Inductance is a fundamental property of an electrical conductor, which quantifies its ability to store energy in a magnetic field when an electric current is flowing through it. Inductance is typically represented by the symbol “L” and is measured in units called henrys (H).

When a current flows through a conductor, it generates a magnetic field around it. If the current changes, the magnetic field also changes, inducing an electromotive force (EMF) or voltage across the conductor, which opposes the change in current. This phenomenon is known as electromagnetic induction and is the basis for the concept of inductance.

## Two types of inductance

- Self-inductance: Self-inductance refers to the inductance of a single conductor or coil, where the changing magnetic field generated by the current flowing through the conductor induces a voltage across the conductor itself. This voltage, known as self-induced EMF, opposes any change in the current.

The self-inductance of a coil is primarily determined by its shape, size, the number of turns in the coil, and the core material (if any) around which the coil is wound.

- Mutual inductance: Mutual inductance occurs when two or more conductors or coils are placed in proximity, and the changing magnetic field generated by the current flowing through one conductor induces a voltage across the other conductor(s). This voltage, known as mutually induced EMF, depends on the relative orientation and distance between the conductors and their individual inductance.

## Mutual Inductance

Mutual induction is a phenomenon in which a change in the current flowing through one coil (called the primary coil) induces an electromotive force (EMF) in another nearby coil (called the secondary coil). This occurs due to the magnetic coupling between the coils, as the magnetic field generated by the primary coil interacts with the turns of the secondary coil.

Mutual inductance (M) is a measure of the effectiveness of this magnetic coupling between the two coils. It is defined as the ratio of the induced EMF in the secondary coil to the rate of change of current in the primary coil:

EMF_secondary = -M * (dI_primary / dt)

Here, M is the mutual inductance, measured in henries (H), and (dI_primary / dt) is the rate of change of current in the primary coil.

The mutual inductance between two coils depends on factors such as the number of turns in each coil, the distance between the coils, the geometry and orientation of the coils, and the core material (if any) shared by the coils.

For two solenoid-shaped coils with a shared core, the mutual inductance can be calculated using the following formula:

M = μ * N1 * N2 * A / l

where:

M = Mutual inductance (H)

μ = Permeability of the core material (H/m)

N1 = Number of turns in the primary coil

N2 = Number of turns in the secondary coil

A = Cross-sectional area of the core (m^2)

l = Length of the coils (m)

It’s important to note that the formula mentioned above is an approximation and assumes that the coils have the same geometry, are closely wound, and share a common axis. It also assumes that the magnetic field is confined to the core material and does not account for leakage flux. For other coil geometries or when the coils are not closely coupled, the calculation of mutual inductance can be more complex and might require numerical methods or finite element analysis.

Mutual induction is the fundamental principle behind transformers, which are used to step up or step down AC voltages in various applications, such as power transmission and signal isolation. The efficiency of a transformer depends on the degree of magnetic coupling between the primary and secondary coils, which is directly related to the mutual inductance.