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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(2ft) (voltage) or Ipsin(2ft) (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(2ft) 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.