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

$V_L+V_C+V_R=0$

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

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

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

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

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

$s=-\alpha \pm \lambda$

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

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

• ${\alpha }^2>{\beta }^2$ ,

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

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

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	$A=e^{(}-\alpha t)$
$\alpha$	$\alpha =\frac{R}{2L}$
$\beta$	$\beta =\frac{1}{LC}$