Inductors

A capacitor stores energy in an electric field between two plates. An inductor stores it in a magnetic field around a coil. The two components are duals of each other — every rule for one has a mirror-image rule for the other, and the AC behaviour of real circuits comes from mixing them.

Structure

At heart an inductor is just a coil of wire. Pass current through the $N$ turns and each turn contributes to a magnetic field that loops through the core and closes around the outside of the coil. Wrap the wire around a lump of ferromagnetic material (iron, ferrite) and the field strengthens dramatically.

Inductance and back-EMF

The total flux threading all N turns is the flux linkage $Ψ = NBA$. For a given coil geometry, Ψ is proportional to the current through it — the constant of proportionality is the inductance $L$:

$L$ is measured in henries (H). One henry stores one weber of flux linkage per ampere of current. Real inductors span microhenries (RF chokes) through millihenries (audio inductors) up to henries (mains transformers and large chokes).

Change the current, and the flux changes with it. A changing flux induces an EMF by Faraday's law — the induced EMF opposes the change that caused it (Lenz's law):

That's the inductor's defining equation. Try to ramp current up quickly and the coil fights back with a big voltage. Try to hold current steady and it offers no resistance at all — just the DC resistance of the copper wire, which is usually tiny.

Energy storage

Building up current in an inductor requires work against the back-EMF; that work is stored in the magnetic field:

When the current eventually falls, that energy has to go somewhere — which is why switching inductive loads abruptly (a relay, a motor winding, an ignition coil) produces voltage spikes. The energy in $\tfrac{1}{2} L I^2$ will find a path, inductive-spike kickback being the classic example. A flyback diode is the standard defence.

Inductors in series and parallel

Combine inductors and the rules mirror resistors (the opposite of capacitors):

Series — the same current flows through every coil, so the induced EMFs add up in the same direction. That's why the inductances add linearly (two 10 mH chokes in series = 20 mH).

Parallel — the coils share the same terminal voltage, so each one carries a current set by its own inductance via $I_k = \int v \, dt / L_k$. The total current is the sum of the branch currents, and dividing through gives the reciprocal-sum rule. Two 10 mH inductors in parallel = 5 mH.

Caveat: the parallel rule assumes the coils are magnetically isolated from each other. If they share a core, mutual inductance $M$ enters the formula — $L_{eq} = L_1 + L_2 \pm 2M$ for series — and the problem gets richer. For this topic we assume no coupling.

Circuit application — flux linkage $\Psi = LI$

The combination rules become concrete once you wire inductors into a real DC loop and trace what each coil stores. A current source forces a fixed supply current $I_s$ into the network.

For series inductors, every coil carries the same current $I_s$. Each stores a flux linkage proportional to its own inductance:

Larger L → more flux, same current. The total flux linkage of the pair equals $L_{eq} \, I_s$ with $L_{eq} = L_1 + L_2$.

For parallel inductors, the dual relationship holds: both coils develop the same flux linkage — just as parallel capacitors share the same voltage. The current, however, divides inversely with inductance (larger L draws less current):

So $\Psi_k = L_k \, I_k = L_{eq} \, I_s$ for every branch — all equal. This is the pure dual of capacitors in series (where all caps carry the same charge).

The stored energy per inductor follows from $W_k = \tfrac{1}{2} L_k I_k^2$. For parallel inductors the larger L draws less current, so it stores less energy despite being bigger — another counterintuitive dual of the capacitor case.

Duality table

Every property a capacitor has, an inductor has the dual of. Memorising the table below short-circuits a lot of AC analysis:

• Capacitor stores energy in an electric field between plates; inductor stores it in a magnetic field around a coil.
• Cap stores charge proportional to V: $Q = CV$. Inductor stores flux linkage proportional to I: $\Psi = LI$.
• Cap current follows dV/dt: $i = C \, dv/dt$. Inductor voltage follows dI/dt: $v = L \, dI/dt$.
• Caps block DC / pass AC. Inductors pass DC / block AC (or more precisely, block high-frequency AC — we'll make that precise with reactance in Topic 6).
• Caps series-reciprocate, parallel-add. Inductors series-add, parallel-reciprocate.

Energising and de-energising

The duality from the table above carries straight into time-domain behaviour. A capacitor can't change its voltage instantly; an inductor, by the same token, can't change its current instantly — that would require infinite EMF across L. So when you close a switch on an L–R loop fed by a DC source $V_s$, the current ramps up rather than jumping.

The current rises exponentially from 0 toward its steady-state value $V_s / R$:

The time constant is the dual of the RC version:

With $L$ in henries and $R$ in ohms, $\tau$ is again in seconds. A 1 H choke fed through 100 Ω has $\tau = 10\,\text{ms}$; a 10 µH inductor through 10 Ω is $\tau = 1\,\mu\text{s}$. The bigger the inductor, the slower it lets current change — large L resists fast di/dt by generating large opposing voltage.

The voltage across the inductor walks the opposite direction: it starts at $V_s$ (because $v_L = L \, dI/dt$ is largest when current is rising fastest) and decays toward zero as the current settles:

Opening the switch is where inductors get spicy. The energy stored in the magnetic field — $\tfrac{1}{2} L I^2$, from the energy section above — has to go somewhere. The inductor refuses to let current drop instantly; it generates whatever back-EMF is needed to keep current flowing through whatever path remains. In a clean L–R loop the current decays exponentially:

But if the path is broken — switch contacts opening across a real coil with no freewheel diode or snubber — the back-EMF can spike to hundreds or thousands of volts, arc across the contacts, and damage things. That's why every relay coil, motor winding, and solenoid you'll meet in industrial wiring has some kind of protection across it.