# 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:

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

or

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

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

with

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

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

${\displaystyle \alpha ^{2}=\beta ^{2}}$  .
${\displaystyle {\frac {R}{2L}}}$  = ${\displaystyle {\sqrt {\frac {1}{LC}}}}$
${\displaystyle R=2{\sqrt {\frac {L}{C}}}}$

Equation above has only one real root

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

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

Equation above has only two real roots

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

${\displaystyle \alpha ^{2}<\beta ^{2}}$  .
${\displaystyle R<2{\sqrt {\frac {L}{C}}}}$

Equation above has only two complex roots

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

## 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.

${\displaystyle Z_{L}=Z_{C}}$
${\displaystyle \omega L={\frac {1}{\omega C}}}$
${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$

### 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

${\displaystyle Z_{L}+Z_{C}=0}$  Or ${\displaystyle Z_{L}=Z_{C}}$
${\displaystyle \omega L={\frac {1}{\omega C}}}$
${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$
${\displaystyle Z=Z_{R}+Z_{L}+Z_{C}=R+0=R}$
${\displaystyle I={\frac {V}{R}}}$

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