Question 1.1The instantaneous values of a-phase voltage and
a-phase current in a balanced star-connected
3-wire load are given. ( (a) ( ( ( ) ) ) )
(b)
(c)
(d) Phasor Diagram
(e)
(f)
Solution:
M a Three-phase Supply b Load (Balanced or Unbalanced)
c M
(
)
(
)
( )
(
) (
( )
)
( (g) ( )
) ( )
Solution:
M a
b
StarConnected Loads
c M Line Voltage and line current are differing by phase angle.
( )
( ( ( ( ( ( ( ( ) ) ) ) ( ) )) )
)
( )
( ) ( )
( [ (
) ) (
( )]
)
(h) Sketch a neat circuit diagram for the one-wattmeter
crossed-coil connection to determine the total reactive power Q in
the above load. Label terminals. Show all quantity symbols.
Solution: (Page 7.5 modified for one watt-meter)
cThree-phase supply Three-phase balanced load
a
bOne-wattmeter crossed-coil connection
(
) ( )
( )
(i)( )
( )
(
)
Solution:
cThree-phase supply Three-phase balanced load
a
b
From the Phasor diagram, ( ( ( ( ) ) ) )
( )
Question 1.2 (a) Sketch and annotate half-wave and full-wave
circuit arrangements of rectifiers with permanent magnet moving
coil instrument for multi-range a.c. voltage measurements. Indicate
the functions of the components. Solution: Half-wave arrangements
of rectifiers with permanent magnet moving coil instrument.
Functions of the components:
The half-wave rectifier circuit with shunt resistance across the
meter lowers the sensitivity for the a.c. range. The rectifier
diode (D1) is not perfect that is, the application of a voltage of
reversed polarity will result in a small leakage current flow
causing a lower average value to be indicated and so the bypass
diode D2 is used to eliminate the effect of this reverse
current.
Full-wave arrangements of rectifiers with permanent magnet
moving coil instrument.
Functions of the components:
There are 4 diodes connected as bridge rectifier. The shunt
resistance improves scale linearity by drawing adequate current
through the diodes toensure their operation in the linear region of
their characteristic.
(b) Explain the limitations, sources of errors and reasons for
different values of d.c. and a.c.sensitivity for a rectifier PMMC
voltmeter.
Solution: Limitation of PMMC voltmeter The half-wave rectifier
circuit with shunt resistance across the meter lowers the
sensitivity for the a.c. range. The rectifier diode (D1) is not
perfect that is, the application of a voltage of reversed polarity
will result in a small leakage current flow causing a lower average
value to be indicated and so the bypass diode D2 is used to
eliminate the effect of this reverse current. To operate in the
linear part of its characteristic, each diode needs a forward
voltage drop ofabout 0.6 V. Therefore, the bridge rectifier may not
work for voltages below 1.5 V or diode currents below 1 m.
(c) Sketch the circuit diagram for, and explain the operating
principle of, a capacitor voltagedivider with magnetic voltage
transformer for high voltage measurements. Show details for 330-kV
input. Write the intermediate voltage equation.
Solution: Primary 330 kV Intermediate Secondary
11 kV CVD
110 V
Magnetic Voltage Transformer
Hence, the final voltage connected across the secondary is 110
V. ( ( ) )
Metering VTs are classed: 0.2M, 0.5M, 1M, 2M or 5M. Protection
VTs are classed: 1P, 2P or 5P. The intermediate voltage equation is
the multiplication of primary voltage to the ratio of capacitors
connected across intermediate, i.e;
[ (
)
]
(d)( )
Solution:
( (
) )
(
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(
)
[(
)
(
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[( [(
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( (
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{
( [(
)
( ) ( (
)
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)
}
( (
) )
(
)
(
)
(e) List and classify the causes of transient high-voltage
spikes in d.c. and a.c. circuits and indicate two simple methods of
suppressing these voltage spikes.
(f) Sketch the waveforms and explain the equations for charging
and discharging of a resistorcapacitor series circuit with
square-wave input. Solution: Charging waveform of a RC series
circuit
Discharging waveform of a RC series circuit
Question 1.3 (a) Sketch a circuit diagram for calibrating a
single-phase energy meter at various power factors.Explain the
principle involved in using a phantom load and state its
advantages. Sketch a phasor diagram to indicate the range of power
factor variation possible and state this range.
Principle involved in using Phantom Load: When the capacity of
the meter to be tested is high considerable power will be wasted if
ordinary loading arrangement is made. For testing of meter of
higher capacities phantom or fictitious loading arrangement is
made. Advantage of using Phantom Load: By interposing transformers
the desired values of voltage and current can be obtained. Such a
setup is useful to calibrate a single-phase energy meter at
different power factor values, without the need for real energy
consuming loads.
C
B
A
(b) Explain the equations and techniques used to separate
transformer eddy-current and hysteresis losses by iron-loss
measurements at two different frequencies. Solution: (Page 6.19)
The iron-loss is the addition of eddy-current loss and hysteresis
loss.
( *
Slope=b
a
0
f1
f2
f
( *
By extracting these two values from iron-loss measurements we
can separate transformer eddy-current and hysteresis losses at two
different frequenciespower-factor wattmeter. By applying correction
factors, the iron loss in the iron sample is determined.
.
The power supplied to the primary from a sinusoidal voltage
source is measured using a low-
Question 1.4 (a) Sketch the frequency response curve and the
equivalent circuit of an audio-frequencytransformer. Label the
inherent capacitances and inductances in the transformer equivalent
circuit and identify them in a table. Indicate how this equivalent
circuit may be simplified. With the aid of sketches, explain how
the transformer reactances affect the frequency response.
Solution:Resonance Peak
10 Frequency f (Hz)
AF Transformer LoadP S
Signal Source
f
The above equivalent circuit is simplified as below;
Capacitances, inductances and resistances in the transformer
equivalent circuitPrimary inter-turn capacitance Secondary
inter-turn capacitance Inter-winding capacitance Primary leakage
inductance Secondary leakage inductance Magnetising inductance
Primary winding resistance Core-loss resistance Secondary winding
resistance
The transformer reactance affects the frequency response when
low frequencies the shunt path is reduced due to leakage reactance.
Again at mid-frequencies the output is constant as reactance
omitted. At the high frequencies the effect of magnetising
reactance is very negligible. Hence at the 3 frequencies the
reactance of the AF Transformer affects.
(b) Sketch a circuit diagram and tabulate the details (voltage,
current, volt-ampere, accuracyclass, VT and CT constants, type,
etc.) of equipment/ accessories necessary for measuring the
voltage, current, power factor, frequency and energy supplied to a
single-phase load rated at 6.35 kV, 500 A, 50 Hz.
Solution: The circuit diagram of single phase high voltage
measurement using CTs and VTs, is as below.
Voltage Current Volt-Ampere
V AThe power requirement must be within the rated VA, , P.F.
Accuracy class Link VT Constant CT Constant Types M
Question 1.5
(a)
.
Solution: a c
b
( ( (
) ) )
(
)
(
)
(
)
The star-connected generator, equivalent circuit below;
(b) Tabulate the most common types of errors in a.c. bridge
measurements and the remedial or precautionary measures taken to
minimise such errors due to nulldetectors, standard components and
signal sources. Solution: The most common types of errors in a.c.
bridge measurements and their remedy are given in the table below;
Error Due to Null-detector Remedial or precautionary measures taken
to minimize error Select suitable detector based on frequency
response, tuning and sensitivity; use matching transformer and
amplifiers as needed. Standard components Good design with adequate
compensation. Selection of components and equipment to suit the
particular test and measurement conditions. Silver contacts;
dust-free enclosure. Proper care and regular maintenance.
Stabilisation of power supply. Temperature compensation. Signal
Sources Use sine-wave generators; use tuned detectors with
filters.
Question 1.6 (a) Sketch the voltage and current resonance curves
for a series R-L-C resonant circuit. Writeexpressions for, and
discuss the relationships between bandwidth,
resonance-frequency-Qfactor, resonance frequency, half-power points
and half-power frequencies.
Solution:
Bandwidth: Bandwidth is the difference between lower half-power
frequency and upperhalf power frequency. Resonance frequency
Q-factor: ( ) ( Resonance Frequency: It is the frequency at which
the power dissipation maximum, in the R-L-C circuit. It is the
geometric mean of upper-half power frequency and the lower half
power frequency. )
Half-power Points: There are 2-half power point, lower half
power point and the upper half power point where the power
dissipated is half of the power dissipated at resonance. The
half-power points describe about leading or lagging of the power
factor. Half-power Frequencies: These are the two frequencies at
the respective half-power points, named as upperhalf power
frequency and the lower half power frequency. The relation between
the above discussed factors is: ( ) ( ) ( )
(
*
(
)
(
*
(b) Sketch, on the one graph, frequency responses showing the
difference between a resonant circuit with a large Q and one with a
low Q. Relate circuit Q-factor, bandwidth and resonance frequency
to circuit resistance and reactance.
Solution:
(c) Explain the differences between single-tuned band-pass and
band-stop R-L-C filter circuits; sketch their circuit diagrams
(show only one circuit diagram of each type) and typical frequency
response curves. Annotate the diagrams with key details.
Question 2.1
(a) Sketch a neat circuit diagram for the above measurement
set-up.
(b)
(
)
(c)
(
)
(d)
(
)
(e)
(
)
(f) ( )
(g) ( )
(h)
Question 2.2 (a) ( )
(b) Sketch the circuit diagram for an Ayrton universal shunt and
state its main advantage. Solution: (Page 1.13) Im Rm mA R2 R3
R4
R1
I1 I0
I2 I3
+
-
Advantage: Switch not needed, due to the series connection of
the resistor across the ammeter. The effective resistance increases
due to this series connection. There is no sudden forceful flow and
no large current due to the selector switch movement.
In the absence of a shunt the meter movement is never left in
the main current circuit. (c) Sketch the circuit diagram and
indicate the operating principles of a T-attenuator.
(d) Define the r.m.s. value of a complex waveform in terms of
the harmonic component peak values. Solution: (Page 4.13)R.m.s
value of a complex waveform is the square root of the sum of the
squares of each individual harmonic components peak values and zero
harmonic.
The r.m.s. values of the harmonic components are:
The r.m.s. value of the non-sinusoidal voltage is then:
(
)
(e) List and briefly define eight key items of information to be
specified for an electrical measuring instrument before selecting
it for purchase.
(f) Explain why the secondary winding of a current transformer
can be safely shortcircuited but not opened, if the primary winding
is energised.
(g) State the equations for the complex forms of impedance of
R-L parallel, R-C series and R-C parallel circuits commonly
encountered in a.c. bridges. Sketch and label the circuits.
(h) Derive the d.c. maximum power transfer theorem using both
theThevenin and Norton equivalent circuits and a resistive
load.
(i) Describe, with the aid of sketches, applications of R-C
circuits involving transients, d.c. blocking and a.c. bypass
characteristics.
Question 2.3 (a) Indicate the various sources of errors in an
electrodynamics wattmeter and briefly explainthe compensation
methods and formula corrections to be applied to determine the true
power in a.c. circuits.
(b) Sketch the circuit diagram of an R-C bridge phase-shifting
network. Sketch, with details, the locus of the detector voltage
when R is variable. Indicate the practical phase-shift and voltage
variation limits of such a circuit and state the main advantages of
the bridge circuit arrangement. State how the phase shift may be
changed from lagging to leading.
Question 2.4
(a) Tabulate a comparison of permanent-magnet moving-coil and
electro- dynamic indicating instruments (graphic symbol,
type/operating principle, uses, control and damping, scaling,
errors, etc.)
(b) Sketch and annotate the circuit arrangement of a three-range
a.c. voltmeter that uses a permanent-magnet moving coil instrument
and bridge rectifier. State the limitations of such instruments.
Explain why the scale is calibrated to read the r.m.s. value of
pure sine-wave voltages.
Question 2.5The energy supplied to a bulk consumer balanced lead
is metered on the 66-kV side of a 1000-kVA, 66/11-kV,
delta-star-connected transformer. Draw a neat metering schematic
that uses appropriately-selected, standard-ratio voltage and
current transformers, precision double-element kWh meter,
cross-coil kvarh meter and indicating instruments, with necessary
selector switches, fuses and test-links. (Use pencil and rule: no
free-hand sketch.) For each measurement instrument, determine and
tabulate the ratings (Voltage, current, volt-ampere, accuracy
class, VT and CT constants, type, etc.).
Note:
Question 2.6 (a) Sketch a series R-L-C circuit diagram, with
details. Write the expression for the resonance frequency. State
the conditions for series resonance. List the methods of achieving
resonance.
(b) Sketch typical graphs of (i) reactances and impedance, and
(ii) voltages and current in a series resonant circuit with fixed
R, L, C and V values but variable frequency f. Indicate the
relationships between circuit resonance-frequency-Q-factor,
bandwidth, resonance frequency, half-power points and half-power
frequencies.
(c) Sketch a parallel R-L-C circuit diagram, with details.
Sketch typical parallel resonance curves in terms of G, Y, Z
and
. Write the expression for the resonance frequency when L
and C have power losses. Indicate how the inherent resistances
of the inductor and capacitor affect the circuit dynamic
impedance.
Question 3.1 In a balanced, star-connected, 3-wire load, the
instantaneous values of c-phase voltage and cphase current are
given: ( ) ( )
Assume phase sequence a-b-c for the balanced supply, with load
star point s.(a)
(b)
(c)
(d)
( (
) )
(e)
(f)
(g)
( ( )
)
(h)
.
(i)
Question 3.2 (a) Sketch circuit, phasor and power diagrams, with
details, for an induction motor and a capacitor connected in
parallel to the same supply.
(b) Explain the trigonometric form of the harmonic Fourier
series. Define the r.m.s. value of a complex waveform in terms of
the harmonic component peak values. Solution: (Page
4.13)Trigonometric form of the harmonic Fourier series: By taking
Fourier series, a complex waveform can be resolved into a number of
sinusoidal and co-sinusoidal harmonic components, including a d.c.
component.
( )The trigonometric form of the harmonic Fourier series is
putting the entire voltage component together; e.g.:
( )( )
R.m.s value of a complex waveform is the square root of the sum
of the squares of each individual harmonic components peak values
and zero harmonic. The r.m.s. values of the harmonic components
are:
The r.m.s. value of the non-sinusoidal voltage is then:
(
)
(c) Define with phasor diagrams and expressions: positive-,
negative- and zero-sequence voltages. Using the sequence
components, write equations for the set of unbalanced phase
voltages.
(d) Indicate the differences in the functional requirements of
metering and protection current transformers. Sketch their B-H
curves, with details. Solution: (Page 8.11) Protection Current
Transformers are used mainly for operation with circuit breakers.
The rating of Current Transformer depends upon the purpose and the
type of fault for which it is used and also on the characteristic
of relay etc.A metering Current Transformer is designed to operate
at lower flux density and exciting current, whereas A Protection
Current Transformer is designed to operate at higher flux densities
with higher range of linear measurement.
Protection
MeteringLinear Part of Curve
0B-H curve of Metering and Protection Metering Current
Transformer operates in the linear part of the B-H curve during
normal primary current.
In Metering Current Transformer when the primary current exceeds
the rated value, the Current Transformer core becomes saturated and
the secondary current Is does not increase abnormally, thus
preventing overload burnout of current coils of meters.
In Protection Current Transformer During a fault on the primary
side equipment, the current is excessive, and the protection
Current Transformer must truly reflect this current on its
secondary side to operate the protective relays promptly.
(e) Sketch and annotate the circuit diagram of an a.c.
Wheatstone bridge with detector D and a.c. voltage source E. State
the null-balance condition. Tabulate and label the forms of
equations for Product Balance and Ratio Balance in terms of the
branch impedances.
(f) Sketch and explain the waveforms, and write the equations,
for charging and discharging of a resistor-capacitor series circuit
with square-wave voltage input.
(g) Sketch and annotate the circuit arrangement of a bridge
rectifier and permanentmagnet moving-coil instrument for
three-range a.c. voltage measurements. Indicate the component
functions. State the limitations of such instruments.
(h) Compare and explain the reasons for the different d.c. and
a.c. sensitivities of the bridge rectifier PMMC voltmeter. State
the main sources of error in the rectifier PMMC voltmeter and
indicate the methods of minimising those errors. Explain why the
scale is calibrated to read the r.m.s. value of pure sine-wave
voltages.
Question 3.3
(a) Tabulate step-by-step the method of applying the
superposition principle to determine the current i in a circuit
with series resistor R and inductor L, if the applied voltage is
given by: ( )
(b) Applying Thevenins theorem to an unbalanced a.c.
Question3.4
( (
) )
Question3.5
(a) Sketch the circuit arrangement of a single-phase
induction-type energy meter and indicate its constructional
features. Briefly explain its principle of operation and state the
methods commonly used to compensate for errors.Solution:
(b) Sketch a circuit diagram for measurement of iron-loss using
an Epstein square. Write the equations and explain the principles
involved. State the power factor calibration requirement for the
wattmeter. Explain the need for a correction to be applied to the
wattmeter reading. Solution: The Circuit diagram for measurement of
iron-loss using an Epstein square.
The iron sample is available as strips of laminations. By
assembling the strips as a square core inside two pre-wound coils
of known number of turns, a transformer is formed. The power
supplied to the primary from a sinusoidal voltage source is
measured using a low-power-factor wattmeter. By applying correction
factors, the iron loss in the iron sample is determined.
Question 3.6(a) A series R-L-C circuit has a.c. source voltage
V. Sketch and fully annotate a neat circuit diagram. Write the
resonance frequency equation. List the methods of achieving
resonance. State the key indicators of the series resonance
condition. Solution:
Series R-L-C Circuit arrangement
(b) A series resonant circuit has fixed R, L and C values, fixed
source voltage V and variable source frequency f. Sketch typical
graphs of (i) reactances, resistance and impedance; and (ii)
voltages and current, as the frequency is varied. Indicate, on
graph (ii), the 3 dB points, the half-power frequencies, the
bandwidth BW, the resonance frequency fr and the lagging and
leading power factor regions.
(c) Sketch, on the one graph, typical current versus frequency
responses of high-Q and low-Q series resonant circuits and identify
their main features. Relate circuit resonance-frequency-Q-factor,
bandwidth and resonance frequency to circuit resistance and
reactance.
(d) Sketch and annotate a parallel R-L-C circuit diagram when
both L and C have significant losses. State the condition for
resonance in such a circuit and write the resonance frequency
expression.
Solution:
Parallel R-L-C circuit diagram when both L and C have
significant losses