6.2 Inductors and inductance

Inductor Fundamentals

An inductor is a coil of wire that stores energy in a magnetic field when current flows through it. While capacitors store energy in electric fields, inductors do the same with magnetic fields. Their key property is resisting changes in current, which makes them critical in power supplies, radio tuning circuits, and many other applications.

Inductor Components and Characteristics

An inductor can be as simple as a coil of wire wound around air, or it can use a core material like iron or ferrite. Core materials with high magnetic permeability concentrate the magnetic field lines, which significantly increases the inductance.

Inductance (symbol $L$) measures how effectively an inductor stores energy in its magnetic field. It's measured in henries (H), named after Joseph Henry, who discovered self-inductance.

$1 \text{ H} = 1 \text{ V·s/A}$

In plain terms: an inductor has 1 henry of inductance if a current changing at 1 ampere per second induces 1 volt across it.

Inductance depends on three main factors:

• Number of turns in the coil (more turns = higher inductance)
• Cross-sectional area of the coil (larger area = higher inductance)
• Permeability of the core material (higher permeability = higher inductance)

Inductor Applications

• Power supply filtering: Smoothing inductors filter out AC ripple, helping deliver steady DC output
• Radio/TV tuning: Tuning inductors pair with capacitors to form resonant circuits that select specific frequencies
• Energy storage: Switched-mode power supplies and boost converters rely on inductors to temporarily store and release energy
• Current limiting: Transformers and electric motors use inductors to control current flow

Magnetic Fields and Induction

Magnetic Field Properties

A magnetic field is the region around a magnet or current-carrying conductor where magnetic forces act. You can visualize it using magnetic field lines, which show both the direction and relative strength of the field. These lines always form closed loops and never cross each other. Magnetic field strength is measured in teslas (T).

When a magnetic field changes over time near a conductor, it induces a voltage in that conductor. This is electromagnetic induction, discovered by Michael Faraday in 1831. The induced voltage is proportional to how fast the magnetic flux is changing.

Faraday's law of induction:

$\mathcal{E} = -N \frac{d\Phi}{dt}$

where $\mathcal{E}$ is the induced EMF, $N$ is the number of turns, and $\Phi$ is the magnetic flux through the coil.

Lenz's Law

The negative sign in Faraday's law comes from Lenz's law: the induced EMF always acts in a direction that opposes the change causing it.

• If the magnetic flux through a coil is increasing, the induced current creates its own magnetic field pointing opposite to the increase.
• If the flux is decreasing, the induced current creates a field that tries to maintain it.

This opposition is what gives inductors their characteristic resistance to current changes. You can also see Lenz's law at work in eddy current brakes (used in trains and roller coasters) and electromagnetic damping in sensitive instruments like galvanometers.

Types of Inductance

Self-Inductance

Self-inductance is what most people mean when they say "inductance." It describes how an inductor opposes changes in its own current. As current through the coil changes, the magnetic field it creates also changes, which induces a voltage back across the coil itself.

The voltage across an inductor is:

$V_L = L \frac{dI}{dt}$

This tells you that the voltage is proportional to how quickly the current is changing. If the current is steady ($dI/dt = 0$), the voltage across an ideal inductor is zero. Rearranging gives the definition of self-inductance:

$L = \frac{V_L}{dI/dt}$

The three factors that control self-inductance are the same ones from earlier: number of turns, coil area, and core permeability. For a solenoid (a tightly wound coil), these combine into:

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

where $\mu$ is the core permeability, $N$ is the number of turns, $A$ is the cross-sectional area, and $l$ is the length of the coil.

Mutual Inductance

Mutual inductance happens when the changing magnetic field from one inductor induces a voltage in a separate, nearby inductor. It's represented by $M$ and also measured in henries.

The voltage induced in the secondary coil is:

$V_2 = M \frac{dI_1}{dt}$

where $I_1$ is the current in the primary coil. The value of $M$ depends on how much magnetic flux from the first coil actually links through the second, which is determined by their geometry, spacing, and orientation.

Key applications of mutual inductance:

• Transformers step AC voltages up or down by using tightly coupled coils with different turn counts
• Coupled inductors in filter circuits and impedance matching networks
• Wireless power transfer, such as smartphone charging pads, where energy transfers between coils without direct contact