Solution HW17 - Problem 9.4 For the circuit shown in Fig P9.4 determine(a the transfer function H = V o V i and(b the frequency ϖ o at which H is

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Unformatted text preview: Problem 9.4 For the circuit shown in Fig. P9.4, determine (a) the transfer function H = V o / V i , and (b) the frequency ϖ o at which H is purely real. C R 1 R 2 L V i + _ V o + _ Figure P9.4: Circuit for Problem 9.4. Solution: (a) KCL at mode V : V o − V i R 1 + V o parenleftbigg 1 j ϖ L + j ϖ C + 1 R 2 parenrightbigg = . Solution leads to H = V o V i = jR 2 ϖ L R 1 R 2 ( 1 − ϖ 2 LC )+ j ϖ L ( R 1 + R 2 ) . (b) Rationalizing the expression for H : H = jR 2 ϖ L R 1 R 2 ( 1 − ϖ 2 LC )+ j ϖ L ( R 1 + R 2 ) × R 1 R 2 ( 1 − ϖ 2 LC ) − j ϖ L ( R 1 + R 2 ) R 1 R 2 ( 1 − ϖ 2 LC ) − j ϖ L ( R 1 + R 2 ) = ϖ 2 R 2 L 2 ( R 1 + R 2 )+ j ϖ LR 1 R 2 2 ( 1 − ϖ 2 LC ) R 2 1 R 2 2 ( 1 − ϖ 2 LC ) 2 + ϖ 2 L 2 ( R 1 + R 2 ) 2 . The imaginary part of H is zero when ϖ = 0 (trivial resonance) or when ϖ = 1 √ LC . Problem 9.12 Convert the following dB values to voltage ratios: (a) 46 dB (b) 0.4 dB (c) − 12 dB (d) − 66 dB Solution: (a) 10 ( 46 / 20 ) = 10 2 . 3 = 199 . 5 ≃ 200. (b) 10 ( . 4 / 20 ) = 10 . 02 = 1 . 047. (c) 10 ( − 12 / 20 ) = 10 − . 6 = . 25. (d) 10 ( − 66 / 20 ) = 10 − 3 . 3 = 5 × 10 − 4 . Problem 9.13 Generate Bode magnitude and phase plots for the following voltage transfer functions: (a) H ( ϖ ) = j 100 ϖ 10 + j ϖ (b) H ( ϖ ) = . 4 ( 50 + j ϖ ) 2 ( j ϖ ) 2 (c) H ( ϖ ) = ( 40 + j 80 ϖ ) ( 10 + j 50 ϖ ) (d) H ( ϖ ) = ( 20 + j 5 ϖ )( 20 + j ϖ )...
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This note was uploaded on 02/21/2011 for the course MSE 231 taught by Professor Bedford during the Spring '10 term at University of Michigan.

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Solution HW17 - Problem 9.4 For the circuit shown in Fig P9.4 determine(a the transfer function H = V o V i and(b the frequency ϖ o at which H is

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