11.3 TRANSFORMER EQUIVALENT CIRCUITS 481in whichI 1 andI 2 are the rms value ofi 1 andi 2 , respectively.
If the flux varies sinusoidally, such that
φ=φmaxsinωt (11.3.4)
whereφmaxis the maximum value of the flux andω= 2 πf,fbeing the frequency, then the
induced voltages are given by
e 1 =ωN 1 φmaxcosωt; e 2 =ωN 2 φmaxcosωt (11.3.5)Note that the induced emf leads the flux by 90°. The rms values of the induced emf’s are given by
E 1 =ωN 1 φmax
√
2= 4. 44 fN 1 φmax; E 2 =ωN 2 φmax
√
2= 4. 44 fN 2 φmax (11.3.6)Equation (11.3.6) is known as theemf equation.
From Equations (11.3.2) and (11.3.3) it can be shown that if an impedanceZ ̄Lis connected
to the secondary, the impedanceZ ̄ 1 seen at the primary is given by
Z ̄ 1 =a^2 Z ̄L=(
N 1
N 2) 2
Z ̄L (11.3.7)Major applications of transformers are in voltage, current, and impedance transformations,
and in providing isolation while eliminating direct connections between electric circuits.
Anonideal(orpractical) transformer, in contrast to an ideal transformer, has the following
characteristics, which have to be accounted for:
- Core losses (hysteresis and eddy-current losses)
- Resistive(I^2 R)losses in its primary and secondary windings
- Finite permeability of the core requiring a finite mmf for its magnetization
- Leakage fluxes (associated with the primary and secondary windings) that do not link both
windings simultaneously.
Now our goal is to develop anequivalent circuitof a practical transformer by including the
nonideal effects. First, let us consider the simple magnetic circuit of Figure 11.2.1(a), excited by
an ac mmf, and come up with its equivalent circuit. With no coil resistance and no core loss,
but with a finite constant permeability of the core, the magnetic circuit along with the coil can
be represented just by an inductanceLmor, equivalently, by an inductive reactanceXm=ωLm,
when the coil is excited by a sinusoidal ac voltage of frequencyf =ω/ 2 π. This reactance is
known as themagnetizing reactance.Thus, Figure 11.3.2(a) showsXm(or impedanceZ ̄=jXm)
across which the terminal voltage with an rms value ofV 1 (equal to the induced voltageE 1 )
is applied.
Next, in order to include the core losses, since these depend directly upon the level of flux
density and hence the voltageV 1 , a resistanceRCis added in parallel tojXm, as shown in Figure
11.3.2(b). Then the resistanceR 1 of the coil itself and theleakage reactanceX 1 , representing
the effect of leakage flux associated with the coil, are included in Figure 11.3.2(c) as a series
impedance given byR 1 +jX 1.
Finally, Figure 11.3.3 shows the equivalent circuit of a nonideal transformer as a combination
of an ideal transformer and of the nonideal effects of the primary winding, the core, and the
secondary winding. Note that the effects of distributed capacitances across and between the
windings are neglected here. The following notation is used: