Inductor: Flux, Energy, V-I Relationship (V = L dI/dt)

Every inductor in a real circuit — a switching regulator's power inductor, a motor winding, a PCB trace with appreciable loop area, a transformer leakage path — enforces one non-negotiable rule: current through it cannot change instantaneously. That single fact is why inductors generate voltage spikes when switches open, why buck converters ripple current instead of stepping it, and why "flyback" exists as both a phenomenon and a converter topology. Understanding flux, energy storage, and the V-I relationship is what lets you predict ringing, size snubbers, and avoid destroying a MOSFET the first time you switch off an inductive load.

Physical Origin: Flux Linkage

An inductor is fundamentally a device that stores energy in a magnetic field. Current I flowing through a coil of N turns produces magnetic flux Φ through the coil, and the two are linearly related for a linear (non-saturating) magnetic core:

Φ = (L/N)·I        or equivalently:      N·Φ = L·I

The quantity N·Φ is called flux linkage, λ. Inductance L is defined as the ratio of flux linkage to current:

L = N·Φ / I     [henries, H]

One henry is the inductance that links 1 weber-turn of flux per ampere of current. Physically, L depends on geometry and core material — number of turns squared, cross-sectional area, path length, and permeability (L ∝ N²µA/l for a simple toroidal core). This is why inductance is a passive, fixed property of the coil (for a given operating point), just as capacitance is a fixed property of a capacitor's geometry.

Faraday's Law and the V-I Relationship

Faraday's law states that a changing flux linkage induces a voltage across the coil:

v(t) = dλ/dt = d(N·Φ)/dt

Substituting N·Φ = L·I and assuming L is constant (linear, unsaturated core):

v(t) = L · dI/dt

This is the defining equation of an ideal inductor. It is the dual of the capacitor's I = C·dV/dt: where a capacitor resists sudden changes in voltage, an inductor resists sudden changes in current. Rearranged for current:

I(t) = I(t₀) + (1/L)·∫[t₀ to t] v(τ) dτ

Current is the time-integral of voltage divided by L — meaning current through an inductor is always continuous (no step discontinuities) as long as the applied voltage stays finite. A voltage step across an ideal inductor produces a ramp in current, not a step.

Consequence for switching: if you force a rapid change in current (e.g., open a switch in series with an inductor), dI/dt becomes very large, and since v = L·dI/dt, the inductor generates a correspondingly large voltage spike — in principle unbounded for an ideal switch opening instantaneously. This is the mechanism behind inductive kickback and is why flyback diodes, snubbers, or active clamps are mandatory across relay coils, motor windings, and switch-mode power stages.

Energy Storage

Instantaneous power delivered to an inductor is p(t) = v(t)·I(t) = L·I·(dI/dt). Integrating power over time to bring current from 0 to I gives the stored energy:

W = ∫ p dt = ∫ L·I·(dI/dt) dt = ∫[0 to I] L·I' dI' = ½·L·I²

W = ½·L·I²     [joules]

Energy is stored in the magnetic field and depends only on the instantaneous current, not its history — same structural result as a capacitor's W = ½·C·V², but with current and inductance swapped for voltage and capacitance. This stored energy is real and recoverable: it's what a flyback converter deliberately dumps into the output every switching cycle, and it's what must go somewhere (snubber, clamp, freewheeling diode) when a current path is suddenly interrupted.

Worked Example: Switching a 10 mH Relay Coil

A relay coil has L = 10 mH and coil resistance R = 50 Ω. It is driven from a 12 V supply through an NPN transistor switch. In steady state (switch closed, dI/dt = 0), the inductor behaves as a short circuit at DC, so:

I_ss = V/R = 12 V / 50 Ω = 240 mA

Energy stored at this current:

W = ½·L·I² = ½ · (10×10⁻³ H) · (0.24 A)² = ½ · 0.01 · 0.0576 = 2.88×10⁻⁴ J = 0.288 mJ

Now suppose the transistor switches off in t_fall = 100 ns with no protection diode, and the coil current is forced to drop from 240 mA to 0 linearly over that interval (a simplified but illustrative assumption):

dI/dt = ΔI/Δt = (0 − 0.24 A) / (100×10⁻⁹ s) = −2.4×10⁶ A/s

v_L = L · dI/dt = (10×10⁻³ H) · (−2.4×10⁶ A/s) = −24,000 V

That is a wildly unrealistic (and lethal to any real transistor) voltage — in practice, the transistor's breakdown voltage or an arcing path limits it, and the actual dI/dt self-limits once the device avalanches. But the calculation shows why an unprotected inductive switch fails: the inductor will generate whatever voltage is needed to try to maintain its current, up to the breakdown limit of whatever is blocking it.

Adding a flyback diode: placing a diode across the coil (cathode to supply, anode to switch node) gives the current a path to freewheel when the switch opens. The coil current now decays through R and the diode's forward drop (V_F ≈ 0.7 V) instead of being forced to zero instantly:

L·dI/dt + I·R = −V_F

This is a standard RL decay (covered in detail in the RL transient response article) with time constant τ = L/R = 10×10⁻³ / 50 = 200 µs. The peak reverse voltage across the switch is clamped to roughly V_supply + V_F ≈ 12.7 V instead of tens of kilovolts — the entire point of the diode is to give the stored energy ½LI² a controlled dissipation path instead of an uncontrolled voltage spike.

Check: at t = 0⁺ after switch-off, I = 240 mA, so v_L = −I·R − V_F = −(0.24)(50) − 0.7 = −12.7 V across the coil, consistent with the diode conducting and clamping the switch node just above the supply rail. Energy 0.288 mJ dissipates in R and the diode over roughly 5τ ≈ 1 ms — orders of magnitude gentler than the unprotected case.

Series and Parallel Inductor Combinations

For inductors with no mutual coupling:

These combine exactly like resistors (not like capacitors) because inductors store energy in current, and series elements share the same current while parallel elements share the same voltage — the same structural argument used for resistors in series/parallel. If windings share a magnetic core, mutual inductance M modifies these formulas and can add or subtract depending on winding polarity — relevant for transformers and coupled-inductor converters, but outside the ideal single-inductor case here.

Frequency-Domain View

For sinusoidal steady state, v(t) = L·dI/dt transforms to phasor form:

V = jωL·I     →     Z_L = jωL

Impedance magnitude |Z_L| = ωL grows linearly with frequency — an inductor looks like a short at DC and an open circuit at high frequency, the exact opposite of a capacitor. This underlies its use as a choke (blocking AC while passing DC) and as the energy-transfer element in switching converters, where ωL at the switching frequency is chosen to set the desired current ripple.

Practical and Non-Ideal Considerations

• DC resistance (DCR): real windings have resistance, causing I²R loss and a finite (not infinite) DC current limit set by V/R, not by the inductance itself.
• Core saturation: the linear relationship Φ = (L/N)·I holds only below the core's saturation flux density. Beyond that, incremental inductance drops sharply, current can spike uncontrollably, and this is a primary failure mode in undersized power inductors.
• Parasitic capacitance: winding-to-winding and winding-to-core capacitance creates a self-resonant frequency above which the inductor behaves capacitively — critical in RF and high-frequency power design.
• Core losses: hysteresis and eddy-current losses in the core dissipate energy beyond simple DCR loss, growing with frequency and flux swing.

Key Takeaways

• Inductance relates flux linkage to current: L = N·Φ/I; Faraday's law then gives the defining relationship v = L·dI/dt.
• Inductor current cannot change instantaneously for finite applied voltage — current is the time-integral of voltage, the dual of a capacitor's voltage continuity.
• Stored energy is W = ½·L·I², dependent only on instantaneous current, not history.
• Interrupting inductor current abruptly forces dI/dt → ∞, generating large voltage spikes — the reason flyback diodes, snubbers, and clamps are mandatory on inductive switching loads.
• In the frequency domain, Z_L = jωL: impedance rises with frequency, making inductors act as chokes at high frequency and shorts at DC — the opposite behavior of a capacitor.
• Series/parallel combination rules mirror resistors (L_eq adds in series, reciprocal-adds in parallel) for uncoupled inductors; mutual coupling modifies this for transformers and coupled coils.