EMF EQUATION OF A TRANSFORMER

Figure 1.22 shows the representation of alternating flux, varying sinusoidally, which increases from its zero value to maximum value (Φ_m) in one-quarter of the cycle, that is in one-fourth of a second where f is the frequency of AC input in hertz.

The average rate of change of flux is given by , that is 4fΦ_m Wb/s or V.

Figure 1.22 Representation of Alternating Flux

This rate of change of flux per turn is the induced emf in V.

Therefore, average emf/turn = 4fΦ_mm V.

Let N₁ and N₂ be the number of turns in primary and secondary.

The rms value of induced emf in primary winding is given by

E₁ = (4.44fΦ_m m) × N₁ = 4.44fΦ_m mN = 4.44f B_mA_rN₁ (1.1)

where is the maximum value of flux density having unit Tesla (T) and A_r is the area of cross-section.

Similarly, RMS value of induced emf in secondary winding is

E ₂ = (4.44fΦ_m )x N₂ = 4.44fΦ_mN₂ = 4.44f B_mA_rN₂ (1.2)

From Equations (1.1) and (1.2), we have

i.e.,

where ‘a’ is the turns ratio of the transformer,

i.e.,

Equation (1.3) shows that emf induced per turn in primary and secondary windings are equal.

In an ideal transformer at no load, V₁ = E₁ and V₂ = E₂, where V₂ is the terminal voltage of the transformer. Equation (1.3) becomes

Example 1.1 The voltage ratio of a single-phase, 50 Hz transformer is 5,000/500 V at no load. Calculate the number of turns in each winding if the maximum value of the flux in the core is 7.82 mWb.

Solution

Here

E₁ = V₁ = 5,000 V

E₂ = V₂ = 500 V

φ_max = 7.82 m Wb = 7.82 × 10⁻³ Wb, f = 50Hz

Let N₁ and N₂ be the number of turns of the primary and secondary windings, respectively.

Since

E₁ = 4.44 f φ_mN₁

i.e.,

Again,

∴