Power Circuit Theory: Tutorial Set 1
Exercise 1
A balanced three-phase Y load has one phase voltage of Vcn = 277∠45◦ V . If the phase
sequence is negative sequence i.e. acb, calculate the line voltages Vca , Vab , and Vbc .
Vca = 480∠15◦ V , Vab = 480∠135◦ V , Vbc = 480∠−105◦ V
Exercise 2
What are the phase voltages for a balanced three-phase Y load, if Vba = 12.47∠−35◦ kV ?
The phase sequence is positive sequence i.e. abc.
Vbn = 720∠−5◦ kV , Van = 7.20∠115◦ kV , and Vbc = 7.20∠−125◦ kV
Exercise 3
A balance Y load of 40 Ω resistors is connected to a balanced, three-phase, three-wire source.
If Vcb = 480∠−35◦ V Calculate the a phase line current. The phase sequence is negative
sequence
Ia = 6.928∠55◦ A
Exercise 4
In a three-phase, three-wire circuit, calculate the line currents to a balanced Y load for which
Zy = 60∠−30◦ Ω and Vcb = 480∠65◦ V . The phase sequence is positive.
Ia = 4.619∠5◦ A, Ib = 4.619∠−115◦ A, and Ic = 4.619∠125◦ A
Exercise 5
Calculate the total average power delivered by a three-phase source with the line to line
voltage 500 V to each of the following balanced Y connected loads with Zy equal to:
a) 30 Ω;
b) (30 + j72) Ω;
c) (30 − j12.5) Ω.
(a) 8.333 kW, (b) 1.233 kW, (c) 7.101 kW
Exercise 6
Calculate the magnitude of the line voltage (VLine−Line ) at the source of the circuit in Figure
E6. As shown, the load phase voltage is 100V and the impedance of each line is 2+j3 Ω.
|VLine−Line | = 179.5 V
Figure E6: Circuit for Exercise 6.
Exercise 7
A 208 V three-phase circuit has two balanced loads, one a ∆ of 21∠30◦ Ω impedances and
the other a Y of 9∠−60◦ Ω impedances. Calculate the magnitude of the line current and
also the total average power absorbed by the two loads.
|Iline | = 21.73 A, Pave = 7.756 kW
Exercise 8
In a 208 V three-phase circuit a balanced ∆ connected load absorbs 2 kW at a 0.8 leading
power factor. Calculate Z∆ .
ZDelta = (41.51 − j31.15) Ω
Exercise 9
Two balanced three-phase motor loads comprising an induction motor and a synchronous
motor are connected in parallel. The induction motor draws 400 kW at 0.8 power factor
lagging and the synchronous motor draws 150 kVA at 0.9 power factor leading. Both motor
loads are supplied by a balanced three-phase 4.160 kV source. If the cable impedance between
the source and load is neglected,
a) Draw the power triangle for each motor and for the combined-motor load.
b) Determine the power factor of the combined-motor load.
c) Determine the magnitude of the line current delivered by the source.
d) A delta connected capacitor bank is now installed in parallel with the combined-motor
load. What value of capacitive reactance is required in each phase of the capacitor
bank to make the source power factor unity?
e) Determine the magnitude of the line current delivered by the source with the capacitor
bank installed.
(b) 0.916 lagging, (c) 81.1 A, (d) j221.3 Ω, (e) 74.3 A
Exercise 10
A balanced 3 phase star connected 400 V system has a per phase impedance of 20∠36.87◦ Ω.
Assuming positive phase sequence and VAB as reference, determine the phase current and
complex power. IAN = 11.55∠−66.87◦ A, ST Y = 8.00∠36.87◦ kVA
If the same impedances are connected in ∆ connection determine the line currents and the
complex power assuming VAB as reference. IAB = 20∠−36.87◦ A, ST ∆ = 24∠36.87◦ kVA
Exercise 11
Consider a single phase AC circuit shown in Figure E11.
Figure E11: Circuit for Exercise 11.
The voltage and impedance values are given in Figure E11. Determine
i) the branch complex powers S1, S2 and S3 and total complex power.
S1 = 7.900∠80.91◦ kVA, S2 = 2.287∠−59.04◦ kVA,
S3 = 6.000∠36.87◦ kVA, ST = 11.89∠52.57◦ kVA.
ii) the supply current and overall power factor.
I59.43 ∠−52.57◦ A, p.f. = 0.608 lagging
iii) the capacitance value that is required to be connected across the load to improve the
power factor to unity, assuming a 50 Hz supply. C = 751.2µF
Exercise 12
Consider a single phase AC circuit shown in Exercise E12.
Figure E12: Circuit for Exercise 12.
The instantaneous power is given by,
p(t) = 960 + 1200 cos(628t − 36.87◦ )W
Determine the value of rms current supplied to the load, the complex power supplied to the
load and the load impedance.
Irms = 7.714 A, S = 1.200∠36.87◦ kVA, Z = 20.17∠36.87◦ Ω
Exercise 13
A 100 kVA, 11000:2200 V, 60 Hz, single-phase transformer has the following equivalent
circuit parameters referred to the high-potential side:
R1 = 6.1 Ω
R20 = 7.2 Ω
Xl1 = 31.2 Ω
Xl20 = 31.2 Ω
Xm = 57.3 kΩ Rc = 124 kΩ
The transformer is supplying at 2.20 kV a load circuit of 50 Ω impedance and a leading power
factor of 0.7. Draw a phasor diagram (not to scale) showing the various phasor magnitudes
and angles for this operating condition. Determine the potential difference and power factor
at the high-potential terminals of the transformer. 10.707 kV, 0.7458 leading
Exercise 14
A 20 kVA, 2200:220 V, 60 Hz, single-phase distribution transformer gave the following test
results:
i) Open-circuit test, low-potential winding excited:
Voc = 220 V, Ioc = 1.52 A, Poc = 161 W
ii) Short-circuit test, high-potential winding excited:
Vsc = 205 V, Isc = 9.1 A, Psc = 465 W
iii) Direct-current winding resistances:
Rlp = 31.1 mΩ, Rhp = 2.51 W
Assume that Xl1 = Xl20 ,determine the equivalent circuit of the transformer referred to the
high-potential side.
(Rc =30.06 kΩ, Xm = 16.51 kΩ, Xl1 & Xl20 =10.91 Ω, R1 = 2.51 Ω, R20 = 3.11 Ω)
Exercise 15
Consider the one-port network shown in Figure E15:
Figure E15: Circuit for Exercise 15.
Show that the graph of the locus of the port impedance, Z, as the resistance R is varied from
0 to ∞ Ω , is a straight line. Show also that the locus of the port admittance, Y, as the
resistance R is varied from 0 to ∞ Ω , is the arc of circle. Determine the centre and radius
of the circle.
Show that the graph of the loci of I, VC and VR as the resistance R is varied from 0 to ∞ Ω,
are arcs of circles. Determine the centre and radius of each circle.
Exercise 16
Consider the one-port network shown in Figure E16:
Figure E16: Circuit for Exercise 16.
Show that the graphs of the loci of VRL , VL0 , VR and VL as the resistance R is varied from 0
to ∞ Ω, are arcs of circles. Determine the centre and radius of each circle.
Exercise 17
Determine the transformer tap ratios when the load voltage is equal to the generating end
voltage, the load voltage is 230 kV and transmits 80 MW at 0.8 pf lagging and the impedance
of the line is (40 + j 150) Ω. Assume tr ts = 1. The arrangement is shown in Figure E17.
Figure E17: Circuit for Exercise 17.
ts = 1.1401
Exercise 18
A wattmeter is connected in a single-phase circuit to
√measure the average power. Show that
the average
power
is
V
I
cos(θ
−
θ
),
given
v(t)
=
2V cos(ωt + θv ) and
v
i
√
i(t) = 2I cos(ωt + θv )
Exercise 19
From a consideration of the instantaneous voltage and currents in a three-phase system, both
star and delta, show that the total power can be measured by means of two wattmeters.
Exercise 20
Show that the power in a three-phase, three-wire system with balanced loads is constant at
every instant. Deduce an expression for the power in terms of the line voltage, line current,
and the power-factor.
Exercise 21
Show that in a balanced system in which the power is measured by the two-wattmeter
method P1 = |Vl ||Il | cos(30 −√φ) and P2 = |Vl ||Il | cos(30 + φ)
Prove also, P1 + P2 = PT = 3|Vl ||Il | cos(φ)
where φ is the power factor angle and denotes a line quantity.
Exercise 22
Given a balanced, positive-sequence,
three-phase
set of voltages in which Van = Vph ∠0, show
√
√
◦
◦
analytically that Vab = 3Van ∠30 = 3Vph ∠30 .
Exercise 23
Given a balanced system of three phase voltages, show analytically vab + vac = 3van
Exercise 24
The maximum power entering a series RL circuit is 500 W and the minimum power is −180
W. The voltage is 230 V, 50 Hz. Determine
i. The values of R and L,
R = 73.22 Ω
L = 437.0 mH
ii. The value of capacitance, connected across the terminals of the network, if the maximum power is to be 360 W.
C=10.83 µF
Exercise 25
The power in a balanced three-phase system in which the power is measured by the twowattmeter
method to be P1 =100 W and P2 = 50 W. Calculate the power factor of the load. The
sequence is positive and wattmeter 1 measures the current in the a phase and wattmeter 2
measures the current in the c phase.
Power factor = 0.866 leading