Problem 2.12
A high-voltage direct-current generating station delivers 10 MW of power at 250 kV to a city, as depicted in Fig. P2.12. The city is represented by resistance R
L and each of the two wires of the transmission line between the generating station and the city is represented by resistance R
TL
. The distance between the two locations is 2000 km and the transmission lines are made of 10-cm–diameter copper wire. Determine (a) how much power is consumed by the transmission line and (b) what fraction of the power generated by the generating station is used by the city.
R
TL
V
0
+
_ R
L
(city)
R
TL
Station
2000 km
Figure P2.12: Diagram for Problem 2.12.
Solution: For copper,
ρ
= 1 .
72 × 10
− 8 Ω
-m .
For each wire of the transmission line, Eq. (2.2) leads to
R
TL
=
ρ ℓ
A
=
1 .
72 × 10
− 8 × 2 ×
π
( 5 × 10 − 2 ) 2
10
6
= 4 .
4
Ω
.
Delivering 10 MW at 250 kV to the city means that
P
L
=
V
2
L
R
L
= 10
7
W, with V
L
= 2 .
5 × 10
5
V
Hence,
R
L
=
V
2
L =
( 2
P
L
Also, the current flowing through R
L
.
5 ×
10 is
10
7
5 ) 2
= 6 .
25 × 10
3 Ω
.
.
I =
V
L
R
L
=
2 .
5 × 10
5
6 .
25 × 10 3
= 40 A .
(a) The power consumed by transmission lines is
P
TL
= 2I
2
R
TL
= 2 × ( 40 )
2
× 4 .
4 = 14080 W .
Total power generated by the source is
P s
= P
TL
+ P
L
= 10
7
+ 14080 = 1 .
001408 MW , and the fraction used by the city is
Fraction =
P
L
P s
=
10
7
1 .
001408 × 10 7
= 0 .
9986
≈ 99 .
86% .
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