Electronics Handbook/Circuits/RLC Series

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

 * $$L \frac{dI}{dt} + IR + \frac{1}{C} \int I dt = 0$$
 * $$\frac{dI}{dt} + I \frac{R}{L} + \frac{1}{LC} = 0$$
 * $$\frac{d^2I}{dt^2} + \frac{R}{L} \frac{dI}{dt} + \frac{1}{LC} = 0$$
 * $$s^2 + \frac{R}{L} s + \frac{1}{LC} = 0$$
 * $$s = -\alpha$$ ± $$\sqrt{\alpha^2 - \beta^2} t$$
 * $$s = -\alpha \pm \lambda t$$


 * $$\alpha = \frac{R}{2L}$$
 * $$\beta = \frac{1}{LC}$$
 * $$\lambda = \sqrt{\alpha^2 - \beta^2} t$$


 * $$\lambda = 0 .  \alpha^2 = \beta^2$$ . There is only one real root,
 * s = -αt
 * $$(\frac{R}{2L})^2$$ = $$ (\frac{1}{LC})^2$$
 * $$R = \sqrt{\frac{L}{C}}$$
 * $$I = A e^(-\frac{t}{T}) $$


 * $$\alpha^2 > \beta^2$$, There are two real roots,
 * $$s = -\alpha$$ ± $$\sqrt{\alpha^2 - \beta^2}$$
 * $$(\frac{R}{2L})^2$$ = $$(\frac{1}{LC})^2$$
 * $$R > \sqrt{\frac{L}{C}}$$
 * $$I = e^-\alpha t [e^\lambda t + e^-\lambda t]$$
 * $$I = A Cos \lambda t$$
 * $$A = \frac{e^-\alpha t}{2}$$


 * $$\alpha^2 < \beta^2$$, There are two complex roots,
 * $$s = -\alpha$$ ± $$\sqrt{\alpha^2 - \beta^2}$$
 * $$(\frac{R}{2L})^2$$ = $$(\frac{1}{LC})^2$$
 * $$R > \sqrt{\frac{L}{C}}$$
 * $$I = e^-\alpha t [e^j\lambda t + e^-j\lambda t]$$
 * $$I = A Sin \lambda t$$
 * $$A = \frac{e^-\alpha t}{2}$$

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

Depends on the value of Resistance the equation has


 * {| class="wikitable" width=100%

! One Real Root !! Two Real Roots !! Two Complex Roots $$R < \sqrt{\frac{L}{C}}$$ I = e^-\alpha [e^j\lambda t + e^-j\lambda t] $$ I = A Sin \lambda t $$ $$ A = \frac{e^-\alpha t}{2}$$
 * One real root $$R = \sqrt{\frac{L}{C}} $$ $$ I = e^-\alpha t$$ || Two real roots $$R > \sqrt{\frac{L}{C}} $$ $$ I = e^-\alpha [e^\lambda t + e^-\lambda t] $$ $$ I = A Cos \lambda t $$ $$ A = \frac{e^-\alpha t}{2}$$ ||
 * One real root $$R = \sqrt{\frac{L}{C}} $$ $$ I = e^-\alpha t$$ || Two real roots $$R > \sqrt{\frac{L}{C}} $$ $$ I = e^-\alpha [e^\lambda t + e^-\lambda t] $$ $$ I = A Cos \lambda t $$ $$ A = \frac{e^-\alpha t}{2}$$ ||
 * }


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