Chapter 3

Electromagnetic induction

Magnetic energy of a two-circuits system

FondamentalOhm's generalized law

Two wire circuits and are in mutual coupling.

So in the absence of other sources of magnetic field :

is the total magnetic flux through the circuit and  the total magnetic flux through the circuit .

Interacting circuits

If the two circuits are rigid and still in the laboratory referential, the inductive electromotive forces are :

The voltage differences between the frames of each circuit are :

FondamentalMagnetic energy of the two-circuits system

The electric power received by the two circuits is :

So :

Finally :

The power dissipated by Joule effect can be recognized :

Another energy can be defined :

This is the magnetic energy of the two-circuits system, in the absence of other sources of magnetic field.

The convention used is that this energy is equal to zero when the currents are equal to zero.

AttentionEnergy of the two circuits system

RemarqueIdeal coupling

It is proven that :

We define :

defines the coupling coefficient between the two circuits.

This coefficient has a value ranged between and .

The case , or , corresponds to the situation where all the magnetic field lines created by one of the two circuits cross the other circuit.

It is the ideal case of perfect coupling.

ExempleExercises on coupled circuits

Consider this circuit :

Coupled circuits

Intensities and voltages are linked :

Kirchhoff's law :

Hence :

Proper modes research :

Suppose (free running).

The solutions researched are harmonic and of same pulsation .

So :

Hence the homogenous system :

This system has one non trivial solution if its determinant is equal to zero :

So :

Hence the two own pulses (with ) :

For the first proper mode, : the two voltages oscillate in accordance.

For the second proper mode, : the two voltages oscillate in opposite accordance. 

The free running corresponds to the linear addition of these two proper modes.

RemarqueAnother definition of self-inductance of a circuit

The circuit is a wire loop (or not) without any interaction with another circuit.

To expand the definition of , we can identify the two expressions of the magnetic energy, hence :

The Laplace rails
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