Download EX: a) Draw a frequency-domain model of the above circuit. Label

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