Magnitude of a function with inductor and capacitor

Question

I have a homework problem with the following circuit:

I have solved the transistor biasing part and then drawn the AC-circuit of it. Then I need to find the transfer function.

These are the calculations I did:

This is the drawing which I drew by hand of what I think the transfer function will be:

I am not sure how we ended up getting infinite for the third case. Would really appreciate some help regarding this.

Similar problems like this I want to be able to solve in just some piece of paper.

Answer 1

Well, I couldn't quite get what you are asking. But, looking at this circuit we can figure \$ Z_{L||C}\$,

^{simulate this circuit – Schematic created using CircuitLab}

From the Fourier transform and circuit equations, we have that

$$ v = i Z_{L||C} = i \left( \left(j\omega L \right)^{-1}+\left(\frac{1}{j\omega C}\right)^{-1}\right)^{-1},$$

$$ Z_{L||C} = \left( \left(\frac{1}{j\omega L}\right) + \left(j\omega C \right) \right)^{-1} = \left( \frac{1- \omega^2 LC }{j\omega L} \right)^{-1} = \frac{j\omega L}{1- \omega^2 LC },$$

$$ \lim_{\omega \rightarrow \infty} Z_{L||C} = 0,$$ $$ \lim_{\omega \rightarrow \sqrt{1/LC}} Z_{L||C} = \infty,$$ $$ \lim_{\omega \rightarrow +0} Z_{L||C} = 0.$$

You could plot the values of \$ ||Z_{L||C} (jw)||\$ and get something similar to \$ ||Z_{L||C} (jw)||\$ ">

Or the log-log plot

\$ ||Z_{L||C} (jw)|| \$">

So \$ ||Z_{L||C} (jw)||\$ is just as you pointed. Now, I don't get where that transfer function \$ \frac{U_1}{U_2} \$ fits in your question at all.