Electronics Fundamentals/Series Circuit/Series LC

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Configuration

Analysis

The total Impedance of the circuit

Z=Z_{C}+Z_{L}

Z={\frac {1}{j\omega C}}+j\omega L

Z=j\omega ^{2}+{\frac {1}{LC}}

Z=j\omega ^{2}+{\frac {1}{T}}

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

Natural Response

The Differential equation of the circuit at equilibrium

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

{\frac {d^{2}i}{dt^{2}}}+{\frac {1}{LC}}=0

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

s=\pm j{\sqrt {\frac {1}{LC}}}t

s=\pm j\omega t

The Natural Response of the circuit

i=ASin\omega 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}}}

$V_{L}+V_{C}=0$ .

V_{C}=-V_{L}

Summary

Series LC can be characterised by

2nd order Differential Equation

{\frac {d^{2}i}{dt^{2}}}+{\frac {1}{T}}=0

T=LC

The Natural Response of the circuit is a Sinusoidal Wave function

i=ASin\omega t

The Resonance Response of the circuit is a Standing Wave function

i=ASin\omega t-ASin\omega t

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