Electronics Handbook/Components/RLC Network

Series RLC

Circuit of three components R, L, C connected in series

Differential Equation of the circuit

At equilibrium the total voltage of the circuit is zero

$V_{L}+V_{C}+V_{R}=0$

$L{\frac {dI}{dt}}+IR+{\frac {1}{C}}\int Idt=0$

${\frac {dI}{dt}}+I{\frac {R}{L}}+{\frac {1}{LC}}=0$

${\frac {d^{2}I}{dt^{2}}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0$

$s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0$

$s=-\alpha$ ± ${\sqrt {\alpha ^{2}-\beta ^{2}}}$

$s=-\alpha \pm \lambda$

With

$\alpha ={\frac {R}{2L}}$ . $\beta ={\frac {1}{LC}}$

When

• $\alpha ^{2}=\beta ^{2}$ .

${\frac {R}{2L}}$ = ${\frac {1}{LC}}$

$R={\sqrt {\frac {L}{C}}}$

The equation has only one real root

s = -α = ${\frac {R}{2L}}$

$I=Ae^{(}-{\frac {R}{2L}})t$

• $\alpha ^{2}>\beta ^{2}$ ,

${\frac {R}{2L}}$ = ${\frac {1}{LC}}$

$R>{\sqrt {\frac {L}{C}}}$

The equation has two real roots

$s=-\alpha \pm \lambda$

$I(t)=e^{(}-\alpha \pm \lambda )t$

$I(t)=e^{(}-\alpha t)(e^{\lambda }t+e^{-}\lambda t)$

$I(t)={\frac {e^{(}-\alpha t)}{2}}Cos\lambda t$

• $\alpha ^{2}=\beta ^{2}$ .

$R<{\sqrt {\frac {L}{C}}}$

The equation has two complex roots

$s=-\alpha$ + $j\lambda$

$I(t)=e^{(}-\alpha t)(e^{j}\lambda t+e^{-}j\lambda t)$

$I(t)={\frac {e^{(}-\alpha t)}{2j}}Sin\lambda t$

Resonance

Resonance occurs when the frequencies components cancels out . Therefore, at resonance

$Z=Z_{R}+Z_{C}+Z_{L}=Z_{R}+0=R$

$I={\frac {V}{R}}$

$Z_{L}=Z_{C}$ . $\omega L={\frac {1}{\omega C}}$ . $\omega ={\sqrt {\frac {1}{LC}}}$

• At $\omega =0I=0$
• At $\omega ={\sqrt {\frac {1}{LC}}}$ . $I={\frac {V}{R}}$
• At $\omega =00I=0$

Impedance

$Z=Z_{R}+Z_{L}+Z_{C}$

$Z=R+j\omega L+{\frac {1}{j\omega C}}$

$Z={\frac {1}{j\omega C}}(j\omega ^{2}+\alpha j\omega +\beta )$

$\alpha ={\frac {R}{L}}$

$\beta ={\frac {1}{Lc}}$

Summary

• The resonance response is in the type of Tuned Resonance Band Pass Filter
• The natural response is a wave of decreasing magnitude

$R={\sqrt {\frac {L}{C}}}$ . One real root . $I(t)=e^{(}-{\frac {R}{2L}})t$

$R={\sqrt {\frac {L}{C}}}$ . Two real roots . $I(t)=e^{(}-{\frac {R}{2L}})t[e^{(}\lambda t)+e^{(}-\lambda t)]$

$R={\sqrt {\frac {L}{C}}}$ . Two complex roots . $I(t)=e^{(}-{\frac {R}{2L}})t[e^{(}j\lambda t)+e^{(}-j\lambda t)]$

Circuit	Series RLC
Configuration
Impedance	$Z={\frac {1}{j\omega C}}(j\omega ^{2}+j\omega {\frac {R}{L}}+{\frac {1}{LC}})$
Differential Equation	$L{\frac {dI}{dt}}+{\frac {1}{C}}\int Vdt+IR=0$
General Differential Equation	${\frac {d^{2}I}{dt}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0$
Natural Equation	$I(t)=A(e^{\lambda }t+e^{-}\lambda t)$
$\lambda$	$\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}$ . $\lambda =0.\alpha ^{2}=\beta ^{2}.({\frac {R}{2L}})^{2}=({\frac {1}{LC}})^{2}$ $I=e^{-}\alpha t=e^{(}-{\frac {R}{2L}})t$ . $\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}>0.\alpha ^{2}>\beta ^{2}.({\frac {R}{2L}})^{2}>({\frac {1}{LC}})^{2}$ $I=A(e^{\lambda }t+e^{-}\lambda t)$ . $\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}<0.\alpha ^{2}<\beta ^{2}.({\frac {R}{2L}})^{2}({\frac {1}{LC}})^{2}$ $I=A(e^{j}\lambda t+e^{-}j\lambda t)$
A	$A=e^{(}-\alpha t)$
$\alpha$	$\alpha ={\frac {R}{2L}}$
$\beta$	$\beta ={\frac {1}{LC}}$