1958 Kawakami: Active Four-Terminal Linear Networks 115

instances and that a large number of alternate equivalent band-pass filter which is capable of achieving a range of configurations exist which require fewer crystals. Two of symmetrical insertion functions which is very much larger the most useful of these are the divided plate crystal than that obtainable with image parameter techniques. fil.ter structure1 and the half-lattice structure.l,ll Among the more desirable insertion functions which may

VII. CONCLUSIONS be achieved are: 1) maximally flat insertion loss; 2) equal ripple insertion loss in pass band and monotonic increase

A theoretical technique has been presented for the in stop band; 3) equal ripple insertion loss in pass and synthesis of the narrow-bandwidth crystal-capacitor stop bands; 4) maximally flat time delay.

Some Fundamental Considerations on Active Four-Terminal Linear Networks*

MASAMITU KAWAKAMIt

INTRODUCTION

-

1

T is well known that the concept of a four-terminal network (4TN) is useful in the analysis of the network performance, and matrix algebra is widely used in

the study of the 4TN. On the other hand, some new kinds of networks such as gyrator, negative impedance con- verter, etc., have been introduced recently. These new networks, which may also be considered as 4TN, have particular characteristics on their matrix representations.

In this paper, we wish to focus our attention entirely upon the matrix representation in order to clarify the general problem as to how ‘this representation may be related to the various kinds of 4TN. For this purpose, simple forms of network matrices in ABCD systems are

. first presented. Some new kinds of 4TN, whose represen- tation matrices have LLelementary” forms, are introduced, and some fundamental relations among these “elementary networks” are established.

In this paper all the discussions are based on the follow- ing definitions. The current from left to right is positive on both terminal pairs of 4TN, and the matrices (F), (Y), (Z), (G), and (H) are defined as follows:

* Manuscript received by the PGCT, July 19, 1957; revised manuscript received, December 4, 1957

t Tokyo Inst. of Tech., Tokyo, Japan.

((3 = 03 =

(4)

ELEMENTAIZY ACTIVE 4TN (EA4TN)

4TN is said to be elementary when 1) three out of four elements of fundamental matrix (F) are zero, or, 2) sub- diagonal elements of (F) are zero, or 3) diagonal elements of (F) are zero.

Table I shows all types of EA4TN, and clearly indicates all active elements abstracted from active networks, including vacuum tubes and transistors.

The brief explanation of this table is stated below.

Voltage-Current Transducer (VCT)

This is a unilateral transducer in which the output current is proportional to the input voltage. The negative sign of proportional constant b means the change in the direction of output current. An example of VCT is an ideal vacuum tube having no interelectrode capacitances and infinite internal (plate) resistance. This is represented simply by VCT, when the left side is input and the right

+ c side is output. We use the notations VCT or VCT, espec- ially to indicate the directions of transmission. Both the input and output impedances of VCT are infinite.

Current-Voltage Transducei (CVT)

This is a dual transducer of VCT, that is, input current quantity is transduced to the quantity of output voltage. This is referred to an ideal dynamo. Both the input and output impedances of CVT are zero.

116 IRE TRANSACTIONS ON CIRCUIT THEORY

TABLE I FAMILY OF EA4TN

TABLE II VCT

-violfayr-ntTolt~pe m Transducer

>VVT

Voltage-Voltage Transducer (VVT)

This circuit transduces the’input voltage to the output voltage; its input impedance is infinite, its output impe- dance is zero. For example, the cathode follower circuit operates nearly as VVT.

Current-Current Transducer (CCT)

This circuit transduces the input current to the output current; its input impedance is zero, its output impedance is infinite. For example, the grounded base transistor operates nearly as CCT.

All of these four transducers are unilateral.

Ideal Voltage Amplifier (IVA)

This circuit transduces the current without any changes and the voltage with amplification of a-‘. Therefore it works as an impedance converter. This impedance converting action is found in vacuum tube and transistor circuits, When a is negative, it is called a negative imped- ance converter (NIC),l and is used in a repeater for compensating line losses, or in RC circuit synthesis.’ When a = - 1, it is called unity negative impedance converter (UNIC).

Ideal Current Amplifier (ICA)

This is a dual transducer of IVA, that is, it transduces the voltage without any changes and the current with a factor of d-‘. Therefore it works as an impedance con-

1 J. L. Merrill, “Theory of the. negative impedance converter,” Bell Sys. Tech. J., vol. 30> pp. 88-109; January, 1951.

2 J. G. Linvill, “RC actwe filters,” PROC. IRh, vol. 42, pp. 555- 564; March, 1954.

1 (F) (G)

O ‘\ 0 bl

June

04

X

X verter, too. An example of ICA is found in grounded emitter transistor circuits.

ICA with negative d is used for the same purpose as NIC. Here, it is called negative admittance converter (NAC)3 in distinction from NIC.

Ideal Power Amplijier (IPA)

This circuit transduces both voltage and current with the factor of A-‘. Thus, it has no impedance coverting action.

IVA, ICA, and IPA are useful in the equivalent trans- formations of active networks.

It is easily seen that VCT, CVT, VVT, and CCT are sub- stantially complete three-terminal networks. Therefore, when we deal with them as four-terminal networks, we have six different cases corresponding to the selection of the earth-terminal and the transmission senses. Tables II to V show all the elements of (F), (Y), (Z), (G)j and (H), for six different cases. Tables II and III show the remarkable effects of the terminal selection on the forms of F matrices. From Table IV, it is seen that we can get NIC by a proper selection of terminals. That is, we see that VVT and NIC are actually the same in nature; only the selection of terminals is different for each of them. By Table V, we obtain NAC, and see that the relation between CCT and NAC is the same as the one between VVT and NIC.

3 T. Yanagisawa, “Current inversion type negative impedance converters,” J. Inst. Elec. Conzmun. Engs., Japan, vol. 39, pp. 933-937; November, 1956.

Kawakami: Active Four-Terminal Linear Networks

TABLE III CVT

TABLE V

117

CCT 1 I

(F) 1 (Y) I

( (F) 1 (Y) (2) 1 (4)

TABLE IV VVT t

and CVT (7). GG with G, = G, is called ideal gyrator (IG). As is well known, the (F) of IG has the relation 1 FJ = - 1. Thus IG is antireciprocal and has the impedance inverting property. But the circuits shown in Figs. 1 and 2 have the same impedance inverting action, although they belong to the reciprocal system. Thus, when 2, is loaded on the right side, the input impedance Zi, of the circuit shown in Fig. 1 becomes X2/Z,, and in the same way Zi, of Fig. 2 becomes -R’/Z,. Thus the circuits shown in Fig. 2 are considered as the T and a equivalent circuits of an ideal negative impedance inverter.

I

1 I.F)

x /i 111 x I I

Matrices of IG are given as follows.

(F)Ig = ’ G-’ , I I G 0

(VI0 = O -G , I 1 G 0 (Z),, = O -G-’ . i 1 G-’ 0

(6)

Generalized Gyrator (GG)

This is obtained by the parallel connection of V??T (It) No (G) and (H) exist for IG.

and VET (=I=), or by the series connection of CTT (A) For the reference, matrices for ideal transformer (IT,)

with the ratio 1:n are shown as follows:

IRE TRANSACTIONS ON CIRCUIT THEORY June

(a) (b) Fig. 2-Negative gyrator.

(@IT, = ’ n , i 1 n 0

(H) IT, =

Neither (Y) nor (2) exists for IT,. The elementary networks described

(7)

above are repre- sented by the points on the (A X D), (B X C) plane (Fig. 3).

RELATIONS AMONG EA 4TN

Relations among the elementary networks mentioned above are considered in this section.

Relations Among Four Unilateral Transducers

Relations among four unilateral transducers are shown in Table VI, where we can see that VCT and CVT are more basic networks than the others, because the latter circuits are derived from the former ones.

Relations Between NIC and NAC

In Table IV VVT and NIC are understood to be the same in nature, but different in the selection of terminals. And the relations between CCT and NAC are the same as the corresponding relations between VVT and NIC. Relations between UNIC and UNAC are: (F> VNIC = x u9..AC,

(8)

(F> VNAC = x U%NI,.

Fig. a--Relative positions of EA4TN on the A X D vs B X C plane.

TABLE VI RELATIONSAMONGFOURUNILATERALTRANSDUCERS

Relations Between IT, and IVA, ICA and IPA

IVA and IT,, IPA:

when n = A-l,

= A2 0 [ I 0 1’

Namely,

(IVA) = (IT,) X (IPA), when n = A-‘. (9)

1958 Kawakami: Active Four-Terminal Linear Networks 119

ICA and IT,, IPA:

: 7

when n = A,

Namely,

(ICA) = (IT,) X (IPA), when n = A.

IPA cna? IVA, ICA:

when a = d

Namely,

a 0 =

I I* 0 a

(IPA) = (IVA) X (ICA).

Relations between VCT, CVT, and IG

From Table II,

parallel connection of T( +) and x( -) = IG.

From Table III,

series connection of z( +) and T( - ) = IG.

Also from Table II, the following (Y) is derived from a

parallel connection of g and z, or z and i?.

(Y> = (13)

This circuit is a’negative gyrator, and thus a reciprocal system as shown in Fig. 2, and also works as a negative impedance inverter. It must be remarked that the dual relations to this circuit are not derived from Table III.

Relations Between IG and IT,

The next three relations are easily derived,

(F)m x V’>m = (F)m, (F)BI, X (F)m, = (F)m,

I

(14)

0’7~ x V%w = U%T.,

where RI, is a reactance inverter, and NG is a negative gyrator; i.e., we can find the cascade connection of two

IG, two RI,, or two NG and make one IT,. In addition, the following relations are interesting and important.

RI, IG

(15)

We shall call the right side of (15) “reactance transformer.”

NG IG

We shall call the right side of (16) “negative transformer.” The right sides of both (15) and (16) are antireciprocal

circuits (i.e., IFI = --I), and impedance converters.

RI, NG

(17)

We shall call the right side of (17) “imaginary trans- former,” because its transformation factor is jR/X. This is reciprocal and works as a negative impedance converter. The relations between the elementary circuits mentioned above are shown schematically in Table VII.

SOME APPLICATIONS OF EA 4TN

Representation of Negative Resistance

Short Circuit Stable Type: This is represented by the two-terminal network which is obtained by a parallel connection of input and output terminals of VCT as shown in Fig. 4. A dynatron is an example of this type.

Open Circuit Stable Type: This is represented by the two-terminal network.which is obtained by a series con- nection of input and output terminals of CVT as shown in Fig. 5. A point contact type transistor is an example of this type.

The physical meaning of the two kinds of negative resistance will be clearly understood by these representa- tions.

Impedance Converter and Inverter

The most general representations of impedance con- verter and inverter are shown in Table VIII, where the necessary condition for the formation of impedance converter is

and the necessary

b=c=O, (1%

condition for impedance inverter is

a=d=O. (19)

120 IRE’ TRANSACTIONS ON CIRCUIT THEORY June

TABLE VII RELATIONS BETWEEN EA4TN

h, i er ‘+ -b Ls’ I

Fig. 4-A representation of the short-circuit stable type negative resistance.

,I ’ i I = -c - 2r

i/c

Fig. 5-A representation of the open-circuit stable type negative resistance.

With the aid of these conditions, Table VII will give some additional knowledge on the concepts of impedance converter and impedance inverter.

Transpositions of 4TN

Ideal Transformer:

That is, an ideal transformer transposes the right side network to the left side network with impedance being multinlied b.y the factor of ‘n.

TABLE VIII GENERAL FORMS OF CONVERTER AND INVERTER

NetworK I Netu’ovK ll

I

I I

If b=c= o , If b=c=0,

con verfev -LIZ 7;, = (a-/)[d-/) z,=$z

Ideal Gyrator:

That is, an ideal gyrator forms a dual transformation circuit to the original circuit.

Unity NIC or Unity NAC:

That is, UNIC or UNAC makes all the circuit elements negative.

Ideal Voltage AmpliJier :

[: :I[: :] = [,a, I:] [:, :I. (23) That is, IVA increases all the impedances of circuit element by k-times.

Ideal Current Amplifier:

That is, ICA increases all the admittances of circuit elements by I= times.

Actions of UNIC and UNAC

1) They make the impedance of 2-terminal network negative.

2) They make 4TN negative. Namely, when a 4TN is san@i~h~ed w.it&,two UNIC or two UNAC, the resultant F matrix is

1958 Kawakami: Active Four-Terminal Linear Networks 121

TABLE IX BASIC VACUUM TUBE CIRCUITS

L I Grounded 1 bi-ounded Grounded 1 teed Boc.+ 7~,De

That is, sandwiching 4TN with UNIC or UNAC makes all the circuit elements negative.

3) Compensation of Loss in 4TN: Sandwiching a UNIC or a UNAC with two identical 4TN, the resultant F matrix is

T-Type Eqlhdcnt circrit

(F)

Thus the loss of 4TN can be compensated with UNIC or _ _ I Basic Il’ransistor Circuits5 UNAC. Further, in the case of symmetrical 4TN (a = d),

a repetition of the above operation makes the resultant F matrix:

Therefore, in this case, the operation is identical to direct c connection of input, and output terminals of 4TN. This viewpoint for 4TN will help to treat loss-compensation of long transmission line.

Basic Vacuum Tube Circuits4

TABLE X BASIC TRANSISTOR CIRCUITS

Table X represents the basic transistor circuits with EA 4TN, and shows clearly their physical meanings.

The conclusions are summarized as follows: 1) Abstracting the activities of active networks, four

unilateral transducers are derived. 2) Of these four transducers, VCT and CVT are

pointed out to be the most basic. 3) All characteristics of unilateral transducers, the

various elementary active networks derived from them, and the relations among them are also described.

4) Some applications of EA 4TN are explained.

ACKNOWLEDGMENT

Table IX represents the basic vacuum tube circuits The author wishes to express his thanks to Shintaro with ‘EA 4TN mentioned above, and shows clgarly the Oshima of Kokusai Denshin Denwa Co., Ltd., for his physical meaning of basic vacuum tube circuits. valuable suggestions.

4 M. Kawakami, “Electronic Circuit,” Kyoritsu Publication h M. Kawakami, “Sourceless equivalent circuits of transistors,”

Co., Ltd., Tokyo, Japan, vol. 1, p. 179; 1953. f.3 Ir&Elec. Commun. Engs., Japan, vol. 37, pp. 525-529; August

, .