ELEC2134 Circuits and Signals
S1 2018 Mid-semester Exam
Total marks: 80 Time: 100 minutes
This paper contains 3 pages and may not be retained
This is a closed-book exam, only UNSW-approved calculators permitted

Fourier Transform pair (for continuous-time signal)





  

  deXtxdtetxX tjtj )(
2
1
)()()(
Fourier Series:
() = 0 + ∑ cos(0) + sin(0)

=1

The fundamental frequency of x(t) is 0 and T is the period of the signal x(t).
...3,2,1)
2/
2/
sin()(
2
)
2/
2/
cos()(
2
00 

  ndtt
T
T
ntx
T
bdtt
T
T
ntx
T
a nn 

Or () = ∑
0+∞
=−∞ where =
1

∫ ()−0
+

for any

QUESTION 1 [22 Marks]
For the circuit in Figure 1, if the source frequency is 15.9 Hz, determine
i) The admittance seen by the source (in phasor form).
ii) The phase angle between the input voltage Vin and current Iin (using the voltage as
reference).
iii) The total power factor.
iv) Sketch the power triangle, clearly labelling the real (average), reactive and apparent
power, and any relevant angles.
v) The type and value of the circuit component which, when connected in parallel with
the components shown in Figure 1, will raise the total power factor to unity.

2.5 50 mH Vin = 20 V
Iin

Figure 1.

Name:___________________
Student ID:_______________
Signature:________________

Page 2 of 3

QUESTION 2 [18 Marks]
(a) Consider the function f(t) shown in Figure 2. Given that
∫ () =
1
2
cos() +

sin() and ∫ () =
1
2
sin() −

cos(),
determine the trigonometric Fourier series of f(t), up to the second harmonic only (i.e. up
to n = 2), showing all working. (10 marks)

Figure 2

(b) If () = −||, derive () using the definition of the Fourier transform. (8 marks)

========== Answer Questions 3 and 4 in a separate answer booklet ==========

QUESTION 3 [26 Marks]
(a) For the series RLC circuit shown in Figure 3, assuming that the source is operating at
frequency = 0, where 0 is the resonant frequency: (6 marks)
i) Sketch an example phasor diagram, clearly labelling the phasor voltages across each
element (including the source) and giving the values of all relevant angles. Note that
you are not required to provide numerical values for the phasor amplitudes.
ii) Explain in words the relationships between the voltage phasors at = 0 in your
phasor diagram.
a
b
R
LC
Vi Vo

Figure 3
(b) For the circuit in Figure 4,
i) Sketch the magnitude response of the voltage gain, i.e. |
()
()
| versus  for  ≥ 0,
labelling all important features, and evaluating at least two values of . (4 marks)

1
10 H
vin
+
-
+
-
vout

Figure 4
t 1
f(t)
A/2
-1
-A/2
2
Page 3 of 3

ii) If a voltage waveform with amplitude spectrum |Vin(ω)| shown in Figure 5 is applied
as the input to the circuit in Figure 4, sketch the amplitude spectrum of the output
voltage |Vout(ω)|. (4 marks)

Figure 5

(c) If the voltage shown in Figure 5 is applied to a 1 resistor only, determine the energy
dissipated by that resistor. (4 marks)
(d) Consider the circuit shown in Figure 6. Determine: (8 marks)
ii) The average power absorbed by the 20 load.

j12
8

j10
1000 V j15
+
-
20
-j12.4 IL

I1

I2

Figure 6.

QUESTION 4 [14 Marks]
(a) If () = ( − 1) + ( + 1), derive () using the definition of the Fourier transform.
(4 marks)

(b) Determine the signal f(t) whose Fourier transform F() is shown in Figure 7. (8 marks)
F(ω)
ω
1
20
-1
10
2-2

Figure 7

(c) Given the definition of the Fourier transform, prove from first principles that
∫ 2() =

−∞
1
2
∫ |()|2

−∞
, i.e. Parseval’s Theorem. Show all working.
(2 marks)

END OF PAPER
 1
2
4
|Vin(ω)|
2 5  