hw9_sol - MEEN 364 Fall 2004 Homework Set 9 Due November 4,...

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MEEN 364 Homework Set 9 Fall 2004 October 28, 2004 Homework Set 9 – Due November 4, 2004 @ 5:00 PM 1. Determine the magnitude and phase of the following transfer functions in terms of the frequency, and draw the asymptotic magnitude and phase Bode plot for each of them. Give a rough estimate of the transitions for each break point. You can use MATLAB only to verify your result. a) 1 () (1 ) ( 10 . 0 2 Gs ss s = −+ ) b) 2 ) . ( 2)( 8 64) s Hs s s = ++ + c) 22 200( 1) . (2 5 ) ( 4 s s + = −− + ) Solution: a) The transfer function is converted to the following form by substituting ω j s = , ( ) 1 . 0 2 1 Gj jj j ωω = This system has a right half-plane pole, though this does not affect its Bode plot, but its phase diagram. From the above relation, the break points can be obtained as . 50 02 . 0 1 , 1 2 1 = = = 1
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MEEN 364 Homework Set 9 Fall 2004 October 28, 2004 The following plots depict the Bode magnitude plot of the individual terms in the transfer function. 1 20log j ω Slope of –20 dB per decade 0 1 ω () 1 1 j 0 1 ω Slope of –20 dB per decade 1 0.02 1 j + 0 50 ω Slope of –20 dB per decade 2
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MEEN 364 Homework Set 9 Fall 2004 October 28, 2004 Combining the above individual Bode plots, the composite asymptotic magnitude curve is obtained as ) ( log 20 ω j G Slope of –20 dB per decade 0 1 50 ω Slope of –40 dB per decade Slope of –60 dB per decade The actual bode magnitude curve is obtained by evaluating the actual magnitude at the break points and joining these points with a smooth curve. For the asymptotic phase curve, Individual terms in the Transfer function Phase when 0 Phase when j 1 -90 -90 1 1 j -180 1 -90 1 02 . 0 1 + j 0 -90 From the above table it can be concluded that, the composite Bode phase curve starts at a phase angle of –270 degrees when ω , and tends to –270 degrees as ω . However, we must consider that the phase increases for frequencies around the break points. The actual bode phase curve is obtained by evaluating the actual phase at the corner frequencies and joining these points with a smooth curve. 0 1 Take into account that 1 1 1 =− has a phase of 180 deg. The phase angle (CCW) increases from 180 deg to 90 deg. 3
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MEEN 364 Homework Set 9 Fall 2004 October 28, 2004 The actual Bode plot for the transfer function is given below. -150 -100 -50 0 50 100 Magnitude (dB) 10 -2 10 -1 10 0 10 1 10 2 10 3 -270 -225 -180 Phase (deg) Bode Diagram Frequency (rad/sec) b) The transfer function is converted to the following form by substituting ω j s = , () 2 2 1 (1 ) ) 128 (2 ) ( ) 86 4 11 28 6 4 j j Hj jj j j j j ωω == ⎛⎞ ++ + ⎜⎟ ⎝⎠ . + This is a non-minimum phase system, because its zero is on the RHS of the complex plane. As shown in section 6.1, page 385 of the textbook, its magnitude plot is the same as if the zero were on the LHS. However, its phase continuously decreases as ω increases. From the above transfer relation, the corner frequencies can be obtained as . 8 , 2 , 1 3 2 1 = = = The following plots depict the Bode magnitude plots of the individual terms in the transfer function.
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This note was uploaded on 04/18/2010 for the course MEEN 364 taught by Professor Staff during the Spring '08 term at Texas A&M.

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hw9_sol - MEEN 364 Fall 2004 Homework Set 9 Due November 4,...

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