self-inductance

self-inductance

1. What Self-Inductance Is

Self-inductance ($L$, henry $H$) = property of a single coil/circuit to induce emf in itself when its own current changes, due to changing flux linking itself:

,

where $φ$ = flux per turn due to its own $I$, $Nφ = λ$ = flux linkage. SI $H = V·s/A = Ω·s$.

2. Why It Matters for Machines

• Energy storage: in magnetic field:

• Opposes change: $L$ tries to keep $I$ constant (Lenz) ⇒ inductive kick when switch opens ($dI/dt$ large ⇒ large $e$).
• In AC, $L$ gives reactance $X_L = ωL$ and phase lag ($i$ lags $v$ 90°).
• In transformer/machines, $L$ determines magnetizing current and leakage.

3. Factors

For solenoid $N$ turns, length $l$, area $A$, $μ=μ_0 μ_r$:

So $L∝N²$, $∝μ$, $∝A$, $∝1/l$. Iron core ($μ_r≈10^3$) ⇒ $L$ 1000× air.

Coefficient of self-induction depends only on geometry and $μ$, not on $I$ (if linear).

4. Worked Examples

(a) Solenoid earlier: N=1000, l=0.1, A=1e-4, μ0 ⇒ L=1.26 mH air, 1.26 H iron (μr=1000) ✓

(b) I changes 2A→0 in 1ms in L=10 mH: e= -0.01·(-2000)=20V spike ✓

(c) Energy: L=1H, I=2A ⇒ W=2 J ✓

(d) Need to limit di/dt? Inductor smooths current.

5. Exam Points

• Define $L=Nφ/I$, $H$, $e=-L dI/dt$, $W=½LI²$ — 5 marks, very frequent (almost certain to ask definition).
• Give $L=μ N²A/l$ — 5 marks, frequent.
• Mention that $L$ opposes change (Lenz) — 3 marks, frequent.
• Distinguish from mutual $M$ — 3 marks.