Electronics/RCL

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RLC Series

RLC series cicuit consists of Resistor, Inductor and Capacitor connect in series

The Differential Equation for the circuit above is

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

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

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

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

s = − α ± $\sqrt{\alpha^2 - \beta^2}$

With

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

α² = β² .

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

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

Equation above has only one real root

s = -α = $\frac{R}{2L}$

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

α² > β² ,

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

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

Equation above has only two real roots

s = − α ± $\sqrt{\alpha^2 - \beta^2}$

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

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

α² = β² .

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

Equation above has only two complex roots

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

s = − α - 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 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

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

Further Reading

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