Homework 5 solutions

Homework 5 solutions - Homework 5 1 For a first order system with = 10 ms up to what frequency will the system attenuate the signal by less than

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1 Homework 5 1. For a first order system with τ = 10 ms, up to what frequency will the system attenuate the signal by less than 2.0%? M( ω ) = 1/[1 + ( ωτ ) 2 ] ½ = 1/[1 + (0.01 ω ) 2 ] ½ The easiest way to do the problem is like this. 1 - M( ω ) 0.020 1 - 1/[1 + (0.01 ω ) 2 ] ½ 0.020 1 - 0.020 1/[1 + (0.01 ω ) 2 ] ½ 0.980 1/[1 + (0.01 ω ) 2 ] ½ 1 + (0.01 ω ) 2 ] ½ 1/0.980 = 1.020 1 + (0.01 ω ) 2 1.041 (0.01 ω ) 2 = 10 -4 ω 2 0.041 ω 2 10,400 ω 102 rad/s f 16.24 Hz Alternatively, we can approach the problem like this. We know that at ωτ = 0.1, M( ω ) 1 M( ω ) = 1/[1 + (0.1) 2 ] ½ = 1/[1 + 0.01] ½ = 1/[1.01] ½ = 1/1.005 = 0.995 The output is attenuated 0.5% (99.5% of the signal amplitude is passed). Since τ = 0.01 sec, this corresponds to ω = 0.1/0.01 = 10, which corresponds to f = ω /2 π = 1.6 Hz. We know that at ωτ = 1, M( ω ) 0.7 M( ω ) = 1/[1 + (1) 2 ] ½ = 1/[1 +1] ½ = 1/[2] ½ = 1/1.414 = 0.707 The output is attenuated almost 30%. Since τ = 0.01 sec, this corresponds to ω = 1/0.01 = 100, which corresponds to f = ω /2 π = 16 Hz. The answer lies between these. Make a plot in this range. M( ω ) > 0.98 below 3.2 Hz.
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2 0.95 0.96 0.97 0.98 0.99 1.00 02468 1 0 1 2 1 4 1 6 f (1/sec) M( ω ) Alternatively, one can guess a few frequencies, approaching the answer. The fastest is to guess in half-way there steps. For example, guess ωτ = 0.3 (halfway between 0.1 and 1 on the log scale), at which M( ω ) = 0.96. Close, but need a smaller number. Next guess ωτ = 0.2, and find that M( ω ) = 0.98. This corresponds to ω = 0.2/0.01 = 20, which corresponds to f = ω /2 π = 3.2 Hz. 2. Plot the following function vs. t, y(t) = A cos( ω t + 45°) for f = 0.16 = 1/2 π , A = ln5. For f = 1/2 π , ω = 2 π f = 1, T = 2 π = 6.28 A = ln5 = 1.6 The phase is positive, so the curve is shifted left .
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This note was uploaded on 11/28/2010 for the course ENES enes100 taught by Professor Staff during the Spring '10 term at Maryland.

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Homework 5 solutions - Homework 5 1 For a first order system with = 10 ms up to what frequency will the system attenuate the signal by less than

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