Chapter 12

# Study of a RLC circuit - Impedances

## Remarque : Importance of sinusoidal currents

• Examples of sinusoidal voltages : voltage of sector ( ) – High voltage lines - The transmission and reception of radio and television signals involve currents vary sinusoidally in time, ...

• Fourier analysis : it shows that any periodic voltage is a sum of sinusoidal functions ; so, if we know how a circuit reacts to a sinusoidal excitation, then (by superposition) the response of this circuit to any periodic voltage is known.

## Fondamental : Intensity in a series RLC circuit

Differential equation circuit (RLC) is (see lecture on transient regime) :

Notations :

The intensity has (once disappeared transient regime) the same pulse as the excitation (low - frequency generator) :

Where is the phase shift over .

Recall that :

: is behind .

In french we say : "Il est toujours positif d'être en avance !" ("It is positive to be ahead !").

Study of a RLC circuit

The figure below shows, by numerical solution of the equation :

the shape of the voltage across the capacitor.

We see the emergence of sinusoidal permanent regime after the disappearance of the transitional regime.

Appearance of permanent sinusoidal regime.

"Resolution" of the series circuit (RLC) in complex notation :

The writing of the KVL in the series RLC circuit, in complex notation and using the notion of impedances :

Whence :

Or :

Where is the impedance of the series circuit (RLC), sum of the impedances of each of its constituents.

We recall the notation :

So :

If we note :

So :

The maximum intensity is therefore :

The argument of the complex impedance cheks :

The phase shift  is known by the relations :

We denote the pulse for which ( and are in phase) :

•  If : the circuit is capacitive and ( is ahead of ).

• If  : the circuit is inductive and ( is behind , we find the effect of Lenz's law).

• For  : there is intensity resonance. The intensity is maximum and :

The following video (by Alain Le Rille) highlights the phenomenon of resonance in a series RLC circuit.

Circuit (RLC) (Vidéo d'Alain Le Rille, enseignant en CPGE au lycée Janson de Sailly, Paris)

## Fondamental : "Solve" a circuit in sinusoidal regime

In complex notation, we can write :

• Across a dipole of impedance (admittance ) :

• Across a complex emf generator and complex internal impedance :

• Across a short-circuit current of the current generator and complex internal admittance :

Thus, we obtained for a linear network in forced sinusoidal regime, expressions identical to those obtained in continuous regime.

The impedances ( and admittances) take the place of resistances (and conductances).

Can be used :

Kirchhoff's Laws (KCL and KVL), the law of voltage and current dividers, series and parallel associations of dipoles, the transform from Thevenin to Norton's representation, ...

## Complément : Average values and RMS (root mean square) values

• The average value of the intensity of an electric current is :

• The RMS (root mean square) intensity is by definition :

Or :

The RMS intensity is the intensity of a direct current that dissipated in a resistor , in a period, the same energy as the AC :

Taking the example of a current sawtooth : considering given by the curve below.

Calculate the average intensity and the RMS intensity of this current sawtooth.

We find :

And :

We recall that for a sinusoidal current,  :

And :