Self-Inductance EMF

One henry is the inductance of a coil in which a current changing at one ampere per second induces one volt. Equivalently, one henry is the inductance that stores one joule when carrying one ampere (from E = ½LI²), or the inductance whose flux linkage is one weber per ampere (L = NΦ/I). In SI base units, 1 H = 1 V·s/A = 1 kg·m²·s⁻²·A⁻². It is a large unit for everyday electronics — most circuit inductors are in the µH to mH range. A coil of about 1,000 turns on a ferrite core can reach a few henries; a large power transformer primary might reach tens of henries.

A relay coil is an inductor. When you switch off the driving transistor, the current through the coil tries to fall to zero very quickly. With a typical coil inductance of 0.1 H and a switching time of 1 µs, dI/dt ≈ 10⁶ A/s, so the induced EMF is around 100 kV — far above what any small transistor can withstand. The flyback diode (also called a freewheeling or snubber diode) provides a path for the current to continue circulating through the coil and the diode after the switch opens. The current then decays slowly through the coil resistance, and the peak voltage across the switch is limited to the diode forward drop plus the supply voltage — typically 0.7 V above the rail. Without the diode, the transistor is destroyed within microseconds.

For an ideal long solenoid, L = μ₀ · μᵣ · N² · A / ℓ, where N is the number of turns, A is the cross-sectional area, ℓ is the length, and μᵣ is the relative permeability of the core. For a toroid the formula is L = μ₀ μᵣ N² A / (2πr). For an air-core single-layer solenoid of finite length, Wheeler's formula L(µH) ≈ (r² N²) / (9r + 10ℓ) with r and ℓ in inches gives a good approximation. Real coils deviate from these formulas due to fringing fields, core saturation, winding capacitance and proximity effects, so the actual inductance is usually measured rather than calculated.

Self-inductance L describes how a changing current in one coil induces an EMF in the same coil. Mutual inductance M describes how a changing current in one coil induces an EMF in a nearby second coil. For two coupled coils, the EMF in coil 2 is ε₂ = −M · dI₁/dt, and vice versa. Mutual inductance is the physical basis of transformers, wireless power transfer, and crosstalk between adjacent PCB traces. The maximum possible mutual inductance between two coils is M = √(L₁·L₂), which occurs when the magnetic coupling is perfect (coupling coefficient k = 1). In transformer design, high M is desired; in PCB layout, low M is desired to avoid crosstalk.

In DC steady state, the current is constant so dI/dt = 0 and the inductor behaves as a short circuit — it has zero voltage across it and passes current freely. In AC, the current oscillates, so dI/dt is non-zero and the inductor presents an impedance X_L = 2πfL, which rises linearly with frequency. This is why inductors are used to block high frequencies (chokes) and pass low frequencies in filters, and why they can't simply be replaced by resistors. The combination of inductive reactance and resistance gives an RL circuit a corner frequency f_c = R / (2πL), below which it behaves resistively and above which it behaves inductively.

In theory, ε = L·dI/dt would blow up if the current could change instantaneously. In practice, this never happens in a real circuit because: (1) every switch has finite switching time, (2) every inductor has inter-turn capacitance that provides an alternative path at high frequencies, (3) every circuit has stray capacitance that limits the rate of voltage rise, and (4) the breakdown voltage of the switch or insulation is reached first. This is exactly why inductive kick is both useful (ignition coils, sparks, camera flashes) and dangerous (destroyed transistors, arcing relay contacts) — the circuit finds a limit one way or another.

An ignition coil is really two inductors on a common iron core: a primary with a few hundred turns and a secondary with tens of thousands of turns. Current flows through the primary, building up a magnetic field. When the contact breaker (or electronic switch) opens, the primary current falls very quickly, generating a primary back-EMF of a few hundred volts (ε = L·dI/dt). By Faraday's law, the same changing flux links the secondary winding, and the induced EMF is multiplied by the turns ratio N_secondary/N_primary. With a ratio of 100:1 and a primary back-EMF of 300 V, the secondary sees around 30,000 V — enough to break down the spark-plug gap. Modern coil-on-plug systems integrate the coil directly on the plug and control the dwell time electronically.