Electronics Handbook/Components/LC Network

LC Series

Circuit of L and C connected in series

Circuit Analysis

Circuit Impedance

$Z=Z_{C}+Z_{L}$

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

$Z={\frac {1}{j\omega C}}(j\omega ^{2}+{\frac {1}{T}})$

T = LC

Circuit Natural Response

At equilibrium the total voltage of the circuit is zero

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

$L{\frac {dI}{dt}}+{\frac {1}{C}}\int Idt=0$

${\frac {dI}{dt}}+{\frac {1}{LC}}\int Idt=0$

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

$s^{2}=-{\frac {1}{LC}}$

s = ± ${\sqrt {-{\frac {1}{LC}}}}$ = ± j ${\sqrt {\frac {1}{LC}}}$

$I=e^{(}j{\sqrt {\frac {1}{LC}}}t)+e^{(}-j{\sqrt {\frac {1}{LC}}}t)$

$I=ASin{\sqrt {\frac {1}{LC}}}t$

$I=ASin\omega t$

At equilibrium or The natural response of the LC series is the Sinusoidal Wave Oscillation

Resonance

Resonance occurs when the frequency components cancel each other out . Therefore, at resonance

$Z_{L}=Z_{C}$

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

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

$V_{C}=-V_{L}$

At resonance the response of the LC series is the Standing Wave Oscillation

Summary

• The natural frequency reponse of the circuit is a Sinusoidal Wave
• At resonance, the frequency reponse of the LC series is the oscillation of a Standing Wave

Circuit	Series LC
Configuration
Impedance	$Z={\frac {1}{j\omega C}}(j\omega ^{2}+{\frac {1}{T}})$
Differential Equation	$L{\frac {dI}{dt}}+{\frac {1}{C}}\int vdt=0$
General Differential Equation	${\frac {d^{2}I}{dt}}+{\frac {1}{T}}=0$
Natural Response	$I(t)=e^{(}j\omega t)+e^{(}-j\omega t)$ $I(t)=ASin\omega t$
T	$T=LC$
$\omega$	$\omega ={\sqrt {\frac {1}{T}}}$
A	$A={\frac {1}{2j}}$