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ELG3311: Solutions for Assignment 1
Problem 2-6:
A 15-kVA 8000/230-V distribution transformer has an impedance referred to the primary
of 80 + j300Ω. The components of the excitation branch referred to the primary side are
RC = 350 kΩ and XM = 70 kΩ.
(a) If the primary voltage is 7967 V and the load impedance is ZL = 3.0 +j1.5 Ω, what
is the secondary voltage of the transformer? What is the voltage regulation of the
transformer?
(b) If the load is disconnected and a capacitor of –j4.0 Ω is connected in its place,
what is the secondary voltage of the transformer? What is its voltage regulation
under these conditions?
Solution:
(a) The turns ratio is
a = 8000 / 230 = 34.78
The load impedance referred to the primary side is
Z LP = a 2 Z L = (34.78) 2 (3.0 + j1.5) = 3629 + j1815 Ω
The referred secondary current is
I SP =
VP
7967∠0°
=
= 1.87∠ − 29.7° A
P
+ Z L (80 + j 300) + (3629 + j1815)
Z iP
The referred secondary voltage is
VSP = I SP Z LP = (1.87∠ − 29.7°)(3629 + j1815) = 7588∠ − 3.1° V
The actual secondary voltage is
VSP 7588∠ − 3.1°
=
= 218.2∠ − 3.1° V
VS =
a
34.78
The voltage regulation is
VP − VSP 7967 − 7588
=
× 100% = 4.99%
VR =
7588
VSP
1
(b) The turns ratio is
a = 8000 / 230 = 34.78
The load impedance referred to the primary side is
Z LP = a 2 Z L = (34.78) 2 (− j 4.0) = − j 4839 Ω
The referred secondary current is
I SP =
VP
7967∠0°
=
= 1.75∠89.0° A
P
+ Z L (80 + j 300) + (− j 4839)
Z iP
The referred secondary voltage is
VSP = I SP Z LP = (1.75∠89.0°)(− j 4839) = 8468∠ − 1.0° V
The actual secondary voltage is
VS =
VSP 8468∠ − 1.0°
=
= 243.2∠ − 1.0° V
a
34.78
The voltage regulation is
VP − VSP 7967 − 8468
=
× 100% = −5.92%
VR =
8468
VSP
2
Problem 2-7:
A 5000-kVA 230/13.8-kV single-phase power transformer has a per-unit resistance of 1
percent and a per-unit reactance of 5 percent (data taken from the transformer’s
nameplate). The open-circuit test performed on the low-voltage side of the transformer
yielded the following data: VOC = 13.8 kV, IOC = 15.1 A, POC = 44.9 kW
(a) Find the equivalent circuit referred to the low-voltage side of this transformer.
(b) If the voltage on the secondary side is 13.8 kV and the power supplied is 4000
kW at 0.8 PF lagging, find the voltage regulation of the transformer. Find its
efficiency.
Solution:
(a) The components of the excitation branch relative to the low-voltage side (secondary
side) are
YEX = GC − jBM =
θ = − cos −1
I OC
15.1
=
= 0.0010942
VOC 13.8k
POC
44.9k
= − cos −1
= −77.56°
VOC I OC
(13.8k )(15.1)
YEX = YEX ∠θ = GC − jBM = 0.0010942∠ − 77.56° = 0.0002358 − j 0.0010685
RC =
1
1
=
= 4240 Ω
GC 0.0002358
XM =
1
1
=
= 936 Ω
BM 0.0010685
The base impedance of this transformer referred to the low-voltage side (secondary side)
is
Z base =
2
Vbase
(13.8k ) 2
=
= 38.09 Ω
5000k
S base
So that
REQ = 1% Z base = (0.01)(38.09) = 0.38 Ω
X EQ = 5% Z base = (0.05)(38.09) = 1.90 Ω
The resulting equivalent circuit is shown below
3
REQ ,S = 0.38 Ω
X EQ,S = 1.9 Ω
RC ,S = 4240 Ω
X M ,S = 936 Ω
(b) The secondary current is
IS =
Pout
4000 kW
=
= 362.3 A
VS PF (13.8 kV)(0.8)
θ = − cos −1 PF = − cos −1 (0.8) = −36.87°
I S = 362.3∠ − 36.87° A
The voltage on the primary side of the transformer (referred to the secondary side) is
VPS = VS + I S Z EQ,S = 13800∠0° + (362.3∠ − 36.87°)(0.38 + j1.9) = 14330∠1.9° V
The voltage regulation of the transformer is
VR =
VPS − VS
14330 − 13800
× 100% =
× 100% = 3.84%
VS
13800
The transformer copper losses and core losses are
Pcopper = I S2 REQ ,S = (362.3) 2 (0.38) = 49.9 kW
Pcore =
(VPS ) 2 (14330) 2
=
= 48.4 kW
RC
4240
Therefore the efficiency of this transformer at these conditions is
4
η=
Pout
Pout + Pcopper + Pcore
× 100% =
4000 kW
× 100% = 97.6%
4000 kW + 49.9 kW + 48.4 kW
Problem 2-8:
A 200-MVA, 15/200-kV single-phase power transformer has a per-unit resistance of 1.2
percent and a per-unit reactance of 5 percent (data taken from the transformer’s
nameplate). The magnetizing impedance is j80 per unit.
(a) Find the equivalent circuit referred to the low-voltage side of this transformer.
(b) Calculate the voltage regulation of this transformer for a full-load current at
power factor of 0.8 lagging.
(c) Assume that the primary voltage of this transformer is a constant 15 kV, and plot
the secondary voltage as a function of load current for currents from no load to
full load. Repeat this process for power factors of 0.8 lagging, 1.0, and 0.8
leading.
Solution:
(a) The base impedance of this transformer referred to the low-voltage side (primary) is
Z base =
2
Vbase
(15k ) 2
=
= 1.125 Ω
S base 200M
So that
REQ = 1.2% Z base = (0.012)(1.125) = 0.0135 Ω
X EQ = 5% Z base = (0.05)(1.125) = 0.0563 Ω
And
X M = (80) Z base = (80)(1.125) = 90 Ω
The equivalent circuit referred to the low-voltage side of this transformer is shown below
REQ , P = 0.0135Ω
X EQ , P = 0.0563Ω
RC , P = not − spcified
X M , P = 90Ω
5
(b) The current on the secondary side of the transformer (referred to the primary side) is
I SP =
Pout
200 MW
=
= 16667 A
VS PF (15 kW )(0.8)
θ = − cos −1 PF = − cos −1 (0.8) = −36.87°
I SP = 16667∠ − 36.87°A
The voltage on the primary side of the transformer is
VP = VSP + I SP Z EQ , P = 15000∠0° + (16667∠ − 36.87°)(0.0135 + j 0.0563) = 15755∠2.24°V
Therefore the voltage regulation of the transformer is
VR =
VP − VSP
VSP
× 100% =
15755 − 15000
× 100% = 5.03%
15000
(c) This problem is repetitive in nature, and is ideally suited for MATLAB. A program to
calculate the secondary voltage of the transformer as a function of load is shown below:
%
%
%
%
%
%
M-file: prob2_8.m
M-file to calculate and plot the secondary voltage
of a transformer as a function of load for power
factors of 0.8 lagging, 1.0, and 0.8 leading.
These calculations are done using an equivalent
circuit referred to the primary side.
% Define values for this
VP = 15000;
%
amps = 0:166.67:16667; %
Req = 0.0135;
%
Xeq = 0.0563;
%
transformer
Primary voltage (V)
Current values (A)
Equivalent R (ohms)
Equivalent X (ohms)
% Calculate the current values for the three
% power factors. The first row of I contains
% the lagging currents, the second row contains
% the unity currents, and the third row contains
% the leading currents.
I(1,:) = amps .* ( 0.8 - j*0.6); % Lagging
I(2,:) = amps .* ( 1.0 ); % Unity
6
I(3,:) = amps .* ( 0.8 + j*0.6); % Leading
% Calculate VS referred to the primary side
% for each current and power factor.
aVS = VP - (Req.*I + j.*Xeq.*I);
% Refer the secondary voltages back to the
% secondary side using the turns ratio.
VS = aVS * (200/15);
% Plot the secondary voltage (in kV!) versus load
plot(amps,abs(VS(1,:)/1000),'b-','LineWidth',2.0);
hold on;
plot(amps,abs(VS(2,:)/1000),'k--','LineWidth',2.0);
plot(amps,abs(VS(3,:)/1000),'r-.','LineWidth',2.0);
title ('\bfSecondary Voltage Versus Load');
xlabel ('\bfLoad (A)');
ylabel ('\bfSecondary Voltage (kV)');
legend('0.8 PF lagging','1.0 PF','0.8 PF leading');
grid on;
hold off;
The resulting plot of secondary voltage versus load is shown below:
7
Problem 2-14:
A 13.2-kV single-phase generator supplies power to a load through a transmission line.
The load’s impedance is Zload = 500 ∠36.87°Ω , and the transmission line’s impedance is
Zline = 60 ∠53.1°Ω .
(a) If the generator is directly connected to the load, what is the ratio of the load
voltage to the generated voltage? What are the transmission losses of the system?
(b) If a 1:10 step-up transformer is placed at the output of the generator and a 10:1
transformer is placed at the load end of the transmission line, what is the new ratio
of the load voltage to the generated voltage? What are the transmission losses of
the system now? (Note: The transformers may be assumed to be ideal.)
Solution:
(a) In the case of the directly-connected load, the line current is
I line = I load =
VG
13.2∠0°kV
= 23.66∠ − 38.6°A
=
Z line + Z load 60∠53.1° + 500∠36.87°
The load voltage is
8
Vload = I load Z load = (23.66∠ − 38.6°)(500∠36.87°) = 11.83∠ − 1.73°kV
The ratio of the load voltage to the generated voltage is
ratio =
Vload 11.83 kV
=
= 0.896
13.2 kV
VG
The resistance in the transmission line is
Rline = Z line cosθ = (60)(cos 53.1°) = 36Ω
The transmission losses in the system are
2
Ploss = I line
Rline = (23.66) 2 (36) = 20.1 kW
(b) In this case, a 1:10 step-up transformer precedes the transmission line a 10:1 stepdown transformer follows the transmission line. If the transformers are removed by
referring the transmission line to the voltage levels found on either end, then the
impedance of the transmission line becomes
′ = a 2 Z line = (1 / 10) 2 (60∠53.1°) = 0.60∠53.1°Ω
Z line
The current in the referred transmission line and in the load becomes
′ = I load =
I line
VG
13.2∠0°kV
=
= 26.37∠ − 36.89° A
′ + Z load 0.60∠53.1° + 500∠36.87°
Z line
The load voltage is
Vload = I load Z load = (26.37∠ − 36.89°)(500∠36.87°) = 13.185∠ − 0.02°kV
The ratio of the load voltage to the generated voltage is
ratio =
Vload 13.185k
=
= 0.9989
VG
13.2k
The current in the transmission line is
I line = aI load = (1 / 10)(26.37) = 2.637 A
The losses in the transmission line are
9
2
Ploss = I line
Rline = (2.637) 2 (36) = 250W
Problem 2-15:
A 5000-VA, 480/120-V conventional transformer is to be used to supply power from a
600-V source to a 120-V load. Consider the transformer to be ideal, and assume that all
insulation can handle 600 V.
(a) Sketch the transformer connection that will do the required job.
(b) Find the kilovoltampere rating of the transformer in the configuration.
(c) Find the maximum primary and secondary currents under these conditions.
Solution:
(a) The common winding must be the smaller of the two windings, and NSE = 4 NC. The
transformer connection is shown below:
(b) The kVA rating of the transformer is
S IO =
N SE + N C
4NC + NC
SW =
(5000) = 6250VA
NC
NC
(c) The maximum primary current for this configuration is
IP =
S IO 6250
=
= 10.4 A
VP
600
The maximum secondary current is
IS =
S IO 6250
=
= 52.1A
VS
120
10
Problem 2-19:
A 20-kVA, 20,000/480-V, 60-Hz distribution transformer is tested with the following
results:
Open-circuit test
Short-circuit test
(measured from secondary side)
(measured from primary side)
VOC = 480 V
VSC = 1130 V
IOC = 1.60 A
ISC = 1.00 A
POC = 305 W
PSC = 260 W
(a) Find the per-unit equivalent circuit for this transformer at 60 Hz.
(b) What would be the rating of this transformer be if it were operated on a 50-Hz
power system?
(c) Sketch the per-unit equivalent circuit of this transformer referred to the primary
side if it is operating at 50 Hz.
Solution:
(a) The base impedance of this transformer referred to the primary side is
Z base, P =
VP2 (20000) 2
=
= 20 kΩ
S
20k
The base impedance of this transformer referred to the secondary side is
Z base,S =
VS2 (480) 2
=
= 11.52 Ω
S
20k
The open-circuit test yields the values for the excitation branch (referred to the secondary
side)
YEX =
I OC 1.60
=
= 0.00333
VOC 480
θ = − cos −1
POC
305
= − cos −1
= −66.6°
VOC I OC
(480)(1.60)
YEX = YEX ∠θ = GC − jBM = 0.00333∠ − 66.6° = 0.00132 − j 0.00306
RC =
1
1
=
= 757Ω
GC 0.00132
XM =
1
1
=
= 327Ω
BM 0.00306
11
The excitation branch elements can be expressed in per-unit as
RC , pu =
RC
Z base,S
=
757
= 65.7 pu
11.52
X M , pu =
XM
327
=
= 28.4 pu
Z base ,S 11.52
The short-circuit test yields the values for the series impedances (referred to the primary
side)
Z EQ =
VSC 1130
=
= 1130
I SC 1.00
θ = cos −1
PSC
260
= cos −1
= 76.7°
(1130)(1.00)
VSC I SC
Z EQ = Z EQ ∠θ = REQ + jX EQ = 1130∠76.7° = 260 + j1100
REQ = 260 Ω
X EQ = 1100 Ω
The resulting per-unit impedances are
R EQ , pu =
R EQ
Z base , P
=
260
= 0.013 pu
20k
X EQ , pu =
X EQ
Z base , P
=
1100
= 0.055 pu
20k
Therefore the per-unit equivalent circuit is show below
REQ , pu = 0.013 pu
X EQ , pu = 0.055 pu
RC , pu = 65.7 pu
X M , pu = 28.4 pu
12
(b) If this transformer were operated at 50 Hz, both the voltage and apparent power
would have to be derated by a factor of 50/60, so its ratings would be
S 50 = (50 / 60) S 60 = (50 / 60)(20k ) = 16.67 kVA
VP ,50 = (50 / 60)VP ,60 = (50 / 60)(20000) = 16667V
VS ,50 = (50 / 60)VS ,60 = (50 / 60)(480) = 400V
(c) The transformer parameters referred to the primary side at 60 Hz are
REQ = Z base REQ , pu = (20k )(0.013) = 260Ω
X EQ = Z base X EQ , pu = (20k )(0.055) = 1100Ω
RC = Z base RC , pu = (20k )(65.7) = 1.31MΩ
X M = Z base X M , pu = (20k )(28.4) = 568kΩ
At 50 Hz, the resistances will be unaffected but the reactances are reduced in direct
proportion to the decrease in frequency.
REQ ,50 = REQ ,60 = 260Ω
X EQ ,50 = (50 / 60) X EQ ,60 = (50 / 60)(1100) = 917Ω
RC ,50 = RC ,60 = 1.31MΩ
X M ,50 = (50 / 60) X M , 60 = (50 / 60)(568) = 473kΩ
The base impedance of the transformer operating at 50 Hz referred to the primary side is
Z base , P ,50 =
VP2 (16667) 2
=
= 16.67kΩ
S
16.67k
13
The resulting per-unit equivalent circuit referred to the primary at 50 Hz is shown below
REQ , pu ,50 =
RC , pu ,50 =
REQ ,50
Z base, P ,50
RC ,50
Z base , P ,50
=
=
260
= 0.016 pu
16.67 k
1.31M
= 78.5 pu
16.67 k
X EQ , pu ,50 =
X M , pu ,50 =
X EQ ,50
Z base , P ,50
X M ,50
Z base, P ,50
=
=
917
= 0.055 pu
16.67 k
473k
= 28.4 pu
16.67 k
14