Environmental Engineering Reference
In-Depth Information
A sine wave is the only waveform that, when differentiated or integrated, results in a replica
of itself. As will be shown later, electric power system components, including transformers,
respond differentially to a sine wave excitation and thus the shape of the waveform is main-
tained throughout the network. Additionally, three-phase electric motors are at their most
effi cient and produce a constant torque when fed from an AC supply of sine wave shape only.
A sinusoidal voltage of frequency f can be written analytically as vV
, where V is
= sin
ω
t
the peak value of the sine wave and
f .
From now onwards, whenever AC is referred to, it is implied that the waveform is of
sinusoidal shape. With AC a measurement dilemma is encountered in expressing how large
or small an AC quantity is. With DC, where quantities of voltage and current are generally
stable, there is no trouble expressing what is the value of the voltage or current in any part
of a circuit.
One way to express the magnitude (or amplitude ) of an AC quantity is to measure its peak
ω
= 2
π
value V . However, when dealing with measurements of electric power, the best way to
express the value of an AC waveform is to label it in terms of its ability to perform useful
work. The way to do this is to assign to the waveform a 'DC equivalent' measurement, i.e.
to label it by the value in volts or amperes of the DC that would have produced the same
amount of heat energy dissipation in the same resistance. The mathematical method of pro-
ducing the DC equivalent value, known as root mean square (RMS), can be found in any
elementary textbook on electricity. In this topic the RMS value of an AC voltage or current
will be denoted by an italic capital letter, for example V, I . It can be shown that for a sine
wave the RMS value is related to the peak by
(A.2)
VV
=
2
=
0 707
.
V
Thus the household voltage of 230 V has a peak of 325 V.
A.5 Response of Circuit Components to AC
Power networks consist mainly of generators, transmission lines, transformers and consum-
ers. Each one of these constituent parts consists in turn of a combination of three basic com-
ponents, namely resistance R , inductance L and capacitance C . Additionally, generators have
internally induced AC voltages caused by the interaction of moving conductors and magnetic
fi elds. These electromagnetic actions are the mechanisms by which mechanical energy, con-
ventional or renewable, is converted into electrical form. Uniquely, in photovoltaic systems
the conversion of solar power is carried out directly into electricity without the aid of
electromagnetics.
The AC source voltages are the drivers of all current fl ows in the network and therefore
partly determine the distribution of power fl ows from the generators to consumers over the
transmission network. The energy fl ows in a network consisting of traditional and/or renew-
able generators can be determined by the application of Kirchhoff's laws, but for this to be
done the 'resistance' to the fl ow of AC current offered by the three basic components must
be known. In other words, the relationship between AC voltage and current (Ohm's law) for
each component must be determined.
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