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Cochrun_Grabel_MOTC

# Cochrun_Grabel_MOTC - IEEE i Method TRANSACTIONS for the D...

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i Method for the IEEE TRANSACTIONS ON CIRCUIT THEORY, VOL. CT-20, NO. 1, JANUARY 1973 D t e ermination OF the Transfer Function OF Electronic Circuits BASIL L. COCHRUN AND ARVIN GRABEL Absfracf-A general method based on the Laplace expansion for determining the transfer function of a wide variety of linear elec- tronic circuits is discussed, The technique developed requires only the calculation of a number of driving-point resistances to specify the coefficients of the transfer function. Dominant-pole techniques are used and extended, making the procedure useful in both analysis and design. As computation only involves resistance networks, complex arithmetic is not required in determination of the response. INTRODUCTION D OMINANT-pole techniques [l]-[3] have been used to approximate both the frequency and time-domain responses of linear active systems. Most of these techniques require that the coefficients of the characteristic polynomial be known in order to be applied. This paper describes a method for determining the coefficients of the characteristic polynomial without the need for evaluating the system determinant. In addition, the method allows the circuit designer to re- late system performance to specific circuit elements and by means of dominant pole techniques to assess their effect on the circuit. For convenience, the low-pass case is developed. The results obtained are readily transformed to the high- pass case by duality and frequency translation. The basic approach to the problem is to generate the charac- teristic polynomial of the form G(s) = Ao 1 + UlS + u& + . . . u,sn (1) The nth-degree polynomial is considered to arise from a system containing n storage elements. By use of the Laplace expansion of a determinant [4], the coefficients are generated. The calculations involved require only that driving-point functions of purely resistive net- works be determined. For a wide variety of electronic circuits, the method significantly reduces the algebra required compared with that required for the evaluation of a determinant. THE X-CAPACITANCE SYSTEM Consider an n-port system, shown in Fig. 1, with n capacitances C,, C2, . . . , C, across ports 1, 2, . . .-, n, respectively. The circle encloses a linear active network with no energy storage elements. The entire network Manuscript received March 17, 1972; revised June 26, 1972. The authors are with the Department of Electrical Engineering, Northeastern University, Boston, Mass. 02115. Fig. 1. n-port with n capacitances. can be represented by a set of node equations with the admittance determinant A given by (2), where the C’s appear only on the principal diagonal: g11 + SC1 02 g13 . . . g1* g21 g22 -I- SC2 g23 . . 0% 1 gn2 The natural frequencies are determined by A(s) = c;; bisi = 0. I n order that A(s) be conveniently com- pared to the denominator of (l), A(s)/bo is formed as A bo= 1 + 2 UiSi = 1 + als + a29 + . . . unP.

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Cochrun_Grabel_MOTC - IEEE i Method TRANSACTIONS for the D...

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