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A Unique Synthesis Method owTransformerless Active RC

• Networks

by

S. K. Mitra

Series No. 60, Issue No. 411G-12142October 12, 1961

Electronics Research LaboratoryUniversity of CaliforniaBerkeley, California

A UNIQUE SYNTHESIS METHOD OF TRANSFORMERLESSACTIVE RC NETWORKS

by

S. K. Mitra

Institute of Engineering ResearchSeries No. 60, Issue No. 411

National Science Foundation Grant G-12142October 12, 1961

ACKNOW LEDGMENT

The author wishes to thank Professor E. S. Kuh for his encourage-

ment and helpful suggestions. Thanks are also due to Messers. J. D.

Patterson and R. A. Rohrer for useful discussions.

ii

ABSTRACT

A unique decomposition of active RC driving-point impedance

functions is presented, which has been obtained by considering the

driving-point synthesis problem in terms of the reflection coefficient.

Application of the decomposition has been shown to guarantee the

realization of the driving-point impedance in Kinariwala's cascade

configurations and Sandberg's special configurations, each contain-

ing one negative impedance converter. The method imposes no

restriction on the impedance function, except that it has only to be

a real rational function. The decomposition technique can be easily

programmed on a digital computer.

iii

TABLE OF CONTENTS

PageI. INTRODUCTION .......................... 1

II. DEVELOPMENT OF THE DECOMPOSITION ........... 2A. The Concept of Reflection Coefficient. . . . . .... . 2

B. Special Case ....... . . . . . . . . . . 5

C. The Decomposition .............. . . . ... 6

D. The Associated Function ................... 7

III. SYNTHESIS PROCEDURES USING ONE ACTJVE ELEMENT... 8

A. The Cascade Method . . . . . . ....... ...... . 8

B. Synthesis Using Special Structures ............... . . . 11

IV. ILLUSTRATIONS. . . . . . . . . . . . . . . . . . . . . . . . . 12

A. Example I . . . . . . . . . . . . . . . . . . . . . . . . . . . 16B. Example 2 . .. .. .. ... . ... . ...... .. 14

C. Example 3 .. .. .. .. .. ... . . . . .. 15

V. CONCLUSION .. .. .. .. .. .. .. .. ... . .. 16

REFERENCES. . . . . . . . . . . . . . ... . .. . . . . . 22

iv

I. INTRODUCTION

Synthesis of active RC networks has received considerable

attention in recent years because of the advantages these networks

possess compared to other types of networks. A one-port active RC

network can be considered to consist of a three-port passive RC net-

work (the so-called associated 1 RC network) to two ports of which

is connected an active device. Usually, the three-port RC network is

a combination of several one-port and/or two-port RC networks. In

general, the steps followed in most of the existing synthesis techniques

are:

Step 1 - Decomposition and partitioning of the given function,

Step 2 - Identification of the associated RC network (or networks)

parameters,

Step 3 - Synthesis of the associated network.

Although several active RC configurations will be found in the literature,

there exists no straightforward decomposition and partitioning method

which would result in unique functions characterizing the associated RC

network.

The purpose of this paper is twofold. First it presents a unique

decomposition of active RC driving-point impedance functions. Secqnd,

it shows that the suggested decomposition can be used to realize im-

pedance functions in existing active RC configurations, where the

associated networks are now easily obtained. In a sense, this paper

proves in a very simple and straightforward way the following:

Theorem: Any real rational function of s, the complex frequency

variable, can always be synthesized as the driving-point impedance of

an active RC transformerless network containing a single negative

impedance converter and associated RC networks which are character-

ized by unique . functions.

The basic idea behind it is to obtain an equivalent active LC

driving-point impedance from the given function by an RC-LC trans-

formation. Then the reflection coefficient corresponding to this

-1-

equivalent LC impedance is obtained. It is shown that the reflection

coefficient can be expressed as a product of two reflection coefficients,

one corresponding to the passive portion and the other representing

the active portion. Following Kelly's approach, 4 the desired decom-

position is obtained upon inverse LC-RC transformation. The suggested

decomposition can be programmed on a digital computer.

II. DEVELOPMENT OF THE DECOMPOSITION

A. The Concept of Reflection Coefficient

Consider an active RC driving-point impedance Z(s) = K N(s)/D(s)

where Z(s) is restricted to be a real rational function of s and the leading

coefficients of N(s) and D(s) are assumed to be unity. Using the RC-LC5

transformation, we can obtain an equivalent active LC driving-point im-

pedance '(s), which is defined as

A 2 s2Z(s) = s z(s 2 ) = Ks N(s ) *(1)

D(s2)

A. AThe reflection coefficient /s) corresponding to Z(s) is given as

En(s) Ain 2 E (s) 1 - Z(s)/R. (

E in(s) 1 + (s)/Rin(

2

according to the notations of Fig. 1. Substituting (1) in (2), we obtain

A(s) D(s 2 ) - s N(s ) - Q(s)P (s) -2 2 (-s)(3)

D(s ) + s N(s 2 )

where the impedance level has been normalized with respect to R.

equal to K. * Now, D(s ) is an even function and s N(s ) is an odd

function. As a result, it is clear from (3) that to each zero of /s)A

in the left-half s-plane there corresponds a pole of /(s) in the right-

half plane symmetrically situated with respect to the jw-axis and vice

versa. We also assume here that if D(s 2 ) has any zero at origin, it is

*Henceforth we will consider only normalized Z(s) and Z(s).

-2-

not cancelled with the factor s in the numerator of Os). Moreover,

since the algebraic sum of an even and an odd polynomial with real co-

efficients cannot have any zeros on the jw-axis unless each of them have

the same jw-axis zeros, it follows that O(s) will not have any jw-axis poles

and zeros except possibly at the origin. Thus we can write

s I -nl m 2 +n2m+n I m2- n2 (4)

where m1 + nI is the unique Hurwitz polynomial HI(s) formed by factoring

the left-half plane poles of A(s) and m 2 + n2 is the unique Hurwitz poly-

nomial H 2(s) obtained by factoring the left-half plane zeros of s) in-

cluding that at origin, if any. The reflection coefficient of an active

driving-point impedance can be thought to consist of two factors, one

representing completely a passive network and the other including the4 Aactive portion. Thus following Kelly's approach, we can factor p(s) as

A(s) = E(s) -I (5)

where

m -nE(s) =- +n (6)

m +n 2

A m2n2r=-n (7)

m-2

Because of the construction procedure outlined above, we note that E(s)

satisfies the properties of the reflection coefficient of a passive LC driving-

point impedance, i.e., E(p), where p = (1 - s)/(l + s), is a "unit function" 6.

* The even part of the polynomials are denoted by the symbol m and

the odd part by the symbol n.

-3-

From (5) and (2) we obtain 4

/ ps)

i+$~s+1 1

r1 ++ (s) + 1

where

A 1-E(s)Zr(s) = 1+ E(s) (9)

1 - /i(s)(s) P1

1 + A1(s)

Substituting the expressions for E(s) from (6) and from (7) in (9)

we obtain

A n1Zr(s) m 1 (10)

AnZ 1(s) m

+ Zr(s) and + 1 are guaranteed to be

in the form of passive LC driving-point impedances because of the properties

of the Hurwitz polynomials. 7 The corresponding circuit representation of4.

(8) as proposed by Kelly is as shown in Fig. 2.

-4-

By making an inverse LC-RC transformation, we obtain a real-

ization for Z(s) using two UMC's + as the active devices, as shown inA

Fig. 3. Z a(S), Zb(S), ZC(s) and Zd(a) of Fig. 3 are related to Zl(s)

and 2r(s) by the following relations:

Za(S) L.=a r

1 1 1

Z b~s) =-b() -r/ s Za(s)

Z c (s)

1 1 1Z d(s)-

(.,)s Z c(s)

It is clear that Z a(s), Zb(s), Zc(s) and Zd(s) are in the form of passive

RC driving-point impedances.

B. Special Case

We will now show that if the given driving-point impedance Z(s)

is a passive RC impedance function, then no active elementswill be

present in the structure of Fig. 3. In this case 1(s), obtained by the

RC-LC transformation of (1), is of the form of passive LC driving-point

impedance. Furthermore, because of the indicated transformatiun

procedure, the numerator of k(s) will be an odd polynomial. Hence

we can write

A negative impedance converter with a unity conversion ratio will be

designed as an UNIC.

-5-

m1 (lZ)

and the corresponding reflection coefficient is

= - (13)rn 1l+n l

Comparing (13)with (4) we note that (13) is a special case of (4) when

n = 0. This implies, in this special case, Fig. 2 and consequently

Fig. 3 are modified to that of Fig. 4a and Fig. 4b, respectively, each

of which, as can be seen, do not contain any UNIC.

C. The Decomposition

From (4) and (8) we obtain

(s) = lm1 - m l n1n1mm -n1n2 (14)

mil, n 2 are even polynomials and n,, n 2 are odd polynomials, Hence

we can express them as

m I 1 al1(s 2

in a.1 (s z

22in2 = a2 (s )

n 1 = s b1(s -

2 (15)n2 : s b2 (s 2 )

Substituting (15) in (14) and then applying the inverse LC-RC transformationwe arrive at the following theorem:

-6-

Theorem - Any real rational function of s, Z(s), can always be

decomposed in the following form,

bI(s)a 2 (s) - a1(s)b 2 (s)

al(s)a2 (s) - s bl(s)b2 (s)

where a1(s)/s b 1(s), b1(s)/a 1 (s), a 2 (s)/s b 2 (s) and b2 (s)/a 2 (s) are unique

functions satisfying the properties of passive RC driving-point imped-

ances.

Identification procedures for a1 (s), etc., can be summarized as follows:

1. From the given Z(s) = N(s)iD(s), obtain the polynomial Q(s)

D(s 2 ) - s N(s 2),

2. Find the zeros of the polynomial Q(s). Form the polynomial

m2 + n2 by factoring the left-half plane zeros of Q(s), including

the zero at origin, if any, and factor out the right-half plane

zeros of Q(s) to form the polynomial m1 - n ,

3. From m 2 + n 2 and m1 - n1 obtain the even and odd parts and

using (15) obtain a1(s), a 2 (s), b1(s) and b2 (s).

D. The Associated Function

Before going into the application of (16) in the synthesis of active

RC networks, we will present an interpretation of (14). Let us define the

function

m +nF ms 2 m+n 2 (7

as the associated function corresponding to Z(s). Though m I + n1 and

m + n 2 are Hurwitz polynomials, F(s) may not be a p. r. function. The

odd and even parts of F(s) are

n m -mn 2Od F(s) = 2 2

m -n 2

(18)

Ev F(s) = m1m 2 -n 1n 22 2n--n-

-7-

Thus we can write

s Z(s 2) 2(s) = Od F(s) (19)

The above results can be summarized as:

Theorem - To every real rational function Z(s), there corresponds

a unique associated function F(s) defined as a ratio of two unique Hurwitz

polynomials and related to Z(s) according to (19).

Now D(s 2 ) corresponds to the numerator of the even part of F(s). If D(s 2 )

is positive on the j-axis, then F(s) will be a p. r. function. In case of

D(s) having odd ordered jw-axis zeros, D(-w ) will not be positive for all

W's. In that case we can multiply both the numerator and the denominator2 2 2 2 2 2of s Z(s ) by the factor fl(s + W. ) so that the augmented D(s ) Ij(s + W.)

1 1 1 1has all even ordered jw-axis zeros and thus stays positive on the jW-axis.

Then we can relate the modified s Z(s 2 ) to a p. r. function G(s) according

to (20), which is given below.

2 2 2s N(s ) (s +w.- dG(s)2 1 d 0~ss Z(s ) M=1 (Z0)

2 2 (20)D (s ) fl (s + Wj vGs

Thus we find that there also exists a p. r. function G(s) corresponding to

z(s).

III. SYNTHESIS PROCEDURES USING ONE ACTIVE ELEMENT

A. The Cascade Method 2

The input impedance Z(s) of the cascade structure of Fig. 5 is

given as

1 - ZLZ(s) 2ll _Z (21)

11z22 ZL

where Z11, z22 and y2 2 are the two-port parameters of the network N.

-8-

Rearranging (16) we obtain,

bl(s) b2 (s)

Z(s) - --(s " a1 (s) b2 (s) (22)

is TMT - a2 (s

Comparing (21) and (22) we identify

at(s)z = z =

I22 a (23)

- ba(s)

L a2 s

From (23) we obtain

2 1 1 (24)12 s b12 (s)

For the set of parameters {z ij} of (23) and (24) to represent a physically

realizable set, the residue condition must be satisfied. Evaluating the

residues of z11, z2 2 and z12 in a pole at s = k' we obtain

kl 11 k 22 (' ..S1 (S) + s b(s)/

(25)

/ a2 ~ 2k12 " bl(s) - s b= a (S)b1 (s ) +

-9-

where hi(s) is the derivative of b1 (s). It is seen that the residue condition

is satisfied with an equal sign.

Next, we will have to show that a rational z12 cii b ujtahd. In2general, the numerator of z12 as given in (24) will not be a perfect square.

If it is not, the following augmentation method which is similar to Darlington's8

technique can be pursued:

Augment the associated function F(s) by a Hurwitz polynomial

m0 + n0 to obtain the modified function

m1 1 on 0 rnnmI+n

F(s) 1m+n 0n 0 r (26)F ms =n +n m +n mI+22 0 0 22

where

m 11 0 0+ nn 0

_M = m m +n n2 2 0 2 0

(27)n 1 = nlm0 + n01

nI = n m +n m2 Z 0 0 2

Recomputing the odd and the even parts of the augmented associated func-

tion we can show that there is no change in the ratio of the odd part to the

even part, both of them being multiplied by the common factor m2 - n0 .

But after augmentation the two-port parameters and ZL of (23) and (24)

will be modified as follows:

a1(s)a 0 (s) + s b1 (s)b 0 (s)Z1l Z 2 2 = s b (s)a 0 (s) + s a (s)b 0 (s)

Zl2 + /a2(s) _ s b 2(s)} {a'(s) - s b2() ( 8z 12 =s bI(s)a0 (s) + s a1(S)b 0 (s) (28)

- b2 (s)a 0 (s) + b 0 (s)a 2 (s)L a 2 (s)a 0 (s) + s b2 b(s)b0 (s

-10-

We can choose a (s) - s b2(s) equal to that factor of a2 (s) - s b2 (s) which

is not a perfect square and thus obtain a rational z 1 2 . It can be shown that

the residue condition is still satisfied with an equal sign.

Furthermore, the decomposition technique guarantees zill z 2 2 ,

1/y22 and ZL to be of the form of passive RC driving-point impedances.

A lattice realization is always possible because of the residues of

the z-parameters being equal in magnitude. Thus any real rational Z(s)

can be realized in the practical Cascade Configuration of Fig. 5 and, in

effect, we have proved the theorem mentioned in the Introduction.

B. Synthesis Using Special Structures 3

The problem of obtaining a rational z1 2 associated with the Cascade

Method is not encountered in the following method. This method uses only

one-port passive RC networks and a generalized type of impedance converter*

in special configurations. The h-matrix of a GIC is given by

0 1(29){zg

Zm

n

where Zm and Z are passive RC driving-point impedances.

Consider the network of Fig. 6, which we will designate as Type I -

Special Configuration. The input impedance of this network is

Z(s) Z 4 Z3 (30)Z4 Z3Z I Z 2

Rearranging (16) we obtain

b1 (s) b 2 (s)

Z s) s bl(s) b 2 (s) (31)

a (S) 7 -a)

*The Generalized Impedance Converter of this section will be

designated as GIC. For idealized circuits and a transistor realizationof GIC, see Re-.3.

-11-

Comparing (30) and (31) we identify

Sb 1 (s) b 2 (s)

i4 a s Z3 =as

(32)

b1(s) ai(s)

Because of the decomposition technique the set of impedances Z1, Z 2 , Z 3

and Z4 are in the form of passive RC driving-point impedances.

The input impedance Z(s) of the Type II - Special Configuration of

Fig. 7 is given as

Z Z ZZ 7 •6Z 5

Z(s) = 8Z8 Z7 5 Z (33)

8 757

8

Comparing (33) with (31), we obtain

- b 2 (s ) - a (s )Za 2 (s)8=

(34)b(s) al(s)

6 a1 s 5 su 1

It is seen that the impedances given in (34) are all passive RC driving-

point impedance functions.

IV. ILLUSTRATIONS

A. Example 1

Consider the realization of an inductance by an active RC network.

Let

Z(s) = s (35)

-12-

We thus have

N(s) = sD(s) = 1

Factorizing

Q(s) =D(s 2) - s N(s = 1- s 3

we obtain

Q(s) = (1- s)(s + s + 1)

Hence we identify

m 2 + n. s 2 + s + 1

m - nI 1 - s

that means

m1 1; n1, s

m 2 s 2 + 1; n 2 s

From this we obtain

az(s) = 1; b1(s) 1 (36I 1 (36)

a 2 (s) s + 1; b 2 (s) = 1

The Cascade Realization Substituting (36) in (24) we find that

12 s

is not a rational function. We thus choose

a2(s) - s b2(S) 1- s

i. e.,

aO(s) = 1; bo(s) = 1 (37)

Finally, from (36), (37) and (28) we obtain the modified z-parameters and

ZL as

-13-

l+sZil = z 2 2 =

_ + 1(38)s+2

- 2 + 1

The complete network considering the positive sign for z12 is as shown in

Fig. 8.

Type I Realization - From (36) and (32) we obtain

~1Z4 =I; Z 3 s'+l

= 1; Z2 IZ1 =I Z 2 8

and the corresponding network is shown in Fig. 9.

Type II Realization - Substitution of (36) and (34) results in

Z1 1z 7 s + z 8 s

Z 1 Z 1

6 5 5s

and the corresponding network is shown in Fig. 11.

B. Example 2

Consider the realization of a negative inductance. It should be noted

from (16) that in case of -Z(s), al(s) and a2 (s), and b,(s) and b 2 (s), as

obtained for +Z(s), are interchanged, respectively. As a result, we obtain

for this example from (36) the following:

al(s) = s +lI; bl1(S) =1I1 1 (39)

a 2 (s) = 1 ; b 2 (s) = 1

Substituting (39) in (24) we obtain

12 s

-14-

Augmenting we obtain a rational z12 and modified zllI z22 and ZL as

given below,

82 + 3s + 1 + 1 + 1--

Z1l z22 = 2s(s +"1) 1+ s T ) I- 1I I I

2 I1l s2+s+1 _ l I (40)Z12 2s(s+l) j2 2(s+l) I + IT- (40

Z s+2

L 2s +

and the final network obtained considering the plus sign for z1 2 is as

shown in Fig. 12.

Instead of realizing the z-parameters in a lattice structure, it is

possible to obtain an unbalanced structure by means of the Fialkow-Gerst9

method. For the given example, the parameters were partitioned as

shown by the dotted lines in (40) and the resulting unbalanced structure is

shown in Fig. 12.

This example was presented to illustrate the possibility of obtaining

unbalanced structures and also to point out that the suggested method does

not always lead to active RC networks with the least number of elements

because of the well known fact that a negative inductance can be realized by

terminating a negative impedance inverter by a positive capacitance.

C. Example 3

As a final example we will realize an impedance function whose

numerator and denominator degrees differ by two. Let

Z(s) = 2 (41)

s +2s+ 2

Following the procedure outlined in Example 1, we obtain

al(s) = s + 1; b1 (s) 1 (42I 1 (42)

a2 (s) s + 2;.bR(S) 1

-15-

and it can be seen that for a cascade realization augmentation is necessary

to in. sure a rational z12. Finally, we obtain a network realization as shown

in Fig. 13, using the formulas of (28).

V. CONCLUSION

It has been demonstrated that any real rational function of the complex

variable can be always decomposei in a unique form and can be realized as

the driving-point impedance of a ne-port transformerless active RC network

containing only one negative impedance converter. The suggested realization

netrorks are practical and the element values of the associated I RC net-woriks are easily obtained. Since the main part of the method consists of

solving a linear equation for its real and complex roots, the complete method

can be programmed easily for a digital computer.

-16-

R.in

+ + N

E. i(s) E0(s)

0

Figuire 1

01

Z ( s(S)

[+L,, +C]

Figure 2

-17-

UNIC

Z (S) -- I

Figure 3

z

z r R. G

[L' C)

(a) (b)

Figure 4

2 3

+L

3'

Figure 5

Figur 6

p GI

z. 2

Figure 6

-59

Z(B)s 1UNIC3/24/3 Values

and farads

Figure 8

1

4I Values

Figre9 L0 1 and faradsz

(a) (b)

GIG

0-z Value s7 8 0 in ohms

LZ7 -and farads

1 1 -* 11L-VA,-

(a) (b)

Figure 10

-20-

Value sin ohms

1UNIC 4/3 3/2 fards

Figure 11 -

Valuesin ohms

/23/2and farads

Z(S) s 1/2UNIC4/

Figure 12

1/24/

i i Value sin ohms

UNIC1/ 25/4 7 10and

Z (S) = /0faradE

i/s2+2+Z1/

Figure 13

REFERENCES

1. DeClaris, N., "Synthesis of Active Networks - Driving-Point Functions,"

1959 IRE Nat. Cony. Rec., Part 2, pp. 23-39.

2. Kinariwala, B. K., "Synthesis of Active RC Networks," B.S. T. J.,

Vol. 38 (September 1959), pp. 1269-1316.

3. Sandberg, I. W., "Synthesis of Driving-Point Impedances with Active

RC Networks," B.S.T.J., Vol. 39 (July 1960), pp. 947-962.

4. Kelly, P. M., "A Unified Approach to Two-Terminal Network Synthesis,"

Trans. IRE, Vol. CT-8, No. 2 (June 1961), pp. 153-164.

5. Kuh, E. S. and Pederson, D. 0., Principles of Circuit Synthesis, New

York: McGraw-Hill Book Co., 1959, p. 149.

6. Caratheodory, C., Theory of Functions, Vol. II, New York: Chelsea

Publishing Co., 1954, p. 12.

7. Kuh, E. S. and Pederson, D. 0., Principles of Circuit Synthesis, New

York: McGraw-Hill Book Co., 1959, pp. 96-98.

8. Guillemin, E. A., Synthesis of Passive Networks, New York: John Wiley

and Sons, 1957, pp. 358-362.

9. Balabanian, N., Network Synthesis, Englewood Cliffs, New Jersey:

Prentice-Hall, Inc., 1958, pp. 144-149.

-22-

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Waseiata 45, D C A01.. Dr. U. Klilo. Varian Associates

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