Electronics/Electronics Formulas/Series Circuits/Series RLC

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Circuit Configuration

Formula

Circuit's Impedance

The total Impedance of the circuit

Z=Z_{R}+Z_{L}

Z=R+j\omega L

Z={\frac {1}{R}}(1+j\omega T)

T={\frac {L}{R}}

Differential Equation

The Differential equation of the circuit at equilibrium

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}}=0

s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0

s=(-\alpha \pm \lambda )t

\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}

\alpha ={\frac {R}{2L}}

\beta ={\frac {1}{LC}}

The Natural Response of the circuit

$\lambda =0$ . $\alpha ^{2}=\beta ^{2}$

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

$\lambda =0$ . $\alpha ^{2}=\beta ^{2}$

i=e^{(}-\alpha t)[e^{(}\lambda t)+e^{(}-\lambda t)]

$\lambda =0$ . $\alpha ^{2}=\beta ^{2}$

i=e^{(}-\alpha t)[e^{(}j\lambda t)+e^{(}-j\lambda t)]

The Resonance Response of the circuit

Z_{L}-Z_{C}=0

.

Z_{L}=Z_{C}

.

\omega L={\frac {1}{\omega C}}

.

\omega ={\sqrt {\frac {1}{LC}}}

V_{L}+V_{C}=0

.

\omega =0

.

\omega =0

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