[SOLVED] How to obtain component values for Chebyshev from the formula?

Does anyone have some resources or mind explaining to me how we can synthesize a Chebyshev filter from the formulas specifically, and not from tables?

Specifically, I did the long form calculations starting with minimum number of poles and so forth, however, when I get to Gvalues G1, Gn I get complex numbers of which I am not sure how to realize into a component value. I was hoping someone could walk me through the design process.

Filter Topology: Chebyshev Low Pass

Gp (Ripple): -1dB, fp: 13.56MHz

Gs: -40dB, fs:27.12MHz (2nd Harmonic)

These values really are arbitrary, and I am only interested in completing the formula for future synthesis and understanding so if you have an example with different values that is fine as long as they are stated.

First off I did the equations for the minimum number of poles:

Npoles = 1/(acosh(fs/fc) * acosh(sqrt((10^-(Gs/10) -1)/(10^-(Gp/10) -1))


Npoles = 1/(acosh(27.12/13.56) * acosh(sqrt((10^-(-40/10) -1)/(10^-(-1/10) -1))

Npoles = 4.53611, Since we cannot realize a filter with fractional poles we round UP. Npoles = 5.

ε^2 = 10^-(Gp/10) -1 = 0.25892541179

We can then verify our transfer function? to make sure the number of poles produces the desired attenuation with the following formula:

|G(f)|^2 = 1/(1+ε^2*cosh^2 [Npoles *acosh(fs/fp)]

|G(f)|^2 = 1/(1+0.258925*cosh^2 [5*acosh(27.12/13.56)] = 2.947108E-5 = 10*log(2.94E-5) = -45.3dB at 27.12MHz which is better than our -40dB minimum.

Now if we take this and turn it into our further values B, Y, A1, G1 etc I get complex values. How do I turn these complex values into component values for systems using only LC components?

I am NOT interested in using precomputed tables I am only interested in learning the actual formula and method behind MANUAL synthesis of the filters. This is purely an exercise in learning and understanding how they are realized using mathematical processes.

Sources:

https://en.wikipedia.org/wiki/Chebyshev_filter

https://www.matheonics.com/Tutorials/Chebyshev.html

https://en.wikipedia.org/wiki/Electrical_resistance_and_conductance

https://en.wikipedia.org/wiki/Electronic_filter_topology#Ladder_topologies

https://web.njit.edu/~gilhc/EE494/tutorial.htm#ChebyshevFilters

Microwave Filters, Impedance-Matching Networks, and Coupling Structures [George L. Matthaei] pg 97 on

ElectricalEngineer said:

Does anyone have some resources or mind explaining to me how we can synthesize a Chebyshev filter from the formulas specifically, and not from tables?

I am NOT interested in using precomputed tables I am only interested in learning the actual formula and method behind MANUAL synthesis of the filters. This is purely an exercise in learning and understanding how they are realized using mathematical processes.

Click to expand...

Do you really expect an answer consisting of the complete mathematical background for computing higher-order Chebyshev parameters?

Hi,

there are online tools for filter values calculation.
Don´t you trust them?

I assume there are the mathematical formulas also.

Klaus

I certainly do trust the filter calculators. One can also use the values and run a simulation to verify the proper calculated values, too.

This is just of educational interest to myself. I'm not sure how to take the numbers and turn them into component values even for filter with fewer poles.

I have read book after book after book, and the Wikipedia pages if you take a look -- https://en.wikipedia.org/wiki/Chebyshev_filter#Implementation

"A passive LC Chebyshev low-pass filter may be realized using a Cauer topology. The inductor or capacitor values of a nth-order Chebyshev prototype filter may be calculated from the following equations:[1]"

However, I've tried running some of the first elements using these equations and I only get complex numbers out and I'm not sure how to turn those complex numbers into the values as stated.

This isn't necessarily "How can I build a Chebyshev filter" per se, more of me wanting to understand the math behind the Chebyshev and how to calculate the component values. This is purely for educational value.

Given that I have spent days searching and all I can find are tables with no worked examples on calculating the values from the formula I was hoping someone here might be able to give me a broad overview "block diagram" approach if you will, or point me in the right direction so that I can find the answers I seek if that makes sense.

ElectricalEngineer said:

Given that I have spent days searching and all I can find are tables with no worked examples on calculating the values from the formula I was hoping someone here might be able to give me a broad overview "block diagram" approach if you will, or point me in the right direction so that I can find the answers I seek if that makes sense.

Click to expand...

I know what you are looking for.
However, I am afraid that a forum like this is not the right place to satisfy your needs.
All I can do is to recommend a book:
"Analog and Digital Filters", Harry Y.-F. Lam, (Bell Tel. Inc., 1979).

I agree with LvW that a complete answer can be found in text books.

My simplified view on "Math behind Chebyshev" is this:

1. You select Chebyshev polynomials for the filter magnitude transfer function because they achieve equiripple.
2. For given order, ripple amount and cut-off frequency, there's a one-to-one relation to the transfer function, respectively poles and zeros. Calculation of polynomial coefficients is straightforward.
3. For a given transfer function, you have many possible implementations, means you have to choose a specific circuit topology realizing the same number of poles and zeros. Calculate its general transfer function and find the component values by coefficient comparison and solving the respective simultaneous equations.

LvW said:

I know what you are looking for.
However, I am afraid that a forum like this is not the right place to satisfy your needs.
All I can do is to recommend a book:
"Analog and Digital Filters", Harry Y.-F. Lam, (Bell Tel. Inc., 1979).

Click to expand...

That book absolutely rocks! It walks you through step by step the actual realization from the transfer function I calculated. This is exactly what I was looking for! Thank you!

I consider this problem now solved. If anyone else is looking for similar answers I recommend getting yourself a copy of "Analog and Digital Filters" by Harry Y.F. Lam