( s) N( s) ( ) The transfer function will take the form. = s = 2. giving ωo = sqrt(1/lc) = 1E7 [rad/s] ω 01 := R 1. α 1 2 L 1.

Transcription

1 Problem ) RLC Parallel Circuit R L C E-4 E-0 V a. What is the resonant frequency of the circuit? The transfer function will take the form N ( ) ( s) N( s) H s R s + α s + ω s + s + o L LC giving ωo sqrt(/lc) E7 [rad/s] ω 0 : 0 7 b. When R 00.0 kω, R : 00.0kΩ L : 0 4 H. What is the damping ratio, ζ? Sketch the Bode plot of the magnitude (db-log) when Vout is the voltage across the inductor H(s) VL(s)/V(s). R α : L α s db α ζ : ω 0 ζ s ζ 50 > α R C ω (rad/s)

2 α δ 0 overdamped ω o d. Sketch the Bode plot of the magnitude (db-log) when Vout is the voltage across the resistor H(s) VR(s)/V(s) db ω (rad/s)

3 (Answer check: 0log( H ) -0dB at s E4) (Answer check: 0log( H ) 0dB at s E7) ( s) H s Zeros: 0 Poles: E5,E9.000E9s +.000E9s + E4 e. When R 000 Ω, what is the damping ratio? Sketch an approximate Bode (db-log) of the magnitude when Voiut across the inductor H(s) VL(s)/V(s) 3

4 db α R/(L) 000/(*E-4) E7 [rad/s] δ E7/E7 ~ ω (rad/s) (Answer check: 0log( H ) -0dB at s E4) (Answer check: 0log( H ) -6dB at s E7, with correction) s H ( s) s + E7s + E4 Zeros: 0 (double) Poles: E7 (double The asymptotes are shown as the dashed red line. The correction to the straight line approximate at the resonant frequency is 0log(/δ) -6dB 4

5 f. Sketch an approximate Bode (db-log) of the magnitude when Vout is the voltage across the resistor H(s)VR(s)/V(s) db ω (rad/s) (Answer check: 0log( H ) -54dB at s E4) (Answer check: 0log( H ) 0dB at s E7) 5

6 E7s H ( s) s + E7s + E4 Zeros: 0 Poles: E7 (double) Applying HW8 concepts to the overdamped circuit The asymptotes are shown as the dashed red line. Additionally, we locate the vertex as 0-0log(/δ)0-(-6dB) 6dB g. When R 00Ω, what is the damping ratio, ζ? Sketch the Bode plot of the magnitude (db-log) when Vout is the voltage across the inductor, H(s) VL(s)/V(s). (Answer check: 0log( H ) -0dB at s E4) (Answer check: 0log( H ) 0dB at s E7) 6

7 db s H ( s) s + E6s + E4 Zeros: 0 (double) Resonant frequency at E7 rad/s Applying HW8 concepts to the overdamped circuit The asymptotes are shown as the dashed red line. The correction to the straight line approximate at the resonant frequency is 0log(/δ) 0dB ω (rad/s) 7

8 H. Sketch the Bode plot of the magnitude (db-log) when Vout is the voltage across the resistor, H(s) VR(s)/V(s) (Answer check: 0log( H ) -80dB at s E4) (Answer check: 0log( H ) 0dB at s E7) E6s H ( s) s + E6s + E4 Zeros: 0 Resonant frequency E7 [rad/s] The asymptotes are shown as the dashed red line. Additionally, we locate the vertex as 0-0log(/δ)0-(0dB) -0dB db 8

9 ω (rad/s) Problem ) RLC Circuit R R Aac 0Adc I L E- C a. Symbolically, determine the transfer function IC(s) / I(s). H ( s) IC I ( s) R + sl ( s) ( R + R) R + + R + sl sc s + R s + s L s + L LC b. Detemine values for the resistors R and R, and the capacitor C, such that the resonant frequency is 0 5 rad/s, a double pole exists at ω 0 5 rad/s and a zero exists at 500 rad/s. double pole: L C α ω o p, p ( ) R + R 0 5 L 9

10 Electric Circuits R Prof. Shayla Sawyer Spring 06 zero: 500 L L 0 ( ) C : R : from resonant frequency from zero R : from alpha c. What type of filter is represented by this transfer function (lowpass, highpass, bandpass, notch?) Bandpass Problem 3) Design problems - Transfer functions For the problem design specifications, determine the transfer funciton that meets the requirements. (There are many designs that may fit the requirements) You should provide a plot that verifies that your transfer function meets the specifications.. Design a. Bandpass filter with a passband of 00 Hz to 00kHz b. In the passband, the gain should be 0<gain<0dB c. The rolloff (slope) in the stopbands should have a magnitude of 40 db/dec a. ω : π ω : π b. add > 6dB gain term to get the required passband range, a gain of meets the requirement c. The rolloff (slope) in the stopbands should have a magnitude of 40 db/dec One second order critically damped HPF and one second order critically damped LPF with an amplifier gain of. H( s) ( ) ( ) s s s 68 s s

11 Problem 4) Design Problem Design a filter that meets the specifications below (draw/show circuit). You need to pick values for any resistors, capacitors or inductors in your circuit. Simulate the circuit in PSpice to verify that your design meets specification. (also show/copy&paste PSpice output). Note, small deviations from the design specifications are allowed and in a real circuit always exist, but they need to be small. Show calculations to justify your design. ω [rad/s] H(s) in db E4 7 E5 0 E6 0 High pass filter with a cutoff frequency at E4 rad/s. 0dB gain in the passband K0. C 0 6 R 00Ω

12 Vac 0Vdc V C E-6 R 00 U + OUT - OPAMP V 0 R3 R k 9k 0 0V.0V 00mV 00Hz.0KHz 0KHz 00KHz.0MHz 0MHz V(R3:) *pi*frequency Problem 5) Design Problem - Multiple Stages Using only first order filters and op amp circuits for each stage, design a filter than meets the specifications below. You need to pick values for any resistors, capacitors, or inductors in your circuit. Simulate the circuit in PSpice to verify

13 that your design meets specifications. Use hte ideal amplifier component called OPAMP in your simulations. Note, small deviations from the design specifications are allowed and in a real circuit always exist, but they need to be small. Show calculations to justify your design. a. Low pass filter with a cutoff frequency of 00 MHz b. In the passband, the gain must be > 0 db c. The asymptotic slope of the stopband should be -60dB/decade -60dB rolloff indicates third order filter, cutoff frequency at 68*0^6 rad/s. Using first order components gives a -9dB correction at the cutoff frequency. Need to add 9dB to meet passband specificaitons, giving a gain of ~0 Using first order RC stages C 0 R.59kΩ R4 Vac 0Vdc V.59k 0 C E- U + OUT - OPAMP R3 R5.59k C E- U + OUT - OPAMP R6.59k C3 E- R k 0k 0 0 0KV.0V 00uV 0nV 0pV.0MHz 0MHz 00MHz.0GHz 0GHz 00GHz.0THz 0THz V(C3:) *pi*frequency 3