EE 223
Final Exam
05/08/2003
Name: ____________________________________
_____ / 100 pts
PART A - Circuit Analysis
You need to show all of your steps in detail to get full credit!
1. For the RLC-circuit given determine ...
+
V(jω) 1Ω
−
(25 pts)
100mH 10µF
2Ω
(a) ω0, f0, Q0, B, ω1 & ω2 (exact), ω1 & ω2 (approximately)
(b) What is the ratio of the magnitude of the capacitors voltage, |VC(jω0)|, at resonance
to the source voltage, |V(jω0)|?
(c) Does this circuit have a high quality factor (yes/no - why)?
Series-resonance circuit with R = 3Ω, L = 100 mH and C = 10µF
(a)
ω0 = 1/√(LC) = 1k rad/s
f0 = ω0/2π = 159.15 Hz
Q0 = ω0L/R = 33.33
B = ω0/Q0 = 30 rad/s
ω1 & ω2 (exact) = ω0 ( √(1+1/4Q02) + 1/2Q0 ) = 985.1 and 1015.1 rad/s
ω1 & ω2 (approximately) = ω0 ± B/2 = 985 and 1015 rad/s
⇒
(b)
|VC(jω0)| = |VL(jω0)| = Q0 |V(jω0)|
(c)
Yes, Q0 ≥ 5 (comparing ω: approximations are close to exact values)
Q0
Page 1 of 7
2. Analyze the following circuit:
(25 pts)
+
2Ω
2A u(t)
2Ω
v(t)
0.1F
vC(0- ) = 2V
−
(a) Transform the circuit into s-domain (draw circuit)
(b) Solve for v(t) initially working in s-domain
(a) Two possibilities to transform circuit:
+
2Ω
I = 2/s
2Ω
+
V(s)
IC0 =0.2
2Ω
I = 2/s
2Ω
VC0 =2/s
10Ω/s
−
10Ω/s
V(s)
−
+
−
(b) use KCL with V(s) as only unknown:
(b) use KCL with V(s) as only unknown:
-2/s + V/2 + Vs/10 - 0.2 = 0
-2/s + V/2 + (V-2/s)s/10 = 0 leads also to
V/10 (s+5) = 2/s + 0.2
-2/s + V/2 + Vs/10 - 0.2 = 0
V = (20 + 2s) / (s (s+5)), using PFE with
A = [s V(s)]|s=0 = 20/5, B = [(s+5) V(s)]|s=-5 = 10/-5
⇒ V = 4/s - 2/(s+5) and
v(t) = 4V u(t) - 2V e−5t u(t)
Page 2 of 7
PART B - Comprehensive Review
Circle the correct answer to the following questions!
1. A 3-phase load is wye-connected and the impedance of each phase in the
load is Zp = R + jXΩ. The system is balanced with small line resistance and
source voltage (line to line) of VS = 208V∠0° volts (rms).
(4 pts)
The current magnitude in each line is approximately (rms amps):
(a) VS / ZP
(b) VS / √3ZP
(c) |VS / √3ZP|
(d) |√3VS / ZP|
(e) VS / 3ZP
Using the current (IS) as determined above: How would you compute the total average
power absorbed by the load?
(a) |IS|2 / R
(b) |IS|2 * R
(c) 3 |IS|2 * R
(d) √3|IS|2 * R
(e) 3|IS|2 / R
2. Answer the questions with respect to the following circuit.
(6 pts)
Note: Assume VS is sinusoidal but do not worry about its magnitude and phase.
R
a
1:4
VS
c
16R
+
~
−
IS
b
d
What element is connected between the terminals a-b and c-d?
(a) Two
inductors
(b) Linear
transformer
(c) Ideal
transformer
(d) Two
separate
elements
(e) None of the
previous
(d) VS / 3R
(e) VS / 5R
In terms of the source voltage VS: What is IS?
(a) VS / R
(b) VS / 1.5R
(c) VS / 2R
What is different between a load impedance as connected to terminals c-d and the
same impedance as seen from terminals a-b?
(a) Magnitude
(b) Phase
(c) Magnitude
&Phase
(d) Reactance
(e) None of the
previous
Page 3 of 7
(6 pts)
3.
t=0
1Ω
i(t)
6A
6V
+
i(t)
~
Element
−
6/e
t
20 ms
The current i(t) vs t for the circuit above is as shown.
What is the circuit Element and what is its value?
(a) Inductor
20mH
(b) Capacitor
20mF
(c) Inductor
200H
(d) Capacitor
200F
(e) None of the
previous
How would you describe the current’s response i(t>0)?
(a) linear
(b) exponential
(c) sinusoidal
(d)
exponentially
damped
sinusoid
(e) None of the
previous
(d) 2J
(e) None of the
previous
What is the initial energy stored in the system?
(a) zero
(b) 2mJ
(c) 20mJ
4. For the given F(s) = (s - 2) / ( s (s + 3)) - Find the time-domain solution f(t)
(s-2)/(s+3) |s=0 = -2/3
(4 pts)
(s-2)/s |s=−3 = -5/-3
-2/3 * 1/s + 5/3 * 1/(s+3) ⇐⇒ f(t) = -2/3 u(t) + 5/3 e−3t u(t)
Page 4 of 7
5. Find F(s) for the pulse-signal f(t) given below.
(4 pts)
f(t)
2
t
1 2 3 4 5 6 7 8 9 10
F(s) = 2/s e−3s - 2/s e−8s = 2/s (e−3s - e−8s)
(4 pts)
6. For the circuit shown:
125.5cos(120πt)V
Load
5Ω
+
~
−
10mF
What is the average power dissipated by the load?
(a) 1570.6 W
(b) 3141.2 W
(c) 1401.4 W
(d) None of the previous
P = (125.5/√2)2 / (52 + (1/(2π60*0.01))2 ) * 5
Page 5 of 7
7. Assume that no energy is stored in the inductor at t=0.
t=0
3Ω
12V
+
~
(4 pts)
6H
i(t)
−
What is the complete response of i(t)?
(a) -2 e-0.5t + 2A
(b) 2 e-0.5t + 2A
(c) -2 e+0.5t + 2A
(d) -4 e-0.5t + 4A
8. The voltage from A to N in a balanced 3-phase wye-connected system is
240∠-30° volts (rms). The phase sequence is positive (abc).
(6 pts)
What is the magnitude of the voltage VBC?
(a) 240/√3 V
(c) √3*240 V
(b) 240 V
(d) √2*240 V
(e) 3*240 V
(d) 90°
(e) 120°
What is the phase of the voltage VBC?
(a) -90°
(b) -120°
(c) -150°
(3 pts)
9.
i(t)
L1
L2
M
L3
If i(t) = iM cos(500 t) in the circuit shown above. What is the correct expression for the
maximum energy stored in the circuit?
(a) ½ iM2 (L1 + L2 + L3)
(c) ½ iM2 (L1 + L2 + L3 + 2M)
(b) ½ iM2 (L1 + L2 + L3 − 2M)
(d) iM2 (L1 + L2 + L3 + 2M)
(e) None of the previous
Page 6 of 7
10. The voltage and current measurements of a 3-phase load are shown in the
following phasor-diagram.
VCA
VCN
(9 pts)
VAB
ICN
IAN
VAN
IBN
VBN
VBC
What type of connection is used for the load?
(a) wye-connection
(b) delta-connection
Is this a phasor-diagram of a balanced system?
(a) yes
(b) no
What type of load is it?
(a) resistive
(b) inductive
(c) capacitive
(d) resistiveinductive
(e) resistive-capacitive
Is the load connected to a system with a frequency of 50 Hz or 60 Hz ?
(a) 50 Hz
(b) 60 Hz
(c) cannot be
determined from the
information given
(d) None of the previous
Page 7 of 7