Homework 4_soln

# Homework 4_soln - ECE 2100 Homework 4 Solution Professor...

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ECE 2100 Homework 4 Solution Due April 1, 2010 Professor Alyosha Molnar Subjects: Sinusoidal steady-state analysis, phasors, complex impedance, complex circuit analysis, AC power transfer, transformers, mutual inductance. 1) Prelab: Looking at the overlaid input/output sinewaves below, extract: -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)

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2) Prelab: 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). a. Pure resistors: analyze the circuit in Fig. 2a i. What are Zac, Zbc and Zab? What are they at f=0Hz? Zac = 2k , Zbc = 3k and Zab= 3k ? at all frequencies ii. What is V 2 if V 1 =1V·cos(2 π ft)? V 2 = R3/(R1+R3)V 1 = V 1 /2 = 0.5V·cos(2 π ft) iii. What is V 1 if V 2 =1V·cos(2 π ft)? V 2 = R3/(R2+R3)V 1 = V 1 2/3 = 0.66V·cos(2 π ft) b. RC: analyze the circuit in Fig. 2b i. What are Zac, Zbc and Zab? What are they at f=0Hz? Zac = 1k +1/(j·f·628nF) b when f=0 Zbc = 2k +1/(j·f·628nF) when f=0 Zab = 3k at all frequencies ii. What is V 2 if V 1 =1V·cos(2 π ft)? In phasors: V 2 = 1/(1+j 2 π f·100 μ s )V 1 In time: V 2 =1V/(1+f 2 ·3.9·10 -7 ) 1/2 cos(2 π ft+atan(-f·628 μ s)) iii. What is V 1 if V 2 =1V·cos(2 π ft)? In phasors: V 1 = 1/(1+j 2 π f·200 μ s )V 2 In time: V 1 =1V/(1+f 2 ·15.6·10 -7 ) 1/2 cos(2 π ft+atan(-f·1256 μ s)) c. Based on the zero-frequency impedances and voltage divider results, how would you extract the values of R1 and R2 for figure 2b? Their series resistance tells you that R1+R2=3k , for frequencies f>>1/(2 π RC), the amplitude of V1 = 1/(2 π R1C), V2 = 1/(2 π R2C), so the ratio of the results of ii and iii above gives the ratio of the resistors: V2/V1=R2/R1=2 R1=1k , R2=2 k .
d. RL: analyze the circuit in figure 2d i. What are Zac, Zbc and Zab? What are they at f=0Hz? ( 29

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Homework 4_soln - ECE 2100 Homework 4 Solution Professor...

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