AC Machines EMF Equation Transformer

Transformer EMF Equation: E = 4.44 f N Φm — Derivation and Worked Example

Author: Abloom Instruction Date: November 07, 2026

Every transformer calculation starts from one equation: E = 4.44 f N Φm. It tells you the RMS voltage induced in a winding from the supply frequency, the number of turns, and the maximum core flux. The number 4.44 is not a magic constant — it comes straight from Faraday's law and the shape of a sine wave, as this post shows.



Figure 1: Flux Φ and induced emf e (a), and a single-phase core-type transformer (b). The emf lags the flux by 90°. (Schematic, not to scale.)

Derivation

Assume the core flux varies sinusoidally: Φ = Φm sin ωt, where Φm is the maximum flux and ω = 2πf. By Faraday's law, the emf induced in a winding of N turns is e = −N × dΦ/dt.

Differentiating: e = −N × ω × Φm × cos ωt = N × ω × Φm × sin(ωt − 90°). So the induced emf is also sinusoidal, with maximum value Emax = N × ω × Φm = 2π × f × N × Φm — and it lags the flux by 90°.

The RMS value is Emax / √2, so E = (2π / √2) × f × N × Φm = 4.44 × f × N × Φm. The factor 4.44 is simply 2π/√2 = 4.443, and it holds whenever the flux is sinusoidal — which is the normal case for a transformer on a sine-wave supply.

Since the same flux links both windings, the same equation applies to primary and secondary: E1 = 4.44 f N1 Φm and E2 = 4.44 f N2 Φm, so E1 / E2 = N1 / N2 — the familiar turns-ratio rule.

Worked example — 230 V to 12 V transformer, 50 Hz

Design the windings of a single-phase 230 V / 12 V, 50 Hz transformer. The core cross-section is A = 25 cm² and the maximum flux density is Bm = 1.0 T.

Step 1 — maximum flux: Φm = Bm × A = 1.0 × 25 × 10⁻⁴ = 2.5 × 10⁻³ Wb (2.5 mWb).

Step 2 — volts per turn: E / N = 4.44 × f × Φm = 4.44 × 50 × 2.5 × 10⁻³ = 0.555 V per turn.

Step 3 — primary turns: N1 = E1 / 0.555 = 230 / 0.555 = 414 turns.

Step 4 — secondary turns: N2 = E2 / 0.555 = 12 / 0.555 = 21.6 → 22 turns (turns must be whole numbers).

Step 5 — check: with 22 turns, V2 = 4.44 × 50 × 22 × 2.5 × 10⁻³ = 12.21 V — close to the required 12 V. The turns ratio is 414 / 22 = 18.8, against the ideal 230 / 12 = 19.2; the small difference is just the rounding of turns to whole numbers.

Key points to remember

  • E = 4.44 f N Φm, where 4.44 = 2π/√2 comes from a sinusoidal flux via Faraday's law.
  • The induced emf lags the core flux by 90°.
  • Same flux links both windings, so E1 / E2 = N1 / N2.
  • Worked example: 230 V, 50 Hz, Φm = 2.5 mWb → 0.555 V per turn → N1 = 414 turns, N2 = 22 turns, giving 12.21 V.
  • Raising the flux density or the core area raises the volts per turn, so fewer turns are needed — that is why transformer cores use high flux densities.