# 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