i
Method
for the
IEEE
TRANSACTIONS
ON
CIRCUIT
THEORY,
VOL.
CT20,
NO.
1, JANUARY
1973
D t
e ermination
OF the Transfer
Function
OF
Electronic
Circuits
BASIL
L. COCHRUN
AND
ARVIN
GRABEL
AbsfracfA
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 drivingpoint
resistances
to specify the
coefficients
of the transfer
function.
Dominantpole
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
OMINANTpole
techniques
[l][3]
have
been
used
to
approximate
both
the
frequency
and
timedomain
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 lowpass
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
nthdegree
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
drivingpoint
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
XCAPACITANCE
SYSTEM
Consider
an nport
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.
nport 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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 Fall '09
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 Determinant, Resistance, Laplace, The Circuit, Laplace expansion

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