# Transfer function and cut-off frequency of cascoded RC-LRC filter

At the moment I'm struggling with the problem, i can calculate the transfer function of the following circuit, but not the cut-off frequency. The transfer function is in my case: $$H(s)=\frac{1}{s^3\cdot (L\cdot R1 \cdot R2 \cdot C1 \cdot C2)+s^2 \cdot (L \cdot R2 \cdot C2)+s \cdot (R1 \cdot (C1+C2))+1}$$ Thanks in advance

• Welcome to SE.EE! First of all: please note cascode =/= cascade. Jun 5, 2019 at 11:03
• It isn't easy to solve a third order manually. Try using a mathematical tools like MatLab, Maxima, etc or simulation tools like LTspiceXVIII Jun 5, 2019 at 11:06
• Modeling this as a (very highly dampened) PI filter, the F_resonance is about 7MHz. Jun 5, 2019 at 13:57

If you want to do this 'by hand' rather than through Matlab or whatever, use the old analogue computing trick of normalising the TF with respect to the s-coefficient. This gives a more manageable set of coefficients that can be re-scaled at the end.

Assuming your TF coefficients are correct:

$$\small H(s)=\frac{1}{(1.225\times 10^{-20})s^3+(2.475\times 10^{-13})s^2+(1.584\times 10^{-7})s+1}$$

Let $$\\small S=(1.584\times 10^{-7})s\$$, and the TF becomes:

$$\small H(S)=\frac{1}{3.083\:S^3+9.864\:S^2+S+1}$$

Now let $$\\small S=j\Omega\$$ and determine the cut-off frequency, $$\\small \Omega_c\$$.

Finally, scale to the actual frequency via: $$\ \omega_c =\frac{\Omega_c}{1.584\times 10^{-7}}\$$

You need to solve:

$$\frac{1}{\sqrt{2}}\cdot\left|\underline{\mathcal{H}}\left(\hat{\omega}\right)\right|=\left|\underline{\mathcal{H}}\left(\omega\right)\right|\space\Longleftrightarrow\space\omega=\dots\tag1$$

Using Mathematica I got (assuming that your transfer function is correct) for $$\\hat{\omega}\$$:

And $$\\left|\underline{\mathcal{H}}\left(\hat{\omega}\right)\right|\$$ gives:

So, solving equation $$\(1)\$$ gives:

Which gives: