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SCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS
SEMESTER 2 EXAMINATIONS 2011/2012
ELECTRICAL ENGINEERING 1
EO129
DR. S.S. SINGH
Time allowed:
THREE hours
Answer:
Any FIVE questions
The total number of questions is EIGHT
Each question carries 15 marks
Items permitted:
Any approved calculator
Items supplied:
None
Marks for whole and part questions are indicated in brackets ( )
May/June 2012
Page 1 of 10
Question 1
(a)
Sketch a graph using suitably labelled horizontal and vertical axes of an
alternating voltage waveform of your choice.
(2 marks)
(b)
Sketch a graph, using clearly marked vertical and horizontal axes, of a
sinusoidal waveform which has a negative DC component and whose average
value is negative. Clearly indicate on it, with labelling and/or markings, the
following waveform features:
(i)
(ii)
(iii)
(iv)
(v)
Peak value
Peak-to-peak value
Average value
Period
One cycle
(5 marks)
(c)
If the period of a sinusoidal waveform is 40 milliseconds, what is its frequency?
Clearly explain how you have arrived at the frequency.
(3 marks)
(d)
If the frequency of a sinusoidal waveform is 100 Hz, derive or calculate its
angular frequency. Indicate the units you have used to express the angular
frequency.
(3 marks)
(e)
Sketch an amplitude-modulated sine wave, which has a DC offset. Label the
horizontal and vertical axes of your sketch.
(2 marks)
EO129 (2011/2012)
Page 2 of 10
Question 2
(a)
Explain what mathematical operations must be carried out on a waveform to
determine its RMS value. Include in your explanation how such a value is
computed by sampling the waveform at equal time intervals.
(2 marks)
(b)
Explain why RMS values are used and state when it is appropriate to use RMS
values.
(2 marks)
(c)
Write down a mathematical expression for a sine-wave voltage whose
frequency is 100 Hz and whose amplitude is 10√2 Volts. Then write down its
RMS value.
(2 marks)
(d)
Write down, using complex notation, a mathematical expression for a sine wave
whose frequency is 100 Hz and whose amplitude is 10√2 Volts.
(2 marks)
(e)
The power dissipated in a resistive load is measured using RMS values. If the
resistor voltage is a sine-wave of amplitude 10 Volts with a DC offset of 10 V
and the amplitude of the sine-wave current is 100 mA, what is the power
dissipated in the resistive load?
Give a full explanation of your answer ensuring that all mathematical steps
used to evaluate the power dissipation are explained clearly. (Note: the RMS
value for a sine wave is the amplitude divided by √2.)
(7 marks)
EO129 (2011/2012)
Page 3 of 10
Question 3
(a)
A triangular AC current wave flows through a purely inductive load. Using the
relationship between voltage and current of an inductor, determine what the
shape of the voltage must be, across the inductor to have given rise to such a
current wave shape.
Clearly sketch both the current and voltage waveforms. Label each waveform
and ensure that the phase relationship between current and voltage is carefully
and correctly represented and labelled.
Indicate the amplitudes of each waveform on the sketch. The sketch must
include clearly labelled horizontal and vertical axes for both waveforms.
State the phase relationship between the two waveforms.
(7½ marks)
(b)
A triangular AC voltage waveform is applied across a purely capacitive load.
Using the relationship between voltage and current of a capacitor, determine
what the shape of the current waveform will be that flows in the capacitor as a
consequence of the applied voltage.
Clearly sketch both the current and voltage waveforms. Label each waveform
and ensure that the phase relationship between current and voltage is carefully
and correctly represented and labelled.
Indicate the amplitudes of each waveform on the sketch. The sketch must
include clearly labelled horizontal and vertical axes for both waveforms.
State the phase relationship between the two waveforms.
(7½ marks)
EO129 (2011/2012)
Page 4 of 10
Question 4
The circuit node in Figure Q4 supports AC currents flowing at mains AC voltages.
The frequency of the mains voltage is 50 Hz. The current i1 splits at the node to form
the currents i2 and i3. All the current waveforms are sinusoidal; however i1 and i2
have a phase angle between them of 90 degrees. The amplitudes of the currents i1
and i2 are √3 Amps and 1 Amp respectively.
i3
i1
i2
Figure Q4
(a)
Write down an expression for the sinusoidal current i1 ensuring that the
amplitude, phase and angular frequency of the current are taken into account in
the expression.
(2 marks)
(b)
Write down an expression for the sinusoidal current i2 ensuring that the
amplitude, phase and angular frequency of the current are taken into account in
the expression.
(2 marks)
(c)
Write down an expression for the nodal current i3 based on the expressions you
have above.
(2 marks)
(d)
Draw a phasor diagram of the nodal currents to scale and use the diagram to
determine the amplitude and phase of the current i3. Write down the
mathematical expression for the current i3.
(9 marks)
EO129 (2011/2012)
Page 5 of 10
Question 5
The circuit diagram in Figure Q5 supports sinusoidal AC currents and AC voltages.
The frequency of the voltage is 50 Hz. The voltages V1 and V2 are produced as a
consequence of the applied supply voltage V. The voltage V1, has an amplitude of
10 V and the voltage V2 has an amplitude of 10 V but has a phase angle of -90
degrees with respect to the voltage V1.
Z1
V1
Z2
V2
V
Figure Q5
(a)
Write down an expression for the sinusoidal voltage V1.
(2 marks)
(b)
Write down an expression for the sinusoidal voltage V2.
(3 marks)
(c)
Draw a phasor diagram of the voltages V1 and V2.
(3 marks)
(d)
Using simple geometry, determine the amplitude of the supply voltage V.
(3 marks)
(e)
Determine the phase difference between the voltages V and V 1 and write down
the full mathematical expression for V.
(4 marks)
EO129 (2011/2012)
Page 6 of 10
Question 6
The circuit diagram in Figure Q6 has two identical capacitors in parallel and is driven
by an AC voltage source V of angular frequency ω. The voltage source causes a
current ic to flow in each capacitor C which gives rise to the capacitor voltage V c.
Is
ic
C
ic
C
Vc
V
Figure Q6
(a)
Write down the equation for the relationship between the rate of change of the
dV
capacitor voltage c and the capacitor current ic for the capacitor of value C
dt
Farads.
(2 marks)
(b)
Using the equation for the relationship between the rate of change of the
dV
capacitor voltage c and the capacitor current ic, derive an expression for the
dt
capacitor current ic, if the expression describing the capacitor voltage is given
by Vc = 100 sin(ωt). Write down the expression for the capacitor current for the
circuit in Figure Q6.
(3 marks)
Question 6 continues on the next page
EO129 (2011/2012)
Page 7 of 10
Question 6 (continued)
(c)
In Figure Q6, for the capacitor of value C = 1 F, if the AC voltage source V is
at a mains voltage of 240 V RMS and the mains frequency is 50 Hz, show,
including all mathematical steps, the following:
(i)
The peak voltage is equal to 339 V.
(1 mark)
(ii)
The angular frequency ω is equal to 314 radians per second.
(1 mark)
(iii)
The supply current is approx. iS = 0.213cos ( 314t ) .
(8 marks)
EO129 (2011/2012)
Page 8 of 10
Question 7
The circuit diagram in Figure Q7 has two identical inductors in parallel and is driven
by an AC voltage source V. The voltage source causes a current iL to flow in each
inductor L which gives rise to the inductor voltage VL.
iS
iL
L
iL
VL
V
Figure Q7
(a)
Write down the equation for the relationship between the inductor voltage V L,
di
and the rate of change of the inductor current L for the inductor of value L
dt
Henries.
(2 marks)
(b)
Using the equation for the relationship between the inductor voltage V L, and the
di
rate of change of the inductor current L , derive an expression for the inductor
dt
voltage VL, if the expression describing the inductor current is given by
iL = 10 sin(ωt).
(3 marks)
(c)
In Figure Q7, for the inductor of value L = 1.0 Henry, if the AC voltage source V
is at a mains voltage of 240 Vrms and the mains frequency is 50 Hz, show,
including all mathematical steps, the following:
(i)
The peak voltage is equal to 339 V.
(1 mark)
(ii)
(iii)
The angular frequency ω is equal to 314 radians per second.
The supply current is approx. iS = 2.160cos ( 314t ) .
EO129 (2011/2012)
(1 mark)
Page 9 of 10
(8 marks)
Question 8
(a)
An operational amplifier can be used to add or subtract voltages. Sketch a
diagram of a circuit configuration, based on the use of an operational amplifier
and resistors, which will enable three independent voltages to be added
together and measured at the output of the circuit.
(6 marks)
(b)
Analyse your circuit and show, using mathematical analysis, that your design
will add together three independent voltages even if any or all of the voltages
are negative in value.
(9 marks)
EO129 (2011/2012)
Page 10 of 10