Electronics/RCL

RLC Series
An RLC series circuit consists of a resistor, inductor, and capacitor connected in series:
 * [[Image:RLC series circuit v1.svg|left|100px]]

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

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

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 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
 * 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