Electronics Handbook/Circuits/LC Series

LC Series Configuration

Picture

Circuit Analysis

Circuit's Impedance

$Z=Z_L+Z_C$

Z = ω L /_90 + $\frac{1}{\omega C}$/_-90

• $Z_L>Z_C$

Z = Z_L - Z_C/_90

• $Z_L<Z_C$

Z = Z_C - Z_L/_-90

• $Z_L=Z_C$

Z = 0 /_ 0

Natural Response

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

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

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

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

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

The Second Ordered Equation has two imaginary roots

$I=e^j\omega t+e^{-}j\omega t$

$I=ASin\omega t$

The natural response of the LC series is a Sinusoidal Wave . Therefore, LC series can be used as a Sin Wave Oscillator

Resonance Response

Resonance occurs in the circuit when $Z_L-Z_C=0.V_C+V_L=0$

• $Z_C=Z_L$

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

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

• $V_L+V_C=0$

$V_L=-V_C$

At Resonance, the LC series has the capability to generate Standing Wave Oscillation

Summary

LC series has a second order equation of current with two imaginaries roots

$I(t)=e^{(}j\omega t)+e^{(}-j\omega t)$

Which generates Sin Wave Oscillation

$I(t)=ASin\omega t$

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

T = LC

At resonance, Impedance of Inductor and Capacitor cancel out

Z_L = Z_C

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

Which generates Standing Wave Oscillation

Template:BookCat