hw8_solutions

hw8_solutions - Homework 8 Solutions ECE 2B Winter 2011...

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Homework 8 Solutions ECE 2B Winter 2011 H.O. #26 Problem 9.26 For the circuit shown in Fig. P9.26: (a) Obtain an expression for H ( ! ) = V o / V i in standard form. (b) Generate spectral plots for the magnitude and phase of H ( ) , given that R 1 = 1 " , R 2 = 2 " , L 1 = 1 mH, and L 2 = 2 mH. (c) Determine the cutoff frequency c and the slope of the magnitude (in dB) when / c ± 1 and when / c ² 1. L 2 L 1 R 1 R 2 + _ V i + _ V o V 1 Figure P9.26: Circuit for Problem 9.26. Solution: (a) At node V 1 , KCL gives: V 1 - V i R 1 + V 1 j L 1 + V 1 R 2 + j L 2 = 0 . Also, voltage division gives V o = R 2 V 1 R 2 + j L 2 . Solution is H ( ) = V o V i = j L 1 R 2 ( R 1 + j L 1 )( R 2 + j L 2 ) + j L 1 R 1 . To express denominator in standard form, we factor out R 1 R 2 and expand terms: H ( ) = j L 1 / R 1 ± 1 + j L 1 R 1 ² ± 1 + j L 2 R 2 ² + j L 1 R 2 = j L 1 / R 1 1 + j ± L 1 R 1 + L 2 R 2 + L 1 R 2 ² + ± j ³ L 1 L 2 R 1 R 2 ² 2 = j K 1 + j 2 #! / c + ( j / c ) 2 , where K = L 1 R 1 , c = ³ R 1 R 2 L 1 L 2 , # = c 2 ´ L 1 R 1 + L 2 R 2 + L 1 R 2 µ . (b) For R 1 = 1 " , R 2 = 2 " , L 1 = 1 mH, and L 2 = 2 mH, K = 10 - 3 , c = 10 3 rad/s , = 1 . 25 . All rights reserved. Do not reproduce or distribute. ©2009 National Technology and Science Press
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Hence, H ( ! ) = j × 10 - 3 1 + j 2 . 5 / c + ( j / c ) 2 , with c = 10 3 rad/s. M [ dB ] = 20log | H ( ) | Spectral plots of M [dB] are " ( ) are shown in Figs. P9.26(a) and (b). 10 1 10 2 10 3 10 4 10 5 10 6 - 80 - 60 - 40 - 20 0 ω M (dB) 10 1 10 2 10 3 10 4 10 5 10 6 - 100 0 100 ω Figures P9.26(a) and (b) ϕ (deg) (c) Low-frequency asymptote ( / c ± 1): H ( ) ² j 10 - 3 = M [ dB ] has slope of + 20 dB/decade . High-frequency asymptote ( / c ³ 1): H ( ) ² - j 10 - 3 2 c = - j 10 3 = M [ dB ] has slope of - 20 dB/decade . All rights reserved. Do not reproduce or distribute. ©2009 National Technology and Science Press
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Problem 9.33 For the op-amp circuit of Fig. P9.33: (a) Obtain an expression for H ( ! ) = V o / V i in standard form. (b) Generate spectral plots for the magnitude and phase of H ( ) , given that R 1 = R 2 = 100 " , C 1 = 10 μ F, and C 2 = 0 . 4 F.
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This note was uploaded on 04/19/2011 for the course ECE 2B taught by Professor York during the Spring '07 term at UCSB.

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hw8_solutions - Homework 8 Solutions ECE 2B Winter 2011...

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