Electronics Handbook/Circuits/RLC Series

RLC Series

Circuit Analysis

Circuit's Impedance

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

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

$Z={\frac {1}{X_{C}}}(j\omega ^{2}+j\omega {\frac {R}{L}}+{\frac {1}{LC}})$

Natural Response

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

${\frac {dI}{dt}}+I{\frac {R}{L}}+{\frac {1}{LC}}=0$

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

$s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0$

$s=-\alpha$ ± ${\sqrt {\alpha ^{2}-\beta ^{2}}}t$

$s=-\alpha \pm \lambda t$

$\alpha ={\frac {R}{2L}}$

$\beta ={\frac {1}{LC}}$

$\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}t$

• $\lambda =0.\alpha ^{2}=\beta ^{2}$ . There is only one real root,

s = -αt

$({\frac {R}{2L}})^{2}$ = $({\frac {1}{LC}})^{2}$

$R={\sqrt {\frac {L}{C}}}$

$I=Ae^{(}-{\frac {t}{T}})$

• $\alpha ^{2}>\beta ^{2}$ , There are two real roots,

$s=-\alpha$ ± ${\sqrt {\alpha ^{2}-\beta ^{2}}}$

$({\frac {R}{2L}})^{2}$ = $({\frac {1}{LC}})^{2}$

$R>{\sqrt {\frac {L}{C}}}$

$I=e^{-}\alpha t[e^{\lambda }t+e^{-}\lambda t]$

$I=ACos\lambda t$

$A={\frac {e^{-}\alpha t}{2}}$

• $\alpha ^{2}<\beta ^{2}$ , There are two complex roots,

$s=-\alpha$ ± ${\sqrt {\alpha ^{2}-\beta ^{2}}}$

$({\frac {R}{2L}})^{2}$ = $({\frac {1}{LC}})^{2}$

$R>{\sqrt {\frac {L}{C}}}$

$I=e^{-}\alpha t[e^{j}\lambda t+e^{-}j\lambda t]$

$I=ASin\lambda t$

$A={\frac {e^{-}\alpha t}{2}}$

Current change with time depends on the value of R L and C

$R={\sqrt {\frac {L}{C}}}$ . Dòng điện giảm dần theo hàm số mủ của e

$R>{\sqrt {\frac {L}{C}}}$ . Dòng điện giảm đến một giá trị âm rồi tăng đến một giá trị dương

$R={\sqrt {\frac {L}{C}}}$ . Mạch điện có có Dòng Điện của Sóng Sin giảm dần theo theo thời gian

Resonance Response

At resonance $Z_{L}-Z_{C}=0.V_{C}+V_{L}=0$

• $Z_{L}-Z_{C}=0$

$Z_{L}=Z_{C}$ .

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

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

Analyze the circuit at

$\omega =0.I=0$ . Capacitor opens circuit

$\omega =00.I=0$ . Inductor opens circuit

$\omega ={\sqrt {\frac {1}{LC}}}.I={\frac {V}{R}}$ . $Z_{L}=Z_{R}.Z=Z_{R}+Z_{L}+Z_{C}=Z_{R}=R$

At resonance, series RLC is capable of select a bandwidth of frequencies where voltage is stable does not change with frequencies. Therefore, can be used as Tuned - Resonance Band Pas Selected Filter

Summary

• The natural Response of the RLC series is a second order differential equation of current

${\frac {d^{2}I}{dt}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}$

Depends on the value of Resistance the equation has

One Real Root	Two Real Roots	Two Complex Roots
One real root $R={\sqrt {\frac {L}{C}}}$ $I=e^{-}\alpha t$	Two real roots $R>{\sqrt {\frac {L}{C}}}$ $I=e^{-}\alpha [e^{\lambda }t+e^{-}\lambda t]$ $I=ACos\lambda t$ $A={\frac {e^{-}\alpha t}{2}}$	$R<{\sqrt {\frac {L}{C}}}$ $I=e^{-}\alpha [e^{j}\lambda t+e^{-}j\lambda t]<br><math>I=ASin\lambda t$ $A={\frac {e^{-}\alpha t}{2}}$

• At Resonance when all the frequency dependent components cancel out RLC series behaves like Tuned Resonance Selected Band Pass Filter