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rootlocus2

# rootlocus2 - The closed–loop characteristic equation is s...

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Root Locus Q UESTION 2 For the system shown in Figure 2 with ( 29 1 3 C G s s = + and ( 29 ( 29 ( 29 2 1 1 2 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 2 There are m = 2 zeros located at – j and j . There are n = 3 poles located at 1, –2, and –3. There are n = 3 branches and n m = 1 branch goes to infinity. Since one branch goes to infinity, the asymptotic angle is 180 0 and the real axis intercept is not required. The Root Locus is on the real axis between 1 and –2, and from –3 to infinity. The closed–loop transfer function is ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 3 2 1 1 4 6 C C K s KG s G s T s KG s G s s K s s K + = = + + +

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Unformatted text preview: -. The closed–loop characteristic equation is s 3 + ( K +4) s 2 + s + ( K –6) = 0. Substituting s = jω ( 29 ( 29 3 2 4 6 j K j K ϖ--+ + +-= (1) Root Locus The real and imaginary parts, respectively, must be simultaneously zero ( 29 2 4 6 K K ϖ-+ +- = (2) 2 1 -+ = (3) Solving equation (3), 0, 1 = ± . For ω = 0, K = 6; therefore, for K = 6, a root locus branch crosses the imaginary axis at the origin. Substituting ω 2 = 1 into equation (2), –2 = 0. While this equation does not make sense, it is known that the root locus branches will touch the imaginary axis at ±1 when K = ∞. The Matlab code for producing the Root Locus Diagram is sys = zpk([-sqrt(-1) sqrt(-1)],[0 0],1); figure; rlocus(sys), title('G_C(s)G(s) = [s^2+1]/s^2'); The Root Locus Diagram is shown below.-6-4-2 2-1.5-1-0.5 0.5 1 1.5 G C (s)G(s) = (s 2 +1)/[(s-1)(s+2)(s+3)] Real Axis Imaginary Axis 2...
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