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

$V_L+V_C=0$

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

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

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

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

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	$A=\frac{1}{2j}$