Problem 1a) 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 1storder lag are: 22KAR1=τ ω +and 1tan()−φ =−τω(S1.1) Thus, applying the principle of superposition we get: 222311AR641411=ω +ω +ω +(S1.2) 111tan( 8 )tan( 2)tan()−−−φ =− ω +− ω +−ω(S1.3) b) Asymptotically as w goes to infinity, the AR is approximated by 311AR82=ωω ω(S1.4) while for ωgoing to zero, AR goes to 3. Thus, the corner frequency will be obtained by solving the equations 31130.39782=→ω =ωω 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 ωand find the corresponding phase angle and amplitude ratios from the above formulae). They are shown in figure 1. 10-710-610-510-410-310-210-11001010.0010.010.1110100ARw-300-250-200-150-100-5000.0010.010.1110100φwProblem 1: Bode plots
has intentionally blurred sections.
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