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72 Power systems engineering ± fundamental concepts

2.11.5 Providing a path for zero-sequence currents

It is generally essential in three-phase transformers to provide a path for zero-
sequence current. A delta winding is used for this purpose. Zero-sequence currents
can flow in the delta without magnetically short-circuiting the entire core.
The delta winding can be either the primary or the secondary, or it can be a tertiary
winding provided specially for the purpose of mopping up residual current (especially
when the primary and secondary are wye-connected). Tertiary windings have other
uses: for example, they may be used to connect local loads, power factor correction
capacitors, or static compensators. The tertiary winding must be designed for the full
fault level at that point in the transmission system, but its continuous thermal rating
is usually less than those of the primary and secondary.

2.12 Harmonics
Ideally the voltage and current in an AC power system are purely sinusoidal. When
the waveform is distorted, it can be analysed (by Fourier's theorem) into components at
the fundamental frequency and multiples thereof. Frequency components other than the
fundamental are called harmonics. The main origins of harmonics are as follows:
(a) non-linear magnetic elements, such as saturated transformer cores
(b) non-sinusoidal airgap flux distribution in rotating AC machines and
(c) switched circuit elements, such as rectifiers, triacs, and other power-electronic
converters.
The main undesirable effects can be summarized as follows:
(i) additional heating of cables, transformers, motors etc.
(ii) interference to communications and other electrical/electronic circuits
(iii) electrical resonance, resulting in potentially dangerous voltages and currents and
(iv) electromechanical resonance, producing vibration, noise, and fatigue failure of
mechanical components.
Fourier's theorem provides the mathematical tool for resolving a periodic waveform
of virtually any shape into a sum of harmonic components: thus an arbitrary periodic
voltage waveform u(t) is written
1
X p

u(t) ˆ u 0 ‡ 2V m cos (mot ‡ f )
m
mˆ1
(2:58)
1
X
ˆ u 0 ‡ [a m cos (mot) ‡ b m sin (mot)]
mˆ1
where u 0 is the average (DC) component. The first form expresses each harmonic in
terms of its RMS value V m and its phase f . Each harmonic is itself sinusoidal and
m
can be considered as a phasor, except that it rotates at m times the fundamental
frequency. The second form expresses each harmonic in terms of cosine and sine
coefficients a m and b m respectively. The main limitation is that the waveform u(t)
must be periodic, that is, it must repeat after a time T ˆ 1/f ˆ 2p/o, where f is the