solution hw7 - Problem 1 a The transfer function of this...

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Problem 1 a) The transfer function of this process can be expressed as the product of three first order lag transfer functions. The AR and phase angles of a general 1 st order lag are: 2 2 K AR 1 = τ ω + and 1 tan ( ) φ = −τω (S1.1) Thus, applying the principle of superposition we get: 2 2 2 3 1 1 AR 64 1 4 1 1 = ω + ω + ω + (S1.2) 1 1 1 tan ( 8 ) tan ( 2 ) tan ( ) φ = − ω + − ω + −ω (S1.3) b) Asymptotically as w goes to infinity, the AR is approximated by 3 1 1 AR 8 2 = ω ω ω (S1.4) while for ω going to zero, AR goes to 3. Thus, the corner frequency will be obtained by solving the equations 3 1 1 3 0.397 8 2 = ω = ω ω ω (S1.5) Taking logarithms in the asymptotic expression for AR, the asymptote slope is –3. c) The Bode plots are obtained computationally (i.e. give an array of values for ω and find the corresponding phase angle and amplitude ratios from the above formulae). They are shown in figure 1. 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 0.001 0.01 0.1 1 10 100 AR w -300 -250 -200 -150 -100 -50 0 0.001 0.01 0.1 1 10 100 φ w Problem 1: Bode plots
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