THE IDEAL TRANSFORMERFew ideal versions of human constructions
exist, and the transformer offers no exception. An idealtransformer
is based on very simple concepts, and a large number of
assumptions. This is thetransformer one learns about in high
school.Let us take an iron core with infinite permeability and two
coils wound around it (with zeroresistance), one with N1 and the
other with N2 turns, as shown in figure 3.2. All the magnetic flux
isto remain in the iron. We assign dots at one terminal of each
coil in the following fashion: if the flux in the core changes,
inducing a voltage in the coils, and the dotted terminal of one
coil is positivewith respect its other terminal, so is the dotted
terminal of the other coil. Or, the corollary to this,current into
dotted terminals produces flux in the same direction.
Fig. 3.2 Magnetic Circuit of an ideal transformerAssume that
somehow a time varying flux, (t), is established in the iron. Then
the flux linkagesin each coil will be 1 = N1(t) and 2 = N2(t).
Voltages will be induced in these two coils:
and dividing:
On the other hand, currents flowing in the coils are related to
the field intensity H. If currentsflowing in the direction shown,
i1 into the dotted terminal of coil 1, and i2 out of the dotted
terminalof coil 2, then:N1. i1(t) N2.i2(t) = H.lbut B = ironH, and
since B is finite and iron is infinite, then H = 0. We recognize
that this ispractically impossible, but so is the existence of an
ideal transformer.Finally:
Equations 3.3 and 3.5 describe this ideal transformer, a two
port network. The symbol of anetwork that is defined by these two
equations is in the figure 3.3. An ideal transformer has an
interesting characteristic. A twoportnetwork that contains it and
impedances can be replaced by anequivalent other, as discussed
below. Consider the circuit in figure 3.4a. Seen as a two port
network with variables v1, i1, v2, i2, we can write:
which could describe the circuit in figure 3.4b. Generally a
circuit on a side 1 can be transferred toside 2 by multiplying its
component impedances by (N2=N1)2, the voltage sources by (N2=N1)
andthe current sources by (N1=N2), while keeping the topology the
same.
Fig. 3.3 Symbol for an ideal transformer
Fig. 3.4 Transferring an impedance from one side to the other of
an ideal transformer
