0195136047.pdf

1.2 LUMPED-CIRCUIT ELEMENTS 37

Mutual flux

Primarywinding Load

(N 1 turns)

Secondary

winding

(N 2 turns)

Ferromagnetic core

of infinite permeability

Resistive load

with resistance R 2

+

−

+

−

+

−

e 1 = v 1

e 2 = v 2

i 1

v 1

i 2

φ

Figure 1.2.12Elementary model of a two-winding core-type transformer (ideal transformer).

Anideal transformeris one that has no losses (associated with iron or copper) and no
leakage fluxes (i.e., all the flux in the core links both the primary and the secondary windings).
The winding resistances are negligible. While these properties are never actually achieved in
practical transformers, they are, however, approached closely. When a time-varying voltagev 1
is applied to theN 1 -turn primary winding (assumed to have zero resistance), a core fluxφis
established and a counter emfe 1 with the polarity shown in Figure 1.2.12 is developed such that
e 1 is equal tov 1. Because there is no leakage flux with an ideal transformer, the fluxφalso links
allN 2 turns of the secondary winding and produces an induced emfe 2 , according to Faraday’s
law of induction. Sincev 1 =e 1 =dλ 1 /dt=N 1 dφ/dtandv 2 =e 2 =dλ 2 /dt=N 2 dφ/dt,it
follows from Figure 1.2.12 that

v 1

v 2

=

e 1

e 2

=

N 1

N 2

=a (1.2.53)

whereais theturns ratio. Thus, in an ideal transformer, voltages are transformed in the direct
ratio of the turns. For the case of an ideal transformer, since the instantaneous power input equals
the instantaneous power output, it follows that

v 1 i 1 =v 2 i 2 or

i 1

i 2

=

v 2

v 1

=

N 2

N 1

=

1

a

(1.2.54)

which implies that currents are transformed in the inverse ratio of the turns.
Equivalent circuits viewed from the source terminals, when the transformer is ideal, are
shown in Figure 1.2.13. As seen from Figure 1.2.13(a), sincev 1 =(N 1 /N 2 )v 2 ,i 1 =(N 2 /N 1 )i 2 ,
andv 2 =i 2 RL, it follows that

v 1

i 1

=

(

N 1

N 2

) 2

RL=a^2 RL=RL′ (1.2.55)

whereRL′ is the secondary-load resistancereferred tothe primary side. The consequence of
Equation (1.2.55) is that a resistanceRLin the secondary circuit can be replaced by anequivalent
resistanceRL′in the primary circuit in so far as the effect at the source terminals is concerned. The
reflected resistance through a transformer can be very useful inresistance matchingformaximum
power transfer, as we shall see in the following example. Note that the circuits shown in Figure
1.2.13 are indistinguishable viewed from the source terminals.