1246

where

PROCEEDINGS OF THE IEEE, AUGUST 1975

On Geometric Progression Ladder RC Networks B. B. BHATTACHARYYA AND M. N. S. SWAMY

Absmcr-The possibility of an approach to the analysis of non- recurrent RC ladder networks by means of recurrent RCG ladder networks is presented.

INTRODUCTION This letter indicates the possibility of an interesting approach to the

analysis of nonrecurrent RC ladder networks by deriving an equivalent circuit of the geometric RC ladder network recently analysed by Pang [ 121 in terms of an RCG recurrent ladder.

Ladder networks have been previously analyzed by a number of methods, such as continued fractions, Chebyshev polynomials, iteration, signal flow graphs, continuants, etc., [ 11 -[ 101 . However, closed-form solutions for the various network functions were obtained, so far, only for the recurrent ladders [ 101 , [ 111. Recently, Pang analyzed a geo- metric progression nonrecurrent RC ladder network by introducing Chebyshev rational functions of the f i s t and second kinds [ 121 . The purpose of this letter is to further point out that an interesting relation exists between a geometric RC and a recurrent RCG ladder. ’This relationship is derived by obtaining the [y] matrix of the geometric ladder in terms of that of a recurrent ladder.

THE [ y ] MATRIX Consider the n-section RC ladder network of Fig. 1. The resistance

and capacitance distributions of this are defined by

( g = 1 , 2 ; - - , p , g # i , k ) . (15)

Summing (15) over all g and rearranging them by using (4), ( 5 ) , and (9) , we obtain

X ( i , h ; h # i, k )

- - - (Ef x Hk +#!?k x R:) * hk &. (16) X(k, h ; h # i , k )

Adding the surface integral on Si n Sk to that in the left-hand side of (16) and the surface integral on Sk n Si to that in the right-hand side of (16) by using (4), (5), and (9), we obtain the reciprocity theorem

-(jieiwr)*ai.E(ri,ik,t)=JkeiWrdk’E ( r k , i i , t ) . (17) -* -

The conclusion is B a t the2eciprocity theorem (17) is valid only when E ~ , p - p , a n d Z t = - Z . =’ f== =t - =

REFERENCES

[ 1 ] K. Kurokawa “Electromagnetic waves in waveguides with wall imnedance.” ;RE Trans. Microwave Theory Tech., vol. MTT-IO, ~~~~

pp:314-320, Sept. 1962. 121 R. B. Dybdal, L. Peters, l r . , and W. H. Peake, “Rectangular wave-

guides with impedance walk,” IEEE Trans. Microwave Theory

[3 ] R. E. Collin, Field Theory of Guided Waves. New York: McGraw- Tech., vol. MTT-19, pp. 2-9, Jan. 1971.

[4 ] A. A. M. Saleh, ‘‘Conservation relation for lossless anisotropic Hill, 1960.

media,”Proc. IEEE (Lett.),vol. 59, pp. 1375-1376, Sept. 1971.

Application of Kirchhoff‘s current law at node k yields the following second-order linear difference equation.

V ( k + l ) = ( s o R C + a + l ) V ( k ) - a V ( k - l ) , k = 1 , 2 , . . - , n . (2a)

Also the current through the resistor in the kth section is

Let us introduce the transformation

Then we have from (2a) and (3)

which may be put in the form

Vu(k + 1) = [(s + uU) RUCu + 21 Vu(k) - Vu(k - 1) (5)

where

R , = & R C,=C (6a)

(I,, = (a - 2 6 + 1 ) ( u R 0 - l . (6b)

The second-order linear difference equation (5) may now be identified Rotating Coordinate Frames’’ with that of the voltage of a recurrent ladder with parameters given by

Corrections to ‘The Electromagnetic Field in

T. SHIOZAWA (6a) and

In the above letter,’ the following corrections should be made. 1) For the notation-of the current densities in (9 ) and (lo), and in

2 ) On ths left-hand side of ( 9 ) , H P y and r p should be replaced by

3) The term p . J in (1 1) should read po J.

G , = auC,. (6c)

the line preceding (9 ) , J should be w-d in place of J. Thus the current in the kth section of the recurrent ladder is given by

gw/c and J p / c , respectively. Iu (k) = [Vu(k- 1)- Vu(k)l (7) RU

Manuscript received April 1 1 , 1975. Manuscript received January 13, 1975; revised February 18, 1975. The author is with the Department of Electrical Communication En- This work was supported by the National Research Council of Canada

H. A.’ Atwater and T. Shiozawa, Proc. IEEE (Lett.), vol. 63, pp. The authors are with the Department of Electrical Engineering, gineering Osaka University, Suita, Osaka 565, Japan.

316-318, Feb. 1975. Concordia Universlty, Montreal, P.Q., Canada.

under Grants 7739 and 7740.

PROCEEDINGS LETTERS 1247

VI01 R i l l 4121 RiK1 RiK+II Wnl

V(21 V IK- I I V iK l V IK+ I l Vh - I1

0

Fig. 1. The geometric progression RC ladder network.

Fig. 2. Equivalent circuit of geometric progression RC ladder network in terms of an RCG recurrent ladder.

Therefore, from (2b) and (7) we obtain

Now, using (3) and (8) we have

and

We also have

where [y ] ,, is the admittance matrix of the n-section recurrent ladder (6). The parameters of [ y l U may be obtained easily from an earlier article using Morgan-Voyce polynomials [ 111 .

Thus from (9) and (lo), we have the [ y ] of the ladder (1) as

The equivalent circuit of the geometric ladder (1) may be obtained in terms of the recurrent ladder (6) from (11) and is shown in Fig. 2.

The ABCD parameters of the geometric ladder may now be deter- mined either from (11) or by using the equivalent circuit of Fig. 2, and are given in the following:

(12a)

where

which are seen to be the same as those obtained by Pang [ 121. In conclusion, it has been shown that the geometric progression

ladder analyzed by Pang [ 121 is equivalent to an RCG recurrent ladder followed by ideal transformers at the input and output ends and terminated by suitable conductances on both ends. The approach is interesting in that it points out the possibility of representing other ’ nonrecurrent ladder networks in terms of the recurrent ladder networks.

REFERENCES

T. C. Fry, “The y,e of continued fractions in the design of electrical networks, Bull. Amer. Math. SOC., vol. 35, pp. 463-

D. -L. Finn, “Continued fraction analysis of tandem networks,”

A. Colombani, “The theory of electric f&ers and the Tchebycheff 1953 Proc. Nat. Electron. Conf., vol. 9, p. 16-25.

polynomials,” J. Eng. Phys (USSR), vol. 7, pp. 231-243, Aug. 1946.

in ladder networks, IRE Tmns. Circuit Theory (Corresp.), vol. V. 0. Mowery, ‘‘Fibopacci numbers and Tchebycheff polynomials

CT-8, pp. 167-168, June 1961. F. F. Kuo and G. H. Leichner, ‘‘e iterative method for deter- mining ladder network functions. Proc. IRE (Lett.). vol. 47.

498, JUly-AUg. 1929.

pp. 1f83-1784, Oct. 1959. T. R. Bashkow, “A note on ladder network analysis,” IRE Trans. Ctrcult Theory (Corresp.), vol. CT-8, pp. 168-169, June 1 9 6 1 .

. ,, ~

- -__- - - --. S. C. Dutta Roy, “Formulas for the terminal impedances and

Inst. Elec. Eng.,vol. 111, pp. 1653-1658, Oct. 1964. transfer functions of general multimesh ladder networks,” Proc.

ladder network functions,” Proc. IRE (Lett.), vol. 48, pp. 1175, G . H. Burchll, “A signal flow graph method for determining

June 1960. M. N. S. Swamy, “Continuants and ladder networks,” Proc. ZEEE (Lett.), vol. 54, pp. 11 10-1 11 1, Aug. 1966. V. 0. ,Mowery, “On hypergeometric functions in iterated net- works, ZEEE Trans. Circuit Theory, vol. CT-11, pp. 232-247, June 1964.

ladders using the polynomials defined by Morgan-Voyce, ZEEE M. N. S. Swamy and B. B. Bhattacharyya, “A study of refwrent

Trans. Circuit Theory,vol. CT-14, pp. 260-264, Sept. 1967. K. K. Pang, “Chebyshev rational functions and ladder RC networks,”IEEE Trans. Circuit Theory, vol. CT-16, pp. 408-410, Aug. 1969.

On the Integration of Unstable Linear Systems FRED TAYLOR

Absrmcr-An efficient approach to the problem of numericaIly computing the solution to a system of ordinary autonomous differ- enthl equations has been proposed by DaPison and Maki In this letter, their alrprithm is modified to effectively treat the problem of integrating in unstable system whose forcing function is a constant.

INTRODUCTION Control and power engineers often use linear autonomous state d e

termined models of the form

i ( 0 = A x ( f ) + Bu(t), ~(0) = xo (1)

where x ( t ) E En and u(t) E E”. For simplicity, a single-channel con- trol problem (i.e., rn = 1) shall be considered. A commonly encoun- tered problem is one of solving (1) when the eigenvalues of A have real components near zero or in the right-hand plane.

A Crank-Nicholson numerical approach to a system of linear autono- mous fiist-order differential equations was proposed by Davison in 1967 [ 11, [2]. For a class of inputs which belong to a class of poly-

Manuscript received January 27. 1975:revised Aoril 1. 1975. , ~, he author is with the Depariment of Electr ih Engineering, h i -

sinhe versity of Texas, El Paso, Tex. 79968.