University of Florida
ECH 4323L / ECH 6326
LABORATORY 2
CALCULATION OF RESIDUES USING MATLAB
Student Name:_______________________________
MATLAB can be used to calculate the residues of a rational function
Y(s) =
(1)
where B(s) is the numerator polynomial of order m, and A(s) is the denominator
polynomial of order n.
The poles of the denominator polynomial are {p
1
, p
2
, p
3
,
..., p
n}
and can be found by taking the roots of the denominator polynomials.
The poles can be
real or complex, distinct or repeated.
When the rational function (1) is strictly proper
(
i.e.,
m < n) basic theory of partial
fractions expansions is as follows.
When the poles are distinct (real or imaginary), the
partialfractions expansion of Y(s) is of the form
Y(s) = +
+
+
...
+
(2)
where r
q
is the residue corresponding to pole p
q
, k=1, 2, …, n.
The residue r
q
is a real
number when pole p
q
is real, and r
q
=
β
+i
γ
is a complex number when p
q
is complex.
If
there are repeated real poles, say p
1
=p
2
=p
3
, then
Y(s) = +
+
+
...
+
(3)
Finally, when the rational function is not strictly proper
, (
i.e.
, m
n), the partialfractions
≥
expansions includes and additional polynomial
K(s), and expansions (2) and (3) must be
respectively modified to the forms
Y(s) = K(s) + +
+
+
...
+
(4)
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 Spring '10
 Crissale
 Complex number, Rational function

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