Homework 4_soln - ECE 2100 Homework 4 Solution Professor Alyosha Molnar Due Subjects Sinusoidal steady-state analysis Phasors frequency-domain

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ECE 2100 Homework 4 Solution Professor Alyosha Molnar Due March 31, 2011 Subjects: Sinusoidal steady-state analysis, Phasors, frequency-domain analysis. 1) Prelab: Looking at the overlaid input/output sinewaves below, estimate: -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 1.2 time, microseconds Voltage Input Output a. The frequency in Hz. F=1/0.5 μ s = 2MHz b. The frequency in radians per second. ω = f·2 π = 12.56Mrad/s c. The change in amplitude between input and output (that is the ratio of output to input amplitudes) |Vout|/|Vin| = 1.4/2 = 0.7 d. The change in phase between input and output (Remember that time delay corresponds to negative phase) ~ - π /4 (-45 degrees) e. Write the transformation from input to output as a phasor, and as a complex number. That is, Vout = (A+jB)Vin. What are A and B? Vout = 0.5(1-j) Vin = Vin( 0.7exp(-j π /4) Prelab: For problems 2-5 , You will be analyzing the impedance and voltage- division characteristics of a several “black box” circuits in lab. Several example circuits are given below. In each case analyze the impedance across each pair of terminals (a complex number, as a function of ω ), and in particular analyze Z for f=0Hz. Also use voltage divider analysis to find the phase and amplitude of V 2 for V 1 =1V·cos(2 π ft), as a function of f, and find it explicitly for f=1kHz and 100kHz. Also, find the frequency (in the range from 1kHz to 100kHz) at which the amplitude of V 2 is maximized. Perform the same analyses for V 1 when V 2 =1V·cos(2 π ft).
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Figure 2: Figure 3: Figure 4: Figure 5: 2) Pure resistors: analyze the circuit in Fig. 2 a. What are Zac, Zbc and Zab? At f=0Hz? Zac = 3k , Zbc = 4k and Zab= 3k ? at all frequencies b. What is V 2 if V 1 =1V·cos(2 π ft)? V 2 = R3/(R1+R3)V 1 = 2/3V 1 = 0.66V·cos(2 π ft) At all frequencies c. What is V 1 if V 2 =1V·cos(2 π ft)? V 2 = R3/(R2+R3)V 1 = V 1 /2 = 0.5V·cos(2 π ft) At all frequencies 3) RC: analyze the circuit in Fig. 2b a. What are Zac, Zbc and Zab? At f=0Hz? Zac = 2k +1/(j·f·628nF) b when f=0 Zbc = 500 +1/(j·f·628nF) when f=0 Zab = 2.5k at all frequencies b. What is V 2 if V 1 =1V·cos(2 π ft)? In phasors: V 2 = 1/(1+j 2 π f·200 μ s )V 1 In time: V 2 =1V/(1+f 2 ·1.6·10 -6 ) 1/2 cos(2 π ft+atan(-f·1.26ms)) At 1kHz: V 2 =0.63Vcos(6.28krad/s·t-0.9) At 100kHz: V 2 =0.008Vcos(6.28krad/s·t-1.56) Max at f = 1kHz c. What is V 1 if V 2 =1V·cos(2 π ft)? In phasors: V 2 = 1/(1+j 2 π f·50 μ s )V 1 In time: V 2 =1V/(1+f 2 ·10 -7 ) 1/2 cos(2 π ft+atan(-f·314 μ s)) At 1kHz: V 2 =0.95Vcos(6.28krad/s·t-0.3) At 100kHz: V 2 =0.032Vcos(6.28krad/s·t-1.53) Max at f = 1kHz d. Based on the zero-frequency impedances and voltage divider results, how would you extract the values of R1 and R2 for figure 2b? Based on resistance, you know component 3 is a capacitor, and R1+R2 =
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This note was uploaded on 06/10/2011 for the course ECE 2100 taught by Professor Kelley/seyler during the Spring '05 term at Cornell University (Engineering School).

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Homework 4_soln - ECE 2100 Homework 4 Solution Professor Alyosha Molnar Due Subjects Sinusoidal steady-state analysis Phasors frequency-domain

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