# Practical Electronics/Series RLC

A circuit of 3 components connected in series ## Circuit's response

### Equilibrium Response

At Equilibrium, the sum of all volatages equal to zero

$v_{L}+v_{C}+v_{R}=0$ $L{\frac {di}{dt}}+{\frac {1}{C}}\int idt+iR=0$ ${\frac {d^{2}i}{dt^{2}}}+{\frac {R}{L}}{\frac {di}{dt}}+{\frac {1}{LC}}i=0$ The equation above can be written as below

${\frac {d^{2}i}{dt^{2}}}=-2\alpha {\frac {di}{dt}}-\beta i$ With

$\alpha ={\frac {R}{2L}}=\beta \gamma$ $\beta ={\frac {1}{LC}}={\frac {1}{T}}$ $T=LC$ $\gamma =RC$ Roots of 2nd ordered differential equation above

• $\alpha =\beta$ $i(t)=Ae^{-\alpha t}$ • $\alpha >\beta$ $i(t)=Ae^{(-\alpha \pm \lambda )t}$ • $\alpha <\beta$ $i(t)=Ae^{(-\alpha \pm j\omega )t}=A(\alpha )\sin \omega t$ ### Resonance Response

The total impedance of the circuit

$Z=Z_{R}+Z_{L}+Z_{C}=R+0=R$ $i={\frac {V}{R}}$ $Z_{L}=Z_{C}$ $j\omega L={\frac {1}{j\omega C}}$ $\omega _{o}=\pm j{\sqrt {\frac {1}{T}}}$ $T=LC$ At $\omega _{o}=\pm j{\sqrt {\frac {1}{T}}}$ the total impedance of the circuit is Z = R . Therefore, current is equal to $i={\frac {V}{R}}$ At $\omega =0.Z_{C}=oo$ , Capacitor opens circuit . Therefore, current is equal to zero
At $\omega =oo.Z_{L}=oo$ , Inductor opens circuit . Therefore, current is equal to zero