Ideal transformer principle and emf equation

A transformer is a static device that transfers electrical energy from one circuit to another at the same frequency by mutual induction. An ideal transformer has no winding resistance, no leakage flux and no core loss.

Operating principle

• When an alternating voltage V1 is applied to the primary winding (N1 turns), a small magnetising current flows and sets up an alternating flux Φ in the laminated core.
• This flux links the secondary winding (N2 turns) as well. By Faraday's law, an emf is induced in each winding: e = −N dΦ/dt.
• The emf in the primary (E1) is the counter-emf opposing V1, so E1 = V1. The secondary emf E2 appears across the load terminals.
• So the voltage is changed in the ratio of turns, V2/V1 = N2/N1 = K, while the frequency stays the same.

Emf equation

Let the flux vary sinusoidally: Φ = Φm sin ωt, ω = 2πf.

e1 = −N1 dΦ/dt = −N1 Φm ω cos ωt
Maximum value        Em1 = N1 Φm ω = 2π f N1 Φm
RMS value            E1  = Em1/√2 = (2π/√2) f N1 Φm
                         = 4.44 f N1 Φm

Similarly for the secondary:

E1 = 4.44 f N1 Φm and E2 = 4.44 f N2 Φm (volts)

where f is in Hz, Φm is the maximum flux in Wb. If the maximum flux density is Bm and the core area is a, then Φm = Bm a, so E = 4.44 f N Bm a.

Example: N1 = 500, f = 50 Hz, Φm = 2 mWb gives E1 = 4.44 × 50 × 500 × 0.002 = 222 V.

From the two equations, E1/N1 = E2/N2 = 4.44 f Φm (the emf per turn is the same in both windings).