## SERIES RLC CIRCUIT WITH FREQUENCY RESPONSE EXPERIMENT

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\(I(s) = \frac{1}{R + Ls + 1/Cs}V_{in}(s) = \frac{s/L}{s^2 + sR/L + 1/LC}V_{in}(s)\)

\(I(s) = \frac{s/L}{s^2 + 2\alpha s + \omega_0^2}V_{in}(s)\)

\(\omega_0 = \frac{1}{\sqrt{LC}} , \alpha = \frac{R}{2L}\)

Band-Pass Filter:

\(V_r(s) = RI(s) = \frac{sR/L}{s^2 + 2\alpha s + \omega_0^2}V_{in}(s) = \frac{2\alpha s}{s^2 + 2\alpha s + \omega_0^2}V_{in}(s) = \frac{2\alpha s}{(s-s_1)(s-s_2)}V_{in}(s)\)

Gain magnitude: \(G_R = \frac{2\alpha w}{|j\omega - s_1||j\omega - s_2|}\)

Phase: \(\Phi_R = \pi/2-\Phi(j\omega - s_1) -\Phi(j\omega - s_2)\)

Low-Pass Filter:

\(V_c(s) = I(s)/sC = \frac{1/LC}{s^2 + 2\alpha s + \omega_0^2}V_{in}(s) = \frac{\omega_0^2}{s^2 + 2\alpha s + \omega_0^2}V_{in}(s) = \frac{\omega_0^2}{(s-s_1)(s-s_2)}V_{in}(s)\)

Gain magnitude: \(G_C = \frac{\omega_0^2}{|j\omega - s_1||j\omega - s_2|}\)

Phase: \(\Phi_C = -\Phi(j\omega - s_1) -\Phi(j\omega - s_2)\)

High-Pass Filter:

\(V_l(s) = sLI(s) = \frac{s^2}{s^2 + 2\alpha s + \omega_0^2}V_{in}(s) = \frac{s^2}{(s-s_1)(s-s_2)}V_{in}(s)\)

Gain magnitude: \(G_L = \frac{\omega^2}{|j\omega - s_1||j\omega - s_2|}\)

Phase: \(\Phi_L = -\Phi(j\omega - s_1) -\Phi(j\omega - s_2)\)

Under-Damped: \(\alpha < \omega_0\)

Complex roots: \(s_{1,2} = -\alpha \pm j\sqrt{\omega_0^2 - \alpha^2}\)

Critically-Damped: \(\alpha = \omega_0\)

Double real root: \(s_{1,2} = -\alpha\)

Over-Damped: \(\alpha > \omega_0\)

Real roots: \(s_{1,2} = -\alpha \pm\sqrt{\alpha^2 - \omega_0^2}\)

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Graph:

- vIN
- vR
- vL
- vC

Listen to:

- vIN
- vR
- vL
- vC

- Time
- Magnitude
- Phase