Electronics/Electronics Formulas/Series Circuits/Series LC

Circuit Configuration

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}}

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}

Series LC can be characterised by

2nd order Differential Equation

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

T=LC

With Natural Response of a Wave function

i=ASin\omega t

With Resonance Response of a Standing Wave function

i=ASin\omega t

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