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Power electronic control in electrical systems 81
inverse of the turns ratio, 1/n. Therefore the ratio of base impedances must be the
2
square of the turns ratio, n . But this is precisely the ratio by which an impedance is
referred from the secondary to the primary. Therefore, if we normalize an impedance
to the base on one side of the transformer, and then refer this per-unit impedance to
the other side, the per-unit value comes out exactly the same. This means that in a
consistent per-unit system, ideal transformers simply disappear. Mathematically, this
can be expressed as follows
2
Z b1 n Z b2 (2:80)
On the secondary base, a load impedance Z (ohms) on the secondary side has the per-
unit value
Z
z p:u: (2:81)
Z b2
2
If we refer Z to the primary it becomes Z n Z. The per-unit value of this on the
0
primary base, is
2
Z 0 n Z
0
z 2 z (2:82)
Z b1 n Z b2
This says that the per-unit value of an impedance is the same on both sides of the
transformer. In other words, in per-unit the turns ratio of the transformer is unity
and it can be removed from the circuit. This is only true if the primary and secondary
2
base impedances are in the ratio n .
These comments apply to ideal transformers only. But real transformers can be
modelled by an ideal transformer together with parasitic impedances (resistances,
leakage reactances etc.) that can be lumped together with the other circuit impedances
on either side. The equivalent circuit of a transformer in per-unit is just a series
impedance equal to r jx, where r is the sum of the per-unit primary and secondary
resistances and x is the sum of the per-unit primary and secondary leakage reac-
tances. The magnetizing branch appears as a shunt impedance. Wye/delta transform-
ers have a more complex representation but still the ideal transformer disappears
from the equivalent circuit. Similarly, tap-changing transformers can be represented
by a simple network.
2.14 Conclusion
This chapter has laid the basic technical foundation for the study of reactive power
control in power systems, with most of the analytical theory required for calculation
of simple AC circuits including three-phase circuits and circuits with reactive com-
pensators. Power-factor correction and the adjustment of voltage by means of
reactive power control have been explained using phasor diagrams and associated
circuit equations. The basic theory of transformers, harmonics, and per-unit systems
has also been covered.
In the next chapter the simple analytical theory of reactive power control is
extended to transmission systems which are long enough to be considered as dis-
tributed-parameter circuits.
