Reza, F. M.

DSpace@MIT

XIX. NETWORK SYNTHESISProf. E. A. Guillemin Dr. M. V. Cerrillo R. A. PucelProf. F. M. Reza P. M. Lewis II M. StriebyA. SYNTHESIS OF RC NETWORKS IN THE TIME DOMAIN1. Introduction. The significance of eliminating inductors wherever possible in thedesign of passive lumped networks has given considerable stimulus to the problem ofdesign of RC networks. In 1949, E. A. Guillemin established certain results of theoreti-cal and practical significance (1). His main result consists of the following:"... Except for a constant loss, any transfer characteristic may be approximatedarbitrarily closely by means of a linear passive network containing only R's and C's,and its physical realization is always possible in an unbalanced form suitable for oper-ation between conventional vacuum tubes."The reader undoubtedly realizes the practical importance of this statement, whichmore or less puts RC networks on an equal basis with RLC networks. In mathematicalterms Guillemin specifically shows that any transfer characteristic of a two-terminalpair passive RLC network with a finite number of lumped elements could be "approxi-mated" by the transfer characteristic of a two-terminal pair passive RC network withlumped elements.12RLC 2 Z12R C (j) 2(Evidently in Guillemin's statement one must exclude the case in whichIZ12 (RLC) ( jw) 2 becomes infinitely large. However, this case is of no importance inpractice. It should be noted, too, that Guillemin's statement can be verified by the useof several different approximation techniques such as those given in references 2 and 3.)2. Purpose. The object of this investigation is to show:"If f(t) is the response of an RLC two-terminal pair, passive network with a finitenumber of lumped elements to a unit impulse applied at time t = 0, then f(t), except fora constant scale factor, can be 'approximated' by the response of a passive two-terminalpair RC network with lumped elements to a unit impulse applied at time t = 0. Thetolerance of the approximation can be made as small as possible, but then one mustallow a more complex RC network." A method for finding the desired RC transfer func-tion is described. Once such a transfer function is obtained, one may consider theexisting synthesis procedures for its realization.3. A Useful Transformation. Let f(t) be the response of an RLC finite, linear,passive, lumped-parameter network to a unit impulse uo(t). Then f(t) is a continuoustranscendental function on the positive real axis of the time. Alsof(t) - 0, t < 0 (1)-96-(XIX. NETWORK SYNTHESIS)RLCI ----- 2f(t)Fig. XIX- 1Unit impulse.Fig. XIX-2Network with finite linear lumpedpassive elements.Fig. XIX-3Assumed responseof N to uo(t)f(t). _ ..0 2 3 4t - DOMAIN- -_ - ]I i I I I [ I EI0 01 02 03 04 05 06 07 08 09 Ix - DOMAINThe transformation of theFig. XIX-4time domain into the x-domain by Eq. 2.We may conveniently use the following transformation:-a tx= 1 - , t>0 (2)where a is a positive constant. (See Figs. XIX-1, XIX-2, XIX-3.)To each point of the positive real axis of the t-domain (Fig. XIX-4), there corre-sponds one point of the positive real axis of the x-domain. The positive real axis ofthe t-domain is transformed into a segment (0, 1) of the x-axis.-97-aot=ou0 (t)t(XIX. NETWORK SYNTHESIS)t = 0 corresponds to x = 0(3)t = oc corresponds to x = 1This changing of the time scale will give us a new function 4(x), instead of f(t), suchthat if xk corresponds to an instant tk then 4(xk) = f(tk). The merit of this transforma-tion will be better understood in the following sections.4. Weierstrass' Approximation Theorem. This approximation theorem is wellknown (4, 5). It states:If p(x) is a continuous function in (a, b) and E is a given positive number, there existsa polynomial P(x) such thatI (x) - P(x) < E; a < x , b (4)Since f(t) and hence 4(x) are continuous functions, within the intervals with which we areconcerned, we may apply inequality 4 by letting a = 0 and b = 1. Thus a polynomial2 nP(x) = a 0 + alx + a2 + . .. an x n  (5)could be found to approximate 4(x) in the sense of inequality 4 as closely as desired.5. Determination of an RC Transfer Function. There are a great number of approx-imation procedures known and used in engineering problems. We are not consideringhere a discussion of approximation techniques. It is fairly simple to select, forinstance, a number of points on the graph of 4(x) and determine the coefficients of apolynomial P(x) passing through those points. Now P(x) should be transformed in a timeresponse P(t) by the use of Eq. 2.P(t) = a 0 + a 1  E- + a 2 1 - + .. an - E(6)or-a t -2a t -na tP(t) = A 0 + A1  o + A 2 E + .A E (7)Now, using the Laplace transformationF(s) = (-st) f(t) dt (8)0one easily derives the transform of each term of Eq. 7, and finallyA A A AP(s) + + + (9)s s + s + 2a "' s + na-98-(XIX. NETWORK SYNTHESIS)The coefficients A k are real, otherwise arbitrary. Equation 9 describes an RC transferfunction whose poles are preassigned on the negative real axis of the frequency domain.This procedure is summarized in the following section.6. Synthesis Procedure Summarized.Step 1. Using transformation 2, graphically or analytically determine (x) from agiven f(t).Step 2. Using existing approximation techniques, determine a polynomial P(x) suchthat I (x) - P(x) I < E for 0 < x < i.Step 3. Using Eq. 2, transform P(x) into a time function P(t).Step 4. Using the familiar Laplace transformation of Eq. 8, find P(s), the Laplacetransform of P(t).Step 5. Use existing RC synthesis procedures such as those given in references 6and 7 to obtain a network realization.F. M. RezaReferences1. E. A. Guillemin, Synthesis of RC-Networks, J. Math Phys. XXVIII, No. 1, p. 22(1949).2. G. L. Matthaei, Filter transfer function synthesis, Proc. I. R. E. 41, 377 (1953).3. C. B. Sharpe, A general Tchebycheff rational function, Proc. I. R. E. 42, 454 (1954).4. E. C. Titchmarsh, The Theory of Functions (Oxford University Press, 1939) p. 414.5. D. F. Tuttle, Jr., Network synthesis for prescribed transient response, DoctoralThesis, Department of Electrical Engineering, M. I. T., (1948) p. 16.6. A. D. Fialkow, Two terminal-pair networks containing two kinds of elements only,Proceedings of the Symposium on Modern Network Synthesis, Polytechnic Instituteof Brooklyn, New York (1952).7. J. L. Bower and P. F. Ordung, The synthesis of RC networks, Proc. I. R. E. 38,263 (1950).8. E. A. Guillemin, A note on the ladder development of RC networks, Proc. I. R. E.40, 482 (1952).-99-

Network Synthesis

http://dspace.mit.edu/bitstream/1721.1/51209/1/RLE_QPR_034_XIX.pdf

Network Synthesis

Abstract

Similar works

Full text

Available Versions

DSpace@MIT