# 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:

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

or

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

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

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

$\alpha ^{2}>\beta ^{2}$  ,
${\frac {R}{2L}}$  > ${\sqrt {\frac {1}{LC}}}$
$R>2{\sqrt {\frac {L}{C}}}$

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 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