Explain the working principle of single phase transformer and derive the EMF equation.

Principle: mutual induction (Faraday's law). When an alternating voltage is applied to one winding, the alternating flux it produces links the other winding and induces an EMF in it.

            laminated iron core (flux phi)
        +-------------------------------+
        |   ---> phi (alternating)      |
  V1 o--|-- N1 turns            N2 turns|--o  E2 / V2 -> load
   ~    |  (primary)           (secondary)|
     o--|--                           --|--o
        +-------------------------------+

Working: the primary winding (N₁ turns) is connected to an AC supply V₁. It draws a small magnetising current which sets up an alternating flux φ in the low-reluctance laminated core. This flux links both windings. By Faraday's law, a self-induced EMF E₁ is induced in the primary (it nearly balances V₁) and a mutually induced EMF E₂ in the secondary. If a load is connected, secondary current I₂ flows; its MMF tends to reduce the flux, so the primary draws extra current I₁' such that N₁I₁' = N₂I₂, keeping the flux nearly constant. Energy is thus transferred from primary to secondary through the magnetic field at the same frequency, without electrical connection. A transformer works only on AC (DC gives constant flux, no induced EMF).

EMF equation: let the flux be sinusoidal, \( \phi = \phi_m \sin \omega t \), with \( \omega = 2\pi f \).

Induced EMF per turn \( e = -\dfrac{d\phi}{dt} = -\omega\phi_m\cos\omega t = \omega\phi_m \sin(\omega t - 90^\circ) \).

Maximum EMF per turn \( E_{m} = \omega\phi_m = 2\pi f\phi_m \). RMS value per turn \( = \dfrac{2\pi f\phi_m}{\sqrt2} = 4.44 f\phi_m \).

(Equivalently: average EMF per turn = flux change/time = \( \dfrac{\phi_m}{1/(4f)} = 4f\phi_m \); RMS = form factor 1.11 × average = 4.44 fφ_m.)

Hence \( E_1 = 4.44 f \phi_m N_1 \) and \( E_2 = 4.44 f \phi_m N_2 \) volts, or with \( \phi_m = B_m A \): \( E = 4.44 f B_m A N \).

Induced EMF lags the flux by 90°.

Also \( E_2/E_1 = N_2/N_1 = K \) (transformation ratio).