Study of a RLC circuit - Impedances
Remarque : Importance of sinusoidal currents
Examples of sinusoidal voltages : voltage of sector (\(50 \;Hz \;– \;220\; V\)) – 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) :
\({u_L} + {u_R} + {u_C} = L\frac{{di}}{{dt}} + Ri + \frac{1}{C}q = e(t)\)
Notations :
\(e(t) = {E_m}\cos \omega t = {E_{eff}}\sqrt 2 \cos \omega t\)
The intensity \(i(t)\) has (once disappeared transient regime) the same pulse as the excitation (low - frequency generator) :
\(i(t) = {I_m}\cos (\omega t + \varphi ) = {I_{eff}}\sqrt 2 \cos (\omega t + \varphi )\)
Where \(\varphi\) is the \(i(t)\) phase shift over \(e(t)\).
Recall that :
\(\varphi >0\) : \(i(t)\) is ahead of \(e(t)\).
\(\varphi<0\) : \(i(t)\) is behind \(e(t)\).
In french we say : "Il est toujours positif d'être en avance !" ("It is positive to be ahead !").
The figure below shows, by numerical solution of the equation :
\(\frac{{{d^2}{u_C}}}{{d{t^2}}} + 2\sigma {\omega _0}\frac{{d{u_C}}}{{dt}} + {\omega _0}^2{u_C} = {\omega _0}^2{E_m}\cos \omega t\)
the shape of the voltage \(u_C\) across the capacitor.
We see the emergence of sinusoidal permanent regime after the disappearance of the transitional 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 :
\({\underline u _R} + {\underline u _L} + {\underline u _C} = R\underline i + jL\omega \;\underline i + \frac{1}{{jC\omega }}\underline i = \underline e\)
Whence :
\(\underline i = \frac{{\underline e }}{{R + jL\omega + \frac{1}{{jC\omega }}}} = \frac{{\underline e }}{{R + j\left( {L\omega - \frac{1}{{C\omega }}} \right)}}\)
Or :
\(\underline i = \frac{{\underline e }}{{\underline z }}\;\;\;\;with\;\;\;\;\underline z = R + j\left( {L\omega - \frac{1}{{C\omega }}} \right)\)
Where \(\underline z\) is the impedance of the series circuit (RLC), sum of the impedances of each of its constituents.
We recall the notation :
\(\underline i = {I_m}{e^{j(\omega t + \varphi )}}\;\;\;\;and\;\;\;\;\;\;\;\underline e = {E_m}{e^{j\omega t}}\)
So :
\({I_m}{e^{j(\omega t + \varphi )}} = \frac{{{E_m}{e^{j\omega t}}}}{{R + j\left( {L\omega - \frac{1}{{C\omega }}} \right)}}\;\;\;\;\;or\;\;\;\;\;{I_m}{e^{j\varphi }} = \frac{{{E_m}}}{{R + j\left( {L\omega - \frac{1}{{C\omega }}} \right)}}\)
If we note :
\(\underline z = R + j\left( {L\omega - \frac{1}{{C\omega }}} \right) = \left| {\underline z } \right|{e^{j\theta }} = Z{e^{j\theta }}\)
So :
\({I_m}{e^{j\varphi }} = \frac{{{E_m}}}{{Z{e^{j\theta }}}} = \frac{{{E_m}}}{Z}{e^{ - j\theta }}\;\;\;\;\;so\;\;\;\;\;\varphi = - \theta \;\;\;\;and\;\;\;\;{I_m} = \frac{{{E_m}}}{Z}\)
The maximum intensity is therefore :
\({I_m} = \frac{{{E_m}}}{{\sqrt {{R^2} + {{\left( {L\omega - \frac{1}{{C\omega }}} \right)}^2}} }}\)
The argument \(\theta\) of the complex impedance \(\underline z\) cheks :
\(\cos \theta = \frac{R}{Z}\;\;\;;\;\;\;\sin \theta = \frac{{L\omega - \frac{1}{{C\omega }}}}{Z}\;\;\;;\;\;\;\tan \theta = \frac{{L\omega - \frac{1}{{C\omega }}}}{R}\)
The phase shift \(\varphi=-\theta\) is known by the relations :
\(\tan \varphi = - \frac{{L\omega - \frac{1}{{C\omega }}}}{R}\;\;\;\;and\;\;\;\;\cos \varphi = \cos \theta > 0\)
We denote \(\omega_0=1/\sqrt{LC}\) the pulse for which \(\varphi=0\) (\(e(t)\) and \(i(t)\) are in phase) :
If \(\omega<\omega_0\) : the circuit is capacitive and \(\varphi>0\) (\(i(t)\) is ahead of \(e(t)\)).
If \(\omega>\omega_0\) : the circuit is inductive and \(\varphi<0\) (\(i(t)\) is behind \(e(t)\), we find the effect of Lenz's law).
For \(\omega=\omega_0\) : there is intensity resonance. The intensity is maximum \(I_m\) and :
\(I_m=\frac{E_m}{R}\)
The following video (by Alain Le Rille) highlights the phenomenon of resonance in a series RLC circuit.
Fondamental : "Solve" a circuit in sinusoidal regime
In complex notation, we can write :
Across a dipole of impedance \(\underline z\) (admittance \(\underline y = 1/\underline z\)) :
\(\underline u = \underline z \underline i \;\;\;\;or\;\;\;\;\underline i = \underline y \underline u\)
Across a complex emf generator \(\underline e\) and complex internal impedance \(\underline z_G\) :
\(\underline u_G=\underline e - \underline z_G \underline i\)
Across a short-circuit current of the current generator \(i_{cc}\) and complex internal admittance \(\underline y_G\) :
\(\underline i = \underline i_{cc}-\underline y_G \underline u\)
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 :
\({I_{moy}} = \left\langle {i(t)} \right\rangle = \frac{1}{T}\int_0^T {i(t)dt} = \frac{Q}{T}\)
The RMS (root mean square) intensity \(I_{eff}\) is by definition :
\({I^2}_{eff} = \left\langle {{i^2}(t)} \right\rangle = \frac{1}{T}\int_0^T {{i^2}(t)dt}\)
Or :
\(I_{eff}=\sqrt{\frac{1}{T}\int_0^T {{i^2}(t)dt}}\)
The RMS intensity is the intensity of a direct current that dissipated in a resistor \(R\), in a period, the same energy as the AC :
\(P = \int_0^T {R{i^2}(t)dt} = RI_{eff}^2T\)
Taking the example of a current sawtooth : considering \(i(t)\) given by the curve below.
Calculate the average intensity and the RMS intensity of this current sawtooth.
We find :
\(I_{moy}=0\)
And :
\(I_{eff}=\frac{I_0}{\sqrt{3}}\)
We recall that for a sinusoidal current, \(i(t)=I_mcos\omega t\) :
\(I_{moy}=0\)
And :
\(I_{eff}=\frac{I_m}{\sqrt{2}}\)