ECE 2300 – CIRCUIT ANALYSIS
HOMEWORK #9
1. A complex number with a phase between 90˚ and 180˚ (in the 2nd quadrant) is added to a
second complex number with a phase between –90˚ and –180˚ (in the 3rd quadrant). What is the
sign of the real part of this sum?
2. A complex number with an arbitrary real part and a positive imaginary part is multiplied by a
second complex number that is imaginary, with a negative coefficient for j. What can you say
about the product?
3. Solve the following equation for m and n. You should assume that m and n are real numbers.
The variable n is an angle, and you should show your units.
 m  67.3 j   4.794  1.875 j
23n
4. Find the nonzero value of ω for which the expression for Z(ω) given is purely real. Give your
answer as a function of R, L, and C.
1

jC 
Z ( ) 
.
 R  j L    1 jC 
 R  j L 
5. In the circuit given, the voltage source vS(t) is made up of the summation of several frequency
components. The circuit is operating in steady state.
a) Find the Thévenin impedance as seen by the load, RL, as a function of the angular
frequency, .
b) Find the Thévenin impedance as seen by the load, RL, at 1[kHz].
5.6[kW]
L1 =
3[mH]
1[kW]
iB
+
vS(t)
-
75iB
15[kW]
1.5[kW]
C1 =
82[F]
RL =
10[kW]
680[W]
9.1
6. A device was connected to a resistor and capacitor, as shown in Figure 1, and the steady-state
voltage that resulted was
v X (t )  65.7 cos 500  rad  t  172 [V] .
 s
The same device was then removed from that circuit, and was connected to a resistor and
inductor, as shown in Figure 2. The steady-state current that resulted was
iX (t )  35.67 sin 500  rad  t  99.8 [mA]. .
 s
Next, the same device was removed from that circuit, and was connected to a resistor, as shown
in Figure 3. Find the steady-state value of vW(t).




R1 = 1[kW]
a
Device
a
+
vX(t)
C1 =
3.3[F]
iX(t)
R2 =
3.3[kW]
Device
L2 =
2.2[H]
b
b
Figure 1
Figure 2
+
b
vW(t)
Device
a
R3 =
1[kW]
-
Figure 3
7. The circuit shown operates in steady-state. Find the numerical expression for the current
iQ(t).
vS (t )  50sin(30  rad  t  23)[V]
 s
iS (t )  12 cos(30  rad  t  48)[A]
 s
50[W]iX(t)
-
+
+
3.3[H]
iQ(t)
560[F]
vS(t)
-
iX(t)
9.2
47[W]
iS(t)
12iX(t)
8. For the circuit shown below:
a) Find the expression for the steady-state current i(t), as a function of R.
b) Find the expression for the steady-state current i(t), for R equal to 500[W], 1[kW], 2[kW], and
4[kW]. (PWA8, No. 1)
L1=0.25[H]
C1=1[F]
+
vS1
-
i(t)
iS1
R
vS2
+
-
vS1(t) = 30 cos (1000[rad/s] t ) [V]
vS2(t) = 25 sin (1000[rad/s] t ) [V]
iS1(t) = 5 sin (1000[rad/s] t + 50º ) [mA]
9. In the circuit below, find the nonzero frequency at which the voltage vS(t) is in phase with the
current iS(t). Problem adapted from (PEQWS8, No. 9).
a
+
vS(t)
R=
180[W]
iS(t)
L=
50[mH]
b
9.3
C=
1[F]
10. The circuit given is in steady-state. The current through an unknown device, iD(t) has been
measured, and its expression is given with the figure.
a) Find the expression for the steady-state voltage vD(t).
b) Find a circuit model in the time domain, including values for the model parameters, for the
unknown device. Assume that it is a passive device. (PWA8, No. 2)
RX =
100[W]
iD(t)
+
vD(t)
Unknown
Device
+
vS(t)
C X=
100[F]
LX =
1[H]
vS(t) = 20 cos (75[rad/s] t + 30º ) [V]
iD(t) = 223 cos (75[rad/s] t + 75º ) [mA]
11. The circuit given is in steady-state. Find a numerical expression for the steady-state current,
i(t). (PWA8, No. 3)
R1=
50[W]
R2=
5[W]
L2 =
0.02[H]
R3 =
15[W]
R4=
10[W]
iS1=
10[A]
C=
2[F]
iS2(t)
i(t)
L3 =
0.06[H]
+
vS3(t)
-
L4 =
0.04[H]
vS3(t) = 100 cos (500[rad/s] t - 30º ) [V]
iS2(t) = 20 cos (1000[rad/s] t - 30º ) [A]
9.4
Selected Numerical Solutions
1. Solution omitted
2. Solution omitted
3. m  97.4 and n  56
or
m  97.4 and n  166.8.
CR 2  L
.
L2C
5. a) Solution omitted, b) 66.5 + 0.203j[W]
6. 45.2 cos(500[rad/s]t + 12.4°)[V]
7. 387 cos(30[rad/s]t – 17°)[mA]
8. Solution omitted
9. Solution omitted
10. Solution omitted
11. Solution omitted
4.  
9.5