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rlocus web - Chapter 7 ROOT LOCUS R E K G C H C K G(s = R 1...

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Chapter 7 ROOT LOCUS

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R + K G C H - E C R K G s K G s H s = + ( ) ( ) ( ) 1
1+ K G(s) H(s) = 0 is called the characteristic equation of the system and its roots are closed loop poles . Root locus is the locus of the roots of the characteristic equation as parameter K is varied ( 0 < K < )

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+ K C - 1 1 s + First order System R Closed loop Pole, s = - 1 - K
K K = 0 × - 1 Root locus for G(s)H(s) = 1/(s+1) head2right closed loop system is stable for all K head2right As K is increased, closed loop pole is moved farther away from imaginary axis; hence the time constant reduces. i.e. Transient response improves.

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K K = 0 × 1 s = 1 - K K = 1 Root locus for G(s)H(s) = 1/(s-1) From the plot , for K < 1, system is unstable KGH = K s - 1 Ex:
+ K - ) 1 + s ( s 1 Second Order C R Ch Eq: s 2 + s + K = 0 closed loop poles at 2 4 1 ± -0.5 K -

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For K = 0 : Roots at 0 and - 1 For 0 < K < 0.25 : Roots are real and equidistant from -0.5. For K = 0.25 : Roots at -0.5 For K > 0.25: Roots are complex conjugate with real part at - 0.5
0.5 × 0 × -1 > > > > Root locus for G(s)H(s) = 1/s(s+1) locus is always symmetric about the real axis

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rlocus web - Chapter 7 ROOT LOCUS R E K G C H C K G(s = R 1...

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