.'R J3'

. ,NETWORK SYNTHESIS,

ESPECIALLY THE SYNTHESIS OF RE,SISTANCELESS 'FOUR:-TERMINAL NETWORKS*)

, by B. D. H. TELLEGEN 537.31

Summary

In considering tw:o-terminal networks the concept of order is intro-duced and defined as the order of the differential equation of the freevibrations of the system which is formed by connecting the terminalsover a resistance. By extending the concept of order to four-terminalnetworks it is possible to indicate the resistanceless four-terminal', networks of a given order. It is found that four different four-terminalnetworks of odd order are possible and five of even order. In fig. 13they are drawn for the zero to thefourth order.

1. Introduction

'Electrical networks, to .which belong the resistanceless four-terminalnetworks especially considered in the following, are systems composed of.coils, condensers and -resistances, The use of such networks involvescertain problems. If, for example, a ,given network contains a number of .sources of voltage or 'current, it may be required to calculate the current, _and the voltage of' the various branches of the network. From' the resultof this calculation it is often possible to determine the optimum values

~for the various elements of the network foi; the attainment of a 'certaintechnical effect. For many technical applications, however, the networkis not given, but the problem is to design a network such that a maximumdesired technical effect may be attained. 'The" part of network theoryconcerned with this is called network synthesis and various investigationshave been devoted to it in the last 20 years. lil network synthesis, therefore,the object of the investigation is the finding of a network; such in contrastto that of the first-mentioned investigation, which may be denoted by the\term 'network analysis, where the network is considered as given. We shallnow consider network synthesis in detail: ', ~hen a network is used for a .certain purpose it must he provided with a

number' of terminals by means of which it can be connected with other partsof the system to.which it belongs and throughwhich it exerts its influenceon that system. In the .simplest case the network has two' terminals and iscalled a two-terminal network. The current which is applied at one terminalof the. network is always equal to the current which is taken off at the otherterminal. The network is therefore said to possess one pair of terminals. Inthe case of a' two-terminal network a relation exists between the, current-I and the voltage Vbetween the terminals (fig.l). 'I'hisrelation may be'written in complex form: " . , '

" V= Z\I, (1)

;

. :.,."

where Z is the impedance of th~ two~ter~inal network.

" *) Published in Dutch in Tijdschr. Nederl. Radiogen. 9, 235, 1942.

VI= ZUIl + Z12I2' ?V2 . Z12Il + Z22I2' S

Due to the reciprocity theorem the coefficient of 12in the first equation .is equal to the coefficient of11in the second equation, The voltage between aterminal .ofthè first pair of terminals and one of the second pair is usually-not considered, S.othat two four-terminal networks are considered to beequivalent when their equations (2) are the same, even though the Iast-mentioned .voltages are different. '

Wcshall consider only passive networks, i.e. networks which containno sources of energy and in which therefore all the resistances, capacities-and self-inductions are positive,

(2)

170 B. D. H. TELLEGEN

10]1

40876 40877

Fig. 2. Four-terminal network.Fig. 1. Two-terminal network *).

If the n~tw.ork has two pairs of terminals, and if it is so used that foreach pair the current which is' applied at the one terminal is equal to the. current taken off at the other terminal, it is. called a four-terminal network.Between the two currents 11 and 12 and the, voltages VI and V2 atthe terminals (fi15. 2) of the four-terminal network there are two relationswhich may be written as follows:

2. Two-terminal. netivorks, ,

Bef.ore considering four-terminal netw.orks,~ detail, we shall review' afew of the characteristics of two-terminal networks. Let us begin byconsidering the network at a single frequency. The impedance' Z of thetwo-terminal .network consists of a real and an imaginary part, so thatwe may write Z = R + jX. Since we are confining ourselves to p'assivenetworks, R will never be negativè: R > 0, while X may be either positiveor negative. The synthesis is veJ;y simple in this case. If Z is given, the two-terminal network can be realized by a resistance with a self-induction or acapacity in séries. It may, however, be realized in many other ways" forexample' by a resistance with a self-induction or a capacity in parallel.,This is an example of a general characteristic of network synthesis. If anenoork is gioen, its equations are uniquely determined, if, Iunoeoer, the equationsare given, many different networks are' possible. Thus when a network with

, ' certain properties is desired, the equations of the desired network shouldfirst be found, since the number of possible equations is smaller than thenumber .ofpossible networks. '

*) In fig.land several of the following figures wc indicate the positive. directi~nof the voltage by a double arrowand plus and minus signs. instead of by, a singlearrowas this is by somepeople drawn in the direction from + to::- and by others from- to +. and is therefore confusing.

NETWORK SYNTHESIS 171

2.1. The concept of order of a two-terminal netw~~k .• I

Let US~hOW consider the properties of a two-terminal network at differentfrequencies. The impedance as a function of the frequency always has thefollowing forme ,',' "1,,'+ 111-1+ +z _ ao IL al JI. • •• , an . (3)

. ,-:- bo },n_+ bI An 1+ ... + b; '

",\There Ä = jw, wlien w is the angular frequency and the a;s and b's are all, real. The number n is called the order of the two-terminal network, since it isequal to the order of the differential equation of the fre~ vibrations of thesystem-which is formed by connecting the terminals of the network over. anarbitrary resistance. In order to understand this, we keep in mind that (3)results from. the differential equation between the current i and the voltagev of the two-terminal network by substituting Ä for dfdt. If we returnto-the differential equation, we maywrite, instead of (3)

dn' , d"-1 ' , " '. dn dn-l ' ..(bo dt"·.+ bI dtn:-l+ ... + b,,) v = (ao dt" + al dtn-1 + ... + an)~ . (4)

. If we connect the terminals over a resista~ce r: then,v = -ri. (5)

The negative sign here is caused by the fact that we have considered v andi as positive with respect to the two-terminal'network ill directions such asindicated in fig. I. By eliminating vor i between (4) and (5)we arrivé at thesame differential equation, which is called the differential equation of thefree vibrations of the system. This reads as follows: '. .

" ~ ~.' '.

(b?r + ao) dtn +_ (blr + al) dtn-l + .•. + b-r + all = o. .~.(6)The order of this differential equation is' actuallyn. With r = 0 this is the

differential equation of the. short-circuited two-terminal network; which isthua obtained by .setting the numerator of Z equal to zero; with r _' 00 itis the equation. of the open two-terminal network, which we obtain byputting the denominator of Z equal to zero. .

The order of a two-terminal network is a measure of 'its ' complexity,If we determine the free vibrations of the two-terminal network connectedto a resistance by' solving equation (6), we obtain ~ solution which containsn constants of integration, so that the initial condition of the system must becharacterized by n independent data in order to determine these constantsof integration. This initial condition is characterized by the distributionof. the electric and magnetic energy at the time t =, 0, thus by the vol-tages on the condensers and the currents through Ithe coils. The order of atwo-terminal network cannot .therefore be higher than the sum of thenumber of condensers and coils occurring in the network. The order may be 'lower than this sum, since the voltages on the condensers and the currentsthrough the coils need not all be independent of each other. If,for instance,. the network contalris three coils in star connection, these coils increasethe order only by two? since, if the currents through two of the coils aregiven, that through the third is thereby determined. The order is thus equalto the number of independent data necessary to determine the distributionof the energy in the initial stilte.

z = p . À (À2 + Wb2) •••, (À2 + Wa2) (À2 +.Wc2) •••

z = P (À2 + Wa2) (À2 + Wc2) •••. À (À2~+·Wb2) •••

where P :2:: 0 and 0 < wa2 <wi <Wc2 •••

(7)

172 B. D. H. TELLEGEN

2.2. The sytuhesis ofresi,stanceless tuio-terminal networksWe may now attack the problem of ascertaining from a given Z = f(A)

of the form of (3), whether it can reprosent the impedance of a two-terminalnetwork, and-if so, of indicating one or more circuits by means of which itcan be realized. For resistanceless two-terminal networks this problçmhas been solved by Foster 1), who arrived at the following result. For"resistancelese two-terminal networks' Z is imaginary and two cases may hedistinguished. Either the numerator contains only terms of odd degree in ~and the denominator only terms of even degree in À, or vice versa. In thesecases Z has the form ,

or

(8) .

(9)At the frequencies 0,' Wa, Wb, Wc, ••. , 00 Z becomes alternately zero a~d

mfirrite,' .or vice' versa. Ifweput Z = jX, X as a function of W is represented, bya curve like that offig. 3 orfig. 4. The resonance frequencies, at which Zbecomes zero or infinite, alternate, while dX/drp is always positi,:e.

w~087a

w

«oers.Figs.3 and 4..Two forms ofthe reactance of a resistanceless two-termlnal riorwork

, as a function of the frequency.

In order' to show that the abo~e conditions for Z are also sufficient for, realizability, Z is expanded into partial fractio~s. Instead of (7) and

(8) we then obtain I

, 1 A 'AZ= ao ' l' + al }.2 + 2 + a2 À2 + 2 + .)..+ an}" (10)

• 11.. Wl _,w2.where ao or an, or both, may he zero. It may he ded~cèd from (9) that none'of the a's can be negative. The first term of (10) can be realized by acapa-city, the Second by a capacity and à self-induction in parallel which are inresonance at the frequency WI' and so on for the followingterms, while the

, last term can be realized' hy a self-. induction. We thus arrive at the cir-o--J ' . --~~ cuit offig. 5 in which for certain ca~es

40aao the single capacity oe the single self-Fig. 5. .Synthesis of a resi~tanccless induction, or both, may be missing.

, two-terminal network, -Besides this circuit many others are of

'"

.---~~~------.-.- ---

NETWORK SYNTHESIS 173'

course possible ID realization of the t,~o-terminaî network 2), the more, thehigher the orde!'. For example the dual circuit with respect to fig. 5 whichis drawn infig. 6 c~n also be used. In these circuitsthe sum of the numberof capacities and self-inductions is equal to '. . ~the order of the network, while the difference ~is unity or zero. In order to find equivalent ' ' j_circuits we may profitably make use 'of the _]geometrical configuratiohs of networks 3).If the two-terminal network contains resist- ----4088'

ances, its synthesis is net' 'so simple. Thisproblem has been investigated by Brune 4), ~but .we shall nöt .gointo it.

"

Fig. 6. Another form of the 'synthesis of a resistanceless

two-terminal network.

3. 'Four;terminal networksWe ~hall now devote our attention to the four-terminal 'network. Whe~

equations of the for~ (2) are 'given; conditions have to be found whichmust be satisfied by Zw Z22·and Z12 if the equations are to represent thoseof.a four-terminal network, and when this is done we must indicate one or

. more networks which realize the four-terminal network desired. The firstpart of this four-terminal network problem can be reduced to the corres-ponding two-terminal network problem by constructing from' the four-terminal network a two-terminal network with the aid of two ideal trans-formers, as is indicated infig. 7. By ideal transformer ~smeant a transforme;rwithout losses, without leakage and without magnetization current, thuswith infinitely large self-induct;ions of both windinga. From this definitionit follows that an ideal transformer does nothing but transform currents :and voltages, and that its properties ~re expressed by the equations

VI = uV2 , 2 (ll)12 = -UIl' S \

where u js the transformation ratio (fig. 8). Although an ideal transformer

40882

Fig, 7. Two-terminal network composed ofa four-terminal network and two ideal trans-

, formers.'

T, ~40663

Fig. 8. Ideal transformer,

cannot be realized, but' only approximated, it is nevertheless the idealelement for general considerations of-networks, like the self-induction,which is an ideal coil, namely free of loss and without capacity, and thecapacity, which is an ideal condenser, ,namely free of loss and withoutself-induction. . .'

In order to determine the impedance Z of the two-terminal network offig. 7, we send a current I through it. The currents ulI and u2I are thenapplied to the four-terminal network, so that according to (2) the voltages

174 B.D.H.TELLEGEN"

on the four-terminal metwork beéome (uIZn + U2Z12) I and (U1Z12 ++ U2Z22)I, and thus .the voltage on the two-terminal network becomesUI (u1Zn + U2Z12) I + U2 (u1Z12 + U2Z22) I. The impedance thus amountsto

,(12). . .

Since UI and Uïi can be chosen quite arbitrarily, we arrive at the necessarycondition for Zw Z22 and Z12' that expression (12°)must be 'able to repre-sent the impedance of a two-terminal network for all values of UI and U2•

If we again consider the four-terminal network first at a single fr.;quen-cy 5) 6) we may put .

Zll = Ru + jXll, Zl2 = R12 + jX12, Z22 = R22' + jX22,and thus

Z =' Ul2Rll + 2nlu2R12 + U22R22 + j(U12Xn + 2nlu2Xl2 + U22X22): (13) 0

Since the real part of Z can never be negative, we must determuî'e theconditions under which Ul

2Rll + 2UlU2R12 + U22R22 >,0 for all values ofUI and u2• This is a so-called quadratic form in UI and U2' and the desiredconditions are the following: . I

Rn :2:: 0, R22 > ° and RnR22 - RIl > 0, .(14)as can easily be proved. _The fact that the conditions (14) are also suffi-

cient follows from the fact that any four-term-inal -network which satisfies (14) can be real-

o ized, for instance with the help of three imped-,ances with a positive real part in T- or IJ-

. connection followed by an ideal transformer, asindicated infig. 9. 0

40884

Fig. 9. Synthesis of a four-terminal network for a sing)e

frequency.

3.1. The synthesis of resista~LCeless[our-terminal. networks

Let us now consider the properties of the four-terminal network atdifferent frequencies. Zw Z22 and ~12 will all have the form of (3), the threedenominators being in general equal and. the numerators different. Thisequality of the denominators can be understood by imagining the way inwhich the equations can be calculated for a given four-terminal' network.By the applieation of Ki:r ch h 0 £f's laws a number of linear equationsin the currents and voltages are obtained, which" upon solution, lead to ~common. numerator determinant for Zn, Z22 and Z12' For resistanceléssfour-terminal networks 7) 8) the Z's all become imaginary and can againbe expanded into partialfractions, so that they may be written in the form. , . . .

o l' ÀZn = ao . 1-1:- al À2+ .W

12 + ... + ~n ~.,

1 . À, Zl2 = ho . 1+ hl À2 + W

l2 +. .. + hn À,

1 À 'Z22 = bo • 1+ bI À2 + W 2' + ... + b~ À.

o I "

(15) .

'.

NETWORK SYNTHESIS 175

Thus (12) ~ecomes

Z = (u12ao +- 2UIU2h~ + U22bo) ~ + (ul2al + 2U1U2hl, + U22bl) .. A2 : Wl2 +.

+ ... -I- (u12an + 2U1U2hn + U22bn)A.· . ,,(16~

Since this must lie able to represent a resistanceless impedance, none ofthe coefficients may he negative, as we have seen above, and since this'must be the case for all values of UI and U2' the following conditions result:

" ai; > 0, bk > 0 and akbl, __:_hk2 > 0 for k from zero to n (17)

similar to (14).The fact that these 'conditions are also sufficient, follows from the fact

that any four-terminal network which satisfies them, can be realized. Forthat purpose, for example', Zn is split up into two parts Zu' and Zu"with the coefficients alo' = ak - hk2fbl' and ak" = hh,2fbk, neither of whichaccording 'to (17) is negative. The part Zu' can be realized separately andcan be concèived as 'an impedance in series with the pair of terminals 1. of the four-terminal network. The part Z1l" forms together\vith Z12 andZ22 a four-terminal network which may be conceived as the connection /in series of a number of strongly coupled partial four-terminal networks., For example, since ao"bo - h02 = 0, the three first terms ao" / A, hol A andbol A can be realized by an ideal transformer with a capacity in parallelwith one of the windings. Similar considerations hold for the other terms"so that for the whole four-terminal network we finally arrive at the circuitof fig. 10. ' .

o"0665

Fig. 10. Synthesis of a resistanceless four-terminal network., .- ,~

3.2. The concept of order ;f a four-terminal. networkThe problem of the resista~celess four-terminal network is hereby

.solved to a certain extent, We have found that the.parameters of the four-terminal network have the form of (15), that their coefficients satisfythe conditions (17). and that the network can be realized according tofig. 10. Nevertheless we are not yet satisfied. When it is required for a. cer-,tain application to construct a four-terminal network which has certainproperties, as a rule the equations of that network cannot be consideredas given, hut they must first be set up. Now we will in general be better

. II,

\ . .: .:

in the way indicated in fig. 11, to a four-terminaLnetwork r made ~p ofresistances whose equations arc , '.

- VI = ruIl + r12I2' )-V2 . r12Il + r2212"~

(19)

176 n. D. H.Tf:LLEGEN

able to satisfy given requirements, the more complicated we are allowed to'make the network. We thus need a classification offour-terminal networksaccording to their degree of complexity, and this can be achieved by exten-ding the concept of' order, discussed- under two-terminal networks, 'tofour-terminal networks. For this, purpose wc, connect the four-terminalnetwork Z whose equations are

. VI = ZUIl + Z1212;, )V2 = Z12Il + Z2212' ~

.(18)

Thenegative signs are caused by the fact that we considered thè V's and. I's as positive with respect to the original four-terminal network in such

directions as indicated in fig. 2. The order of the differentlal equation of

\

Fig. 11. The four-terminal network Z,conneeted to a four-terminal network r consisting, of resistances. I

the free vibrations of the system so :Èor~ed is then called the order of thefour-terminal network. By eliminating VI' V2 and 11 or 12 betwe-tn (18)a~d (19) we arrive at .

(Zu + rll) (Z22 + ~22)_ (Z12 + r~)2 = 0 or

, ZllZ22 - ~122+ r22Z11+ rnZ22'- 2r12Zl2 + rn~22 - ;122= O •. (20)

In ord~r'fr6m'this to obtain the differential equation ó{the free vibrations,. we place ZnZ22 - Z122, Zw Z22 and -?'12all over the saI!1edenominator andput, for example, . .

Z· _'A Z- - B Z _HZ'Z Z 2i_,D (21)'n.- C' l!2- C' 12 -. C' 11 22- 12' - C '. -

A, B, C,D and H are thus polynomials in it among which there is the follo-wing relation:

AB - H2 = CD. (22)

By suhatittrting (21) m (20); multiplying' by C and replacing A by d/dl,',/

NETWORK SYNTHESIS 177

(23)

. the differential' equation of the free vibrations is ~btained. The' order of the

. four-terminal network is thus equal' to the highest degree in. A. of A..B,C~D and H. For four-terminal as for. two-terminal networks, the ordercannot ne- higher. :than· the sum of the- number- of qondensers and coil?,,'in certain cases, however, it may he.Iower. ' .

3.3. The: synthesis' of resistanceless fóur-tenniTFal networks of a given. orderWe shall now investigate what resistanceless four-terminal networks of a

, given order are possible, beginning with the zero order, then, the fust order,second order, etc. In doing this we-shall make use-of the followingpro.perties.From the absence of resistance in a four-terminal network it follows thateach of the quantities A, B, C, D and H contains either only terms of evendegree or only terms of odd degree in A. Since ~1l' Z22 and Z12 in this caseare all imaginary, we can. distinguish. two. casea. namely either A,.B and Hare of..even degree in À and C and D of odd degree, 0.1' vice :versa. For thezero order this means either A, Band H of zero. degree, i.e. constant, andC and D zero, os vice versa. Furthermore we must pay attention to ,thèrelation (22) which exists amongçthe quantities. Finally we must keep inmind that the partial fractions into- which- every impedance or- admittancecan be split up, have positive coefficients.\ ,'.

3'.31. Zero order

~) A,B andH constant, CandDzero. From (22)itfollowsthatAB-H2- O.In order to see what this represents-we substitute t21) in (IS) and obtain

I

This is brought into ft. different form by expressing VI and 12in termsof 11and V2, and together with (22) we obtain '

, (24) .

When we put C and D equal to. zero. and Hand B constant, we see that·the equations correspond-tö equations (11) of an ideal transformer,

h) A, B ,and !l zero, C and D 'constant, From (22) it now follows that ,~CD = 0, thus either C = 00.1' D = O.

bI) A, B, C and H zero, Jj constant. In order to find out what this represents.: .(23) is put in a different form by expressing 11 and 12in terms of VI and.~V2• We then obtain . . , '

BH'11=' D V1- D V2'l

" H . A (25)12='-=- D ~1 + D V2•

178 r Ó: D. D. H. TELLEGEN

- With. A, Band H .zero and D constant this leads to 11= 0 and 12= 0,. so that in this case the four-terminal network consists of two openpairs of terminals;

b2) A, B, D, H zero, C constant. This substituted in (23) gives ,VI = 0'. and V2 = 0, so that in this case, the network consists of two short-circuited pairs of terminals.

3.32. First order

a) A, Band H first degree, C and D constant, We thus put A = aA, B = b?,H = hA, C = c, D d. Then' acèording to (22) abA2-h2A2 = cd. forall values of A, thus ab-h2 = 0 and cd = 0, and therefore either c .= 0or d = O. -

al) ab:_h2 = 0, c = O. Substituted in (24) (his gives

VI' bdA11+~V2 ~ ~

12 . -: 11, .' ~

. These are the equations of a four-terminal network consisting of an- .ideal transformer with a capacity in series with .one of the windings.

a2) ab~h2 . 0, d = O.Substituted in (24) this gives

VI= .. ,iV2'!

~ h· C12= ~1/1+ bA V2•

This is an ideal transformer with a ~elf-inductioD in p;rallel with oneof the windings.

b) A, Band H -constant, C and D first degree. We set A = a, B = b,H = h, C = cA, D = dA. For every value of A, ab-h2 = cdA2, thusab-h2 = 0 and cd = 0 and so either c = 0 or d = O.

. b:1)ab--:h2 = 0, c = O.Substituted in (24) this gives

VI= d; IJ +~V2, !h .

12 = --:-b 11 • - ~

This is an ideal transformer with a self-induction in series with one ofthe windings. .

, . b2) ab-h2 = 0, d = O. Suhstituted in (24) this gives, h .

V1= b V

2'lh . CA

12= - b 11+b J[2'

This is an ideal transformer with -; capacity in parallel with one ofthe windings.

h) A, Band H first degree, C and D second degree. We put A = a_Ä,B = bA, H = hA, C= coA2 + Cl' D = doA2+ dl' Then for every value'of A: abA2-h2A2 = (coA2 + Cl) (do,12_-+, dl)' and, thus codo = 0,ab-h2 = codl + cldo, cldl '= 0.There are then four cases: Co= 0, Cl - 0,'or do = 0,dl = 0, or Co= 0, dl = 0, or do === 0, Cl = 0. '_

,hl) ab-~2 = 0, Co='0, Cl = 0.Substitutedin (24) this gives

, V""= do,12+ dl1."+ ~ •1 bA 1. b 2'

h12= -,/1, .

This is 'an ideal transformer with a capacity and a self-induction inseries with 0I,le of the windings.

b2) ab-h2 = 0, do , 0, dl = O. Suhstit,uted in (24) thjs gives• h-VI= '"ij V2, lI = - ~I + C0,12+ Cl V. "2 b 1 b,1 2'

This is an ideal. transformer with a capacity and a self-induction in, parallel with one of the windings.h3) ab-h2 = cldo, co'=' 0, dl = 0. This in (24) gives

doA h lVl= TIL + b V2.'

,h Cl12= - bIL + bA V2•

. This is an ideal' transformer with a self-induction ill parallel and a,self-induction in series with one of the windings.

•

..NETWORK SYNTHESIS ·179.

,3.33'., Second order,a) A,','B and H sec~nd degree, C. and D :first degree. We put

A = ao,12+ al' B = boA2.+ bI' H = hoÄ~+ hl' C= cÄ, D = dA. Then(ao,12,+ al) (bo,12+ bI) - (hoA2 + hl)2 = cd,12 for every value of A andthus aobo-ho2 = 0, aObl + a1bo-2hohl = cd and- a1b1-h12 = 0. Sub-stituted in (23) this 'gives

s r--z-: ',--V = aoA2+ al I + ± l'aobo ,12± 1- albl I !'1 CA 1 c,1 2'

± 11aobo A2 ± 11albl I + bo,12/+bl I 'V2 CA 1 CA 2'

This may he realized hy the circuit offig.12, con-sisting of a capacity, a self-induction and two ideal

. transformers. '

Fig. 12. A resistanceless four-terminalnetwork- of the second order.

40887

180 n; D. U. TELLEGEN

b<t) "ab-h2 = cOd!, do '= 0, Cl = O. This in'(24) gives. d h'

V! =. b.~'11+ b. V

2 '.l. h Co?"

12 = -bIl +T V2• I

This is an ideal transformer with a capacity in parallel and a capacityin series with one of the windings.

3.34. Higher order. I

In quite a similar way we mayalso investigate four-terminal networksof higher orders. This will be found in the appendix. The results of thisinvestigation are shown in fig. 13, in which four-terminal networks fromthe zero order, to the fourth order are drawn. From this figure it is clearhow it ma; be extended to higher orders. Except for the zero order, thereare five kinds of four-terminal networks of even order and four kinds ofodd _order. One of the networks of even order. contains no capacity or self-induction in parallel or in series witli the terminals. These networks may be

. ,considered as basic types, from which the others can be derived. The four-terminal networks of odd order are formed from the basic type of oneorder lower by connecting a capacity or a self-induction in parallel or in

! -

o ][ J[' <>-- -0

0-- -<>

1

2

3

40688'Fig. 13. The resistanceless four-terminal networks from the zero to the fourth order.

';'" ..NETWORK SYNTHESIS 181

series with the terminals. Four of 'the networks of even órder are formedfrom, the basic type of two orders lower by connecting a capacity and aself-induction in series, or a capacity and a self-induction in parallel, or a .: r

capacity in parallel and a capacity in series, or a self-induction in paralleland a self-induction in series, with 'the terminals. In the circuits drawn, thesum of the number of capacities and self-inductions is equal to the orderofthe network, whilethe difference is unity in the case of -the odd.orders,'and zero or two in that of the even orders. In addition to the circuits drawnmany; others of the same order are also possible. All these will, however; ,have equations which 'are the same as, or are special cases of, the equationsof the circuits .drawn ill fig. 13. For example, it is possible to replace everycircuit of third or higher order in fig. 13 containing a capacity and/or a self-induction in parallel with the terminals, by a circuit derived from it by ,connecting this capacity and/or self-induction, instead of in párallel withthe terminals, in parallel with the self-induction and/or capacity of the'basic type incorporated in this circuit. In that case all the circuits arespecial cases of the circuit of fig. 10. However, no parallel circuit in series'with the ,terminals "as in fig. 10 occurs.' "

. 3.4. ConclusionSummarising, it may_be said that when a ;resistanceless four-term~al '

network is required for a certain purpose, we ~)1ouldbegin by determiningthe permissible order of. the network. If the order' is given, i~,is sufficient"to examine the networks of this order, as given in fig. 13 for their usefulness ' , "for the desired purpose, since these networks include all those of that order.When one has found in that way which network is best suited for the pur-'pose in view, it can be investigated whether that network could not betterhe realized practically by an equivalent circuit: In order to find' suchcircuits use may be made of the geometrical configurations of networks 3).If practical difficulties are encountered in the realization, so that the de-'sired four-terminal network can only he more or less closely approximated,we may compare the results attainable with every practicable networkwith those which should be attained with, the theoretical ideal network',and thus detcrmine whether it is worth while to attempt to approximatethe latter \still more closely. , ' , " '

Finally we may mention that the synthesis of four-terminal networks.containing resistance has been studied hi Gewertz9). The problem ofindicating all the networks of a given order in this case has not yet been solved.

4. AppendixThe followjng is an investigation of four-terminal networks of higher

orders. We have already seen that the equations of an arbitrary resis~., tanceless four-terminal network can be givenfn the form (15), and thatthe network can be realized with -the circuit of fig. 10. The terms

al Á2:' 2' ~ .Á2: 2 and bI .Á2: 2 ,vin in generalincrease the order of the, COl ,COl ,COl • .

networkbyfoli~, sinceZllZ22-Z.;.l~villcont;~ina term (ulbl - hl'.!) ~Á2:2~12)i 'and this will therefore increase the deglee of the denominator by four.Only when albl - hl2= 0 win the order be increasedby two. In the circuitof fig. 10 this is manifested by the' fact that i.n general, in addition to the. "

,

,,/

182 D. D. H. TELLEGEN

parallel circuit tuned to Wl from the basic type, a parallel circuit tunedto Wl is present in series with the terminals. The terms in question are thusrepresented in the figure by two self-inductions and two capacities, thus bythe same number of elements as the increase of the order by these terms.'If albl - h12 = 0, the last parallel circuit is absent. If the parallel circuitin series with the terminals is: tuned to a different frequency than Wl'',the onder of the network willnot thereby he altered. ,The network firstconsidered is thus a special case of this new network.The latter network is,.however, again a special case of another network of the same order, derivedfrom ir'by the addition' of an ideal transformer, one of whose windings is inparallel with the parallel circuit in question and the other in series with theother transformer windings connected in series. By this addition of an .

, ' 'ideal transformer the parallel circuit is, as it were, included in the basictype of the circuit. From these considerations it follows that in the investi-gation of resistanceless four-terminal networks of any desired order we may"disregard networks with parallel circuits in series, with the terminals,because they may he considered as special cases of four-terminal networksof the same order having 'no such circuits. -

4.. 1. Even order

We shall first examine the four-terminal networks of the even order 2n,and in particular those.for which A, B and IJare of degree 2n and C and Dare of degree 2n-I. From AB-H2 = CD it follows that AB-H2 containsno terms of degree zero and no terms of degree 4n. When we split upA/C, HjC and BIC into partial fractions, the latter contain not only a termwith ljÀ but also a 'term with À, so thac the network parameters can hewritten in the form (15). Since, as we have seen above, we may disregardnetworks with parallel circuits in series with the terminals, we may confineourselves to the four-terminal networks for which a"bh-hk2 = 0 for valuesof k from 1 to n-I. Since AB-H2 "contains no terms of degree zcro or4n, aobo-h02 = 0 and allb,,-h,.2 = 0 also. The network thus belongs to thebasic type and can be realized with circuits of the form of the first circuitin fig. 13 of the second order and of the fourth order. .

4.2. Odd ~rder -"We shall now examine networks of the odd order 2n + I.

a) 'A, B and Hof degree 2n+ 1, C and D of degree 2n. From AB-H2 = CD> it follows that AB-H2 contains no term of degree 4n,+ 2, and CD"none of degree zero. From this it follows that either C or D containsno term of degree zero.' " "

al) C contains no term of degree zero. In order to investigate this we 'start from equations (23). A, B, Hand C have a common factor J..Upon expanding into partial fractions we may therefore write

A 1 I À ,c= ao' J: + al À2+ W 2 + .;. + an_X,

, 1

H h 1+ t: ).. + h 1,C;_ 0'). lÀ2+Wl2+ .. ·. n1'.,

B 1 ÀC = bo'" + bI À2 2+ ...+ b« À.

1', + Wl. '

(26)

! ..

NETWORK SYNTHESIS 183

A~ above,.a"bk-"-:h,,2 = 0 forvalues of le from 1 to n~l, and since AB-H2 .contains no term of degree 4n + 2, a;:b~-hn2 = Oalso. However aobo=-h02will not be zero here and thus the four-terminal network consists of thebasic type of the order 2n with a-capacity in series with the terminals.

a2) D contains no term of degree zero. Here we start from equations (25),'and A, B; Hand, D have a common factor À.. In an analogous way we ,arrive at a four-terminal network which consists of the basic type of.'-the order 2n with a self-induction in patallel with the .terminals,

b) A, B:and H ofdegree 2n, C and D of degree 2n + 1. From AB-H2 = CD_;..it now follows that AB-H2 contains no term of degree zero and CDnone, of degree 4n + 2. From the latter it follows that either. C or D ': contains no term of degree 2n + 1. ..,.

bI) C contains no. term of degree 2n+ 1.We again start from equations (23). and may again write equations (26). Here too a"bk..,-h,,2 _:_ 0 for valuesof k from 1 to n-l, while, since AB-H2 contains no term of degreezero, aobo-ho2 = 0 also. However anbn-h,.;2 will not be zero here,' andthus the four-terminal network consists of the basic type of the order2n with a self-induction. in series with the terminals.

b2) D contains no term of degree 2n+ 1. In an analogous way this leadsto a four-terminal network which consists of the basic type of the order2n with a capacity in parallel with the ~·erminals. .

-,4.3. Even order {cotuituuuion]

. We have still to deal, with part 'of the four-terminal networks of, evenorder, namely those for which A, Band H. are of degree 2n + 1 and Cand D of degree 2n'+ 2. From AB-H2 = CD it follows that CD containsno term of degree 4n + 4 and none of degree zero. This gives rise tofour cases: ~... . 'I ,

1) C contains no term of degree 2n + 2 a~d nonè of degree zero,' I2) D contains no term of degree 2n + 2 and nonè of degree zero,3) C contains no term of degree 2n + 2 and D none of degree zero,4) :D contains no term of degree 2n + 2 and C none of degree zero.1) C, contains no term of degree 2n + 2 and none of degree zero. We

start from equations (23). A, B, Hand C have a common factor À. and,we may again write equations (26). Here again akbk.-h,,2 = 0 for values'of k from 1 to n:""'1. Now, however, neither aobo-h02 nor a"bn-hn2 isequal to zero, so that the four-terminal network consists of the basictype of the order 2n with a capacity and a self-induction in series,withthe terminals. ,

,2) D contains no term of degree 2n· + 2 and none of degree zero. Startingfrom equations (25) we arrive at a network which consists of the basictype of order 2n with a capacityand a self-induction in parallel withthe terminals.

3,) C. contains no term of degree 2n + 2 and D none of degree zero.We start from equations (23). When we expand A/C, H/C and B/Cinto partial fractions, they contain a term with À. but none with I/À.,so, that we may -write ., ' " . " ..

A À. À. ()C = ao 12+-2 + ; .. + all-1 À.2 + '2 + an À. 27 .

I\. Wo W n-1 _

./

184 D. D. H. TELLEGEN

. .

and corresponding expressions for HI C arid BIC.Hete again akb/;-lu,2 = 0.for values' of k from 1 to n-l, while anbn-hn2 is not equal to zero. Thenetwork thus consists of the four-terminal network for which anb,.-h,.2 = 0,also, with a self-induction in series with the terminals. Since for 'the latter network anb,.-h~2 = 0, its D contains no term of degree 2n + 2,either, so that the network is of the order 2n + 1, namely the four-terminal network for which A, Ban? H are of degree 2n+ 1, C and Dof degree 2!Land D contains no term of degree ,zero. This is the net-work of the order2n + 1'which is discussed above under az and whichconsists ,of the basic type of'the order 2n with a self-induction in parallelwith the terminals. The desired four-terminal network of order 2n + 2-thus consists of the basic type of the order 2n with a self-induction inparallel and a self-induction in series with the terminals.

4) D contains, no term of degree 2n + 2 and q none of degree zero.A, B, If and C have a common factor À. When AIC, HIC and BIC areexpanded into partial fractions, they contain a term with 11.1 but nonewith À, so that we may write , '

"

(28)... . ., and corresponding expressions for HI C and BIC.Here again akbk-h,,2 = 0for values ofk from 1 to n.,--l,while aobo-ho2 is not equal to zero. The net-,work thus consists ofthe four-terminal network for which aobo-ho2 = 0'also, with a capacity in series with the terminals. Since for the latternetwork aobo-ho2 = 0, its D contains no term of degree zero either,so that A~B, H, C and D all 'have a common factor A. and the networkis thus of the order 2n + 1, namely the four-terminal network forwhich A, Band H are of degree 2n, C of degree 2n + 1 and D of degree2n-1. This is the-network of degree 2n + 1 dealt with above under b2,which consists ofth~ basic type of the order 2nwith a capacity in parallelwith the terminals. The desired four-terminal network of order 2n + 2thus consists of the basic type of order 2n with a capacity in paralleland a capacity in series with the terminals. , "

, ,

REFERENCES

1) R. M. Foster, .À reactance theorem, Bell. Syst, Techn. Journ. 3, 259, 192.1,.2) W. Cauer, Die Verwirklichung von "Wechselstromwiderständen vorgeschriebener

Frequenzabhängigkeit, Arch. f. Elektrot. 17, 355, 1926.3) B. D., H. Tellegen, Geometrical configurations and duality of electrical net-

works, Philips Techn. Rev. 5; 324, 1940. . ' ,4) O. Brune, Synthesis of a finite two-terminal network whose driving-point impe-

dance is a prescribed function offrequency, Journ. Math. Phys. 10, 191, 1931.5) M. Vaulot, Sur les constantes du quadripêle passif, Rev. Gén. Electr. 22, .f.93,1927.6) R D. H. Tellegen, SUl'les constantes du quadripöle passif, Rev. Oën, Elcctr, 2-1"

211, 4-10, 1928. _ _ •7) W. Cauer, Ein Reaktanztheorem, Sitzungsber. Preuss, Akad. Wiss. IT, 30/32,

673, 1931. .' ' ,8) E. A. Guillemin, Communication networks VoL H, 1935, p. 216. •9) C. Gl."wertz, Synthesis of a finite four-terminal network from its prescribed driving

point functions and transfer function, Journ. Math. Phys. 12, 1,1933.