# Help demonstrating mathematically the behaviour of a passive filter

Consider the following circuit:

By inspection, it can be said that the complex impedance of the circuit will be $$\R_2\$$ for very low frequencies and $$\R_1\$$ for very high ones, as the inductor will behave as an open circuit for very high frequencies and the capacitor will do the same for very low ones.

I have tried to demonstrate the second statement (the impedance will be $$\R_1\$$ for very high frequencies) mathematically but I can't. Below are my calculations (let $$\s\$$ be $$\j\omega\$$):

$$Z_t = \frac{1}{\frac{1}{R_1 + \frac{1}{sC}} + \frac{1}{R_2 + sL}} =$$

$$\frac{1}{\frac{sC}{sCR_1 + 1} + \frac{1}{R_2 + sL}} =$$

$$\frac{(sCR_1 + 1)(R_2 + sL)}{sC(R_2+sL) + sCR_1 + 1} =$$

$$\frac{sCR_1 R_2 + s^2CR_1 L + R_2 + sL }{s^2L + 1 + s(CR_1 + CR_2)} =$$

$$\frac{-w^2CR_1 L + R_2 + j\omega(L + CR_1 R_2)}{-w^2L + 1 + j\omega(CR_1 + CR_2)}$$

Then, I calculate the module of $$\Z_t\$$ as:

$$|{Z_t}| = \frac{\sqrt{(R_2 - \omega^2CR_1L)^2 + (\omega(L+CR_1R_2))^2}}{\sqrt{(1-\omega^2L)^2 + (\omega(CR_1+CR_2))^2}}$$.

For very high frequencies, I do:

$$\lim_{\omega \to \infty}\frac{\sqrt{(R_2 - \omega^2CR_1L)^2 + (\omega(L+CR_1R_2))^2}}{\sqrt{(1-\omega^2L)^2 + (\omega(CR_1+CR_2))^2}}$$

As $$\(\omega^2)^2 = \omega^4\$$, I cancel out the squared terms to the right, leaving the following:

$$\lim_{\omega \to \infty}\frac{\sqrt{(R_2 - \omega^2CR_1L)^2}}{\sqrt{(1-\omega^2L)^2}}$$

which yields

$$\frac{\sqrt{C^2R_1^2L^2}}{\sqrt{L^2}} = CR_1$$

But it should be $$\R_1\$$. What am I missing?

• Try and concentrate on being an EE (rather than trying to find a mathematical solution for something that basic-EE-common-sense tells us is true). By the way, it's only a filter if it has an input node and a different output node (and a common point). Jan 13 at 16:08
• Where is the ouput node for the filter?
– LvW
Jan 13 at 16:34
• Probably @Verbal Kint could give a good answer with FACTs. Jan 13 at 16:45

the term $$\s^ 2 L\$$ should be $$\ s^2 L C \$$ -> $$\-w^2LC\$$.
NB: you don't need passing to $$\w\$$.
Just search the "limit" when s-> infinity = $$\(s^2 C R1 L)/(s^2 L C) = R1\$$.