three terms of Equations (32) and (33) 2 l ♦ 4-^ + ^ Then X i,” 4 -£ (v4) So for unit length, the propagation constant becomes V - =<+ i / 3 With this in mind, tha attenuation constant and the mve-length constant can be derived by -perfOnslag.ths ep em tien of extracting the square root of ZY in the polar form and reducing the result to the rec tangular form, The distance between two positive maxima along the lino would bo termed a "wave-length", A, and would occur for the number of sections where the lag was 27T*radians* So Ay5 = ZtT and A = SlJL lgg| - / 3 Also V m '±L (Mi f t Because the wave length Is derivable from , the latter obtains its mme of the "wave-length constant", and is expressed in radians per unit length. The units of o( are called nepers per unit length. * A is the wavelength expressed in number of sections CHAP5ER VII IKHKJDOGflOl fO FILftSS As has been previously Indicated, oomaualcation systems often tend to discriminate between frequency bands, accepting one band and reject ing all others or perhaps rejecting one band with the transmission of a ll others. Combinations of electrical networks which accomplish this purpose are called '’filters*. She ordinary resonance circuit also tends to make such distinctions, but this is usually for only a very narrow band of frequencies. The filter, on the other hand, can be so designed as to transmit frequency bands of any width. Such bands are termed "transmission" bands, and tiioso tiiich are not passed are called "atten uation" bands. Theoretically a filter should produce no attenuation in the bend desired and an infinite attenuation at all other frequencies. In practice such an ideal case is never realized. I t Is often possible, however, to come relatively close to such a situation by increasing tho number of elomonts in the filte r# On a commercial basis, tho "law of diminishing returns" tends to lim it tho number of elements so as to produce tho re quired attenuation with the least possible number of olocents so as to keep the oast a t a minimum. Figure 11 shows curves illustrating tlio actual and. ideal cases. - If resistance could bo eliminated, tho construction of filtors might -2 9 - be a more almple matter; especially since only Inductances am. capaci tances are needed to construct such filters* 'fne visual procedure Is, ' *4 deal filte r ^-tr nor isoion bard-# ?ig. 11. --Attenuation vcr Ideul un Artual v; L ‘ " - - ..... - - " - -...... - ^7^ therefore, to confute all filter constants as though no resistance were present anti later if it becomes necessary, to make corrections from the theoretical ease* I f only pure reactances /ere used there would be four possible combinations of signs for the open- and short-circuited impedances*~ 2* W* L* Bverltt, Communication linearint Condition 1. K*) = f s3C = -jXa Condition 2. = -jX& Zso = > jXa Condition Z, Zqq ” +^Xa S00 = +jX& Condition 4e Zqq ™ “JXa Zgg = — jXg If either condition 1 or condition 2 o zists, ’ a0 -V -j% x & (571 Then the ohar actor is tic impedance wonld bo a pur o r os is tance . If, hen?- •wer, the condition for either 3 or 4 "holds, %0 ( 58) the characteristic impedance will be a pfliro roactanco. From tliese conditions it is possible to gonerallao. If the im pedances are reactances of opposite sign the characteristic impedance is a pure resistance; if of the same sign, the dharactoristlo Impedance Is a pure reactance. It is, of course, possible for difficulties to enter due to the fact that the open-■and short-circuit impedances of a network may cliange with frequeney, so that over a certain rargo the charac teristic impedance may be a pure resistance while over another it may be a pure reactance. c n ip ra m ...... losses i r slem si^ Insertion Losses • ' ' ' r It is Important at this tins to consider losses In eleetrloal net works. Consider Insertion Iocs, which is the loss resulting when tho network is introduced between a supply and a load. She to tal insertion loss of a filter is composed of the attenuation, plus interaction and reflection losses. So if these three components could bo found, the lossoo in a f ilte r can be determined. Interaction losses are small compared to tho other two and are, therefore, often neglected in practice. The attenuation constant CC5h<* ’ 1 * id F rrn , 1391 wlioro a ^ /l- (low pa««) or —2 (hlsli paai), at - —l. (low pass) • :fo or -SL- (hig^ pass), f = frequency at which attenuation constant is to - f: be computed, fc » cut-off frequency, and - frequency o f ;infinite - ' attenuation. The accompanying chart of Figure 12 gives the attenuation constant 3. A good discussion on "Insertion Loss in Filters" is given by J. Krlts and 3. L. Oruenborg, Floctronios, 14, 45 (March, 1941}• Equation 39 in its present, form has been taken from here and also Figaro 12. The sig nificance of m will be discussed later. - 3 2 - Insertion Loss in Filters — 19 _— 70 is — — IS r- 3.0 — 5S — 4.0 — so -----00 O /wo »r y «rr /» Of a simple 7T or T section, and either high pass or low pass, in terms of a and or m« From this it is possible to do term in© any one of the three variables, provided the other fc?o are given and a straight edge Is laid across them* The point at the Intersection of the third line is the value In question. Reflections It lias previously been Intimated that If a line is terrain ted in its characteristic impedance, no reflections occur; w'nereas if it is terminated in some other impedance, reflections will be present* Con sider the latter case as shown in Figure 13* I s I R Zo E s Z s ? ig . 1 3 __ Analysis of line not U-rr:inated ir 5V. 34- Both the toput ani terminating linpodanoe are shewn as being composed of two parts .4 One part Is the characteristic impedance and the other ether Impedance which may be either positive or negative and any phase relationship. As a result, it is evident that . % - V ZB (40) and ' , . -s * 20« s (41) By means of the compensation theorem®. Figaro 13 can be modified by replacement of generators in place of impodances as shewn In Figure 14. Lot Ia be the omponent of the sending-end owreat dm@ to Eg oM Ea (initial wavo). ’ I Ijj be the oompomnt of the receiving-end current due to 3g , aM Ea (initial wave). Lg be the oonponont of the receiving-end current due to % " , (reflected wave). . 1 ■ | | ’ '■ .7 ' " • ' : ' ’/ ' ' . I 5 Lo the oomponont of tho sondlng-end current duo to • ' (reflected mve). ' ■ " ' ■ ' ; . - ■ . . , Throughout this entire discussion, if the subscript o Is added to an impedance! i.e., 2S) it refers to the sending end. If the subscript E is added, it refers to the terminating or receiving end. 6 * The compensation theorem states tlxat any impedance in a network may bo replaced by & generator of sore internal impodanbo, wiioso generated voltage at every instant is equal to the instantaneous x>ctcntlal difference produced across the replaced impedance by tho current flowing through i t . . 14— Circuit equivalent to network of Pi . I ? . 1£' and IjJ are due to the initial wave traveling from left to right in Figure 13 and I:,” and IE are due to the reflected wave traveling from right to left# Then ft (42a) • -36- l * = gfiffa = sfr 1sBs * = 143a I 22, v - 1 .Q-^» lk_ o-^ (43b) li 5 pyfso i0n =_& »- M ai - - fe(zr M (430) ** 2%. 22„ - V *1 - - (43d) V = % * 2Z„ From tiiese aquations it is possible to set %i single equations to obtain the sending and I’eoeiving currents* Shea W - ( '/ —2 1 T — T_ t «__ Ia = is. t is" , (44a) 2ZC. IB - 1 ^ 4 = D v-tZ 5-Zr,llJ o ~ So by the solution of Equations (28a) and (28b) as sinailtaneous eguBtions, (45) (Zo^H)(Zg^2o)e +(Z0-ZE)(Zg—Zo)o ^ B 2o+ZalG^t(Zo-zE)o -^ J i . ■- (2o+Zh) (Z8+20) O^riZo-ZB) (Za-Zo )«“* ^ With reflections so tliat Wo waves are traveling in opposite dlreotions, it is often rach more convenient to express relations in terns of hyperbolic functions, since algebraically and in th® coBqjlex form they represent Wo waves traveling in opposite directions. -57- By definition these are alrih'-ac « ...... (4fj ■' . 2 cosh x = X,m (40) 2 tanh x = .&Wl,Z = cosh x ' Oxfo-x (49) If the functions are applied. Equations (45) and (46) ______.V ■■■■ ' ( 60) (zHtZs )cosh2f^t (Z0t 2s5&J s lrily ^ Eg( ZpOOShy^ 4Zt;S inh tJ] ( 51 ) Z0( ZH+Z®) c oshfj+( Zo^ Z rZ s )sinh )tJ. Often It is much more useful to laio?/ t’m sending- aM reeeivlng-ead currents In tense of the voltage at the sending-end Eg rat tor than the generated voltage. If the impedance of the generator is zero, the gen erated voltage Is equal to Eg. If in Equations (50) am (51) Zs is equal to zero. • & - (52) Zjic o sh +2 0s inh 'tJL" I - ^(ZoCOshJrlfZxtSln-hyi F (63) Z0 (ZBco s h y ^ 20sin h ijf) By use of the last equation, tho input impedance dan be found. Zin « Ss a Z&(2Beoshjtft-Z08lahy^) (54) n Is (ZodwhX^fZgminhY^) -3 6 - Bils gives the input lnqpedanoe of a line terminated In Zg. I t i® Often desirable, however, to find the input ispedance «hen the lino is either open- or short-circuited at the receiving end*' In such oases % becomes infinite and zero respectively and so^*7 2 se-= Z^ -% « 30taah (55) cosh (56) zoc - V 111 Sow multiplying those two equations ^so^bc “ 2o • and - ' 2o * This is identical: with Equation (2 2 ) whioh was derived for a 2 section. In a similar way tanh ( 6 8 ) "ttm tanh ^ (59) low if it would be possible to express the components of T as real functions, of and/? could then be determined from tables of these 6 * The subscripts "sc" and "oc" refer to short-circuit and epea-circult conditions respectively. 7. Equation (56) is derived by changing Equation (54) into the form ^ . .Minting 2r -QO. sirih Tf+Zo/Zn cosh values. To do this lot tsnh Y a '^ jb (601 ■ "'CC then A -f JB g s InlioC opg/J t j cosho< sln/5 ooshK oosy^-j sinhofBin/S A t jB = (slnW ooB hO (ooG ^?t8ln^ ) t j (s iiydbo^d) (oosh%r-s lhhS<) oosh^cos^Stslnh^slnyg A t jB 8lnha equating tho real and imaginary ports of this equation. A = sinhofooshef sinh 2 of sirih2e< toos^cr 2 (sinh2*arfcoSyS ) P sln^ cog/ g sin ZjS slnh^of ^ oos^@ 2 ( to o s^ y ) A24B2 - | 8lrih^ U sirih2g<.» sln2 >g I oosh^ ¥ I slnh o ■i'">-2— * 4-i cos^/S— 2 1 + A2 t B2 = .-8?.M & .t l -_ g SM" 2 « s in h e r e oa einh2oC*coB^9 1- (a24 r2> . ow^-sin2^ _ ops 2 /g cinh^teos2^ sinli^+cos2^ Therefor® tanh 2 o< 2A ( 61) 1 4 (a2 4 B2) -4 0 - ^ x - u W 1621 From the la st two e^mtlenc real functions of the conponente ©f Jf can be cosputed, and e< and yS are then deterninod from values found in tables of these fmxstions. r ’ If the absolute valuo of Equation (60) Is used. It ©an be teen tint by substitution into Equation (58), A* 2 T B2 » / Zso i : ^ •' . (63) Pool :: ■■ It can, therefore, be seen that all tho lino constants can bo determined by measurements of only the open- sad shert-olrsuited impedances.^ ; ' ' /. : : - " , - As has been shovm before, the propagation constant of a no Work is given by the equation trnih V « a O sc” (64) So if either condition 1 or condition 2 holds, tahh T - ^ (65) 7. If 20 and y' are known, 2 and Y con be determined from the relations Of Equations (31) and (34), i.e. and T = ^27” ." *lhw 2 » 20¥ and Y = JL . As indicated in Equation (57), Zo=1/%oo2sc whereas if condition 3 or condition 4 holds, tarih T So if the characteristic tapodsnoe is a paro rosistanco tho hyper- boll© tangent of the propagation constant is a pure Imaginary, while if tho characteristic tapodanoe is a puro reactance, tho hyperbolic tangent is a real number. From definition before . tanhy« tanhU 4 1 ^ 1 g sirih y , sihh«Kcos^+j coshKsln/f cosh y cosh If the denominator is rationalised and sim plified, tanh tf- sihhXoeshX^j sin^cos/g 6 inh2<<* cos^yd' t67) If tatih y is a pure imaginary, them sinhe this can be true only when e< = 0 , so there can be no attenuation when the characteristic impedance is a pure resistance. If, on the other hand, Utah Y is a re a l number, SinyticOSyS = 0 (6t) Hence, if tahh V has a value, siahoceeshoC must have a value greater than zero and. there mat be attenuation. Therefore, in a symetrloal network of pure reactances, if tlio charac teristic Impedance is a pure resistance the attenuation constant is zero, whllo if the character istie impedance is a pure reactance there must be a value for the attenuation constant. CHAPTER IX FILTERS Reaetanco Curves is has been shovm—if in any case, the characteristic impodanco can be expressed as the square root of a product of two terms (i.@. Z0»fZocZso or any other such form), when the reactances are of opposite sign, there will bo no attenuation. For such a case, there will bo a transmission band. If on the other hand, the reactances arc of equal signs, the signal will be in an attenuation band. Such an example is shown in Figure IB. The points marked f^, fg, f 3 » and so forth are critical froqueaoies and are points whore a curve goes either through aero or through infinity. It is at stxih points where the conditions may be fulfilled so as to change from a transmission band to an attenuation band or vice versa. I t can be seen tliat tiiore can be no attenuation between fj. su^d f^i since neither fg or fg is a cut-off frequency and at every point the reactance curves arc opposite in sign. Between zero and thoro will bo attenuation, and again between £4 and fg. It can bo seen from this discussion and the figure, that cut-off frequencies occur when there is a critical frequency for either curve which does not coincide with a c ritic a l frequency of the other curve. Often, it may be difficult to determine the open- and short-circuited Inqpedanoes. I t then becomes more desirable to use another equation to -4 5 - determlne reactance curves. Elie equation jireviously derived for the dhsraoteriBtio impedance of a T section would bo ideal. ZoT " V W ^ . (70) This may be rew ritten in a form so as to contain only a product . ■ • ■ ' : / ’ - ^ \ ^ - ' Of two terms under the rad ical. The equation would then be - ' ■ / ' ' ' , . ■ ' - 2q2 « • (71) Since both 2 q aixl 2g can easily bo measured, the reactance curves can be drawn and the transmission and attenuation bands determined. Cut-off frequencies would occur under my of the following condi- tions$ Zq » 0 (Zg / 0 or eo ) (72) 2q 2g 4 m 0 (Zq, 3^ 0 or oo ) (73) Z2 ** cx3 (Zq / 0 or«X> ) (74) For the case when Zq «=oO , there wuld be x10 cut-off fr^uenoy ’a, . ' . ’ since Z2* -“ would also equal infinity and the critical freqtaemies - - ' 4 - would coincide. Low- and High-Pass F ilters, If reactance curves are made from low-and hi^-pass filters, they will appear as shown in Figure 16. The cut-off frequency.in each case, could be determined if tlie curves were drawn to scale. They might also X Trensn issi on l*Attemiaticr>e”Tr' nsr issi cr C• 15.--Determination of transmission and attenuati'-n b nds of f ilt e r network fron open- end short-'*!roui ted reactance curves. 6f4 r ’W t ~ A r, 6 r, r, 2C, L,k L,n 2 C, 4I- T-4I- Q T / jw p a ss High p a s s Two band pass (a) ' a>) (c) Fig. / 6 . —Analysis of filter sections by reactance sketches. -46*' be oaloalafced. Gonaldor tho lav-pass filter of Figure 16 a Then (75) ttVZT which Is tho cut-off frequency of a low-pass f ilte r . In case of a high-pass flltw , = - 4J(4)L w G _ 1 " ' ' V- " f. (76) It Is therofm*® posslblo to celoulate out-^f frequancy of a single soot Ion low- or high-pass filter if L and 0 ere givon and, con versely to find either tho value of L or G if the other Is given and the desired cut-off frequency known* ; I t is somewhat unfortunate, but Is found that the ol^zaracteristlo linpodance of either a low-pass or hl^i-pass filter varies wl#i frequency. In the caso of a low-pass filter (l.e.f Figure 16a)e s luco L = - l — G 2ot * Ye C1 * Zq^ refers to 20 of a T section. oT SI (77) <.2 7T (78) 1/ f02 If valuos ire applied to trvose relations it is found that the characteristic iwpedance varies as ehcsvii in Figure 17. -4 7 - As lias been Indicated previously^ a simple low-pass or high-pass filter will not have a sharp cut-off frequency in practice. It is, • J howover, possible to greatly Improve the response by addl% another ! : - V-*^- : element;as shown in Figure 18b. Figure 18 shows the contrast in res- - y ;c > i ponse of the two oases. The second case has an extra inductance in the f '" ' ^ i • "connecting" branch. Tills causes this .part of the circuit to be resonant j : - A j I at a cer tain" froquemiy,. and Gierefore offer a very small impedance. This ! : produces the sqim effect as a;short circuit across the line, and there- i fore thore-is an infinite attenuation at that frequency. In order to calculate the value of the now impedance which is to be derived and added, o onsldor tlio characteristic! Impedance of a T | i ^ o section. f;. h! In tlio derived type le t tlio ibw branches bo called 3£ and' 2^ and its cliaracteristlc impedance Sq*. Consider thatA^ and 2q* arc related by the equation ! i J h ' (79) Since :tte characteristic impedance must not be changed, the problem is to findf a 'configuration of Zg* such that 20 * 20*. If , tint is the case 2 ,* 2 m23 2 % ^ 2 j_*2 n* •t * m2q2 2 * . 4 L‘/l L'/l l-'/i. L,/x i Jl3 vifr. 19__Variation of attenuation with frequency in two types of low-oass filter. -4 9 - ^ (80) m If this is tlo case, tine diiar iCtor is tic impedance and the cat-off frequency of tW derived struotcre are identical to that of the original one. It can be seen that toe new impedance is coErposed of txto parts as was to be expected. ‘fiie ordinal T configuration siiall be called the prototype and the new ono will be kncra as the derived type. Figure 19 tiiows both typos with the proper values of JUapedance. Z1 '2 Z1 /c Pro ♦ o e "Deri ved Tvne Fir. 1Q.--T sections .vhic'n he ye identic'1 characteristic irrpedances. Networks with Identical characteristic impedances and cut-off fre quencies can bo connected to^cthor and be froo from reflections. It Is .tierofore possible to aad sections to tbe orl^lnol prototype and In : : ^ ... V- . . ' ttiat my improve tho notion 'of-the f ilte r , ospecially around the edges ' of tie transmission band. - , - . ' ' . ; ' - ' ' ■■ ■ .■ " ' ' ; . ■ . : for a lew-pass filter, the frequency of .infinits attenuation will occur at the frequency at idiich the shunt arm becomes resonant* The frequency of Infinite attomtatlcn is then given by f . - Z-rM LC i r y (l-m^lLO = ( 81 ) ” 4 - % r For the high-pas;; filte rs foo » ■C Z . 410^15” I t - ( 82) ' i From these relations it is possible to obtain thovaluo of m for the derived section; since only the cut-off frequency nni the frequency of infinite attenuation need be kncwii. .’.Filter' sections can bo dorivod Just as easily in the fern of Tf - b i sect ions as T, In that case it bo cocas much easier to us e acmi tta o c es than impedances .nd co. verting back in t o l;*> • or to 7T configuration H ' maZl * z2 (83) The circuit would appear as sliewn in figure 20. xAAAAAA ?' 2 0 .--IT seotions (fhieh hpye identic®! For oost cases, a single f i l t e r section provides enough attenuation It is, however, possible to join together a number of m-derived sections with different values of m and still have no reflection. Tiie values of -5 2 - m be so ohosuii that tie re ere a series of Infinite attonufitlon freiuenoios ozxi Ven for all tracLiopJ purposes, there w ill be infinite attenuation ever a ran,je of frequenoiese It if difficult to terminate a series of sections propsrly, but if an m-derived unit is divided into half sections and used to termini;te the filter, it is possible to maintain the in^iedance match to the filter figure 21 sows tin variation of the character Is tie impedance of a low-pass filter with terminating tialf sections of different m-valuese Tr t '' .er?y in r Cent of Cut-f f f ? lp . ? l ___Che.rrcteri ti- if •• f of 1o*-: ■?s f i t t - r e . It can be seen tfiat tiiO most oonstsat charzxcteristio i;Kvdance occurs if fete filter is ternimted in an n-dorlvod TT-eection viith : a « 0 *6 . fhorefior® no m tter v/iat tto different sections may be* the a-derlvod tormina ting balf sections shall liavo an m o.mol to 0 . 6. ' If L- asul O', are 'th® -vaMee'-fW the $rototypo, and Zg for the tw^md low^ss type will-bo ' - ' ' - ' ' ■ V ' . \ Zp* = jntoL Fran Equation (76) Oiiae 20y is the sane for the prototype and the derived type arranged as a 2 section, Bamfelon (77) may be used to dotormino the characteristic Impedance of the derived type rearranged as a IT soetion. (84) From Figure 2i i t can bo coen that if m Is equal to 0,6 for ter minating half sootlons, the character!#tie impedazsce, (in the ease of a Tf section), is constant unt il close to tM cut-off frequency. As a result, if tie filter Is terminated in a constant res is fcance, .there will be no reflection unt il very near cut-off if tie characteristic impedance is yl/O , which is the value at mto frequency. It is therefore obvious that the purpose of the terminating half sections in filters is to change the characterietio impedance of the filter from a value equal to that of tin 2 section, to the acre desirable value obtained from certain "77* sections. With these facts in mind, it is now possible to compute the values of low- end high-pass f ilte r s . For the low-pass f ilte r , 1 (85) ____ 0 » 7T fcR ( 8 6 ) For the high-pass filter. R * 1 - 65- L » R 477 £-0 1 . C |8 8 ) 47TfQlt If it beoomas necessary to worlc vory close to the cut-off frociusncy, the terminating half section may havo to bo terminated with a mm#lor valve of to so as to give a larger margin in vdiioh to wo*. Band Pass Filters Low- or high-pass filters have only a limited mazdber of uses* Some times i t is desirable to pass two narrow bands of freqnenoies as shown in Figure 16c. In such a case, there are two transmission bands separated by an attenuation band, and mother attenuation band outside the two. Such a filter can be made into a single band-pass filter if the frequency at which 2 % equals zero aid the frequency at which Zg equals infinity are made to coincide. Shis can bo true if L1G1” l2°2 (89} She filter then becomes a constant K filter, for *2 Op j WLg 22 (1-te 2 L202) j(WL2“ * Whenever a filter is built of S sections, the terminating section is charged to a TT section and sp lit into two parts. For a rr filte r the opposite is the case. *56** *- • 2 % m i: (90) Cl ge At the mid-frequency, i.e ., the ©no vvhoro 2^ = 0, the t^iarecterlstto iinpedanco is oq^ual to and tii® bend-pass f ilte r is terminated in its constant K vain© as is the case fa? the low- and high-pass types* At the two cut-off frequencies, for the T section. Z 2 - 2 12 2 = - B2 4 (91) where B is the terminating resistance for iftiich the filter is to be designed and equal to K. From Equation (91), at the cut-off frequencies. (91a) If tu q * the lower cut-off frequency ' - - . - ' ujg ” the upper cut-off freqwmy m « the nidfroquency where 2 q «• 0 , “ 00 2 a —i --- = __L (92) &1G1 i*2g2 From Equations (91) and (90), 2R « 2 V2 122 ” = *jZq (at cut-off frequency) - <^ili (93a) A constant K refers to a constant product of Zq and Zg* 1 1 ' i f ° " = k . - a,iLi (93b) From Equation (95b), ^ i0! - 2 U,ir^ 2®i - 1 ■ 0 (94a) (94b) W22l101 “ 2 &,2 ^ ^ 1 “ 1 s 0 By the binomial theorem ^ ^ 201+^ 101 % A/jL^LiiC^ - VhgOi (95a) Ll ° l ' and similarly, Y (Ii24-Li )C^ •» YlgOi (95b) L1C1 - (L2~L1^G 1 - L2° l _ 1 (96) L12c12 Therefore (96a) ” "m2 The mid-frequency is therefore the geemtrie mmn of the two cut- ®£f frequencies. i ' The data v&ioh are usually given for the design of a band-pass filter are the eut-off frequencies and the resistance Into which it is to operate. From these data the values of Lls Clf Lg, hnd Gpoan bo cosputed. From Equation (92) the follewlng relations can be deducedt 1971 Substitute Equation (97) in Equation (93b). 2R = ' L ,_B. . ' ■ (98) T T l f ^ l Substitute Equation (98) In Equation (97) and obtain o j.-l/.r /.11..;;' (99) 47rf1f2R From Equation (%)) and the following explaimtion, a 2 - i 2: = h . . °2 ° i Therefore B(fg-fi) (1(X» 2 ^ 1*2 -6 9 - (1011 2 TTlfg-filB The banl-paes prototype can now be designed from : [nations (98) f (99), (100), and (101). From the prototype an mrtierived type can also be obtained. In tij^ure 22 are shown the arrange went and relations of the m-derlved band filter. becomes zero at a frequency in the lower m L, *£, / m m jZz z -C, /-/ — 7nT v_<—» P?. --rr-^ierlved hr' filter attenuation band and again passes throu^i zero in the upper band. At these frequencies the attenuation will become infinite. To solve for these frequencies of infinite attexmation, equate tiie shunt arm of figure 22 to zero. , Let W** be the notation for 6u at the frequencies of infinite attenuation. l= n £ / ^ oqL 2 b 0 .*■ X W eepi / n (l- OWLgOg) Stoo® LqGi * LgOg this becomes (l-m2)( dUee^iO!-!)2^ CUe^L^ « 0 (102) From Equations (99) and (100) T ..^- (fg-fl)2 „ 4TT2(f3-fi )g . (W2-6JX)2 (103) 167r2f 12f22 64 TT* ^ 2 4Wn4 Substitute Equation (103) in Equation (102) and substitute — <*»o2 for L^Oj. Then . c y w y IM4I (l-m2 |lt*J«»2- Factoring V lv ( tu^-cu/) t W^(Wg-W^) m 0 (104a ) 2 2 yl-nr [OJoo-C*^ ) - (Wg-W^) * 0 (104b) —61— Solving for W/) a V^)2-2^2^1-»Ol\4U-m2)^lt02 z i i ^ *(Wg-W%) ~ 4n^&)%Wg (104ol a V I ^ - Since CUe^ moot be positive, only positive roots have any waning, fho two infinite attenuation frequencies will be CJ - yi^2-^l>* “ -(OJo-cJi) ( 10W „ tf{<*>2rWl)* ~ 4m2WiC02 4 (Wg-Wi) 00 2 “ " ------(105b) 2 yi-ia2 By multiplying Equations (105a) and (105b) i t w ill bo soon that the mid-frequency is again the geometric mmn of the froquenoies of infinite attenuation. ^ u,w2 „ OJ22-2Cc>2fcUi- ^22-2qJ2UJ1-t012 4(l-m^) C-u^ l » CV1Cu)2 « tJm2 (106) So by proper determination of m any one frequency of infinite attenuation may be selected, and the other will be fixed by Equation (106). If W^, W2, and one tU w are given, the value of m may be deter mined, From Equations (104a) and (104b) the following can be obtained: 12.07) From these preceding relations the values of the filter elements can be calculated# A complete filter circuit terminated with half TT sections and with another m-derived section for a second frequency of infinite attenuation is shown in Figure 25. The inductances and capacitances are given in terms of the values obtained In Equations (98) to (101) and the constant, m. The value of m will be constant for any one section and will depend upon the particular design desired, however, for the terminating half sections it will usually be 0.6. If sharp out-off is required a small value of m must be used for one section. L-^vl L ■-derived t ectio n . CHAPTER X IHDUCfAlGE OOBSTRDOTIOm ■ So far in tho consideration of filter components, it has boon assumed that the Inductance of all colls was constant. It is found in the design of audio filters that the values of inductance must in some cases be as high as one henry. In such a case, it becomes necessary to use some magnetic core so as t© increase tho permeability. As is well known, the permeability of most magnetic materials varies with flux density so that the inductance depends somewhat upon the current flowing through the windings. Furthermore, if the current is alternating or varying as in speech, the Inductance varies throughout the cycle, which causes tho production of harmonics. Another factor to be considered is that eddycuirent losses increase very rapidly with frequency causing the so il to have a lower effective inductance. These two factors immediately rule out most of the ordinary magnetic substances which are usually used in inductance coils. If air-core coils were used, tho amount of wire would be tremendous, and the cost would also bo high. Eddy current losses con bo reduced, to a certain extent, by l aminating tho core. This is quite satisfactory at lew frequencies, but for high frequencies (even in the audio range), the losses again become largo. It has been found that much better results can be obtained if tho core is ground Into a fine powder, mixe curve 1 (iron core) 21---vor: •' ' r etizatinr. " rve Curve 1 is the normal magnetization curve of iron# uurve 2 is the mag netisation curve of a permalloy dust core made by A os tern lectric.^ i. igure 2b shows tW resultant permeability of the preceding curves". G & 9. From actual laboratory tests by L# H. Alldredge and author. 10. Suggestions for use of such cores were received from B.L.iiveritt by personal correspondence. —66- In order to build the Inductances so that they can be used for dif ferent filters, they were built in cascade form just as ordinary resistance ron cere J A' 25.--?* " e'ih;' \ dial boxus are constructed# This makes them of value In many electrical experiments other than filters# The permalloy dust cores which were used were toroidal In shape with 2.26 Inch outside diameter and 1#4 Inch inside diameter# First a coll was constructed having taps from 1 to 10 m#h. In steps of 1 m#h. This coll was made by winding #24 double silk-covered wire on a single core# The exact position of taps was determined by the use of an Inductance bridge# Next a coll was constructed having taps from 10 to 100 m#h# in steps of —67— IQte.h. This was made by winding the wire on tm cores tied together# Then a third coil was made having .taps from 100 to 1000 m#h. in steps of 100 m#h. Thfee cores were used for this, .;j%e combination of the three coils was mounted Inside a box 6^" s 4" x 4", and a bokellte top flttwi with regular radio dials. This combination provides any indttotanco from 0 to 1110 m.h# in steps of 1 m.h. CHAPl’iH XI . BAKD-iAS3 i'lUSm DSS IGI It was then proposed to design a filter, to get actual response and to oho cl: it with theoretical valm@@. She proposed characteristics were: .. - " - - ' ■ fl ” 1000 cycles fg ■'2000'cycles fool « 800 cycles B = 500 olms Terminated by Wing half TT sections with m = 0.6 According to Biimtion (I06j it 1$ found ttmt the nidfreqnency is 1414 cycles. This w ill also bo the midfrequency for the two infinite attenuation frequencies. Bow since one is already defined, the other frequency of Infinite attenuation is found to be 2495 cycles. With this in mind and Equation (107), the value of m which would be used in a derived section to obtain this other infinite attenuation frequency is found to be 0.81. V The fundamental values of Inductance and capacitance to be used are found from Equations (98) to (101)• If these are then sinstituted into the derived sections of Figure 23, the actual values of inductance and capacitance are known. This theti gives all that is needed to build the f ilte r . I t is , however, desirable from an economic point of view to combine any inductances or capacitances if two happen to be In series, which - 69- will in no ;ay change the cirouit* Tills then requires less elements and therefore less time and money to complete. As an example of such a com bination, refer back to FI nire 23. Consider the elements in the left branch of the prototype and back into the terminating half section. This Involves two inductances and the same uaxsber of capacitances. These are, from left to right, oLi/2, 2Cp/m, 2Ci, L^/2. Since ;all four are in series, the two inductances can be adued algebraically and the two capa citances by the law of addition of capacitances in series. These four elements are then replaced by only two. In this manner the filter can be simplified. The first filter constructed was a prototype designed as has been indicated. The resultsare shown in Figure 26. 11. o( = iog0 li/lg, where Ii is the input current to the filter and *2 Is the output current• iSAtibQ -7 0 - 1 or comtjar Ison, a filter with orderIved tormina ting half sections was then built# fhe circuit is shown in l i^uro 27# R e su lts of this f i l t e r are shown in figure 28# Qie circuit is id e n tic a l to that of Figure 23 except that the central m-dorived section was omitted due to lack of sufficient inductances• hie to this omission tnere was no infinite attenuation at 800 cycles. As a result, a small amount of current was transm itted through the f i l t e r i t frequencies below 1000 cycloo# -7 1 - “500' ” 1000 1500 2000 T?ig. 2°. - - A tte n v tio n ^ or Ooicpleted f i l t e r GMifBB XII SDUHAKY la deslsnizy filters, it Is first necessary to know the character istics of the circuit la idiioh it is to be used, and reasily the allow able range of frequencies around the cub-off frequencies, between zero and infinite attenuation, fills will date mine the number of section# to be used. The m *s will be selected to properly arrange the values of Infinite-attenuation frequencies. If a sharp cut-off is required a small value of m must be used for one of the sections. The end of the filter should be terminated in a half section of m = 0.6. A filter constructed in this manner, consisting only of a prototype and one m-derlved section arranged as terminating half-sections, prove# to be very satisfactory. BIBLIOGRAPHY Borst, John, "Filter Design Charts—I"; Slootronlost 13,8 {1940} Borst, John, "Filter Design Charts—II"; Slaatronias♦ 13,10 (1940) Borst, John, "Filter Design Charts—III"; Flnctronioa* 13, 11 (1940) Crandall, I.B., and KaeKemsle,D. "Analysis of the Energy Distribution In Speech"; Fhnleal Review; 19. 3 (1922) Everitt, will last L«, Cormmnication Engineer in;? ? licOraw-Hill Book Co., Ino; lew York and London (1937) HrIts, J., and Graenberg, B.L., "Insertion Loss in Filters"; Electronic** 14, 3 (1941) Stewart, George W., Introduotor? Acoustics: D. Van lostrand Co., Inc.; lew York (1937) Ter man, ?• E. Radio En.glnderlng: McGraw-Hill Book Co, Inc.; Kew York and London (1937) E5751. 1=141 -b l CE / ? < / / ^ / a 3900 1 00 1 28448 1 b J V/a C>d jf t V V s'