E.M.F. equation of transformers

E.M.F. equation of transformers

1. What the E.M.F. Equation Is

E.M.F. equation relates induced emf per turn to flux and frequency for sinusoidal flux. For sinusoidal flux

where $Φ_m$ = peak flux (Wb) = $B_m·A$ ($B_m$ peak flux density, $A$ core area), the rms induced emf per winding is:

where $f$ = frequency (Hz), $N$ = turns, $Φ_m$ = peak flux. This is the transformer e.m.f. equation (derived from Faraday).

2. Derivation (exam expects steps)

1. Flux

2. Faraday:

So peak emf

3. RMS for sine:

(since $2π/√2≈4.44).

If flux is not sine but form factor $F=1.11$ for sine used: general

⇒ $4.44$ for sine.

3. Why It Matters

• Design: given $E$, $f$, $Φ_m$ (chosen $B_m≈1.0–1.5$ T for iron to avoid saturation), find required turns

.

• Voltage ratio:

as earlier.

• Flux density check:

must be < saturation ($B_m=Φ_m/A$).

4. Worked Examples

(a) Single-phase transformer E1=230V, f=50Hz, Φ_m=0.01 Wb: N1=230/(4.44·50·0.01)=230/2.22≈104 turns ✓

(b) Core A=0.01 m², B_m=1.2T ⇒ Φ_m=0.012 Wb, f=50Hz, N=200 ⇒ E=4.44·50·200·0.012=533V ✓

(c) Need to find B_m: B_m= E/(4.44 f N A)

(d) For 3-phase? Same per phase.

(e) If frequency changes to 60Hz, E increases 20% for same Φ_m (since E∝f).

5. Exam Points

• State and derive $E=4.44 f N Φ_m$ (from $e=-N dφ/dt$) — 7-10 marks, very frequent (the most common transformer equation derivation).
• Show $E_m=2πf N Φ_m$ and $E=4.44 f N Φ_m$ — 5 marks, frequent.
• Use to find $N$ or $Φ_m$ or $B_m$ — 5-7 marks, very frequent.
• Mention $4.44 =2π/√2$ comes from RMS — 3 marks.