Aylesford Co-generation Plant
The plant produces 220 MW heat and 98 MW of electrical energy [National Power PLC]
I.1 Introduction
In all public electricity supply systems, the mains voltage alternates at 50 or 60 cycles per second (Hz) and when a load is connected it draws an alternating current.
An alternating current or voltage changes its polarity periodically to give a wave-like shape on an oscilloscope trace. Sine, square and even saw-tooth waves are used in different electronic circuits, but this chapter considers only the sine wave, which is the shape of the AC mains.
A generator produces a three-phase alternating voltage and this voltage is increased (stepped-up) for long distance transmission (Figure 1.1, shown on page 2). The transmission voltages are then stepped-down and the power dis-tributed to loads. Low voltage final distribution circuits typically use four wires (in Europe, three phase wires each at 230 V and a neutral wire that provides the zero volt reference). Single-phase loads (e.g. houses) are connected across two wires (to one phase wire and the neutral wire) and three-phase loads (industry and com-mercial buildings) are connected to all four wires.
I.2 Alternating current (AC)
A periodic voltage or current waveform is described by its:
1. Period (T)
The time taken for one cycle of voltage, or current, is known as the period, with the symbol T and is measured in seconds. The period can be measured from any point on one cycle to the corresponding point on the next (Figure I.1).
2. Frequency (f )
The number of cycles that a waveform completes in 1s is its frequency. Fre-quency has the symbol f and is measured in Hertz (Hz). Period and freFre-quency are related by: f ¼ 1=T Hz; i.e. a longer period results in a lower frequency, while a shorter period gives a higher frequency.
3. Peak value (Vm)
The positive or negative maximum value is known as the peak value. The peak value is measured in volts or amperes.
4. Peak-to-peak value (2Vm)
The magnitude between the negative peak and positive peak is called the peak-to-peak value.
(t) T
T
–Vm
Peak value
Vm
Peak-to-peak value
Time
Figure I.1 Sinusoidal voltage waveform and definitions
I.3 Root mean square value of voltage and current
One measure of a sinusoidal voltage or current is obtained by considering the average power dissipated in a resistor.
Applying Ohm’s law to the two circuits shown in Figure I.2:
AC current
Figure I.2 Resistor fed by DC and AC
For DC: VDC¼ IDC R. Therefore, the power dissipated as heat is: P ¼ VDC IDC¼ I2DC R.
For AC: If the current through the resistor is iðtÞ ¼ ImsinðwtÞ, then vðtÞ ¼ ImR sinðwtÞ.
The instantaneous power dissipation is: pintðtÞ ¼ vðtÞ iðtÞ ¼ I2mR sin2ðwtÞ.
The average power dissipation: Pave¼ 1=TRT
0 I2mR sin2ðwtÞdt.
Substituting sin2ðwtÞ ¼ ð1 cosð2wtÞÞ=2, the following equation is obtained:
Pave¼1
The average heating effect in both circuits is equal, if I2DC R ¼I2mR
2 IDC¼ Imffiffiffi
p2 ðI:2Þ
The value of the AC current, that gives the same heating effect as a DC current into a resistor, is termed the root mean square (rms) current and has the value Im= ffiffiffi
p2
for a sine wave. Alternating voltages and currents are usually expressed by their rms values.
I.4 Phasor representation of AC quantities
Consider a vector of length Vm rotating anti-clockwise at angular velocity w (Figure I.3). As the vector OA rotates, its projection on the y axis will describe a sinusoidal signal (Va). Similarly when the vector OB rotates, its projection on the
y axis describes Vb. The length of the vector, OA or OB, corresponds to the peak value of the sinusoidal signal (Vm) and the angle between the vectors OA and OB corresponds to the angle between any identical points in the two waveforms.
If the rotation of vectors OA and OB are atw rad/s, then the angle q (from the origin) of the sinusoidal waveform Vaat time t is equal towt rad. Therefore, the equation for the sinusoidal signal Vais vaðtÞ ¼ Vmsinq ¼ VmsinðwtÞ. Similarly the equation for the sinusoidal signal Vbis vbðtÞ ¼ Vmsinðwt þ fÞ.
From Figure I.3: The period of the signal Vais 2p=w.
If the frequency of the signal is f, then: 1=f ¼ 2p=w; i.e. w ¼ 2pf .
The vectors OA and OB which rotate at the same angular velocity as the sinu-soidal signal and which represent the magnitude and the angle of the sinusinu-soidal signals are called phasors. The angle between two phasors describes how far ahead or behind a point on one sinusoidal signal is to the same point of the other signal and is called the phase angle. If the phasors of two signals coincide with each other, then the two signals are in phase. If one is ahead of the other signal by an anglef (measured anti-clockwise) then it leads the second one by anglef. If one is behind the other signal by an anglef (measured clockwise) then it lags the second one by angle f.
Two phasors (OA and OB) which represent signals Vaand Vbcan be written in two forms as given in Table I.1. Even though the lengths of phasors OA and OB in Figure I.3 are equal to the peak value of sinusoidal signal, in phasor representation, rms values are more usually used.
Table I.1 Time and phase representation of AC quantities
Signal Time representation Phasor representation
Va Va¼ Vmsinðwt þ 0Þ Va¼Vm
ffiffiffi2 p ff0
Vb Vb¼ Vmsinðwt fÞ Vb¼Vm
ffiffiffi2 p ff f Degrees 90º
2
180º 270º
3π2
360º 2π Radians
O A
B O
Va Vb
Vm
p p
f f q
w
Figure I.3 Development of a sinusoidal waveform by revolving a phasor