mutual inductance

mutual inductance

1. What Mutual Inductance Is

Mutual inductance ($M$, henry $H$) = property of two coils where changing current in one induces emf in the other via common flux:

where

($φ_{21}$ = flux through coil 2 due to $I_1$). SI $H$ same as $L$.

2. Coefficient of Coupling ($k$)

Not all flux links both coils; coupling coefficient

where $0≤k≤1$ ($k=1$ perfect coupling all flux common, $k=0$ no coupling). For tightly coupled (iron core transformer) $k≈0.95$–$1$; for air-core loose $k$ small.

Energy:

(sign depends on dot convention).

3. Factors

for solenoid sharing same core (ideal $k=1$). So $M∝N_1 N_2$, $∝μ$, $∝A/l$, depends on geometry and separation.

4. Dot Convention

Dots on coils indicate reference polarity: if both currents enter dotted terminals, fluxes aid ($+M$); if one enters dot and other leaves, subtract. Determines sign of induced emf.

5. Worked Examples

(a) Two coils L1=0.4H, L2=0.6H, k=0.8 ⇒ M=0.8√(0.24)=0.39H ✓

(b) Transformer primary N1=100, N2=1000, same core l,A, μ ⇒ M=k·μ·N1·N2·A/l ⇒ voltage ratio N2/N1.

(c) Need to find induced emf: M=0.1H, dI1/dt=100A/s ⇒ e2= -10V ✓

(d) Why k<1? Leakage flux not linking both.

6. Exam Points

• Define $M=N_2 φ_{21}/I_1$, $e_2=-M dI_1/dt$, $H$ — 5 marks, very frequent (almost certain).
• Give $k=M/√(L1 L2)$ and $0≤k≤1$ — 3-5 marks, frequent.
• Mention dot convention — 3 marks, frequent.
• Formula $M=μ N1 N2 A/l$ for solenoid — 3-5 marks, frequent.