a note on computer-aided synthesis of resistive n-ports

319 SHORT PAPERS

ample) are

2 4 lr 0 -11

fl. 2 1 0

0 cz=3 1 -1 1 1 Tt’ =

--I 1J 4 0 1 5 -1 1

Furthermore, the solution matrix corresponding to CZ can be obtained from that corresponding to Ci through the following transformation:

(C&G)-l = (Tz’)-‘(cltzC,)-‘(T~~)I-~. (31)

Using (31), we can effect the removal of branch 5 which becomes a link in CZ.

V. SBN~ITIYITY

The solution matrices and their respective converses provide a direct method in evaluating the partial derivatives of the solution matrices with respect to the variations in the impedance or admittance parameter of any branch in the network. Such partial derivatives are needed in the sensitivity study. Consider the case of the tree solution matrix. Let zi be the impedance of the link element in question. Upon taking the partial derivative ‘of the solution matrix in (13) with respect to zi (see, for instance, [S, p. 161]), ‘we obtain

az, ;; = ZTCT[(CT)LZTCT + G-l $ [(CT)~ZTCT + ZL]-~(C~)~~T. (32)

1

By virtue of (13) and (16) after a straightforward manipulation, (32) becomes .

az, azL -- azi

= Z*CTZL--l __ zL-1(cT)tz8. azi

It is noted that aZJazi is a matrix with only one nonvanishing element equal to unity in the position occupied by zi in ZL. In turn, the product matrix Z~i(Z~/zi)z~-i yields a matrix of which the only nonvanishing element is l/zi2 in the zc positition in ZL. Mindful of the similar effect the latter has on the rest of the product, (33) can thus be written as

az, 1 --= ; Z,CT"(CTi) tz, azi t2 (34)

where Gi is the column matrix in CT relating to zi. Repeated use of (34) provides a natural extension to the evaluation of the partial derivatives of higher orders. By taking the partial derivative of (34) with respect to zj, the second derivative becomes

azz,

zazj = 2, g Z~@(CT? tZc~i(C~i) tZ,

where the summation is done over the permutation PZ of i and j. The third derivative can readily be shown as

where PI indicates the summation over the permutation of i, j, and k. The extension to higher orders is apparent. Nevertheless, the procedure holds only for the link elements. ‘.

Similarly, the partial derivative of the noda! solution matrix with respect to the link impedance zi is

az,,’ 1 -= ; Zla(ALf)tAL~Z1* azi 2

where Ar,i is the column matrix in AL relating to zi and the partial derivative of the mesh solution matrix with respect. to the tree branch admittance yG is ,

av, --= -!- Y,DL”(DL”)~Y. a% R=

where DL~ is the column matrix in DL relating to y,,. The extension to the higher order partials follows the same,procedure as above.

Equation (37) can also be obtained from (34) by premultiplying (34) by (&)t and postmultiplying (34) by Br, upon recalling the fact that Zi, = (BT)~Z& and CT = -Br(Ar,)t. The same expression is obtained earlier by Branin by using an additional link in the ith branch [3].

VI. CONCLUSION

In this short paper, the solutions of the converse problems of Kron’s “factorized form” of the network solution matrices are found. Some interesting symmetries are observed in the converse solutions as against the solution matrices. For those who are familiar with Kron’s special theory ‘of ,diakoptics, these converse problems are akin to Kron’s tearing concepts.

The stepwise algorithm used to construct the solution matrices from the partial solution matrices is also applicable in recovering the partial solution matrices fromthe solution matrices. Such procedure used both days makes it possible to modify the solution matrix in simple steps when the network elements are modified. It should be pointed out that fo’r this to work, the mutual couplings, if existed, can exist only among the tree ,branches in the case of the tree or nodal method, or’ the link elements in the mesh case. In the most general case where the mutual couplings exist among all the elements,’ the stepwise algorithm concerning the removal of one link or tree’branch at a time is not possible. However, it is always possible in obtaining the solution matrices, even in the presence of the mutual couplings, provided that all the impedances or admittances are up- dated’ at each step.

Unlike other methods in which either additional elements [3] or current ratios [6] are needed, the converse and the solution matrices provide a simple and direct means in evaluating the partial derivatives of all orders of the solution matrices with respect to the variations of the impedances or admittances of the appropriate branch elements.

ACKNOWLEDGMENT

The author wishes to thank Prof. C. T. Tai for his valuable sug- gestions.

REFERENCES

111 G. Kron, Diakopfics. London, England: Macdonald. 1963. [Zl -, Tensor Analysis 0-f Nefwork (rev. ed.). London. England: Macdonald.

1965. (31 F. H. Branin. Jr., “The relation between Kron’s method and the classical meth-

ods of network analysis,” Ma!rir and Tensor Quart.. vol. 12, pp. 69-105. Mar. 1962.

[4] -, “An abstract mathematical basis for network analogies and its significance in physics and engineering,” M&ix and Tensor Quart.. vol. 12. pp. 31-49. Dec.

[Si kq6~~Uman Inlroduclion LO M&ix Analysis. New York: McGraw-Hill, 1960. [6] P. Penfield,’ Jr.. R. Spence. and S. Duinker. Tellegen’s Theorem and Electrical

Nelworks. Cambridge. Mass.: M.I.T. Press, Res. Monogr. 58. 1970.

A Note on Computer-Aided Synthesis of Resistive n-Ports

M. R. KARIM AND PAUL M. CHIRLIAN

Abstrac’-A technique of synthesizing a resistive n-port is de- scribed. The port structure is first determined using some necessary conditions. Branch conductances are obtained by solving a system of equations on a computer.

I. INTRODUCTION

The problem of synthesizing a resistive n-port network with more than n+l nodes from a prescribed short-circuit admittance or open-

Manuscript received February 11. 1972; revised September 11. 1972. M. R. Karim was with the Department of Electrical Engineering, Stevens Insti-

tute of Technology. Hoboken, N. J. 07030. He is how with Bell Laboratories. Whip- pany, N. J. 07981.

P. M. Chirlian is with the Department of Electrical Engineering. Stevens Insti- ture of Technology. Hoboken. N. J. 07030.

320 IEEE TRANSACTIONS ON CIRCUIT THEORY, MAY 1973

circuit impedance matrix has not been solved completely. Director and Rohrer [l] have developed a computational algorithm. In this procedure, a port configuration of 2n nodes is assumed. Branch admittances are determined by minimizing a function with a computer using the concept of adjoint networks. If a matrix is not extremali or the inverse of an extremal impedance matrix, more than one realization with the same, or even different, port configuration may be possible [2]. In that case the algorithm converges rapidly. Some matrices, on the other hand, may not at all be realizable except with a specific port configuration. One such matrix is the following:

rlo 5 5 0.11

(1)

For these matrices the algorithm fails to converge. The synthesis problem must, therefore, be solved in two parts: one involving the determination of a port configuration and the other that of branch admittances. The purpose of this short paper is to describe a synthesis technique wherein the port configuration is first determined using some necessary conditions of an earlier paper [3]. The branch conductances are then obtained by solving a set of algebraic equations with a digital computer.

II. SYNTHESIS TECHNIQUE

To describe the synthesis procedure, we will realize the following real matrix

r7 4 -0.75 0 1

L 0 1.2 3 8 d

as the short-circuit admittance matrix of a transformerless 4-port network composed only of positive resistors. First, we determine the port structure. It can be shown [3] that (2) cannot be realized on a linear-tree port configuration. Suppose it is partitioned as in (3) below:

-0.75 0 1 1.2

------

8 3

3 > 8

(3)

such that the principal submatrices Yn and Y22 satisfy the necessary conditions of [3, theorem 11. It may, therefore, be possible to realize (2) with the port structure of Fig. l(a). With this port configuration we may proceed to determine branch conductances in the following way.

It should be noticed that there may be a resistor between each pair of nodes. Thus there may be 15 resistors in the 4-port. Let xi, . . . rib represent conductance values of these resistors. In Fig.

l(a), ‘we assume that a port is added between nodes 3 and 4. Let Y,(X) be the short-circuit admittance matrix of the resulting S-port, called the augmented network, where X = (3~1, . . . , 3~15). Y,(X) is the short-circuit admittance matrix of the resulting S-port, called the augmented network, where X = (xi, . . . , x15). Y,(X) is partitioned as in (4):

where Yn.(X) and Yz.(X) are short-circuit admittance matrices of the augmented S-port with, respectively, the addedand original port(s) short-circuited. If the added port of the augmented S-port is now open-circuited, the result is a 6-node 4-port network with admittance matrix Y’ given by

1 A real symmetric matrix Y =[y<j] of order n is extremal if for some index k, lyikl =yii, for i=l, . . . , 72.

I k

I

+

2

c

2 +

3

(a)

+ 4

>

5

3+

4

1.6

(b) Fig. 1. Realization of matrix (2). (a) Port configuration. (b) Complete realization.

Values are in mhos. Functional F(X) has been reduced to 0.0409.

Clearly, if there is a value of the vector X with all its components nonnegative such that matrices (2) and (5) are equal, the required 4-port will be realized. Hence, equating these two matrices and noting that they are symmetric, we obtain the following nonlinear algebraic equations:

&j(X) = Yij, i=1;..,4,j>i. (6) Equations (6) can be solved in a number of ways. One of them is to define a functional

F(X) = 2 2 (Oij(X) - Yij)’ (7) i-l j-i

or some other increasing real function of the absolute values of the errors cij(X) -j’<j. The problem of solving (6) is then approximately equivalent to minimizing (7). If F(X) can be minimized below an acceptable value, the matrix Y can be considered to be realizable with the predetermined port structure. If we can make F(X) zero, the solution is exact. It is minimized on a digital computer using a direct search method. The search is initiated with the conductance between nodes 3 and 4 set to 1 mho, all the rest being zero. The realization is given in Fig. 1 (b). Matrices of order up to ten have been realized with great accuracy using this technique. Impedance matrices can be handled similarly.

III. REMARKS

1) If a matrix is realizable with a certain port structure, the search converges rapidly (for all examples tested). For instance, in the realization of matrix (2) the functional F(X) was reduced from 305 to 0.0409 in 12 s of central processing unit (CPU) time (IBM 360/67). Four-ports having the same admittance matrix (2), but with conductances different from those of Fig. l(b), may be obtained by choosing a different step size for the optimization process or initiating the search with different values of conductances between adjacent port subtrees.

2) Only necessary conditions [3] have been used in the determination of port configurations. Nevertheless, every example tested that satisfies these conditions has been realized with arbitrarily small errors. This, of course, does not prove the sufficiency of the conditions, but indicates that the procedure works in a great many cases.

3) As mentioned earlier, there are matrices that have been proved to be realizable, but which cannot be realized with 2n nodes. To see

SHORT PAPERS 321

(a) v) (c)

Fig. 1. Transformation of a noisy two-port into a noisefree twc-port with two Fig. 2. Port configuration with which matrix (1) can be realized. uncorrelated noise sources and with two correlation impedances.

this, consider matrix (1) again. In an attempt to realize it with 8 nodes, (7) could be reduced from 529.02 to 72.07 in less than 25 s, to port 3.51 in approximately 90 s, to 1.44 in 3 min, and to 1.33 in 4 min. Similar results were obtained with the conjugate gradient method. One can, therefore, conclude that matrix (I) is not realizable with eight nodes. It can be shown [3] that it is not realizable with six nodes either. Suppose it is partitioned as in (8) below:

10 I 5 5 I 0.1 Pen

--_I ----I ---

I [YGl = z i

15 5 I 0.1

5 71 4 Fig. 2. Interreciprocity for the original and the adjoint network.

--I ---_I ---

0.1 I 5418 -

$2; fif. Z]. (8)

Since Yu, YZZ, and Ys3 satisfy the necessary conditions of [3, theorem 1 I, a port configuration may be as shown in Fig. 2. In fact, calculations have shown that the numerical process rapidly converges with this port configuration.

Fig. 3. Symbols and sign conventions used for two corresponding terminals.

REFERENCES .-.-. ,.-. . _

111 S. W. Director and K. A. ~ohrer, “un the design O* reastar ice n-port networks 1 (a)] is transformed into a noisefree two-port with two external corre-

bv dicital commuter.” IEEE Trans. Circuit Thaw. vol. C .^. i?;p.. i969: .- _

~~. -T-16. pp. 337-346. lated noise sources [Fig. l(b)]; then it is changed according to Fig. ^ ..- _

,L, A. Lempel and 1. Cederbaum, ‘~ermlnal conhguratmns 01 - n-port networks,” 1 (c) with two uncorrelated noise sources and two correlation imped-

IEEE Trans. Circuit Theory. vol. CT-15 pp. 50-53, Mar. 15 b68. antes. Now the application of the procedure of [2] would be possible. [3] M. R. Karim and P. M. Chirlian, “Synthesis of resistive i

linear P-tree port structures,” n-port networks on

19. nn 7RL-75(6 M-v 1077 IEEE Trans. Circuit Theory ( Corresp.). vol. CT-

It is obvious that this transformation is very time consuming. In --. rr .--- ___, _.__, _, .-. addition, the original .network has to be changed. By this reason, a

new procedure detailed in the following sections is given leading to a considerable simplification.

II. INTERRECIPROCITY TO THE ADJOINT NETWORK

The interreciprocity is valid for linear time-invariant circuits that are interreciprocal relative to each other. Bordewijk defined the interreciprocity in his paper [3] as follows (see Fig. 2) :

Computerized Determination of Electrical Network Noise Due to Correlated and Uncorrelated

Noise Sources

KARL HARTMANN, WILLY KOTYCZKA, AND MAX J. 0. STRUTT

Abstract-A new computation method is given connected with the analysis of equivalent circuits including uncorrelated as well as correlated noise sources, thus saving computer time.

I. INTRODUCTION

Noise calculations in microwave circuits have to be as efficient as possible, because minimization of the noise factor is often a design objective. For these devices there are many equivalent circuits with correlated or partially correlated noise sources [4]-[7]. In [2] an efficient noise computation method for only uncorrelated noise sources is given. For this procedure the correlated sources have to be trans formed into uncorrelated ones. This transformation is laborious. As an example, the application of [ZJ is discussed with a microwave bipolar transistor (Fig. 4). The noise sources i,a and i,, separated by the capacitance CT are correlated. At first, the noisy two-port [Fig.

Manuscript received April 3, 1972; revised September 11 1972. The authors are with the Department of Advanced Electrical

Federal Institute of Technology, Zurich, Switzerland. Engineering. Swiss

2 zk’ vk’ = 2 zk’. vk. k-l k=l

For the validity of these equation s, a sign convention has to be intro duced between the original network and the topologically identical adjoint network. This sign convention is represented by Fig. 3. Rohrer et al. [2] do not use these signs because they are only inter- ested in the squared voltages at the output. For the correlated noise sources, however, this sign convention is essential.

If an ideal current source Z is put at the port k of the original network (Fig. 2), the value V,, is measured at the port n (the sign included). If the same ideal current source Z is put at the port n’ of the adjoint network, the value VW is measured at the port k’. By (1) it can be proved that the voltages V,, and Yk, are equal (the sign included). The following section shows the application of this result.

III. EFFICIENT COMPUTATION OF THE NOISE VOLTAGES AND NOISE FACTORS

The theory is applied to a microwave bipolar transistor. In [4]-[7] the noise equivalent circuits of microwave transitors are shown. In [l] the original network and the adjoint network of a bipolar transistor are derived. Fig. 4 shows the original network with the noise sources. In Fig. 5 the adjoint network without the noise sources but with the unit current source Z at the output and the voltages, which have to be calculated, are given. The noise sources are

a note on computer-aided synthesis of resistive n-ports

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