Electronics/RCL

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:

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

or

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

$s^2 + \frac{R}{L} s + \frac{1}{LC} = 0$
$s = -\alpha$ ± $\sqrt{\alpha^2 - \beta^2}$

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 generates Standing wave when it operates in resonance . At Resonance

$Z_L = Z_C$
$\omega L = \frac{1}{\omega C}$
$\omega = \sqrt{\frac{1}{LC}}$

R ≠ 0 ZL = ZC

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

ω = 0 . Capacitor opens circuit . I = 0
ω = oo . Inhductor opens circuit . I = 0

Plot the three value of I at three ω above we have a graph I - ω . 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 halved then circuit has a stable current $I = \frac{V}{2R}$does not change with frequency over a Bandwidth of frequencies ω1 - ω2 . 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