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

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

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

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

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

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