Homework 6_soln - ECE 2100 Spring 2010 Homework assignment...

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Unformatted text preview: ECE 2100 Spring 2010 Homework assignment 6 Due May 8, 2010 Covered: Chapters 11-13: 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 circuit. a) Derive a voltage transfer function H(s)=Vout/Vin for the circuit. Where are its poles and zeros? ( 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 || - + =- + =- + = + + = + + = + + + = + + + = + = 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 600 , 90 9000 1 2 1 2 1 : : 2 j LC RC RC s poles s zero =- - = = b) Add a series resistance for the inductor of 2.3 and derive the voltage transfer function H(s) including the effects of the inductor series resistance. Where are the poles and zeros now? ( 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 || - + + =- + + + + = + + + + + = + + + + + + + = + + + = 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 900 , 90 10000 1 2 1 2 1 : 2000 / : 2 j LC RC RC s poles L R s zero L =- - = = = c) Now, in the frequency domain (s = j ), plot on an expanded linear frequency scale (about resonance) the magnitude of the voltage transfer function [|H(j )|] in decibels with and without the inductor series resistance. What is its effect here? On these plots illustrate the circuit's bandwidth and resonant frequency. -12-9-6-3 10000 15000 20000 Frequency, Hz dB w/o R with R The resistor reduces amplitude by ~1dB, and slightly increases bandwidth. w/o the resistor, 3dB BW is 27kHz, center frequency, 14.5krad/s w the resistor, 3dB BW is 32.5 krad/s, center frequency, 14.5krad/s d) Plot with a normal Bode magnitude presentation the voltage transfer function magnitude of the circuit! The frequency range should be from 100 to 10 5 Hz. -50-40-30-20-10 10 100 1000 10000 100000 Frequency, Hz dB 2. Prelab : consider the active RC filter shown. a) Derive the transfer function H(s).=Vout/Vin ( 29 ( 29( 29 ( 29 ( 29 ( 29( 29 j s krad j s krad s j 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 + +- = + +- = + + = + +- = / 5 / 10 40 1 1 1 1 1 1 1 1 2 2 1 2 1 1 2 2 2 1 1 1 1 2 2 2 b) What are the locations of the poles and zeros of this circuit? Zero at =0, poles at p1 =1/R 1 C 1 = 5krad/s, p2 =1/R 2 C 2 = 10krad/s, c) Plot the Bode plot of H(j ) for the frequency range 100 to 10 5 Hz. Show both magnitude and phase. Name this circuit....
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Homework 6_soln - ECE 2100 Spring 2010 Homework assignment...

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