Calculating impedance and transfer function of arbitrary length ladder filter?

[bitrex]

Hi everyone - I'm trying to figure out this "tack hammer" method of determining the impedance and transfer function of an arbitrary length RCL ladder filter. This method is briefly explained in the second chapter of Wes Hayward's book "Introduction To Radio Frequency Design," however it is gone over very briefly and no worked examples are given, so I'm at a loss. I guess the basic idea is that for each "plane" in the ladder filter one determines the admittance, calculates the impedance by doing a polar to rectangular conversion, changing the signs, converting back to rectangular coordinates, and then proceeding up the line. Or something like that. With all the reactances in the diagram in terms of s-parameter variables I'm having trouble seeing how to go about doing all the polar to rectangular conversions. As I mentioned, a link to a reference with a worked example would be much appreciated if anyone knows of one. Sometimes attempting to learn anything from this particular book feels like pulling teeth, as it often feels as if it were written for hardcore mathematicians rather than students.

 

[The Electrician]

Some questions.

Are you trying to get a transfer function in terms of symbolic components, such asl L1, C1, R1, L2, etc., or do you want a numerical transfer function?

Do you know some linear algebra, and do you have a fancy calculator that can do matrix arithmetic such a TI89, or HP50, or a program like Matlab?

Also, can you post a schematic of a typical ladder you want to solve?

 

[bitrex]

Hi Electrician, thank you for your reply. What I'd like to be able to do is get a polynomial transfer function, such as $$\frac{1}{1+jwRC}$$ for a one pole RC low pass filter, and extend that to a series-shunt ladder filter of arbitrary length. I don't have a schematic available to post at the moment, but an example would just be RCRCRC series-shunt-series-shunt-series-shunt, or LCLCLC in the same manner. I'd like to be able to obtain a transfer function and then plot the frequency and phase response for various component values. I could of course find this out using SPICE, but I'd like to really understand the analytical method of doing it before relying on computer simulations. I imagine for filters containing more than 3 reactive elements I will have to use numerical methods to find the poles and zeros as there will be a polynomial with an order greater than 3 in the denominator of the transfer function?

I have a TI-82 graphing caculator, but I don't believe it's capable of doing matrix operations with complex numbers. I do however have some experience with linear algebra, and access to the Maple CAS which can do many of the things MATLAB can do. I also have a good linear algebra reference if I need to refresh my memory on anything. Thank you so much for your time!

 

[f95toli]

Any reason why you just can't write down the ABCD matrices for each filter, multiply them and THEN calculate the S-parameters?

I presume you already know that finding the transfer function is just half the story? Determining which parameters gives the best response for your application is considerably more difficult; unless of course you use a "standard" filter (Butterworth etc) and simply look up the values in a table.

 

[The Electrician]

Any reason why you just can't write down the ABCD matrices for each filter, multiply them and THEN calculate the S-parameters?

I would guess that he wouldn't be asking if he already knew about ABCD parameters.

To bitrex:

Go have a look at this:
http://en.wikipedia.org/wiki/Two-port_network

and this:
http://bmf.ece.queensu.ca/mediawiki/index.php/Network_Parameters

You can create a chain matrix for a single series element (R or L), create another for a shunt C and then multiply the first times the second, resulting in a chain matrix for the combination.

Then if you want to cascade N such sections, just multiply N of the matrices for the two-element combination.

When you're done, the voltage transfer ratio, Vo/Vi, will be the reciprocal of the (1,1) element of the final matrix.

You can deal with a resistive load, for example, by post multiplying by another chain matrix created from a single shunt resistance before you extract the (1,1) element.

Be aware that if the elements are symbolic variables rather than numbers, the transfer function will get large and unwieldy very quickly. However, Maple can handle that sort of thing.

 

[bitrex]

That's a really elegant method for calculating transfer functions that I hadn't seen before. Because of health problems, I've never been able to enroll in a proper academic program for EE or take any EE courses, so I've been largely consigned to learning the best I can through what I can get off the Internet or find in the local library. The problem with that, aside from the obvious motivational issues, is that one often finds oneself jumping ahead in the material too soon, then realizing one has missed some critical concept required and having to backtrack and figure out the missing information. I had actually started thinking about the transfer functions of ladder filters not in a RF frequency context exactly, as I was doing some reading about phase-shift oscillators at audio frequencies and I was having trouble calculating the transfer function for a 3 pole RC ladder filter to see how it satisfies the Barkhausen condition assuming that the stages weren't buffered.

This will give me a good excuse to backtrack and brush up a bit on my linear algebra!

 

[The Electrician]

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