Practical Electronics/Series RLC

A circuit of 3 components connected in series

Circuit's response

Equilibrium Response

At Equilibrium, the sum of all voltages 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