Exam Final B Solutions December 3, 2001
ECE 221: Electric Circuits Dr. McNames
Write the first letter in your last name, your 6-digit identification number, and your student identification number below. Do not begin the exam or look at the problems until instructed to do so. You have 100 minutes to complete the exam. Do not use separate scratch paper. If you need more space, use the backs of the exam pages. If you have extra time, double check your answers. If you run out of time, write a note describing your strategy and equations that can be used to help solve the problem.
Problem 1:______ / 13 Problem 2:______ / 15 Problem 3:______ / 11 Problem 4:______ / 12 Total:______ / 50
First Letter in Last Name:_____________ 6-Digit Identification Number:_____________ Student Identification Number:_____________
ECE 221 Final Exam B 1. Fundamental Concepts (13 pts) Circle True or False for each of the questions below or fill in the blank.
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a. (1 pt) What is e j 90 equal to in rectangular coordinates? j b. (1 pt) We use Phasor analysis because it enables us to apply Kirchhoffs laws and our standard analysis techniques by converting differential equations into algebraic relationships. True True False False c. (1 pt) The energy stored in an inductor can change instantaneously. d. (1 pt) The resistance of real, physical inductors is large enough that it can be safely ignored. True True False False e. (1 pt) Ideal models of operational amplifiers assume the output resistance is infinite. f. (1 pt) If the passive sign convention is satisfied, then current enters the positive terminal of a circuit element. True True h. False False g. (1 pt) The energy stored in a pair of magnetically coupled coils must be non-negative. (1 pt) For linear transformers, if current leaves a dotted terminal, it induces a positive voltage at the dotted terminal of the other coil. True False
i. (1 pt) Phasor analysis only enables us to solve for the transient component of circuits operating in a sinusoidal steady-state. True j. True True False False False (1 pt) The charge stored in a capacitor is directly proportional to the current.
k. (1 pt) The current passing through a capacitor can change instantaneously. l. (1 pt) The time constant is a measure of how quickly the voltages and currents in a first-order circuit settle to their final values. True False m. (1 pt) If two independent sinusoidal sources are operating at different frequencies in a circuit, one must use superposition to apply phasor analysis. True False
ECE 221 Final Exam B 2. Nodal Analysis (15 points)
25 mH
io
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19
100 F
+
15 sin(500t) V
28
4io
vo
-
57
a. (5 pts) The phasor-domain equivalent of the circuit above is shown below. Label the phasor-domain value of each circuit element. Note there are 7 labels missing.
j12.5
+
1
Io
19
+
2
-j20
+
1590 V
V1
-
28
V2
-
4 Io
Vo
-
57
b. (6 pts) Use a nodal analysis to write two independent equations in terms of the phasors and V1 V2. You do not need to simplify your equations. Node 1:
V1 15 90 V1 V1 V2 + + =0 j12.5 28 19
V2 V1 V2 V + 3 1 + =0 25 33 72 j 20
Node 2:
c. (2 pts) Solve for the the phasors V1 and V2.
V1 = 8.03 137 V
V2 = 7.7139.5 V
d. (1 pt) Write an expression for the phasor Vo. If you could not solve for V2 in part c., assume V2 = 4.3-122 V.
57 Vo = V2 = 8.33043.3 V 57 j 20
e. (1 pt) Use your answer from part d. to write the time-domain expression for expression for vo(t).
vo(t) = 8.33cos(500t + 43.3 ) V
ECE 221 Final Exam B 3. Thevenin/Norton Equivalents (11 pts)
j11 k
I1 I2
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5 k 300 mA a j7 k b
-j4 k
2 k
a. (1 pt) Find the equivalent impedance between the terminals labeled a and b.
Zab = (5 j 4)k || (2 + j 7)k = (5.98 + j1.29)k
b. (2 pts) Find the phasor currents I1 and I2.
2 j4 I1 = 30m = 17.6 86.6 mA (2 j 4) + (5 + j 7) 5 + j7 I2 = 30m = 33.931.3 mA (5 + j 7) + (2 j 4)
c. (2 pts) Find the phasor voltage between the terminals a and b if the nodes are left unconnected (open circuit voltage), as shown. Hint: use your answer to part b. Voc = I1 j 7 k I 2 2k = 70.9 23.2 V d. (4 pts) Draw both the Thevenin and Norton equivalents of the circuit in the phasor domain as seen from the nodes a and b. Clearly label these nodes.
5.98 k j1.29 k a 5.98 k 70.923.2 V b 9.446.71 mA j1.29 k b a
e. (1 pt) If the sinusoidal frequency is 500 rad/s, what is the value of the energy storage element (inductor or capacitor) in the Norton equivalent circuit?
j L = j1.29k ; L =
1290 = 2.58 H 500
f. (1 pt) If the current source in the original circuit was changed to 150 mA, what would the Thevenin equivalent voltage be? VTh = 1 Voc = 35.5 23.2 V (by linearity) 2
ECE 221 Final Exam B
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4. Operational Amplifiers (12 pts) For the circuit below, assume that the operational amplifier is ideal and operating in the linear region.
j3 k V1 9 k V2 -j2 k V4 5 k j5 k 11 k 2 k V3
Vo
a. (1 pt) Does the 11 k resistor affect the value of Vo? (Circle one) Yes No b. (1 pt) Does the inductor with an impedance of j5 k affect the value of Vo? (Circle one) Yes No c. (1 pt) Write a simple expression for V3 in terms of V2.
j 2k V3 = V2 = (0.5 j 0.5)V2 ; Voltage Divider 2k j 2k d. (1 pt) Write a simple expression for V4 in terms of Vo. 5k V4 = V2 = (0.5 j 0.5)V2 ; Voltage Divider 5k + j 5k
e. (1 pt) Write a simple expression for V4 in terms of V3. V4 = V3 f. (3 pts) Use Kirchoffs current law (KCL) to write a single equation that relates V1, V2, and Vo. V V V2 V V + 2 o =0 Equation: 2 1 + j 3k 9k 2k j 2k g. (2 pts) Use the equations above to eliminate V2 and write an expression for Vo in terms of V1. 1 1 11 1 + V4 = V3 = (0.5 j 0.5)V2 = (0.5 j 0.5)Vo ;V2 = Vo Vo + = V1 , Vo 9 j3 9 2 j2 = 0.253 34.7 V h. (2 pts) If v1(t) = 36 cos(500t - 30), user your answer above to write an expression for vo(t). If you were unable to answer part g., assume Vo = (0.23 j 0.89) V1. vo(t) = 9.11cos(500t 64.7 ) V