Download Electrical Power generation and transmission

Electrical Power generation and transmission
1. Alternating Voltages and Currents
These are voltages and currents which repeatedly change direction. The most
usual species of them can be represented mathematically by V psin(2ft)
(voltage) or
Ipsin(2ft) (current), where f is the frequency in hertz, t is the time in seconds,
and Vp and Ip are the peak values of voltage and current respectively. Their
significance is best appreciated from the following graph.
Voltage
400
300
200
100
0
-100
-200
-300
-400
0
0.01
0.02
0.03
0.04
Time, second
0.05
0.06
0.07
0.08
This graph - actually of the normal British mains voltage of 240 volt - shows
how the voltage varies with time.
Several questions are often asked by students at this stage.
1. But my TI-85 (or commonsense - equally correctly) says that the
average value of sin(x) is zero, so how can an alternating voltage
actually do any good?
Consider the following two circuits.
+
10 V d.c.
5 ohm
-
10 V d.c.
5 ohm
+
In each of these circuits, a current of 10/5 = 2 A will flow in the 5-ohm resistor.
In the first one the resistor current will flow upwards and in the second one
downwards, but the same power will be dissipated in the resistor in each case
- 10  2 = 20 w. (We would have got the same answer by V 2/R = 100/5 = 20
w.) The power is the important thing!
2. If it is a 240-volt supply, why does the voltage go from -339 volt to
+339 volt?
The answer to this one develops from that of Question 1. It is simple to give
the "effective" value of a direct voltage! It is a bit more awkward with an
alternating one - especially with its average value being zero! The value we
use is that of the direct voltage or current which would give the same power in
the circuit. We remember (or have forgotten) that the power in a resistance is
either V2/R or I2R, so it does seem to have to do with the square of the
voltage or current. It is noteworthy that (240)2 has the same value as (-240)2
(try it on the calculator!). It proves that the average value of sin 2(x) is 0.5, so
the average value of Vp2sin2(2ft) is 0.5Vp2. The effective value of the
voltage is the square-root of this (so V2/R will still work), so it will be the
square root of 0.5Vp2 ... which is 0.707 Vp .. and 0.707  339 = (near
enough) 240!
So ... for a sinusoidal alternating voltage or current .. the effective or
root-mean-square (r.m.s.) value is 0.707 times the peak value.
3. This is all very well, but why not just use d.c. and save all this bother?
Two main reasons in respect of SINGLE-PHASE a.c.:
a) It is much easier, especially at highish voltages and currents, to generate
a.c. than d.c. - the reason why will emerge when we consider electrical
machines.
b) An alternating voltage can be easily and efficiently stepped up or down by
a device called a transformer which has no moving parts. Its operation will be
explained when we consider electro-magnetism. Direct voltages cannot be
stepped up and down in this convenient fashion.
c) More reasons will emerge when we examine three-phase a.c. !
2. Three-Phase alternating current
The general idea is illustrated by the following diagram and graph.
Red
Yellow
Generator
Load
Blue
Neutral (0 V)
Voltage from line to neutral
400
Red
Yellow
Blue
300
200
100
0
-100
-200
-300
-400
0
0.01
0.02
0.03
Time, second
0.04
0.05
0.06
The advantages gained by using three-phase rather than single-phase are
considerable.
1. A major saving on copper because, if the loads on the three phases are
equal, the returning currents in the neutral wire will add up to zero and we will
need a much smaller neutral wire – and incur lower resistive (I2R) losses
2. More power can be produced from a given size of three-phase generator
(‘alternator’) than from the same size of single-phase machine.
3. It is possible to use highly-efficient (and relatively cheap) three-phase
induction motors if a three-phase supply is available.
For those reasons, nearly all practical electrical distribution systems are threephase.
3. Generation and transmission -- Why so many voltages ?
In Britain (and many other countries) electricity is generated at up to 11 kV,
transformed up to anything between 132 kV (the original National Grid
voltage) and 400 kV for transmission, transformed back down again to 11 kV
for more local distribution, and finally transformed down again to 240 V to
enter our homes and offices (industry often takes it at 11 kV).
Two arguments:
Safety and technology limitations (suggesting a lower
voltage)
Efficiency (suggesting a higher one)
The efficiency argument is based on the power lost in the cables being I 2R, so
reducing the current I will reduce the power loss. We also know that the
power depends on voltage times current, so more voltage will allow us to
have less current for the same power. This is why the transmission voltages
in the National Grid are so high ! The lines are carried on high towers
(‘pylons’) out of harm’s way, so safety is not a problem; the highest voltage of
400 kV is because of the practical limitations of insulation and transformer
technology. This would not do in the home, where safety enforces the use of
240 V.
A few examples
1. The domestic electricity supply system in the U.S.A. is 110 volt r.m.s. at 60
Hz. Calculate:
a) The time for one cycle of the voltage.
b) The peak value of the voltage.
c) The current taken by a 100-watt light bulb (resistive).
2. Complete the following table for a voltage 339sin(314.2t).
Time, second
0
.001 .002 .003 .004 .005 .006 .007
.008 .009 (please note that these last two times should be at the end)
Voltage
0
104.8
Voltage2
0
10977
and work out the average value of the voltage2. Is its square root
approximately 240?
3. By taking values from the three-phase graph (and assuming it to be of
current), demonstrate that the three currents add up to zero at any particular
time.