1223-1 ELECTENG 291 (17/06/2022 17:00) Fundamentals of Electrical Engineering (Exam)

ELECTENG 291: Fundamentals of Electrical Engineering
THE UNIVERSITY OF AUCKLAND
SEMESTER ONE, 2022
Campus: City, NZ Online, Offshore Online
ELECTRICAL AND ELECTRONIC ENGINEERING
Fundamentals of Electrical Engineering
(Time Allowed: 3 hours and 30 minutes additional time)
(Allowable materials: Open Book)
NOTE
Attempt ALL questions.
All questions are of equal mark value.
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Write your answers in dark blue or black PEN.
Cross out any work you do not want marked.
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appropriate engineering
assumptions, state them clearly, and continue with your
answer.
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document.
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1223-1 ELECTENG 291 (17/06/2022 17:00) Fundamentals of Electrical Engineering (Exam)
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1223-1 ELECTENG 291 (17/06/2022 17:00) Fundamentals of Electrical Engineering (Exam)
1 There are FOUR questions in the PDF document shown.
Instructions:
Attempt all questions
All questions are of equal mark value.
Show ALL working unless instructed otherwise.
Write your answers in dark blue or black PEN.
Cross out any work you do not want marked.
If you believe you need further information than that provided, make some appropriate
engineering assumptions, state them clearly, and continue with your answer.
Submit your scanned or electronic hand-written solutions as a single PDF document below.

Upload your file here. Maximum one file.
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Maximum marks: 40
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Question 1
Attached
ELECTENG 291: Fundamentals of Electrical Engineering
QUESTION 1 (10 marks)
A resistive
Fig. 1.
DC
circuit comprising various independent and dependent sources is shown in
ix
4V
(a)
(b)
(c)
1Ω
vx A
ix V
3A
vx
2Ω
vOUT
Fig. 1: A resistive DC circuit.
Use node-voltage analysis to determine the output voltage 𝑣𝑣OUT of the circuit. [4 marks]
Find the Thévenin equivalent of the circuit as ‘seen’ from its output.
[4 marks]
If the output of the circuit in Fig. 1 is shorted (by an ideal wire), determine the current
through the short circuit.
[2 marks]
Page 1 of 4
ELECTENG 291: Fundamentals of Electrical Engineering
QUESTION 2 (10 marks)
A first-order circuit is shown in Fig. 2. The switch in the circuit has remained open for a very
long time before it is closed at 𝑡𝑡 = 0.
ix
t=0
12 V
2Ω
4Ω
0.25 F
6V
Fig. 2: A first-order circuit.
By means of circuit analysis, determine:
(a)
(b)
(c)
(d)
the maximum and minimum value of the supply current 𝑖𝑖𝑥𝑥 (𝑡𝑡) for 𝑡𝑡 > 0,
[3 marks]
the differential equation governing the supply current 𝑖𝑖𝑥𝑥 (𝑡𝑡), and
[3 marks]
the initial rate of change of the supply current 𝑑𝑑𝑖𝑖𝑥𝑥 (0+ )⁄𝑑𝑑𝑑𝑑,
[2 marks]
the analytical expression for the supply current 𝑖𝑖𝑥𝑥 (𝑡𝑡) for 𝑡𝑡 > 0.
[2 marks]
Page 2 of 4
ELECTENG 291: Fundamentals of Electrical Engineering
QUESTION 3 (10 marks)
A second-order circuit with an unknown resistor 𝑅𝑅 is shown in Fig. 3. The switch in the
circuit has remained open for a very long time before it is closed at 𝑡𝑡 = 0.
ix
6V
1Ω
t=0
0.5 F
2H
R
Fig. 3: A second-order circuit.
The supply current 𝑖𝑖𝑥𝑥 (𝑡𝑡) was measured for 𝑡𝑡 > 0 and has the following 𝑠𝑠-domain equivalent
representation:
𝐾𝐾(𝑠𝑠 2 + 𝑠𝑠 + 1)
(As).
𝐼𝐼𝑥𝑥 (𝑠𝑠) = ℒ[𝑖𝑖𝑥𝑥 (𝑡𝑡)] =
1
𝑠𝑠 �𝑠𝑠 2 + 𝑠𝑠 + 3�
for some constant 𝐾𝐾 > 0.
(a)
(b)
(c)
By considering the initial and final values of 𝑖𝑖𝑥𝑥 , explain why the constant 𝐾𝐾 and the
resistor 𝑅𝑅 must both have a numerical value of 2.
[4 marks]
Sketch the 𝑠𝑠-domain equivalent representation of the circuit. Make sure to indicate all
component values.
[2 marks]
Analyse your circuit in (b) and show that the expression for 𝐼𝐼𝑥𝑥 (𝑠𝑠) is as given. [4 marks]
Page 3 of 4
ELECTENG 291: Fundamentals of Electrical Engineering
QUESTION 4 (10 marks)
A speaker is connected to the output of an amplifier (represented by its Thevenin equivalent)
via a capacitor as shown in Fig. 4. The output of the amplifier is a sinusoidal signal given by
𝑣𝑣𝑆𝑆 (𝑡𝑡) = 5√2 cos(𝜔𝜔𝜔𝜔) V at some tunable frequency 𝜔𝜔.
50 Ω
100 nF
i(t)
400 mH
vS(t)
v(t)
RΩ
Amplifier
Speaker
Fig. 4: A simplified audio system.
(a)
Explain why there exists a specific frequency 𝜔𝜔 for 𝑣𝑣𝑆𝑆 (𝑡𝑡) at which the speaker will
receive maximum average power from the amplifier and determine the value of this
frequency.
[2 marks]
(b)
At the frequency calculated in (a), if the speaker was measured to absorb an average
power of 0.25 W, determine the value of the resistance 𝑅𝑅 in the speaker.
[2 marks]
(c)
(d)
At the frequency calculated in (a) and the resistance value calculated in (b), determine
an expression for the steady-state voltage 𝑣𝑣(𝑡𝑡) across and current 𝑖𝑖(𝑡𝑡) through the
speaker.
[3 marks]
At the frequency calculated in (a) and the resistance value calculated in (b), sketch a
phasor diagram of the speaker system in Fig. 4. Show all voltage and current values for
each of the components.
[3 marks]
Page 4 of 4