Electronics/RCL

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RLC Series

An RLC series circuit consists of a resistor, inductor, and capacitor connected in series:

By Kirchhoff's voltage law the differential equation for the circuit is:

$L\frac{dI}{dt}+IR+\frac{1}{C}\int Idt=V(t)$

or

$L\frac{d^{2}I}{d{t}^2}+R\frac{dI}{dt}+\frac{I}{C}=\frac{dV}{dt}$

Leading to:

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

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

with

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

There are three cases to consider, each giving different circuit behavior, ${\alpha }^2={\beta }^2,{\alpha }^2>{\beta }^2,or{\alpha }^2<{\beta }^2$ .

${\alpha }^2={\beta }^2$ .

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

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

Equation above has only one real root

s = -α = $\frac{R}{2L}$

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

${\alpha }^2>{\beta }^2$ ,

$\frac{R}{2L}$ > $\sqrt{\frac{1}{LC}}$

$R>2\sqrt{\frac{L}{C}}$

Equation above has only two real roots

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

$I=e^{(}-\alpha +\sqrt{{\alpha }^2-{\beta }^2})t+e^{-}(\alpha +\sqrt{{\alpha }^2-{\beta }^2})t$

$I=e^{(}-\alpha )e(\sqrt{{\alpha }^2-{\beta }^2})t-e^{-}(\sqrt{{\alpha }^2-{\beta }^2})t$

${\alpha }^2<{\beta }^2$ .

$R<2\sqrt{\frac{L}{C}}$

Equation above has only two complex roots

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

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

$I=e^j(-\alpha +\sqrt{{\beta }^2-{\alpha }^2})t+e^j(-\alpha +\sqrt{{\beta }^2-{\alpha }^2})t$

Circuit Analysis

R = 0

If R = 0 then the RLC circuit will reduce to LC series circuit . LC circuit will generate a standing wave when it operates in resonance; At Resonance the conditions rapidly convey in a steady functional method.

$Z_L=Z_C$

$\omega L=\frac{1}{\omega C}$

$\omega =\sqrt{\frac{1}{LC}}$

R = 0 Z_L = Z_C

If R = 0 and circuit above operates in resonance then the total impedance of the circuit is Z = R and the current is V / R

At Resonance

$Z_L+Z_C=0$ Or $Z_L=Z_C$

$\omega L=\frac{1}{\omega C}$

$\omega =\sqrt{\frac{1}{LC}}$

$Z=Z_R+Z_L+Z_C=R+0=R$

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

At Frequency

I = 0 . Capacitor opens circuit . I = 0

I = 0 Inductor opens circuit . I = 0

Plot the three value of I at three I above we have a graph I - 0 At Resonance frequency $\omega =\sqrt{\frac{1}{LC}}$ the value of current is at its maximum $I=\frac{V}{R}$ . If the value of current is half then circuit has a stable current $I=\frac{V}{2R}$does not change with frequency over a Bandwidth of frequencies É1 - É2 . When increase current above $I=\frac{V}{2R}$ circuit has stable current over a Narrow Bandwidth . When decrease current below $I=\frac{V}{2R}$ circuit has stable current over a Wide Bandwidth

Thus the circuit has the capability to select bandwidth that the circuit has a stable current when circuit operates in resonance therefore the circuit can be used as a Resonance Tuned Selected Bandwidth Filter

Further Reading

1. RCL circuit analysed in the time domain
2. RCL circuit analysed in the frequency domain