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Unformatted text preview: Chemical Engineering Department
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
B(s)
Y(s) = A(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 {p1, p2, p3, ..., pn}
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 partialfractions expansions is as follows. When the poles are distinct (real or imaginary), the
partialfractions expansion of Y(s) is of the form
r1
r2
r3
rn
Y(s) = sp + sp + sp + ... + sp
1
2
3
n (2) where rq is the residue corresponding to pole pq, k=1, 2, …, n. The residue rq is a real
number when pole pq is real, and rq=β+iγ is a complex number when pq is complex. If
there are repeated real poles, say p1=p2=p3, then
r1
r2
r3
rn
Y(s) = sp + (sp )2 + (sp )3 + ... + sp
1
n
2
3 (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 © Oscar D. Crisalle 19942009 ECH 4323L Lab 2  Calculation of residues using MATLAB 2 r1
r2
r3
rn
Y(s) = K(s) + sp + sp + sp + ... + sp
1
2
3
n (4) r1
r2
r3
rn
Y(s) = K(s) + sp + (sp )2 + (sp ) 3 + …+ sp
1
n
2
3 (5) and Partialfraction expansions of rational functions are done by the MATLAB function
residue. Given the numerator and denominator polynomials B(s) and A(s), residue
calculates polynomial K(s), the residues {r1, r2, …, rn} and the poles {p1, p2, …, pn} of the
function Y(s) = B(s)/A(s). More specifically, the function residue is used with the syntax:
>> [ r, p, K ] = residue( B, A )
where
B
A
r
p is a row vector representing the numerator polynomial of Y(s)
is a row vector representing the denominator polynomial of Y(s)
is a column vector containing the residues r1, r2, …, rn
is a column vector containing the poles p1, p2, …, pn K is a row vector representing the residue polynomial K(s)
( Note: when K(s) = 0 then MATLAB returns the empty
vector K = [ ] ) Study the description of the MATLAB function residue found in the MATLAB User's
Guide. You can type help residue to get information online.
1.0 Use MATLAB to find the residues of the following rational function. Provide all the
intermediate information requested.
3
Y(s) = (s2 + 6s + 8) poles = { 2, 4 } 1.1 Identify polynomials B(s) and A(s) using the MATLAB notation
>> B = . >> A = . ECH 4323L Lab 2  Calculation of residues using MATLAB 3 1.2 Use function residue to find residues r1 and r2 and the residue polynomial K(s) such
r1
r2
that Y(s) = K(s) + sp + sp
1
2
r1 = r2 = K(s) = 1.3 Find y(t) = L1 [Y(s)] (Note: there is no need to use MATLAB for this inversion: do
it analytically, i.e., "by hand"). . y(t) = 2.0 Use MATLAB to assist you in finding the partialfraction expansion and the inverse
Laplace transform of the function
2
Y(s) = (s + 2)2
2.1 Use function residue to find poles p1 and p2, residues r1 and r2, and the residue
polynomial K(s).
p1 = p2= r1 = r2 = K(s) = 2.2 Write the partialfractions expansion of Y(s) . Y(s) = 2.3 Find y(t) = L1 [Y(s)] (Note: there is no need to use MATLAB for this inversion: do
it analytically, i.e., "by hand"). . y(t) = 3.0 Use MATLAB to assist you in finding the partialfraction expansion and the inverse
Laplace transform of the function
2
Y(s) = (s + 2)2 (s2+2) ECH 4323L Lab 2  Calculation of residues using MATLAB 4 3.1 Use function residue to find the poles, the residues, and the residue polynomial. . poles = { } residues = { } K(s) = . . 3.2 Write the partialfractions expansion of Y(s). . Y(s) = 3.3 Find y(t) = L1 [Y(s)] (Note: there is no need to use MATLAB for this inversion: do
it analytically, i.e., "by hand"). . y(t) = 4.0 Use MATLAB to assist you in finding the partialfraction expansion and the inverse
Laplace transform of the function
2e2s
Y(s) = (s + 2)2 (s2+2)
Hint: Take advantage of the the Real Translation Theorem: define the undelayed part of
the rational function, use MATLAB to calculate all poles and residues for the undelayed
rational function, invert the undelayed rational function to get a timedomain result, and
then delay the timedomain result by shifting the time variable by the value of the delay.
4.1 Use function residue to find the poles, the residues, and the residue polynomial. . poles = { } residues = { } K(s) = . . ECH 4323L
4.2 Lab 2  Calculation of residues using MATLAB 5 Write the partialfractions expansion of Y(s). . Y(s) = Hint: the partialfractions expansion of Y(s) is equal to the partialfractions
expansion of its undelayed part times a Laplacedomain delay factor of the form e! t o s . 4.3 Find y(t) = L1 [Y(s)]. Note: there is no need to use MATLAB for this inversion: do
it analytically, i.e., "by hand"). . y(t) = 5.0 Use MATLAB to assist you in finding the partialfraction expansion of the nonproper
rational function
Y(s) = s3– 6s2+11s –6
s+1 5.1 Use function residue to find the poles, the residues, and the residue polynomial. . poles = { } residues = { } K(s) = . . 5.2 Write the partialfractions expansion of Y(s).
Y(s) = . ...
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This note was uploaded on 07/14/2011 for the course ECH 4323 taught by Professor Crissale during the Spring '10 term at University of Florida.
 Spring '10
 Crissale

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