Transformer EMF Equation Formulas

Transformer EMF equation formulas: the voltage per turn from frequency, flux density and core area, the turns for a voltage and the flux density at a frequency.

The formulas

f is the frequency in hertz, Bm the peak flux density in tesla, A the net core area in m², Φm the peak flux in webers and E the RMS voltage of the winding. The constant 4.443 is √2 × π, and it is often rounded to 4.44. The equation is for a sinusoidal voltage. The flux density rises when the frequency falls or the voltage rises, so a transformer that runs at a low frequency or at an over-voltage can saturate, and its magnetising current then rises sharply. The core area, the flux density and the voltage here are examples, and the real values come from the design.

Worked example: a core of 0.012 m² at 1.5 T and 50 Hz, with windings for 11,000 V and 415 V

1. Φm = 1.5 × 0.012 = 0.018 Wb
2. Et = 4.443 × 50 × 0.018 = 3.999 V per turn
3. N(HV) = 11,000 / 3.999 = 2,751, so 2751 turns
4. N(LV) = 415 / 3.999 = 103.8, so 104 turns

Flux density of the HV winding of 2751 turns at 11,000 V by frequency

Frequently asked questions

Why does the flux density rise at a low frequency?

For a fixed voltage, the flux needed to produce it is inversely proportional to the frequency. A transformer for 50 Hz therefore sees a higher flux on a 40 Hz supply.

What happens when the core saturates?

The magnetising current rises sharply and is full of harmonics, the transformer runs hot and the iron loss rises. Protection against over-fluxing watches the ratio of volts to hertz.

Why is the turn count rounded?

A winding has a whole number of turns, so the voltage per turn and the flux density shift a little from the design value.

Does a bigger core help?

Yes. A larger area gives more voltage per turn at the same flux density, so fewer turns are needed, but the core costs more.

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