Exam Solutions 41-50

Exam Solutions 41-50 - which is obtained as follows: 1 + (s...

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Unformatted text preview: which is obtained as follows: 1 + (s + 1) (2s + 1)( s + 1) = 0 (2s + 1)( s + 1) + (s + 1) = 0 s(2s + 1) + (3s + 2) = 0 (3s + 2) = s(2s + 1) (3s + 2) s(2s + 1) = s(2s + 1) (3s + 2) = 1 and therefore: F(s) = s(2s + 1) (3s + 2) 1. Draw the axes of the s-plane to a suitable scale and enter an on this plane for each pole of F ( s ) and a for each zero of F ( s ) . The poles and zeros of F ( s ) are poles: p 1 = 2 3 zeros: z 1 = 0, z 2 = . 5-1.5-1-0.5 0.5-1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1 Real axis Imaginary axis 2. Find the real axis portions of the locus. The real axis portion of the root locus is located between z 1 and z 2 and left of the p 1 . Since there is one pole and there are two zeros, one pole will come from infinity.-1.5-1-0.5 0.5-1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1 Real axis Imaginary axis 3. Draw the asymptotes for large values of . | n m | = number of asymptotes = p i z i | n m | l = 180 + 360 ( l 1) | n m | , l = 1 , 2 , ..., | n m | The number of asymptotes is equal to the absolute difference in the number of poles and the number of zeros. Since there is one pole and there are two zeros, the number of asymptotes is equal to one. For > the pole p 1 moves towards one zero, while another pole comes from infinity to the other zero. To compute the origin of the asympote we use the following expression = n i=1 Re(p i ) m j=1 Re(z j ) | n m | = 2 3 + 0 . 5 | 1 2 | = 1 1 = 1 where n is the number poles and m is the number of zeros of F ( s ). The departure angles of the asymptote is computed as follows (take absolute values for l and n m ) 1 = 180 + 360 1 = 180 1 = 180 4. Compute locus departure angles from the poles and arrival angles at the zeros q dep = summationdisplay i summationdisplay i 180 360 l q arr = summationdisplay i summationdisplay i +180 +360 l where q is the order of the pole or zero and l takes on q integer values so that the angles are between 180 . Since all the poles and zeros are located on the real axis, there is no need to compute the departure and arrival angles.-1.5-1-0.5 0.5-1-0.8-0.6-0.4-0.2 0.2 0.4 0.6 0.8 1 Real axis Imaginary axis 5. Estimate (or compute) the points where the locus crosses the imaginary axis This is not the case for : 0 + > ....
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Exam Solutions 41-50 - which is obtained as follows: 1 + (s...

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