PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

Impedance Transformations in Four-ElementBand-Pass Filters*

R. 0. ROWLANDSt

Summary-In a recent paper by Belevitchl it was shown how afour-element band-pass filter with all its frequencies of peak attenua-tion located on the same side of the pass band, could be transformedinto a structure having, with the exception of the terminating halfsections, the configuration of a high- or low-pass filter, with a conse-quential saving in components. Although this was proved possiblefor particular cases, it was assumed to hold for the general case.

In the present paper it will be shown that, by using a differenttype of basic section for constructing the filter, a similar result maybe obtained with slightly greater economy, and it will be proved thatthis can be achieved in the general case.

INTRODUCTION

T HE THEORY will be developed for filters havingtheir frequencies of peak attenuation locatedabove the pass band. It can be extended to filters

having their frequencies of peak attenuation located be-low the pass band by the principle of duality.

Consider, therefore, the networks of Fig. 1. Theyhave been drawn as half sections so that the terms threeand four element become apparent. Fig. 1(a) is a three-element filter from which the four-element filters are de-

where R is the nominal impedance of the filter andfi andf2 are the cutoff frequencies.

Fig. l(b) is a series derived four-element filter inwhich

/f2 - f22m=

/ f2 fi2

f2so being the frequency of peak attenuation.This is one of the configurations used by Belevitch in

his analysis. Fig. 1 (c) is a shunt derived filter having anattenuation characteristic similar to Fig. 1 (b). Fig. 1 (d)is identical to Fig. l(c), except that the arrangement ofthe series arm is different. Its elements are given by

mLaCa2=

(Ca + (1 - m2)Cb)2

(1 -m2)C6(Ca + (1 - M2)Cb)

McaC2 = (C. + (1 -m2)Cb)/m.

Ca.¢<Law Ca. t-La m -mLa LI C2

mCLCb~~~~~~~~T2Cb C1CbnCv(L-)Cb -m Cb -mCb

(a) (b) (c) (d)Fig. 1-Basic 3- and 4-element filters.

rived. It has an attenuation peak at infinite frequency,and its elements are given by

RLa =

2X(f2 - fl)

Ca _ f2 - flIa =-

27rf,2R

Cb =27r(f2 + fl)R

* Decimal classification: R386.1. Original manuscript receivedby the Institute, November 17, 1948; revised manuscript received,April 27, 1949.

t British Broadcasting Corporation, Evesham, Worcestershire,England.

' V. Belevitch, 'Extension of Norton's method of impedancetransformations to band pass filters," Elect. Commun., vol. 24, pp.59-65; March, 1947.

This type of structure is the one used in the filters dis-cussed below. The full section is formed by connectingthe half section to its mirror image on the left-hand side.

THEORYSuppose that a composite filter is constructed by con-

necting in tandem a number of sections corresponding toFig. l(d), each having a frequency of peak attenuationwhich may or may not be different from the others. Itdoes not affect the argument how the filter is terminatedbut as it is in general desirable for a band-pass filter tohave a terminating image impedance of the second or-der, it will be assumed that the filter is terminated ateach end in a three-element half section. The filter willthen appear as shown in Fig. 2.To simplify the calculations which come later, each

resonant circuit has been designated Z. and the remain-

1949 1337

C, =

PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

Fig. 2-Composite filter.

Fig. 3-Transformed composite filter.

ing capacitors Spq, S being the reciprocal of the capaci-tance and pq being the meshes to which the capacitor iscommon. If meshes 1 and n are called the externalmeshes and the remainder the internal meshes, then theproblem becomes that of retaining the ladder structurewhile altering the impedance of the internal meshes insuch a manner that the capacitors S22 * * * S dis-appear.The first part of the problem is fulfilled by pre- and

postmultiplying the matrices for the circuit componentsby a matrix in which all the elements except those in themain diagonal are zero and those at the extremities ofthe diagonal are unity.2 If this operation is carried outon the capacitors we have

Since none of the Z's appear as a mutual impedance be-tween two meshes the effect of the operation is simplyto multiply each Z, by x2p. The transformed networkwill then be as shown in Fig. 3, where

Sl = Sll + (1 -X2)S12S2 = - X2S12 + X22(S12 + S22 + 523) -X2X3S23S3 = -X2X3S23 + X32(S23 + S33 + S34) -X3X4S34Sn = (1 - Xn-l)S(n-l)n + Snn.

These equations are obtained by summing the terms ineach row of the capacitor matrix. Since the shunt ca--pacitors, being common to two meshes, will appeartwice in each row, once with a positive sign and once

Sll + S12- 812

X 0

0

-S12

S12 + S22 + S23

- 823. . . . . . . 0

0

0 ...-0

-S23 . . .0

S23 + S33 + S34 *.* 0

. . . . . . . . . . . .

O ... Sin-l)n + Sen

S11 + S12 -X2S12 0 ... 0

-X2S12 X22(S12 + S22 + S23) -X2X3S23 0' ' °

= 0 -X2X3S23 X32(S23 + S33 + S34) ' * * 0............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 0 0 S(n-l.n + Snn

2 E. A. Guillemin, 'Communication Networks," vol. II, p. 234; with a negative sign these will disappear in the sum,John Wiley and Sons, New York, N. Y., 1931. leaving only the series capacitors.

I

0

0

0

0

X2

0

0

0

0

X3

0

x

1

0

0

0

0

X2

0

0

0

0

X3

0

1338 November

Rowlands: Impedance Transformations in Four-Element Band-Pass Filters

The conditions that the series capacitors in all theinternal meshes vanish lead to (n -2) simultaneous equa-tions in (n-2) unknowns.Namely,

- S12 + (S12 + S22 + S23)x2 -S23X3

-S23X2 + (S23 + S33 + S34)X3 - S34X4

eliminated. The terminating half sections are identical,and are shown as a general four-element type. One of theinternal sections containing the impedance xaZt isshown in full.

- O=O

(1.1)

( 1.2)

- S(n-2)(n-l)Xn-2 + (S(n-2)(n-1) + S(n-l)(n-1) + S(n-l)n)Xn-1 - S(n-l)n = 0

These may be solved and the values of the x's obtained.By inspection of Fig. 3 it will be seen that for the physi-cal realizability of the filter every x must be positive andsince in particular cases Si, and Snn may be zero, bothX2 and xn_i must be less than unity.

First, assume that X2 is negative. From (1.1) we findthat

S22 + S23 S12X3= X2 + ~ (X2 - 1), (2.1)

S23 S23

i.e., X3 is also negative and mod. x3>mod. X2. From (1.2)we get

S33 + S34 S23

X4 = X3 +- (X3 - X2) (2.2)S34 S34

i.e., X4 is also negative and mod. x4>mod. X3, and soon,until we finally have from (1. (n -3)) that X.-, is nega-tive and mod. x-i> mod. Xn_2. These results are incon-sistent with (1. (n -2)) as the left-hand side does notvanish and so these equations cannot yield negative val-ues for X2, xs, etc.Next assume that X2 is greater than or equal to 1.

Equations (2.1), (2.2), * * *, (2.(n-3)) lead to the re-

sults that

Xn-l> Xn-2 . . X3 > X2>l.

These again are inconsistent with (2. (n - 2)), and so x2

must be less than unity.Starting with (2. (n -2)), we can similarly prove that

the only values of Xn.l consistent with (2. 1) are positivevalues less than unity. The necessary conditions are

therefore satisfied, and so the transformation is alwayspossible.

EXTENSION OF THEORY

The theory will now be extended to include the seriescapacitors of the terminating half sections among thosewhich are made to disappear. Consider the filters ofFig. 4.

Fig. 4(a) shows a filter in which all the series capaci-tors associated with the internal sections have been

In Fig. 4(b) this section has been bisected, thus divid-ing the filter into two parts which have been designatedA and B.

In Fig. 4(c) the positions of A and B have been inter-changed, and in Fig. 4(d) the two similar impedances Z,have been combined as also have the capacitors S. Thisshows that not only can the capacitors in the terminat-ing half sections of a symmetrical filter be equally well

Zi Si Zlt SntSi Zn2Zl

Z, I SijZK +ZK S I Z1s,,-S_ OS_Tt,,, s(,| Stn-w)n

A B

1z s, , Z (c)

SW+I) Stn-1)n Sit- _SVK,_-0K

B A

I ZK 22Z 22S, ZK

I-NTW%,

Fig. 4-Showing the rearrangement of the parts ofa transformed filter.

made to disappear, but that an added advantage is to begained, in that n -1 capacitors may be eliminated in thiscase against n-2 in the other. In obtaining the elementvalues of such a filter it is, of course, only necessary to

equate n-2 of the first n-1 S's to zero. Sn will then

(1. (n-2))

13391949

1PROCEEDINGS OF THE I.R.E.-Waves and Electrons Section

automatically be zero.It is interesting to note that the above transforma-

tions hold for any assembly of elements having a similarconfiguration to the above filters. The only condition isthat all the elements represented by the S's are similarin kind. The Z's may consist of any combination of im-pedances.The transformed circuits for a filter with attenuation

peak frequencies below the pass band are shown in Fig.5. The formulas for the elements are the same as for thefilter with attenuation peaks above the pass band, pro-vided that capacitance C is substituted for S, and ad-mittance Y for impedance Z.

XeC,2t X X3Cts n-IC"-nFg 5-Ft1 ---1wita pek bl th pasbad

Fig. 5-Filters with attenuation peaks below the pass band.

Extension o the Planar ]Diode ransit-Time Solution*

NICHOLAS A. BEGOVICHt, ASSOCIATE, IRE

Summary-Llewellyn's small-signal theory for the parallel-planediode is extended to include a closed-form second- and third-ordersolution for complete space-charge operation. Comparison with Ben-ham's conservation of charge method of treating the electronic equa-tions shows that additional terms not given by Benham are obtainedin the present solution.

I. INTRODUCTION

LEWELLYN'S1'2 TREATMENT of the finite-transit-time behavior of the generalized diode isthe extension of the Lagrangian method of solving

the electronic equations first introduced by Muller.3 Inhis paper, Llewellyn gives the closed-form and seriesfirst-order solution for the generalized diode. The analy-sis to follow will extend the solution to a closed- andseries-form second- and third-order solution for com-

plete space-charge operation.The important limitation of Llewellyn's theory and

the extension here presented is the assumption of a sin-gle-valued electron velocity for all electrons crossing anyplane parallel to the cathode surface; that is, electronsnever pass each other in their transition from cathode toanode.4 This assumption leads to the dc potential and

* Decimal classification: R134. Original manuscript received bythe Institute, February 1, 19409; revised manuscript received, July 15,1949.

t Hughes Aircraft Co., Culver City, Calif.'Frederick B. Llewellyn, "Operation of ultra-high-frequency

vacuum tubes," Bell. Sys. Tech. Jour., vol. 14, pp. 632-665; October,1935.

2 Frederick B. Llewellyn, "Electron-Inertia Effects," CambridgeUniversity Press, London, England 1941.

8 Johannes Muller, "Elektronenschwingungen im hochvakuum,"Iochfreq. und Elektroak., vol. 41, pp. 156-167; May, 1933.

4 An excellent paper on the implications of a single-valued velocityassumption has been given by L. Brillouin, "Influence of space chargeon the bunching of electron beams," Phys. Rev., vol. 70, pp. 187-196;August, 1946,

current being related by Child's Law. It neglects impor-tant properties of the potential minimum that usuallyexist in front of a cathode which ejects electrons with aMaxwellian velocity distribution. Only when the distancefrom the cathode to the potential minimum is very smallcompared to the cathode-anode spacing in the diode willLlewellyn's, or any other solution based on single-valueelectron velocity, give a correct answer. A very usefulrecent paper by Kleynen gives tables for the calculationof the potential minimum distance.5

Units used throughout the analysis will be cgs prac-tical units, the same as those used by Llewellyn.' 2

II. ZERO TRANSIT ANGLE COMPLETE SPACE-CHARGE DIODE SOLUTION

In the calculation of the higher-order complete space-charge diode' solution, it is convenient to consider twodistinct modes of operation. These are:

Mode AThe independent variable in the analysis is the cur-

rent through the diode which is composed of a dc andsmall single-frequency ac component.

Mode BThe independent variable in the analysis is the poten-

tial across the diode which is composed of a dc and smallsingle-frequency ac component. Mode A operation ofthe diode requires the calculation of the applied voltages;

6 P. H. J. A. Kleynen, 'Extension of Langmuir's (Q, 7) tables for aplane diode with a Maxwellian distribution of the electrons" PhilipsRes. Rep., vol. 1, pp. 81-96; January, 1946.

e Zero electric-field and electron-emission velocity at the cathode.

1340 November