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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
CochrunGrabel 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 frequencyindependent
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, S221 00 Lund, Sweden (email: [email protected]).
Publisher Item Identiﬁer S 10577122(99)027531.
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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