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Download EX: a) Draw a frequency-domain model of the above circuit. Label
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PRACTICE EXAM 4 PROBLEM 3 SOLUTION 1250 F 14 EX: a) Draw a frequency-domain model of the above circuit. Label all components and the current measurement with numerical values. b) Write a numerical expression for i(t). SOL'N: a) We convert the voltage source and the current measurement to phasors, and we convert R and L to impedances. The phasor for vs(t) captures the magnitude of 3 and zero phase shift of the cosine waveform: P[3cos(1kt)V] = 3e j0° V The impedance of R is R, and the impedance of L is jωL where ω = 1k is found in vs(t). b) To solve for i(t), we use the frequency-domain circuit to find the phasor I. The method of solution is the same as it would be for a DC circuit but with complex numbers for voltages or currents and complex numbers for impedances instead of resistances. Here, we may use the equivalent impedance of R in parallel with jωL and Ohm's law to find I. zTot = 40 Ω || j30 Ω = 10 Ω ⋅ 4 || j3 = 10 Ω ⋅ 4( j3) 4 + j3 or zTot = 40 Ω || j30 Ω = j120 Ω − j 120 Ω ⋅ = 4 + j3 − j 3 − j4 Now we use Ohm's law. I= Vs 3V 3 − j4 = = A 120 Ω zTot 40 3 − j4 In preparation for converting back to the time domain, we convert the numerator to polar form. I= 5e− j53° A = 0.125e− j53° A = 125e− j53° mA 40 We take the inverse phasor to get i(t). i(t) = 125 cos(1kt − 53°)mA