Lab 02-Residues - Chemical Engineering Department...

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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 partial-fractions expansion of Y(s) is of the form r1 r2 r3 rn Y(s) = s-p + s-p + s-p + ... + s-p 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) = s-p + (s-p )2 + (s-p )3 + ... + s-p 1 n 2 3 (3) Finally, when the rational function is not strictly proper, (i.e., m ≥ n), the partial-fractions expansions includes and additional polynomial K(s), and expansions (2) and (3) must be respectively modified to the forms © Oscar D. Crisalle 1994-2009 ECH 4323L Lab 2 - Calculation of residues using MATLAB 2 r1 r2 r3 rn Y(s) = K(s) + s-p + s-p + s-p + ... + s-p 1 2 3 n (4) r1 r2 r3 rn Y(s) = K(s) + s-p + (s-p )2 + (s-p ) 3 + …+ s-p 1 n 2 3 (5) and Partial-fraction 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 on-line. 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) + s-p + s-p 1 2 r1 = r2 = K(s) = 1.3 Find y(t) = L-1 [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 partial-fraction 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 partial-fractions expansion of Y(s) . Y(s) = 2.3 Find y(t) = L-1 [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 partial-fraction 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 partial-fractions expansion of Y(s). . Y(s) = 3.3 Find y(t) = L-1 [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 partial-fraction expansion and the inverse Laplace transform of the function 2e-2s 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 time-domain result, and then delay the time-domain 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 partial-fractions expansion of Y(s). . Y(s) = Hint: the partial-fractions expansion of Y(s) is equal to the partial-fractions expansion of its undelayed part times a Laplace-domain delay factor of the form e! t o s . 4.3 Find y(t) = L-1 [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 partial-fraction 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 partial-fractions 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.

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