Inductance - Coupled Inductors

Coupled Inductors

Mutual inductance occurs when the change in current in one inductor induces a voltage in another nearby inductor. It is important as the mechanism by which transformers work, but it can also cause unwanted coupling between conductors in a circuit.

The mutual inductance, M, is also a measure of the coupling between two inductors. The mutual inductance by circuit i on circuit j is given by the double integral Neumann formula, see calculation techniques

The mutual inductance also has the relationship:

where

is the mutual inductance, and the subscript specifies the relationship of the voltage induced in coil 2 due to the current in coil 1.

N₁ is the number of turns in coil 1,

N₂ is the number of turns in coil 2,

P₂₁ is the permeance of the space occupied by the flux.

The mutual inductance also has a relationship with the coupling coefficient. The coupling coefficient is always between 1 and 0, and is a convenient way to specify the relationship between a certain orientation of inductors with arbitrary inductance:

where

k is the coupling coefficient and 0 ≤ k ≤ 1,

L₁ is the inductance of the first coil, and

L₂ is the inductance of the second coil.

Once the mutual inductance, M, is determined from this factor, it can be used to predict the behavior of a circuit:

where

V₁ is the voltage across the inductor of interest,

L₁ is the inductance of the inductor of interest,

dI₁/dt is the derivative, with respect to time, of the current through the inductor of interest,

dI₂/dt is the derivative, with respect to time, of the current through the inductor that is coupled to the first inductor, and

M is the mutual inductance.

The minus sign arises because of the sense the current I₂ has been defined in the diagram. With both currents defined going into the dots the sign of M will be positive.

When one inductor is closely coupled to another inductor through mutual inductance, such as in a transformer, the voltages, currents, and number of turns can be related in the following way:

where

V_s is the voltage across the secondary inductor,

V_p is the voltage across the primary inductor (the one connected to a power source),

N_s is the number of turns in the secondary inductor, and

N_p is the number of turns in the primary inductor.

Conversely the current:

where

I_s is the current through the secondary inductor,

I_p is the current through the primary inductor (the one connected to a power source),

N_s is the number of turns in the secondary inductor, and

N_p is the number of turns in the primary inductor.

Note that the power through one inductor is the same as the power through the other. Also note that these equations don't work if both transformers are forced (with power sources).

When either side of the transformer is a tuned circuit, the amount of mutual inductance between the two windings determines the shape of the frequency response curve. Although no boundaries are defined, this is often referred to as loose-, critical-, and over-coupling. When two tuned circuits are loosely coupled through mutual inductance, the bandwidth will be narrow. As the amount of mutual inductance increases, the bandwidth continues to grow. When the mutual inductance is increased beyond a critical point, the peak in the response curve begins to drop, and the center frequency will be attenuated more strongly than its direct sidebands. This is known as overcoupling.