# Electronics Handbook/Components/RLC Network
## Series RLC

Circuit of three components R, L, C connected in series

## Differential Equation of the circuit

At equilibrium the total voltage of the circuit is zero
    $V_{L}+V_{C}+V_{R}=0$
    $L{\frac {dI}{dt}}+IR+{\frac {1}{C}}\int Idt=0$
    ${\frac {dI}{dt}}+I{\frac {R}{L}}+{\frac {1}{LC}}=0$
    ${\frac {d^{2}I}{dt^{2}}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0$
    $s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0$
    $s=-\alpha$ ± ${\sqrt {\alpha ^{2}-\beta ^{2}}}$
    $s=-\alpha \pm \lambda$
With
    $\alpha ={\frac {R}{2L}}$ . $\beta ={\frac {1}{LC}}$
When
* $\alpha ^{2}=\beta ^{2}$ .
    ${\frac {R}{2L}}$ = ${\frac {1}{LC}}$
    $R={\sqrt {\frac {L}{C}}}$
The equation has only one real root
    s = -α = ${\frac {R}{2L}}$
    $I=Ae^{-{\frac {R}{2L}})t$
* $\alpha ^{2}>\beta ^{2}$ ,
    ${\frac {R}{2L}}$ = ${\frac {1}{LC}}$
    $R>{\sqrt {\frac {L}{C}}}$
The equation has two real roots
    $s=-\alpha \pm \lambda$
    $I(t)=e^{-\alpha \pm \lambda )t$
    $I(t)=e^{-\alpha t)(e^{\lambda }t+e^{-}\lambda t)$
    $I(t)={\frac {e^{-\alpha t)}{2}}Cos\lambda t$
* $\alpha ^{2}=\beta ^{2}$ .
    $R<{\sqrt {\frac {L}{C}}}$
The equation has two complex roots
    $s=-\alpha$ + $j\lambda$
    $I(t)=e^{-\alpha t)(e^{j}\lambda t+e^{-}j\lambda t)$
    $I(t)={\frac {e^{-\alpha t)}{2j}}Sin\lambda t$

## Resonance

Resonance occurs when the frequencies components cancels out . Therefore, at resonance
    $Z=Z_{R}+Z_{C}+Z_{L}=Z_{R}+0=R$
    $I={\frac {V}{R}}$
    $Z_{L}=Z_{C}$ . $\omega L={\frac {1}{\omega C}}$ . $\omega ={\sqrt {\frac {1}{LC}}}$
* At $\omega =0I=0$
* At $\omega ={\sqrt {\frac {1}{LC}}}$ . $I={\frac {V}{R}}$
* At $\omega =00I=0$

## Impedance

    $Z=Z_{R}+Z_{L}+Z_{C}$
    $Z=R+j\omega L+{\frac {1}{j\omega C}}$
    $Z={\frac {1}{j\omega C}}(j\omega ^{2}+\alpha j\omega +\beta )$
    $\alpha ={\frac {R}{L}}$
    $\beta ={\frac {1}{Lc}}$

## Summary

* The resonance response is in the type of Tuned Resonance Band Pass Filter
* The natural response is a wave of decreasing magnitude
       $R={\sqrt {\frac {L}{C}}}$ . One real root . $I(t)=e^{-{\frac {R}{2L}})t$
       $R={\sqrt {\frac {L}{C}}}$ . Two real roots . $I(t)=e^{-{\frac {R}{2L}})t[e^{\lambda t)+e^{-\lambda t)]$
       $R={\sqrt {\frac {L}{C}}}$ . Two complex roots . $I(t)=e^{-{\frac {R}{2L}})t[e^{j\lambda t)+e^{-j\lambda t)]$
    CircuitSeries RLC Configuration Impedance$Z={\frac {1}{j\omega C}}(j\omega ^{2}+j\omega {\frac {R}{L}}+{\frac {1}{LC}})$ Differential Equation$L{\frac {dI}{dt}}+{\frac {1}{C}}\int Vdt+IR=0$ General Differential Equation${\frac {d^{2}I}{dt}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0$ Natural Equation$I(t)=A(e^{\lambda }t+e^{-}\lambda t)$ $\lambda$$\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}$ . $\lambda =0.\alpha ^{2}=\beta ^{2}.({\frac {R}{2L}})^{2}=({\frac {1}{LC}})^{2}$ $I=e^{-}\alpha t=e^{-{\frac {R}{2L}})t$ . $\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}>0.\alpha ^{2}>\beta ^{2}.({\frac {R}{2L}})^{2}>({\frac {1}{LC}})^{2}$ $I=A(e^{\lambda }t+e^{-}\lambda t)$ . $\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}<0.\alpha ^{2}<\beta ^{2}.({\frac {R}{2L}})^{2}({\frac {1}{LC}})^{2}$ $I=A(e^{j}\lambda t+e^{-}j\lambda t)$ A$A=e^{-\alpha t)$ $\alpha$$\alpha ={\frac {R}{2L}}$ $\beta$$\beta ={\frac {1}{LC}}$