TF EIM Chapt3

gain

Loop Theorem

[math]\displaystyle{ \Rightarrow V= I(R+X_{tot}) = I \left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right ) }[/math]

or

[math]\displaystyle{ I= \frac{V_0 e^{i \omega t}}{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} }[/math]

Notice

When [math]\displaystyle{ \omega \approx \omega_{LC} = \sqrt{\frac{1}{LC}} }[/math] then the AC signal is attenuated.

Looking at the Voltage divider aspect of the circuit

[math]\displaystyle{ V_{AB}=V_{out} = \frac{X_{tot} }{R + X_{tot}}V_{in} }[/math]

[math]\displaystyle{ \left |\frac{ V_{out}} {V_{in}}\right | = \sqrt{ \left [ \frac{X_{tot} }{R + X_{tot}} \right ] \left [ \frac{X_{tot} }{R + X_{tot}} \right ]^*} }[/math]

[math]\displaystyle{ = \sqrt{ \left [ \frac{\frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ] \left [ \frac{\frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} }{\left (R+ \frac{i \omega L}{1- \frac{\omega^2}{\omega_{LC}^2}} \right )} \right ]^*} }[/math]

[math]\displaystyle{ = \sqrt{ \frac{ \omega^2 L^2 \omega_{LC}^4}{R^2(\omega_{LC}^2 - \omega^2)^2 - \omega^2L^2 \omega_{LC}^4}} }[/math]

[math]\displaystyle{ = \sqrt{ \frac{ \omega^2 }{C^2R^2(\omega_{LC}^2 - \omega^2)^2 - \omega^2}} }[/math]

[math]\displaystyle{ = \sqrt{ \frac{ \omega^2 }{\frac{1}{\omega_{RC}^2}(\omega_{LC}^2 - \omega^2)^2 - \omega^2}} }[/math]

[math]\displaystyle{ L = 33 \mu H }[/math] and [math]\displaystyle{ C = 1 \mu F }[/math] and R=100 [math]\displaystyle{ \Omega }[/math]

[math]\displaystyle{ \Rightarrow \omega = 174077.66 }[/math] rad/s

Forest_Electronic_Instrumentation_and_Measurement