EMF EQUATION OF A TRANSFORMER
The magnetic flux ϕ set up in the core of a transformer when an alternating voltage is applied to its primary winding is also alternating and is sinusoidal.
Let ϕm be the maximum value of the flux and f be the frequency of the supply. The time for 1 cycle of the alternating flux is the periodic time T, where T = (1/f) seconds
The flux rises sinusoidally from zero to its maximum value in the (1/4) cycle, and the time for the (1/4) cycle is (1/4f) seconds. Hence the average rate of change of flux = (ϕm/ (1/4f)) = 4f ϕm Wb/s, and since 1Wb/s D 1 volt, the average emf induced in each turn = 4f ϕm volts. As the flux ϕ varies sinusoidally, then a sinusoidal emf will be induced in each turn of both primary and secondary windings.
For a sine wave,
Form Factor = r.m.s Value / Average Value
Hence r.m.s. value = form factor*average value = 1.11 * average value Thus r.m.s. e.m.f. induced in
=1.11 * 4fϕm volts
Therefore, r.m.s. value of e.m.f. induced in primary,
E1 = 4.44 f ϕmN1 volts
and r.m.s. value of e.m.f. induced in secondary,
E2 = 4.44 f 8ϕN2 volts
Dividing E1 by E2
AIR CORE TRANSFORMER
Some small transformers for low-power applications are constructed with air between the two coils. Such transformers are inefficient because the percentage of the flux from the first coil that links the second coil is small. The voltage induced in the second coil is determined as follows.
where N is the number of turns in the coil, dϕ/dt is the time rate of change of flux linking the coil, and ϕ is the flux in lines.
At a time when the applied voltage to the coil is E and the flux linking the coils is ϕ lines, the
the instantaneous voltage of the supply is:
Since the amount of flux ϕ linking the second coil is a small percentage of the flux from the first coil, the voltage induced into the second coil is small. The number of turns can be increased to increase the voltage output, but this will increase costs. The need then is to increase the amount of flux from the first coil that links the second coil.
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