Page 105 - Electric Machinery Fundamentals

TRANSFORMERS 81

Vp(t) t!..e.
-- = =a (2- 28)
vS<t) Ns
The smaller the leakage fluxes of the transformer are, the closer the total transformer
voltage ratio approximates that of the ideal transfonner discussed in Section 2.3.

The Magnetization Current in a Real Transformer
When an ac power source is connected to a transfonner as shown in Figure 2-8, a
current flows in its primary circuit, even when the secondary circuit is open~
circuited. This current is the current required to produce flux in a real ferromag-
netic core, as explained in Chapter 1. It consists of two components:
1. The magnetization current i , which is the current required to produce the
M
flux in the transformer core, and
2. The core-loss current i + > which is the current required to make up for hys-
h e
( teresis and eddy current losses in the core.
Figure 2-11 shows the magnetization curve of a typical transformer core. If
the flux in the transformer core is known, then the magnitude of the magnetization
current can be found directly from Figure 2- 1l.
Ignoring for the moment the effects of leakage flux. we see that the average
flux in the core is given by
I J
- </> p = Np v p(t)dt (2- 18)

If the primary voltage is given by the expression vp(t) = V cos wI V, then the re-
M
sulting flux must be
I J
- </>p = N V cos wtdt
p M
V M .
= --sm wi Wb (2-29)
wNp
If the values of current required to produce a given flux (Figure 2- 1 la) are com-
pared to the flux in the core at different times, it is possible to construct a sketch
of the magnetization current in the winding on the core. Such a sketch is shown in
Figure 2-11b. Notice the following points about the magnetization current:
1. The magnetization current in the transformer is not sinusoidal. The higher-
frequency components in the magnetization current are due to magnetic
saturation in the transfonner core.
2. Once the peak flux reaches the satlll'ation point in the core, a small increase
in peak flux requires a very large increase in the peak magnetization current.
3. The fundamental component of the magnetization current lags the voltage ap-
plied to the core by 90°.
4. The higher-frequency components in the magnetization current can be quite large
compared to the fundamental component. In general, the fmther a transfonuer
core is driven into saturation, the larger the harmonic components will become.