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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. I NTRODUCTION 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 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. T HE C OCHRUN –G RABEL M ETHOD 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 (1) where is the complex frequency as usual. It is clear that the role of the ’s and the ’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 (2) In the following, we will consider the normalized polynomial: (3) Manuscript received September 14, 1998. This paper was recommended by Associate Editor N. Josef.
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