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Chapter 9: Transmission Lines
9-1. Calculate the dc resistance in ohms per kilometer for an aluminum conductor with a 3 cm diameter.
SOLUTION The resistance per meter of aluminum conductor is given by Equation (9-2):
r DC (^) (^2) A r
ρ ρ
π
where ρ = 2.83 × 10
8 5 DC (^2)
2.83 10 -m 4.004 10 /m 0.015 m
r π
− −
× Ω
= = × Ω
Therefore the total DC resistance per kilometer would be
5 DC
1000 m 4.004 10 /m 0.040 /km 1 km
R
− §^ ·
= × Ω ¨ ¸= Ω
9-2. Calculate the dc resistance in ohms per mile for a hard-drawn copper conductor with a 1 inch diameter.
(Note that 1 mile = 1.609 km).
SOLUTION The resistance per meter of hard-drawn copper conductor is given by Equation (9-2):
r DC (^) 2 A r
ρ ρ
π
where ρ = 1.77 × 10
- Ω-m. Note that the radius in this equation must be in units of meters. This value is
8 5 DC 2
1.77 10 -m 3.493 10 /m
0.0254 m 0.5 in 1 in
r
− −
× Ω
= = × Ω
« ¨^ ¸»
Therefore the total DC resistance per mile would be 1000 r DC = 0.03367Ω.
5 DC
1609 m 3.493 10 /m 0.0562 /mile 1 mile
R
− §^ ·
= × Ω ¨ ¸= Ω
Problems 9-3 through 9-7 refer to a single phase, 8 kV, 50-Hz, 50 km-long transmission line consisting of two
aluminum conductors with a 3 cm diameter separated by a spacing of 2 meters.
9-3. Calculate the inductive reactance of this line in ohms.
SOLUTION The series inductance per meter of this transmission line is given by Equation (9-22).
1 ln H/m 4
D
l r
where
7
μ μ 0 4 π 10 H/m
− = = ×.
7 0 1 2.0 m^4 10 H/m^1 2.0 m^6 ln ln 2.057 10 H/m 4 0.015 m 4 0.015 m
l
μ π
π π
− § · × § · − = (^) ¨ + (^) ¸ = (^) ¨ + (^) ¸= × © ¹ © ¹
Therefore the inductance of this transmission line will be
( ) (^ )
6 L 2.057 10 H/m 50,000 m 0.1029 H
− = × =
The inductive reactance of this transmission line is
X = j ω L = j 2 π fL = j 2 π ( 50 Hz ) ( 0.1029 H )= j 32.3Ω
9-4. Assume that the 50 Hz ac resistance of the line is 5% greater than its dc resistance, and calculate the series
impedance of the line in ohms per km.
SOLUTION The DC resistance per meter of this transmission line is given by Equation (9-22).
r DC (^) (^2) A r
where ρ = 2.83 × 10
8 5 DC 2
2.83 10 -m 4.004 10 /m 0.015 m
r π
− −
× Ω
= = × Ω
Therefore the total DC resistance of the line would be
( ) (^ )
5 R DC (^) 4.004 10 /m 50,000 m 2.
− = × Ω = Ω
The AC resistance of the line would be
R AC = ( 2.0 Ω ) ( 1.05) = 2.1Ω
The total series impedance of this line would be Z = 2.1 + j 30.5Ω , so the impedance per kilometer
would be
Z = ( 2.1 + j 32.3 Ω ) / 50 km( ) = 0.042 + j 0.646 Ω/km
9-5. Calculate the shunt admittance of the line in siemens per km.
SOLUTION The shunt capacitance per meter of this transmission line is given by Equation (9-41).
ln
c D
r
12 12
8.854 10 F/m 5.69 10 F/m
ln
c
− −
×
= = ×
Therefore the capacitance per kilometer will be
9 c 5.69 10 F/km
− = ×
The shunt admittance of this transmission line per kilometer will be
9 6
y sh j 2 π fc j 2 π 50 Hz 5.69 10 F/km j 1.79 10 S/km
− − = = × = ×
Therefore the total shunt admittance will be
( ) (^ )
6 5 Ysh j 1.79 10 S/km 50 km j 8.95 10 S
− − = × = ×
9-6. The single-phase transmission line is operating with the receiving side of the line open-circuited. The
sending end voltage is 8 kV at 50 Hz. How much charging current is flowing in the line?
SOLUTION Although this line is in the “short” range of lengths, we will treat it as a medium-length line,
because we must include the capacitances if we wish to calculate charging currents. The appropriate
transmission line model is shown below.
(c) The voltage regulation of the transmission line is
10000 8000 VR 100% 100% 25% 8000
S R
R
V V
V
= × = × =
Problems 9-8 through 9-10 refer to a single phase, 8 kV, 50-Hz, 50 km-long underground cable consisting of two
aluminum conductors with a 3 cm diameter separated by a spacing of 15 cm.
9-8. The single-phase transmission line referred to in Problems 9-3 through 9-7 is to be replaced by an
underground cable. The cable consists of two aluminum conductors with a 3 cm diameter, separated by a
center-to-center spacing of 15 cm. As before, assume that the 50 Hz ac resistance of the line is 5% greater
than its dc resistance, and calculate the series impedance and shunt admittance of the line in ohms per km
and siemens per km. Also, calculate the total impedance and admittance for the entire line.
SOLUTION The series inductance per meter of this transmission line is given by Equation (9-22).
1 ln H/m 4
D
l r
where
7
μ μ 0 4 π 10 H/m
− = = ×.
7 0 1 0.15 m^4 10 H/m^1 0.15 m^6 ln ln 1.021 10 H/m 4 0.015 m 4 0.015 m
l
− § · × § · − = (^) ¨ + (^) ¸ = (^) ¨ + (^) ¸= × © ¹ © ¹
Therefore the inductance of this transmission line will be
( ) (^ )
6 L 1.021 10 H/m 50,000 m 0.0511 H
− = × =
The inductive reactance of this transmission line is
X = j ω L = j 2 π fL = j 2 π ( 50 Hz (^) ) ( 0.0511 H (^) )= j 16.05Ω
The resistance of this transmission line is the same as for the overhead transmission line calculated
previously: R AC (^) = 2.1Ω. The total series impedance of this entire line would be Z = 2.1 + j 16.05Ω , so
the impedance per kilometer would be
Z = ( 2.1 + j 16.05 Ω ) / 50 km( ) = 0.042 + j 0.321 Ω/km
The shunt capacitance per meter of this transmission line is given by Equation (9-41).
ln
c D
r
( )
12 11
8.854 10 F/m 1.21 10 F/m
ln
c
− −
×
= = ×
Therefore the capacitance per kilometer will be
8 c 1.21 10 F/km
− = ×
The shunt admittance of this transmission line per kilometer will be
( ) ( )
8 6
y sh j 2 π fc j 2 π 50 Hz 1.21 10 F/km j 3.80 10 S/km
− − = = × = ×
Therefore the total shunt admittance will be
( ) (^ )
6 4 Ysh j 3.80 10 S/km 50 km j 1.90 10 S
− − = × = ×
9-9. The underground cable is operating with the receiving side of the line open-circuited. The sending end
voltage is 8 kV at 50 Hz. How much charging current is flowing in the line? How does this charging
current in the cable compare to the charging current of the overhead transmission line?
SOLUTION Although this line is in the “short” range of lengths, we will treat it as a medium-length line,
because we must include the capacitances if we wish to calculate charging currents. The appropriate
transmission line model is shown below.
The charging current can be calculated by open-circuiting the output of the transmission line and
calculating I (^) S :
S S S
Y
Z
Y
V V
I
( ) (^ )
4
4
1.90 10 S 8000 0 V 8000 0 V
1.90 10 S / 2
S
j
j j
−
−
× ∠ ° ∠ °
×
I
I S = 0.760 ∠ 90 ° A + 0.762 ∠ 90 ° A = 1.522 ∠ 90 °A
Since the shunt admittance of the underground cable is more than twice as large as shunt admittance of
the overhead transmission line, the charging current of the underground cable is more than twice as large.
9-10. The underground cable is now supplying 8 kV to an 800 kVA, 0.9 PF lagging single-phase load.
(a) What is the sending end voltage and current of this transmission line?
(b) What is the efficiency of the transmission line under these conditions?
(c) What is the voltage regulation of the transmission line under these conditions?
SOLUTION At 50 km length, we can treat this transmission line as a “short” line and ignore the effects of
the shunt admittance. (Note, however, that this assumption is not as good as it was for the overhead
transmission line. The higher shunt admittance makes its effects harder to ignore.) The transmission line
is supplying a voltage of 8 kV at the load, so the magnitude of the current flowing to the load is
800 kVA 100 A 8 kV
S
I
V
(a) If we assume that the voltage at the load is arbitrarily assigned to be at 0° phase, and the power
factor of the load is 0.9 lagging, the phasor current flowing to the load is I = 100 ∠ − 25.8 °A. The
voltage at the sending end of the transmission line is then
V S = V R + Z I
V S = 8000 ∠ ° 0 V + ( 2.1 + j 16.05 Ω ) ( 100 ∠ − 25.8 °A)
V S = 8990 ∠8.7 °V
(b) The complex output power from the transmission line is
S OUT (^) = V I R = 8000 ∠ ° 0 V 100 ∠ − 25.8 ° A = 800,000 ∠25.8 °VA
Therefore the output power is
P OUT (^) = 800 cos 25.8 ° =720 kW
The complex input power to the transmission line is
4 C 3.286 10 90 S
− = × ∠ °
( 10.3^ 52.5^ ) ( 0.00033 S)
ZY^ j^ j D
(d) The phasor diagram is shown below:
(e) The rated line voltage is 138 kV, so the rated phase voltage is 138 kV / 3 = 79.67 kV, and the
rated current is
out 200,000,000 VA^ 837 A 3 3 138,000 V
L LL
S
I
V
If the phase voltage at the receiving end is assumed to be at a phase angle of 0°, then the phase voltage at
the receiving end will be V R (^) = 79.67 ∠ ° 0 kV, and the phase current at the receiving end will be
I (^) R = 837 ∠ − 25.8 °A. The current and voltage at the sending end of the transmission line are given by
the following equations:
V S = A V R + B I R
V S = ( 0.9913 ∠0.1 °) ( 79.67 ∠ ° 0 kV) + ( 53.5∠ 78.9 ° Ω ) ( 837 ∠ − 25.8 °A)
V S = 111.8 ∠18.75 °kV
I S = C V R + D I R
( ) (^ )^ (^ ) (^ )
4 S^ 3.286^10 90 S^ 79.67^0 kV^ 0.9913^ 0.1^837 25.8^ A
− I = × ∠ ° ∠ ° + ∠ ° ∠ − °
I S = 818.7 ∠ − 24.05 °A
(f) The voltage regulation of the transmission line is
111.8 kV 79.67 kV VR 100% 100% 40.3% 79.67 kV
S R
R
V V
V
= × = × =
(g) The output power from the transmission line is
P OUT = S PF = ( 200 MVA ) ( 0.9) =180 MW
The input power to the transmission line is
P IN = 3 V φ , S I φ, S cos θ= 3 111.8 kV ( ) ( 818.7 A cos 42.8) ( ° =) 201.5 MW
The resulting efficiency is
IN
OUT
180 kW 100% 100% 89.3% 201.5 kW
P
P
η = × = × =
9-12. If the series resistance and shunt admittance of the transmission line in Problem 9-11 are ignored, what
would the value of the angle δ be at rated conditions and 0.90 PF lagging?
SOLUTION If the series resistance and shunt admittance are ignored, then the sending end voltage of the
transmission line would be
V S (^) = V R (^) + jX I R
V S = 79.67 ∠ ° 0 kV + (^) ( j 52.5 Ω) ( 837 ∠ − 25.8 °A)
V S = 106.4 ∠ 21.8 °kV
Since δ is the angle between V S and V R , δ = 21.8°.
9-13. If the series resistance and shunt admittance of the transmission line in Problem 9-11 are not ignored,
what would the value of the angle δ be at rated conditions and 0.90 PF lagging?
SOLUTION From Problem 9-11, V S = 111.8 ∠18.75 °kV and V R = 79.67 ∠ ° 0 kV, so the angle δ is
18.75° when the series resistance and shunt admittance are also considered.
9-14. Assume that the transmission line of Problem 9-11 is to supply a load at 0.90 PF lagging with no more
than a 5% voltage drop and a torque angle δ ≤ 30 °. Treat the line as a medium-length transmission line.
What is the maximum power that this transmission line can supply without violating one of these
constraints? Which constraint is violated first?
SOLUTION A MATLAB program to calculate the voltage regulation and angle δ as a function of the
power supplied to a load an a power factor of 0.9 lagging is shown below.
% M-file: prob9_13.m
% M-file to calculate the voltage drop and angle delta
% for a transmission line as load is increased.
% First, initialize the values needed in this program.
v_r = p2r(79670,0); % Receiving end voltage
v_s = 0; % Sending end voltage (will calculate)
r = 0.103; % Resistance in ohms/km
x = 0.525; % Reactance in ohms/km
y = 3.3e-6; % Shunt admittance in S/km
l = 100; % Line length (k)
% Calculate series impedance and shunt admittance
Z = (r + j*x) * l;
Y = y * l;
% Calculate ABCD constants
A = Y*Z/2 + 1;
B = Z;
C = Y(YZ/4 + 1);
D = Y*Z/2 + 1;
% Calculate the transmitted power for various currents
% assuming a power factor of 0.9 lagging (-25.8 degrees).
% This calculation uses the equation for complex power
% P = 3 * v_r * conj(i_r).
i_r = (0:10:300) * p2r(1,-25.8);
power = real( 3 * v_r * conj(i_r));
% Calculate sending end voltage and current for the
% various loads
v_s = A * v_r + B * i_r;
i_s = C * v_r + D * i_r;
9-15. The transmission line of Problem 9-11 is connected between two infinite busses, as shown in Figure P9-1.
Answer the following questions about this transmission line.
Infinite Bus 1 Infinite Bus 2
Figure P9-1 A three-phase transmission line connecting two infinite busses together.
(a) If the per-phase (line-to-neutral) voltage on the sending infinite bus is 80∠ 10 ° kV and the per-phase
voltage on the receiving infinite bus is 76∠ 0 ° kV, how much real and reactive power are being
supplied by the transmission line to the receiving bus?
(b) If the per-phase voltage on the sending infinite bus is changed to 82∠ 10 ° kV, how much real and
reactive power are being supplied by the transmission line to the receiving bus? Which changed
more, the real or the reactive power supplied to the load?
(c) If the per-phase voltage on the sending infinite bus is changed to 80∠ 15 ° kV, how much real and
reactive power are being supplied by the transmission line to the receiving bus? Compared to the
conditions in part (a) , which changed more, the real or the reactive power supplied to the load?
(d) From the above results, how could real power flow be controlled in a transmission line? How could
reactive power flow be controlled in a transmission line?
SOLUTION
(a) If the shunt admittance of the transmission line is ignored, the relationship between the voltages and
currents on this transmission line is
V S (^) = V R (^) + R I + jX I
where I (^) S = I (^) R = I. Therefore we can calculate the current in the transmission line as
S R
R jX
V V
I
265 0.5 A
10.3 j 52.
I
The real and reactive power supplied by this transmission line is
P = 3 V (^) φ , RI φcos θ= 3 76 kV ( ) ( 265 A cos 0.5) ( ° =) 60.4 MW
Q = 3 V (^) φ (^) , RI φsin θ= 3 76 kV ( ) ( 265 A sin 0.5) ( ° =) 0.53 MVAR
(b) If the sending end voltage is changed to 82∠ 10 ° kV, the current is
82,000 10 76,000 0 280 7.7 A 10.3 j 52.
I
The real and reactive power supplied by this transmission line is
P = 3 V (^) φ , RI φcos θ= 3 76 kV ( (^) ) ( 280 A cos 7.7) ( ° =) 63.3 MW
Q = 3 V (^) φ (^) , RI φsin θ= 3 76 kV ( ) ( 280 A sin 7.7) ( ° =) 8.56 MVAR
In this case, there was a relatively small change in P (3 MW) and a relatively large change in Q (
MVAR) supplied to the receiving bus.
(c) If the sending end voltage is changed to 82∠ 10 ° kV, the current is
80,000 15 76,000 0 388 7.2 A 10.3 j 52.
I
The real and reactive power supplied by this transmission line is
P = 3 V (^) φ , RI φcos θ= 3 76 kV ( (^) ) ( 388 A cos) ( −7.2 ° =) 87.8 MW
Q = 3 V (^) φ (^) , RI φsin θ= 3 76 kV ( ) ( 388 A sin) ( −7.2 ° = −) 11 MVAR
In this case, there was a relatively large change in P (27.4 MW) and a relatively small change in Q (11.
MVAR) supplied to the receiving bus.
(d) From the above results, we can see that real power flow can be adjusted by changing the phase angle
between the two voltages at the two ends of the transmission line, while reactive power flow can be
changed by changing the relative magnitude of the two voltages on either side of the transmission line.
9-16. A 50 Hz three phase transmission line is 300 km long. It has a total series impedance of 23 + j 75 Ω and
a shunt admittance of j 500 μS. It delivers 150 MW at 220 kV, with a power factor of 0.88 lagging.
Find the voltage at the sending end using (a) the short line approximation. (b) The medium-length line
4 C 4.953 10 90.2 S
− = × ∠ °
( 23 75 ) ( 0.0005 S) 1 1 0.9813 0. 2 2
ZY^ j^ j D
The receiving end line voltage is 220 kV, so the rated phase voltage is 220 kV / 3 = 127 kV, and the
current is
( (^) )
out 150,000,000 W^ 394 A 3 3 220,000 V
L LL
S
I
V
If the phase voltage at the receiving end is assumed to be at a phase angle of 0°, then the phase voltage at
the receiving end will be V R (^) = 127 ∠ ° 0 kV, and the phase current at the receiving end will be
I (^) R = 394 ∠ − 28.4 °A. The current and voltage at the sending end of the transmission line are given by
the following equations:
V S = A V R + B I R
V S = 148.4 ∠8.7 °kV
I S = C V R + D I R
I S = 361 ∠ − 19.2 °A
(c) In the long transmission line, the ABCD constants are based on modified impedances and
admittances:
sinh d Z Z d
tanh (^) ( / 2)
d Y Y d
and the corresponding ABCD constants are
Z Y
A B Z
Z Y Z Y
C Y D
§ ′ ′^ · ′ ′
The propagation constant of this transmission line is γ = yz
6 500 10 S 23 75 0.00066 81. 300 km 300 km
j j
γ yz
− § × ·§ + Ω· = = (^) ¨ ¸= ∠ ° ¨ ¸ © ¹©^ ¹
γ d = (^) ( 0.00066 ∠81.5 °) ( 300 km) = 0.198∠ 81.5 °
The modified parameters are
( )
sinh sinh 0.198( 81.5) 23 75 77.9 73 0.198 81.
d Z Z j d
tanh (^) ( / 2) (^4) 5.01 10 89.9 S / 2
d Y Y d
− ′ = = × ∠ °
and the ABCD constants are
Z Y
A
B = Z ′= 77.9 ∠ 73 ° Ω
1 4.97 90.1 S
Z Y
C Y
§ ′^ ′ ·
Z Y
D
The receiving end line voltage is 220 kV, so the rated phase voltage is 220 kV / 3 = 127 kV, and the
current is
out 150,000,000 W^ 394 A 3 3 220,000 V
L LL
S
I
V
If the phase voltage at the receiving end is assumed to be at a phase angle of 0°, then the phase voltage at
the receiving end will be V R (^) = 127 ∠ ° 0 kV, and the phase current at the receiving end will be
I (^) R = 394 ∠ − 28.4 °A. The current and voltage at the sending end of the transmission line are given by
the following equations:
V S = A V R + B I R
V S = 148.2 ∠8.7 °kV
I S = C V R + D I R
I S = 361.2 ∠ − 19.2 °A
The short transmission line approximate was rather inaccurate, but the medium and long line models were
both in good agreement with each other.
9-17. A 60 Hz, three phase, 110 kV transmission line has a length of 100 miles and a series impedance of
0.20 + j 0.85Ω/mile and a shunt admittance of
6 6 10
− × S/mile. The transmission line is supplying 60
MW at a power factor of 0.85 lagging, and the receiving end voltage is 110 kV.
(a) What are the voltage, current, and power factor at the receiving end of this line?
(b) What are the voltage, current, and power factor at the sending end of this line?
(c) How much power is being lost in this transmission line?
(d) What is the current angle δ of this transmission line? How close is the transmission line to its
steady-state stability limit?
SOLUTION This transmission line may be considered to be a medium-line line. The impedance Z and
admittance Y of this line are:
Z = ( 0.20 + j 0.85 Ω/mile ) (100 miles ) = 20 + j 85 Ω
6 Y j 6 10 /mile 100 miles j 0.0006 S
− = × =
The ABCD constants for this line are:
( 20 85 ) ( 0.0006 S)
ZY j j A
B = Z = 20 + j 85 Ω = 87.3∠ 76.8 ° Ω
( 20 85 ) ( 0.0006 S)
1 0.0006 S 1 0.00059 90.2 S
ZY^ j^ j C Y j
§ · ª^ +^ Ω º
