MOTC_inductors

# MOTC_inductors - IEEE TRANSACTIONS ON CIRCUITS AND...

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 46, NO. 4, APRIL 1999 481 Transactions Briefs Extension of the Cochrun–Grabel Method to Allow for Mutual Inductances Pietro Andreani and Sven Mattisson Abstract— The Cochrun–Grabel (C–G) method, an algorithm for ﬁnd- ing the characteristic polynomial of a circuit containing reactances, has so far been restricted to circuits not employing mutual inductances. In this paper we present an intuitive, yet rigorous, proof of the Cochrun-Grabel method for a general RLC circuit, and we extend the method to allow the analysis of an RLC circuit containing mutual inductances. Index Terms— Circuit theory, Cochrun–Grabel method, method of time constants, mutual inductances. I. INTRODUCTION The Cochrun–Grabel (C–G) approach [1] to the problem of ﬁnding the characteristic polynomial of a reactive circuit has two main features: ﬁrst, it only requires the analysis of frequency-independent subcircuits derived from the circuit under study and second, it shows clearly how the polynomial coefﬁcients depend on the circuit reactances. The method may also be used for estimating the dominant pole of a multistage ampliﬁer (see, e.g., [2]–[4]) where typically only the coefﬁcient of the linear s term in the polynomial is calculated. The approximation so obtained is often very close to the actual pole value. Extensions to the method have been presented in [5]–[7]. In this paper we give a more physical, yet rigorous, proof of the method, 1 and we extend it to handle general RLC circuits with mutual inductances. II. THE COCHRUN–GRABEL METHOD It is well known that the characteristic polynomial of an RC circuit is a linear function of each capacitive admittance. The characteristic polynomial of an RLC circuit is also a linear function, but of each capacitive admittance and each inductive impedance. To see this it is sufﬁcient to apply the modiﬁed nodal analysis [8] to a circuit, resulting in the equation G C C s R C L s V I a i i v i (1) where s is the complex frequency as usual. It is clear that the role of the Cs ’s and the Ls ’s in the determinant expansion for the matrix in (1) is identical. The characteristic polynomial [i.e., the determinant of the matrix in (1)] can be written in general as p H @ s Aa a H H C a H I s C a H P s P C C a H n s n : (2) In the following, we will consider the normalized polynomial: p @ s p H @ s A a H H aIC a I s C a P s P C C a n s n : (3) Manuscript received September 14, 1998. This paper was recommended by Associate Editor N. Josef. The authors are with the Department of Applied Electronics, Lund Univer- sity, S-221 00 Lund, Sweden (e-mail: [email protected]). Publisher Item Identiﬁer S 1057-7122(99)02753-1. 1 The original proof was limited to RC and RL circuits only.

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## This note was uploaded on 12/29/2011 for the course ECE 594A taught by Professor Rodwell during the Fall '09 term at UCSB.

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MOTC_inductors - IEEE TRANSACTIONS ON CIRCUITS AND...

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