Chapter 3

# Magnetic energy of a two-circuits system

### Fondamental : Ohm'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 :

### Fondamental : Magnetic 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.

### Remarque : Ideal 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.

### Exemple : Exercises on coupled circuits

Consider this circuit :

Coupled circuits

Intensities and voltages are linked :

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.

### Remarque : Another 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 :

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