Two-port network: transformation of low-pass to band-stop filter

[Master1022]

TL;DR Summary

Topic: Two port network filter synthesis; filter transformation from low pass to band-stop equivalent. This question is specifically about transforming an inductor and capacitor to their band stop equivalent circuit components. I have a question about the derivation of the values of these circuit components

Hi,

Context:
I was looking through some circuits material in the topic of filter synthesis for two port networks. For simplicity, these networks have been synthesized for normalized conditions: 1 rad/s cut-off frequency and a termination resistor of 1 $\Omega$. An example filter is shown in the image below - the 1 $\Omega$ termination resistor can be seen.

I was reading about the transformation of LPF (filter) circuits to band stop (filter) circuits. The transformation is given by: (where $s_n$ is normalised, and $\omega_0$ is the target cut-off value)

Question 1:
Then below I read the following (shown in the image below). For the inductor transformation, am I correct in thinking that the penultimate expression should instead read:
$$\frac{1}{s L_{e1} + \frac{1}{s C_{e1}}}$$
That is, the $L_{e1}$ and $C_{e1}$ should be swapped around. When I asked about it, I was told that I was "definitely wrong", but I am struggling to see why.

Question 2:
If we wanted to transfer both the filter type (LPF --> Band stop) and the filter values (i.e. change from normalized values of 1 rad/s cut-off to arbitrary value), how would I transform the circuit values? For the inductors, would I:
1. Transform the inductor via $L_{new} = \frac{R_0 L_n}{\omega_0}$
2. Then transform $L_{new} s_n$ as shown in the image above

or have I misunderstood this?

Any help would be greatly appreciated. I can provide more information if required to clarify the post.

 

[Achy47]

Thank you for your post. I am a scientist who specializes in circuit design and I would be happy to help clarify your questions.

Question 1:
You are correct in thinking that the expression for the inductor transformation should be \frac{1}{s L_{e1} + \frac{1}{s C_{e1}}}. This is because in a band stop filter, the inductor and capacitor are swapped compared to a low pass filter. The mistake in the expression shown in the image is likely a typo or error in the source material. It is important to always double check equations and expressions to ensure accuracy.

Question 2:
To transfer both the filter type and values, you would first transform the inductor using the equation L_{new} = \frac{R_0 L_n}{\omega_0} as you mentioned. Then, you would transform the inductor again using the expression shown in the image above. This will give you the new inductor value for your desired cut-off frequency.

I hope this helps clarify your questions. If you need any further assistance, please do not hesitate to ask. Good luck with your circuit design!