root locus practice 2

# root locus practice 2 - + . The closedloop characteristic...

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Root Locus Q UESTION 5 For the system shown in Figure 5 with ( 29 1 C G s = and ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 2 s s G s s s - - = + + , sketch the Root Locus Diagram by hand and using Matlab. How many zeros does the system contain and what are their values? How many poles does the system contain and what are their values? What are the total number of branches and how many branches will go to infinity? Where is the Root Locus on the real axis? If necessary, calculate the asymptote angles and real axis intercept, imaginary axis crossing, and the value of K at the imaginary axis crossing. G(s) KG C (s) R(s) + - Y(s) Figure 5 There are m = 2 zeros located at 1 and 2. There are n = 2 poles located at –1 and –2. There are n = 2 branches and n m = 0 branches go to infinity. The Root Locus is on the real axis between –1 and –2, and between 1 and 2. The closed–loop transfer function is ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 3 2 1 1 3 3 2 2 C C K s s KG s G s T s KG s G s K s K s K - + = = + + + - +

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Unformatted text preview: + . The closedloop characteristic equation is ( K +1) s 2 + (33 K ) s + (2+2 K ) = 0. Substituting s = j ( 29 ( 29 ( 29 2 1 3 3 2 2 K K j K + +-+ + = (1) Root Locus The real and imaginary parts, respectively, must be simultaneously zero ( 29 2 1 2 2 K K -+ + + = (2) ( 29 3 3 K-= (3) One solution is K = 1 and = 2 ; therefore, for K = 1, root locus branches cross the imaginary axis at 2 . Another solution is K = 1 and = 0; however, this is not a valid solution since K should be positive. The Matlab code for producing the Root Locus Diagram is sys = zpk([1 2],[-1 -2],1); figure; rlocus(sys), title('G_C(s)G(s) = [(s-1)(s-2)]/[(s+1)(s+2)]'); The Root Locus Diagram is shown below.-2-1 1 2-1.5-1-0.5 0.5 1 1.5 G C (s)G(s) = [(s-1)(s-2)]/[(s+1)(s+2)] Real Axis Imaginary Axis 2...
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## root locus practice 2 - + . The closedloop characteristic...

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