Power system fundamentals - Wind Power Integration: Connection and System Operational Aspects - page 27

The voltages are related as follows:

v 2
v 1 ¼

N 2
N 1

ð 2 : 12 Þ

It may be seen that the voltage is transformed in proportion to the turns ratio.
Note that the transformer's operation depends on flux variation, as Faraday
observed. In particular, an alternating primary voltage will ensure a varying flux,
and hence an alternating secondary voltage in phase with it.
The relation between the currents may be derived by considering the magnetic
circuit formed by the iron core. The m.m.f. acting on the core is given by

l m a

F ¼ N 1 i 1 N 2 i 2 ¼ S F ¼

The sign of the secondary current reflects the fact that it opposes the assumed
flux direction. l is the core length, a its cross-sectional area and m its permeability.
Assuming that the core permeability is infinite, the m.m.f. F must be zero for a
finite flux. Hence

i 2
i 1 ¼

N 1
N 2

ð 2 : 13 Þ

The power output from the secondary is given by

v 1 i 1 ¼ v 1 i 1 ¼ p in

N 2
N 1

N 1
N 2

p out ¼ v 2 i 2 ¼

The power delivered by the transformer equals the power supplied. This is
unsurprising, given that we assumed an ideal core with no leakage, and took no
account of copper losses in the windings.
A real transformer differs from the ideal in the following respects:

Leakage flux - some primary flux does not link the secondary and vice versa . ●

The windings have resistance, leading to a power loss under load. ●

The core permeability is finite; hence a magnetising current is drawn. ●

There are iron losses in the core due to magnetic hysteresis and eddy currents. ●

The most significant of these effects in the system context is flux leakage. The
leakage flux links one coil only, and therefore manifests itself as an inductance in
series with each coil, known as leakage inductance - see L 1 , L 2 in Figure 2.3. The
winding resistances ( R 1 , R 2 ) are easily included in series with the leakage induc-
tances. These model transformer copper losses, which are proportional to the
square of current. The effect of these series components is to create a full-load
series voltage drop of up to 10 per cent of rated voltage at power frequencies. The
predominance of the leakage inductance over winding resistance at power fre-
quency ensures that the drop is largely at right angles to the line to neutral voltages.
For this reason the change in the transformer voltage ratio over the loading range
is minimal.