# Electronics/RCL

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## 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
    $L{\frac {d^{2}I}{dt^{2}}}+R{\frac {dI}{dt}}+{\frac {I}{C}}={\frac {dV}{dt}}$
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 = ${\frac {R}{2L}}$
    $I=Ae^{(-{\frac {R}{2L}})t}$

### $\alpha ^{2}>\beta ^{2}$ ,

    ${\frac {R}{2L}}$ > ${\sqrt {\frac {1}{LC}}}$
    $R>2{\sqrt {\frac {L}{C}}}$
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 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
    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

## Further Reading

* RCL circuit analysed in the time domain
* RCL circuit analysed in the frequency domain