8.5 Inductance

Inductance fundamentals

Inductance describes how a circuit element opposes changes in current by generating a voltage in response to that change. This property sits at the heart of Faraday's law in action: a changing current produces a changing magnetic flux, which induces an EMF that resists the original change. Understanding inductance is essential for analyzing AC circuits, designing transformers, and working with any system where currents vary in time.

Definition of inductance

Inductance is the property of a conductor or coil that quantifies how much voltage is induced for a given rate of change of current. More precisely, it's the ratio of magnetic flux linkage to the current producing that flux.

The defining relationship comes directly from Faraday's law applied to a coil's own flux:

$V = -L \frac{di}{dt}$

where $L$ is the inductance, $V$ is the induced EMF (back-EMF), and $\frac{di}{dt}$ is the rate of change of current. The negative sign reflects Lenz's law: the induced voltage opposes the change in current.

Equivalently, inductance can be written as:

$L = \frac{N\Phi}{I}$

where $N\Phi$ is the total magnetic flux linkage through the coil and $I$ is the current.

Units of inductance

The SI unit of inductance is the henry (H), named after the American physicist Joseph Henry. One henry is the inductance that produces one volt of EMF when the current changes at one ampere per second.

In practice, most inductors you'll encounter have values much smaller than one henry:

• millihenry (mH) = $10^{-3}$ H (common in power electronics)
• microhenry (μH) = $10^{-6}$ H (common in RF circuits)
• nanohenry (nH) = $10^{-9}$ H (relevant at very high frequencies, including parasitic inductance in PCB traces)

Factors affecting inductance

The inductance of a coil depends on its geometry and the magnetic properties of its surroundings:

• Number of turns (N): Inductance scales as $N^2$, so doubling the turns quadruples the inductance.
• Cross-sectional area (A): A larger loop area captures more flux, increasing $L$.
• Length of the coil (l): A longer solenoid spreads the turns out, reducing the flux density and lowering $L$.
• Core permeability ($\mu$): Wrapping the coil around a high-permeability material (like iron or ferrite) concentrates the magnetic field and dramatically increases $L$.

These factors combine in the solenoid formula discussed in the self-inductance section below.

Inductors

Inductors are passive components designed to provide a specific inductance in a circuit. They store energy in their magnetic field (not in an electric field, like capacitors) and are used for filtering, energy storage, impedance matching, and signal processing.

Inductor construction

An inductor is typically a length of conductive wire (usually copper) wound into a coil around some core. The key design choices are:

• Core material: Air, ferromagnetic materials (iron, ferrite), or non-magnetic materials (plastic, ceramic). The core determines the permeability and therefore the inductance per turn.
• Number of turns and wire gauge: More turns means higher inductance; thicker wire means lower DC resistance and higher current handling.
• Winding geometry: Solenoid (cylindrical), toroid (doughnut-shaped), or multilayer. Toroids confine the magnetic field well, reducing stray flux and electromagnetic interference.

Types of inductors

• Air core inductors have no magnetic core material. They offer very stable inductance (no saturation effects) and are used at high frequencies where ferromagnetic losses would be problematic.
• Ferromagnetic core inductors use iron or ferrite cores to boost inductance significantly. The tradeoff is that the core can saturate at high currents, causing the inductance to drop.
• Variable inductors allow adjustment of the inductance value, typically by moving a ferrite slug in or out of the coil or by changing the effective number of turns. These are useful in tunable RF circuits.

Inductor applications

• Power supply filtering: Inductors smooth out current ripple in DC power supplies.
• RF circuits: Used for impedance matching, tuning (in LC resonant circuits), and filtering specific frequency bands.
• Switch-mode power supplies (SMPS): Inductors store energy during each switching cycle and release it to the load, enabling efficient voltage regulation.
• Energy harvesting and pulsed power: Inductors store magnetic energy and can release it rapidly for high-power pulses.

Self-inductance

Self-inductance describes how a single inductor's own changing current induces a voltage across itself. This is the most direct application of Faraday's law to a coil: the coil's own changing flux creates a back-EMF that opposes the current change.

Self-inductance concept

When current through an inductor changes, the magnetic flux through the coil changes proportionally. By Faraday's law, this changing flux induces an EMF across the inductor. By Lenz's law, this self-induced EMF (back-EMF) always acts to oppose the change in current.

If the current is increasing, the back-EMF acts to slow the increase. If the current is decreasing, the back-EMF acts to sustain it. This is why inductors resist sudden changes in current.

Self-inductance formula

The self-inductance is defined as:

$L = \frac{N\Phi}{I}$

where $N$ is the number of turns, $\Phi$ is the magnetic flux through each turn, and $I$ is the current. The product $N\Phi$ is called the flux linkage (sometimes written $\Lambda$).

The induced EMF is then:

$\mathcal{E} = -L\frac{di}{dt}$

Self-inductance calculations

For a solenoid (long cylindrical coil), the self-inductance has a clean analytical form:

$L = \frac{\mu_0 \mu_r N^2 A}{l}$

where:

• $\mu_0 = 4\pi \times 10^{-7} \text{ H/m}$ is the permeability of free space
• $\mu_r$ is the relative permeability of the core material
• $N$ is the total number of turns
• $A$ is the cross-sectional area of the solenoid
• $l$ is the length of the solenoid

This formula assumes the solenoid is long compared to its diameter (so the field inside is approximately uniform). For a toroid of mean radius $r$ with a small cross-section $A$:

$L = \frac{\mu_0 \mu_r N^2 A}{2\pi r}$

For more complex geometries (multilayer coils, planar inductors), numerical methods or empirical formulas are typically needed.

Mutual inductance

Mutual inductance quantifies the magnetic coupling between two separate coils. When the current in one coil changes, it induces a voltage in a nearby coil through their shared magnetic flux. This is the principle behind transformers and all magnetically coupled circuits.

Mutual inductance concept

Place two coils near each other. Current $I_1$ in coil 1 (the "primary") generates a magnetic field. Some fraction of that field's flux passes through coil 2 (the "secondary"). If $I_1$ changes, the flux through coil 2 changes, inducing a voltage in coil 2 even though there's no electrical connection between them.

The strength of this effect depends on:

• How close the coils are and their relative orientation
• The geometry of both coils (number of turns, area)
• The permeability of the medium between and around them

Mutual inductance formula

The mutual inductance $M$ is defined by:

$M = \frac{N_2 \Phi_{21}}{I_1}$

where $\Phi_{21}$ is the flux through each turn of coil 2 due to current $I_1$ in coil 1, and $N_2$ is the number of turns in coil 2.

A key result from electromagnetic theory (the Neumann formula) guarantees that mutual inductance is symmetric:

$M_{12} = M_{21} = M$

This means the voltage induced in coil 2 by a changing current in coil 1 uses the same $M$ as the voltage induced in coil 1 by a changing current in coil 2. The induced EMF in the secondary is:

$\mathcal{E}_2 = -M\frac{di_1}{dt}$

Mutual inductance calculations

For coaxial solenoids (one inside the other), the mutual inductance can be calculated analytically. If a short solenoid of $N_2$ turns and area $A_2$ sits inside a long solenoid of $n_1$ turns per unit length:

$M = \mu_0 n_1 N_2 A_2$

For more complex arrangements (coplanar loops, non-coaxial coils), you may need numerical methods such as finite element analysis (FEA). Mutual inductance can also be measured experimentally using bridge circuits or by driving one coil with a known AC current and measuring the open-circuit voltage on the other.

Inductance in circuits

Inductors combine in circuits following rules analogous to resistors, but with an important difference: mutual coupling between inductors can complicate things. The formulas below assume no mutual coupling unless stated otherwise.

Inductance in series circuits

For inductors in series with no mutual coupling, the total inductance is simply the sum:

$L_{\text{total}} = L_1 + L_2 + L_3 + \cdots$

If mutual coupling exists between series inductors, the total inductance also includes mutual inductance terms. For two coupled inductors in series:

$L_{\text{total}} = L_1 + L_2 \pm 2M$

The sign depends on whether the coils are wound so their fluxes add ("+", aiding) or oppose ("-", opposing).

Inductance in parallel circuits

For inductors in parallel with no mutual coupling:

$\frac{1}{L_{\text{total}}} = \frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{L_3} + \cdots$

For two inductors in parallel, this simplifies to:

$L_{\text{total}} = \frac{L_1 L_2}{L_1 + L_2}$

The parallel combination always has a total inductance smaller than the smallest individual inductor.

Equivalent inductance

In complex circuits, you find the equivalent inductance the same way you'd find equivalent resistance: identify series and parallel groups, combine them step by step, and reduce until you have a single equivalent value. Watch for mutual coupling between inductors, which adds cross-terms that don't appear in simple resistor networks.

Energy storage in inductors

An inductor carrying current stores energy in its magnetic field. Unlike a capacitor (which stores energy in an electric field), an inductor's stored energy depends on current, not voltage.

Energy storage formula

The energy stored in an inductor is:

$W = \frac{1}{2}LI^2$

where $W$ is the energy in joules, $L$ is the inductance in henries, and $I$ is the current in amperes.

This can be derived by integrating the instantaneous power $P = Vi = Li\frac{di}{dt}$ delivered to the inductor as the current ramps from zero to $I$:

$W = \int_0^I Li \, di = \frac{1}{2}LI^2$

Notice the energy scales with $I^2$: doubling the current quadruples the stored energy.

Energy density in inductors

The energy stored per unit volume inside the magnetic field of an inductor can be expressed as:

$u = \frac{B^2}{2\mu}$

where $B$ is the magnetic flux density and $\mu$ is the permeability of the medium. This is a general result for magnetic field energy density and applies beyond just inductors. For a solenoid with a uniform internal field $B = \mu n I$, you can verify that integrating $u$ over the solenoid volume recovers $W = \frac{1}{2}LI^2$.

Inductor energy applications

• Switch-mode power supplies (SMPS): The inductor stores energy during the "on" phase of the switching transistor and delivers it to the load during the "off" phase, enabling efficient DC-DC conversion.
• LC filters in renewable energy systems: Inductors paired with capacitors smooth the output of solar inverters and wind turbine generators, reducing voltage and current ripple.
• Pulsed power systems: Radar transmitters and particle accelerators charge inductors slowly, then discharge the stored energy in short, high-power bursts.

RL circuits

RL circuits contain resistors and inductors and exhibit time-dependent (transient) behavior when voltages or currents change. They're governed by first-order linear differential equations.

RL circuit analysis

Consider a series RL circuit driven by a DC voltage source $V_0$ switched on at $t = 0$. Applying Kirchhoff's voltage law:

$V_0 = iR + L\frac{di}{dt}$

Solving this first-order ODE gives the current as a function of time:

$i(t) = \frac{V_0}{R}\left(1 - e^{-t/\tau}\right)$

where $\tau = L/R$ is the time constant. The voltage across the inductor decays exponentially:

$V_L(t) = V_0 \, e^{-t/\tau}$

For the decay case (source removed, inductor discharging through $R$):

$i(t) = I_0 \, e^{-t/\tau}$

where $I_0$ is the initial current.

RL time constant

The time constant $\tau = \frac{L}{R}$ sets the timescale for transient behavior:

• At $t = \tau$, the current has reached about 63.2% of its final value (charging) or decayed to about 36.8% of its initial value (discharging).
• After $5\tau$, the transient is essentially complete (current within ~0.7% of its steady-state value).

A larger inductance means a longer time constant (the inductor "resists" the change more). A larger resistance means a shorter time constant (the resistor dissipates energy faster, so the transient dies out sooner).

RL circuit applications

• DC motor modeling: The armature winding of a DC motor behaves as an RL circuit. The time constant $\tau$ determines how quickly the motor current (and therefore torque) responds to voltage changes.
• Audio crossover networks: RL filters separate audio signals into frequency bands directed to appropriate speaker drivers (tweeters, woofers).
• Switching power supply output stages: RL filtering at the output smooths the pulsed waveform into a steady DC voltage with low ripple.

Inductance in AC circuits

When driven by a sinusoidal source, inductors produce a voltage that leads the current by 90°. This phase relationship and the frequency-dependent opposition to current flow are central to AC circuit analysis.

Inductive reactance

Inductive reactance is the effective opposition an inductor presents to AC current:

$X_L = \omega L = 2\pi f L$

where $f$ is the frequency in Hz, $\omega = 2\pi f$ is the angular frequency, and $L$ is the inductance. The units are ohms (Ω).

Unlike resistance, inductive reactance depends on frequency. At DC ($f = 0$), $X_L = 0$ and the inductor acts like a short circuit (just a wire). At high frequencies, $X_L$ becomes very large and the inductor blocks current. This frequency dependence is what makes inductors useful as filters.

Inductance in series AC circuits

In a series RL circuit driven by an AC source, the impedance is:

$Z = R + jX_L = R + j\omega L$

The magnitude of the impedance is:

$|Z| = \sqrt{R^2 + X_L^2}$

The phase angle between the source voltage and the current is:

$\phi = \arctan\left(\frac{X_L}{R}\right)$

The current lags the applied voltage by this angle $\phi$. For a series circuit with multiple inductors (no mutual coupling), the total reactance is $X_{L,\text{total}} = X_{L1} + X_{L2} + \cdots$

Inductance in parallel AC circuits

For a parallel RL circuit, it's often easier to work with admittance $Y = 1/Z$. The total admittance is:

$Y = \frac{1}{R} - \frac{j}{X_L} = G - jB_L$

where $G = 1/R$ is the conductance and $B_L = 1/X_L$ is the inductive susceptance. The magnitude of the impedance for a parallel RL combination is:

$|Z| = \frac{R \cdot X_L}{\sqrt{R^2 + X_L^2}}$

In parallel circuits with multiple inductors, the individual susceptances add.

Coupled inductors

Coupled inductors share magnetic flux between two or more coils. The degree of coupling determines how effectively energy transfers from one coil to another, which is the operating principle behind transformers.

Coupling coefficient

The coupling coefficient $k$ measures what fraction of one inductor's flux links with the other:

$k = \frac{M}{\sqrt{L_1 L_2}}$

• $k = 0$: No shared flux (completely independent inductors).
• $k = 1$: All flux from one coil passes through the other (ideal, perfect coupling).
• In practice, $k$ falls between 0 and 1. Well-designed power transformers with high-permeability cores achieve $k > 0.99$. Air-core coupled coils might have $k$ values of 0.01 to 0.5 depending on geometry and spacing.

The quantity $k^2$ represents the fraction of magnetic energy in one coil that is coupled to the other.

Transformer basics

A transformer consists of two (or more) coils wound on a shared magnetic core. For an ideal transformer ($k = 1$, no losses):

$\frac{V_s}{V_p} = \frac{N_s}{N_p}$

$\frac{I_p}{I_s} = \frac{N_s}{N_p}$

where subscripts $p$ and $s$ denote primary and secondary, and $N$ is the number of turns. A step-up transformer has $N_s > N_p$ (increases voltage, decreases current). A step-down transformer has $N_s < N_p$.

Power is conserved in an ideal transformer: $V_p I_p = V_s I_s$. Real transformers have losses from core hysteresis, eddy currents, and winding resistance, but well-designed power transformers can exceed 95% efficiency.

Coupled inductor applications

• Flyback and forward converters (SMPS): Coupled inductors provide electrical isolation between input and output while enabling efficient energy transfer.
• RF impedance matching: Coupled coils in RF transformers match source and load impedances to maximize power transfer.
• Common-mode chokes: Two windings on a single core cancel differential-mode signals while presenting high impedance to common-mode noise, reducing electromagnetic interference (EMI).
• Baluns: Balanced-to-unbalanced converters use coupled inductors to interface between balanced transmission lines and unbalanced circuits (e.g., connecting a dipole antenna to coaxial cable).

Inductance measurement

Accurate inductance measurement is necessary for verifying component specifications and troubleshooting circuits. Several methods exist, each suited to different accuracy requirements and frequency ranges.

Inductance measurement methods

• LCR meters: Apply a small AC test signal at a selectable frequency and directly read out $L$, along with resistance and quality factor $Q$. This is the most common bench method.
• Impedance analyzers: Sweep across a wide frequency range, measuring complex impedance at each point. Useful for characterizing inductors that will operate over a range of frequencies.
• AC bridge methods: The Maxwell bridge and Hay bridge compare an unknown inductance against known reference components to achieve high-accuracy measurements.
• Resonant method: Place the unknown inductor in a series or parallel LC circuit with a known capacitor. Measure the resonant frequency $f_0$, then calculate:

$L = \frac{1}{(2\pi f_0)^2 C}$

This method is particularly useful at RF frequencies where direct impedance measurement becomes difficult.

Inductance meters

Dedicated LCR meters (handheld or benchtop) are the standard tool for inductance measurement. Key features to consider:

• Test frequency selection: The measured inductance can vary with frequency due to core losses, parasitic capacitance, and skin effect. Choosing a test frequency close to the inductor's operating frequency gives the most relevant result.
• DC bias capability: Some meters can apply a DC bias current during measurement, which is important for characterizing inductors under realistic operating conditions (since ferromagnetic cores can saturate at high currents, reducing the effective inductance).
• Accuracy and range: Benchtop instruments typically offer better accuracy (0.1% or better) and wider measurement ranges than handheld units.