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Unformatted text preview: ECE 2100 Spring 2010 Homework assignment 6 Due May 8, 2010 Covered: Chapters 1113: Laplace transforms, impulse functions, partial fractions, transfer functions, initial, final value theorems, poles/zeros, passive filters, active filters, Bode plots. 1. Prelab: Considering the following RLC filter circuits. a) Derive a voltage transfer function H(j ϖ )=Vout/Vin for each circuit. Parallel: ( 29 ( 29 2 9 2 2 2 2 2 2 18000 10 3 . 8 18000 1 1 1 1 1 1 1 1 1 1 1 1  1  ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ + × = + = + = + + = + + = + + + = + + + = + = j j RC j LC RC j LC R L j R L j j H LC s R L s R L s RLC s sL R sL LC s sL R LC s sL sL sC R sL sC sC sL R sC sL s H Series: ( 29 ( 29 2 9 2 9 2 2 2 2 467000 10 3 . 8 10 3 . 8 1 1 1 1 1 1 ϖ ϖ ϖ ϖ ϖ ϖ ϖ + × × = + = + + + = + + + = j L R j LC LC j H LC s sRC LC s sC sL R sC sL s H b) For the circuit on the left (parallel RLC) add a series resistance for the inductor of 2.3 Ω and derive the voltage transfer function H(j ϖ ) including the effects of the inductor series resistance. Why is this less important for the circuit on the right? ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 9 7 2 2 2 2 20000 10 37 . 8 10 9 . 3 18000 1 1 1 1  1  ϖ ϖ ϖ ϖ ϖ ϖ ϖ + × × + = + + + + = + + + + + = + + + + + + + = + + + = j j RC L R j RLC R R RLC R RC j j H RLC s sL C sRR R R R sL LC s C sR R sL R LC s C sR R sL sC R sL R sC R sL s H L L L L L L L L L L L L This is less important for the series circuit because that Resistance is in series with a much larger resistor, and so has little effect on resonance. c) Plot on an expanded linear frequency scale (about resonance) the magnitude of the voltage transfer function [H(j ϖ )] in decibels of the parallel (left) circuit with and without the inductor series resistance. What is its effect here? On these plots illustrate the circuit's bandwidth and resonant frequency. The resistor reduces amplitude by ~2dB, and slightly increases bandwidth. w/o the resistor, 3dB BW is 10 krad/s, center frequency, 91.5krad/s w the resistor, 3dB BW is 12 krad/s, center frequency, 91.5krad/s d) Plot with a normal Bode magnitude presentation the voltage transfer function magnitude of each circuit (one set of axes) and compare their response. Name each circuit! The frequency range should be from 100 to 10 5 Hz. 2) Prelab: consider the active RC filter shown. a) Derive the transfer function H(j ω ).=Vout/Vin ( 29 ( 29( 29 ( 29 ( 29 ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ j C R j C R R C j C R j C R j R C j C j C R j C R j R j H + + = + + = + + = 1 1 2 2 1 2 1 1 2 2 2 1 1 1 1 2 2 2 1 1 1 1 1 1 b) What are the locations of the poles and zeros of this circuit? Zero at ω =0, poles at ω p1 =1/R 1 C 1 = 5krad/s, ω p2 =1/R 2 C 2 = 10krad/s, c) Plot the Bode plot of H(j ω ) for the frequency range 100 to 10 5 Hz. Show both magnitude and phase. Name this circuit....
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 Fall '05
 KELLEY/SEYLER
 RC circuit, RL circuit

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