Electromagnetic induction
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Mutual coupling between two coils |
When two coils are placed close to each other (as shown in figure) with an AC source connected to one of the coils, electrical energy is transferred to the second coil through electromagnetic induction.
Magnetic flux produced by the first coil (primary coil) links with the second coil (secondary coil), thereby energy transfer from the primary side to the secondary side of the transformer.
To enhance the flux linkage between the two coils, a transformer will generally have a core made of high permeability magnetic material, onto which the two coils will be wound as shown in the figure below:-
For a sinusoidal supply voltage, the primary supply current (magnetizing current Im) is fundamentally a sinusoidal quantity given by:-Im = Imax . sin(ωt)
where, ω = 2𝜋f radian/s ; f is the supply frequency in Hz
N1 = Number of turns in primary coil
In primary coil a sinusoidal voltage source is supplied and a sinusoidal current Im (magnetizing current) flows in the primary winding. This current produces a MMF=N1 .Im in the primary coil that in turn produces a magnetic flux in the core with reluctance S, given by:
Ⲫ = N1 .Im / S
The flux created by this sinusoidal magnetizing current (Im) is also a sinusoidal quantity and is given by:-
Ⲫ = Ⲫm .sin(ωt)
Since this flux is time varying, according to Faraday's law of electromagnetic induction, a self-induced EMF is produce in the primary coil itself given by:-
e1 = -N1 . dⲪ/dt
e1 = -N1 . d{Ⲫm .sin(ωt)} / dt
e1 = -N1 .ω. Ⲫm .cos(ωt)
e1 = -N1 .ω . Ⲫm .sin(90° - ωt)
e1 = N1 .ω . Ⲫm .sin(ωt - 90°)
This equation shows that the self-induced EMF in primary coil is also a sinusoidal signal and the peak value of this equation is given by:-
Peak value of e1, Em1 = N1 .ω . Ⲫm = 2𝜋f . N1 . Ⲫm
RMS value of the self-induced EMF in primary coil is given by:-
E1 = Em1 / 2^(1/2) = 2^(1/2) 𝜋. f . N1 . Ⲫm
E1 = 4.44 f N1 . Ⲫm
The same flux after passing through the core, links with the secondary coil having N2 number of turns, and according to Faraday's law of electromagnetic induction, induces a mutually induced EMF in the secondary coil given by:-
Proceeding in the same manner, final expression for the instantaneous value of EMF induced in the secondary coil is given by:-
e2 = N2 .ω . Ⲫm .sin(ωt - 90°)
The secondary induced EMF is also sinusoidal. Thus, electrical energy is first converted to magnetic energy in the core and then back to electrical energy in the secondary coil. The secondary coil can now drive an electrical load connected across its terminals.
Peak value of e2, Em2 = N2 .ω. Ⲫm = 2𝜋f . N2 . Ⲫm
RMS value of the induced EMF in secondary coil is given by:-
E2 = Em2 / 2^(1/2) = 2^(1/2) 𝜋. f . N2 . Ⲫm
E2 = 4.44 f N2 . Ⲫm
Ratio of the primary to secondary voltage is given by
E1 / E2 = 4.44 f N1 . Ⲫm / 4.44 f N2 . Ⲫm
E1 / E2 = N1 / N2 = a = turns ratio of the transformer
The voltage ratio of a transformer is thus in direct proportion to the turns ratio.
When I1 and I2 are the input and output currents respectively, then,
Input power = E1 I1 ; Output power = E2 I2
In an ideal transformer, with no internal power losses, the input and output powers can be assumed to be equal. Thus, E1 I1 = E2I2
E1 / E2 = I2 / I1 = N1 / N2 = a
The current ratio of a transformer is thus in inverse proportion to the turns ratio
By properly selecting relative number of turns in primary and secondary windings, a transformer can be used either to raise the voltage or reduce the voltage.
Step-up.....................E2 > E1....................N2 > N1
Step-down................E2 < E1....................N2 < N1
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