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S 800 kVA = = 100 A V 8 kV
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(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 09 lagging, the phasor current flowing to the load is I = 100 258 A The voltage at the sending end of the transmission line is then
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VS = VR + ZI VS = 8000 0 V + ( 21 + j 323 )(100 258 A ) VS = 10000 164 V
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(b) The complex output power from the transmission line is
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SOUT = VR I* = ( 8000 0 V )(100 258 A ) = 800,000 258 VA
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Therefore the output power is
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POUT = 800 cos 258 = 720 kW
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The complex input power to the transmission line is
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S IN = VS I* = (10000 164 V )(100 258 A ) = 1,000,000 422 VA
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PIN = 1000 cos 422 = 741 kW
The resulting efficiency is
POUT 720 kW 100% = 100% = 972% PIN 741 kW
The voltage regulation of the transmission line is
VR =
VS VR 10000 8000 100% = 100% = 25% VR 8000
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 D + ln H/m 4 r
(9-22)
where = 0 = 4 10 7 H/m
0 1 015 m 4 10 7 H/m 1 015 m 6 l= + ln = + ln = 1021 10 H/m 4 4 0015 m 0015 m
Therefore the inductance of this transmission line will be
L = 1021 10 6 H/m ( 50,000 m ) = 00511 H X = j L = j 2 fL = j 2 ( 50 Hz )( 00511 H ) = j1605
The inductive reactance of this transmission line is
The resistance of this transmission line is the same as for the overhead transmission line calculated previously: RAC = 21 The total series impedance of this entire line would be Z = 21 + j1605 , so the impedance per kilometer would be
Z = ( 21 + j1605 ) / ( 50 km ) = 0042 + j 0321 /km c=
The shunt capacitance per meter of this transmission line is given by Equation (9-41)
( 8854 10 12 F/m )
015 ln 0015
D ln r
(9-41)
= 121 10 11 F/m
Therefore the capacitance per kilometer will be The shunt admittance of this transmission line per kilometer will be
c = 121 10 8 F/km
y sh = j 2 fc = j 2 ( 50 Hz ) 121 10 8 F/km = j 380 10 6 S/km
Therefore the total shunt admittance will be
Ysh = j 380 10 6 S/km ( 50 km ) = j190 10 4 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 181
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 :
IS =
IS =
YVS VS + 1 2 Z+ Y /2 j190 10 4 S ( 8000 0 V )
8000 0 V
( 21
+ j 323 ) +
1 j190 10 4 S / 2
I S = 0760 90 A + 0762 90 A = 1522 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, 09 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
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