Electronics Handbook/Components/RLC Network

Series RLC
Circuit of three components R, L, C connected in series
 * [[Image:RLC series circuit.png|left|100px]]

Differential Equation of the circuit
At equilibrium the total voltage of the circuit is zero
 * $$V_L + V_C + V_R = 0$$
 * $$L \frac{dI}{dt} + IR + \frac{1}{C} \int I dt = 0$$
 * $$\frac{dI}{dt} + I \frac{R}{L} + \frac{1}{LC} = 0$$
 * $$\frac{d^2I}{dt^2} + \frac{R}{L} \frac{dI}{dt} + \frac{1}{LC} = 0$$
 * $$s^2 + \frac{R}{L} s + \frac{1}{LC} = 0$$
 * $$s = -\alpha$$ ± $$\sqrt{\alpha^2 - \beta^2}$$
 * $$s = -\alpha \pm \lambda$$

With
 * $$\alpha = \frac{R}{2L}$$ . $$\beta = \frac{1}{LC}$$

When
 * $$\alpha^2 = \beta^2$$.
 * $$\frac{R}{2L}$$ = $$ \frac{1}{LC}$$
 * $$R = \sqrt{\frac{L}{C}}$$

The equation has only one real root
 * s = -α = $$\frac{R}{2L}$$
 * $$I = A e^(-\frac{R}{2L}) t$$


 * $$\alpha^2 > \beta^2$$ ,
 * $$\frac{R}{2L}$$ = $$\frac{1}{LC}$$
 * $$R > \sqrt{\frac{L}{C}}$$

The equation has two real roots
 * $$s = -\alpha \pm \lambda $$
 * $$I(t) = e^(-\alpha \pm \lambda)t$$
 * $$I(t) = e^(-\alpha t)(e^ \lambda t + e^ -\lambda t)$$
 * $$I(t) = \frac{e^(-\alpha t)}{2} Cos \lambda t$$


 * $$\alpha^2 = \beta^2$$.
 * $$R < \sqrt{\frac{L}{C}}$$

The equation has two complex roots
 * $$s = -\alpha$$ + $$j \lambda $$
 * $$I(t) = e^(-\alpha t)(e^ j\lambda t + e^ -j\lambda t)$$
 * $$I(t) = \frac{e^(-\alpha t)}{2j} Sin \lambda t$$

Resonance
Resonance occurs when the frequencies components cancels out. Therefore, at resonance
 * $$Z = Z_R + Z_C + Z_L = Z_R + 0 = R$$
 * $$I = \frac{V}{R}$$
 * $$Z_L = Z_C $$ . $$ \omega L = \frac{1}{\omega C}$$ . $$\omega = \sqrt{\frac{1}{LC}}$$


 * At $$\omega = 0 I = 0$$
 * At $$\omega = \sqrt{\frac{1}{LC}}$$ . $$I = \frac{V}{R}$$
 * At $$\omega = 00 I = 0$$

Impedance

 * $$Z =Z_R + Z_L + Z_C$$
 * $$Z = R + j\omega L + \frac{1}{j\omega C}$$
 * $$Z = \frac{1}{j\omega C} (j\omega^2 + \alpha j \omega + \beta) $$
 * $$\alpha = \frac{R}{L} $$
 * $$\beta = \frac{1}{Lc} $$

Summary

 * The resonance response is in the type of Tuned Resonance Band Pass Filter
 * The natural response is a wave of decreasing magnitude
 * $$R = \sqrt{\frac{L}{C}}$$ . One real root . $$I(t) = e^(-\frac{R}{2L}) t$$
 * $$R = \sqrt{\frac{L}{C}}$$ . Two real roots . $$I(t) = e^(-\frac{R}{2L}) t [e^(\lambda t) + e^(-\lambda t)] $$
 * $$R = \sqrt{\frac{L}{C}}$$ . Two complex roots . $$I(t) = e^(-\frac{R}{2L}) t [e^(j\lambda t) + e^(-j\lambda t)] $$


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

!Circuit!! Series RLC . $$\lambda = 0. \alpha^2 = \beta^2. (\frac{R}{2L})^2 = (\frac{1}{LC})^2$$ $$I = e^-\alpha t = e^(-\frac{R}{2L}) t$$
 * Configuration || RLC_series_circuit.png
 * Impedance || $$Z =  \frac{1}{j\omega C} (j\omega^2 + j\omega \frac{R}{L} + \frac{1}{LC} ) $$
 * Differential Equation || $$L\frac{dI}{dt} + \frac{1}{C} \int V dt + IR = 0$$
 * General Differential Equation|| $$\frac{d^2I}{dt} + \frac{R}{L} \frac{dI}{dt} + \frac{1}{LC} = 0$$
 * Natural Equation || $$I(t) = A (e^\lambda t + e^-\lambda t)$$
 * $$\lambda$$ || $$\lambda = \sqrt{\alpha^2 - \beta^2}$$
 * General Differential Equation|| $$\frac{d^2I}{dt} + \frac{R}{L} \frac{dI}{dt} + \frac{1}{LC} = 0$$
 * Natural Equation || $$I(t) = A (e^\lambda t + e^-\lambda t)$$
 * $$\lambda$$ || $$\lambda = \sqrt{\alpha^2 - \beta^2}$$
 * Natural Equation || $$I(t) = A (e^\lambda t + e^-\lambda t)$$
 * $$\lambda$$ || $$\lambda = \sqrt{\alpha^2 - \beta^2}$$
 * $$\lambda$$ || $$\lambda = \sqrt{\alpha^2 - \beta^2}$$

. $$\lambda = \sqrt{\alpha^2 - \beta^2} > 0. \alpha^2 > \beta^2. (\frac{R}{2L})^2 > (\frac{1}{LC})^2$$ $$I = A (e^\lambda t + e^-\lambda t)$$

. $$\lambda = \sqrt{\alpha^2 - \beta^2} < 0. \alpha^2 < \beta^2. (\frac{R}{2L})^2 (\frac{1}{LC})^2$$ $$I = A (e^j\lambda t + e^-j\lambda t)$$


 * A || $$A = e^(-\alpha t)$$
 * $$\alpha$$ || $$\alpha = \frac{R}{2L}$$
 * $$\beta$$ || $$\beta = \frac{1}{LC}$$
 * }
 * $$\beta$$ || $$\beta = \frac{1}{LC}$$
 * }
 * }
 * }