solmidterm05_mae143b

As gs is stable the output y t to a sinusoidal input

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Unformatted text preview: os( √ t) = 2 2 2 √ 1 1 1 −√ t = e 2 (( 2 + 1) sin( √ t) − cos( √ t)) 2 2 y (t) = 1 −√ t 2e 2 1 3. As G(s) is stable, the output y (t) to a sinusoidal input signal u(t) = cos ωt is given by y (t) = M (ω ) cos(ωt + φ(ω )) (2) for large values of t. Examine the approximate (asymptotic) behavior of M (ω ) and φ(ω ) for the low (ω < 1 rad/s), and high (ω > 1 rad/s) frequency range. Comment on the slope of log10 M (ω ) when plotted against log10 ω over these frequency ranges. [20pt] √ 1 + ω2 1 − jω √ M (ω ) = = −ω 2 + 2jω + 1 (1 − ω 2 )2 + 2ω 2 √ √ 1 + ω2 ω2 ω 1 lim M (ω ) = lim = lim = lim 2 = lim = 0. ω →∞ ω →∞ ω →∞ ω (1 − ω 2 )2 + 2ω 2 ω→∞ (ω 2 )2 ω→∞ ω Taking log10 of M (ω ) at very high frequencies gives 1 log10 (M (ω )) = log10 ( ) = log10 1 − log10 ω. ω From this equation it can be seen that M (ω ) will have slope of −1 in log-scale at high √ √ frequencies. 1 + ω2 1+0 = lim =1 lim M (ω ) = lim 2 )2 + 2ω 2 ω →0 ω →0 ω →0 (1 − ω (1 − 0)2 Since M (ω ) approaches a constant value at low frequencies, the slope will be zero. √ 1 − jω 2ω −ω √ ) φ(ω ) = ∠ )...
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