TF EIM Chapt3

RLC circuit

An RLC circuit is a Resistor, an Inductors, and a Capacitor in series with an electromotive force.

Effective impedance

$X_{out} = R_L + X_C + X_L = R_L + \frac{1}{i \omega C} + i \omega L$

$\left | X_{out} \right | = \sqrt{\left [ R_L + i \left (\frac{-1}{\omega C} + \omega L\right ) \right ]\left [ R - i \left (\frac{-1}{\omega C} + \omega L\right ) \right ]^*}$

$= \sqrt{ R_L^2 + \left ( \omega L - \frac{1}{\omega C} \right )^2}$

Gain

Loop Theorem

$V_{in} = I (R+ X_{out})$

Voltage Divider

$V_{AB}=V_{out} = \frac{X_{out}}{R + X_{out}}V_{in}$

$\left | \frac{V_{out}}{V_{in}}\right | = \sqrt{\left [ \frac{X_{out}}{R + X_{out}}\right ]\left [ \frac{X_{out}}{R + X_{out}}\right ]^*}$

$R_L + i \left ( \omega L - \frac{1}{\omega C}\right)$

$\left | \frac{V_{out}}{V_{in}}\right | = \sqrt{\left [ \frac{R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}{R + R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}\right ]\left [ \frac{R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}{R + R_L + i \left ( \omega L - \frac{1}{\omega C}\right)}\right ]^*}$

$= \sqrt{\frac{R_L^2 + \left ( \omega L - \frac{1}{\omega C}\right)^2}{(R + R_L)^2 + \left ( \omega L - \frac{1}{\omega C}\right)^2}}$

$= \sqrt{\frac{R_L^2 + \left ( \frac{\omega^2 LC - 1}{\omega C}\right)^2}{(R + R_L)^2 + \left ( \frac{\omega^2 LC - 1}{\omega C}\right)^2}}$

Let

$\omega_0 = \sqrt{\frac{1}{LC}}$

Then

$\left | \frac{V_{out}}{V_{in}}\right | = \sqrt{\frac{R_L^2 + \left ( \frac{\omega^2 - \omega_0^2}{\omega_0^2 \omega C}\right)^2}{(R + R_L)^2 + \left ( \frac{\omega^2- \omega_0^2}{\omega_o^2 \omega C}\right)^2}}$

When$\omega = \sqrt{{1}{LC}} = \omega_0$

Then

$\left | \frac{V_{out}}{V_{in}}\right | =\frac{R}{R+R_L}$