Inductor Theory
Part of Electrical Theory
How coils store energy in magnetic fields, why they oppose changes in current, and their role in motors, generators, and filters.
Why This Matters
Inductors are everywhere in electrical systems — they’re the core of every motor, generator, transformer, and relay. Understanding how they work at the theoretical level explains behaviors that otherwise seem mysterious: why motors draw enormous current when starting, why sparks appear when you disconnect a coil from a battery, why certain frequencies pass through a circuit while others don’t.
In rebuilding contexts, you’ll be winding coils for motors, generators, and transformers. Getting the inductance right determines whether your motor has the right torque, whether your transformer operates efficiently, whether your relay fires reliably. The theory here converts rule-of-thumb winding into informed engineering.
What an Inductor Does
An inductor is simply a coil of wire — sometimes wound around a core, sometimes air-core. When current flows through the wire, it creates a magnetic field around the coil. When that current changes, the magnetic field changes, and by Faraday’s law, the changing magnetic field induces a voltage in the coil itself.
This self-induced voltage opposes the change in current — it’s the electrical embodiment of inertia. Just as a heavy flywheel resists changes in rotational speed, an inductor resists changes in current. This property is called inductance, measured in henries (H).
The core principle: An inductor cannot change its current instantaneously. Any attempt to change current is resisted by a back-EMF proportional to the rate of change.
The governing equation: V = L × (dI/dt)
- V = induced voltage (volts)
- L = inductance (henries)
- dI/dt = rate of current change (amps per second)
Energy Storage in Magnetic Fields
When current flows through an inductor, energy is stored in the magnetic field surrounding the coil. When current is interrupted, this energy must go somewhere:
Energy stored: E = ½ × L × I²
A coil with L = 1H carrying I = 10A stores: E = ½ × 1 × 100 = 50 joules
When the circuit is opened, these 50 joules try to maintain the current. With no path available, voltage spikes to whatever level forces current through the available resistance — often thousands of volts in a fraction of a millisecond. This is the spark you see when disconnecting a DC electromagnet.
Practical consequences:
- Relay coils and solenoids need “flyback diodes” (or spark suppressors) across them — otherwise the voltage spike destroys the switching transistor
- Ignition coils in engines exploit this deliberately: a large inductor suddenly interrupted produces the high voltage that fires spark plugs
- In motors and generators, the inductance of windings limits how fast current can change, smoothing the commutation process
Inductance: What Affects It
L = μ × N² × A / l
Where:
- μ = permeability of the core material (how well it supports magnetic flux)
- N = number of turns
- A = cross-sectional area of the core
- l = length of the coil
Key relationships:
| Factor | Effect on inductance |
|---|---|
| Doubling turns N | 4× inductance (N² relationship) |
| Doubling core area A | 2× inductance |
| Halving coil length | 2× inductance |
| Iron core vs. air core | 100–10,000× more inductance |
| Ferrite core vs. iron | Can be higher at high frequency |
The N² relationship is crucial: Doubling turns quadruples inductance. This is why transformer and motor winding calculations always involve squared terms.
Core Materials and Their Properties
The core material dominates inductance because permeability (μ) varies enormously:
Air core: μ = 1 (reference). Air-core coils are used in radio-frequency applications where iron cores would have excessive losses at high frequencies. Inductance is low but stable.
Iron core: μ = 1,000–10,000 depending on grade and frequency. Soft iron or silicon steel dramatically increases inductance and allows smaller, more powerful coils. Used in power transformers, large inductors, motors.
Ferrite core: μ = 100–15,000 depending on composition. Ferrite is a ceramic made from iron oxide and other metals. It maintains high permeability at higher frequencies than iron (iron has eddy current losses above ~1kHz; ferrite can work to MHz). Used in radio transformers, switch-mode power supplies, EMI filters.
Lamination: Iron cores for AC use must be laminated — thin sheets insulated from each other. This limits eddy currents (circulating currents induced in solid cores by changing flux, which waste energy as heat). The thinner the laminations, the lower the eddy current loss.
Building core materials from salvage:
- Transformer E/I laminations salvaged from old transformers
- Stacked silicon steel strips, insulated with shellac between layers
- Ferrite toroids from old computer power supplies (EMI chokes)
- Soft iron rod for DC electromagnets (eddy currents don’t matter for DC)
Mutual Inductance and Transformers
When two coils are placed near each other, the magnetic field of one passes through the other. Current change in coil 1 induces voltage in coil 2 — this is mutual inductance, the operating principle of transformers.
M = k × √(L1 × L2)
Where k is the coupling coefficient (0 = no coupling, 1 = perfect coupling). Tightly wound coils on the same core have k ≈ 0.95–0.99.
Transformer voltage ratio: V2/V1 = N2/N1
Transformer current ratio: I2/I1 = N1/N2 (opposite — more turns on secondary means less current)
Transformer power: V1×I1 ≈ V2×I2 (conservation of energy, minus losses)
Time Constant and Transient Response
When a voltage is suddenly applied to a series RL circuit, current doesn’t jump instantly to its final value. It rises exponentially with time constant:
τ = L / R
After one time constant τ, current has reached 63% of its final value. After 5τ, it’s essentially at 100%.
Practical examples:
| L | R | τ | Time to full current |
|---|---|---|---|
| 1 mH | 10 Ω | 0.1 ms | 0.5 ms |
| 100 mH | 10 Ω | 10 ms | 50 ms |
| 1 H | 10 Ω | 100 ms | 500 ms |
Motor starting current: A large motor with L = 500mH and winding resistance R = 1Ω has τ = 0.5s. During the first 0.5 seconds, while inductance is building up its magnetic field and before the motor builds back-EMF, current is limited mainly by resistance — which is why starting current can be 5–10× running current.
Inductors in Filters
Inductors and capacitors work together as frequency-selective filters because their reactances vary with frequency in opposite directions:
Low-pass filter (inductor in series): High frequencies see high X_L — blocked. Low frequencies pass. Used to smooth rectified DC (choke filter), remove high-frequency noise from power supplies.
High-pass filter (inductor in shunt to ground): Low frequencies are shunted to ground through low X_L. High frequencies pass. Used in audio crossover networks.
Bandpass filter (LC resonant circuit): Series resonance creates low impedance at one frequency, allowing that frequency to pass while others are blocked. Parallel resonance creates high impedance at resonance, allowing surrounding frequencies to bypass while the resonant frequency is blocked.
Building inductors for filters:
- Determine required inductance: L = X_L / (2πf)
- Choose core material (air for HF, iron for LF)
- Calculate turns: N = √(L × l / (μ × A))
- Wind carefully — even winding, no crossed turns
- Measure inductance by resonance method (pair with known capacitor, find resonant frequency)
The inductor is the fundamental building block of electromagnetic energy — motors, generators, and transformers are all variations on the coil-of-wire principle. Mastering inductor theory gives you the foundation to design, wind, and repair all electromagnetic machinery.