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

$v_L+v_C+v_R=0$

$L\frac{di}{dt}+\frac{1}{C}\int idt+iR=0$

$\frac{d^{2}i}{d{t}^2}+\frac{R}{L}\frac{di}{dt}+\frac{1}{LC}i=0$

The equation above can be written as below

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

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

• $\alpha =\beta$

$i(t)=A{e}^{-\alpha t}$

• $\alpha >\beta$

$i(t)=A{e}^{(-\alpha \pm \lambda )t}$

• $\alpha <\beta$

$i(t)=A{e}^{(-\alpha \pm j\omega )t}=A(\alpha )\sin \omega t$

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