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Stepped impedance lowpass filter - how to compensate lengths?

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I have just fallen in love with experimenting with microwave stuff  ;D Just out of my stupid curiosity, I am trying to design a stepped impedance lowpass filter. I have tried to obtain some useful sources of information round the web. One of them, a lecture from a Prof. T.L.Wu mentions "compensation of parasitic elements". However I do not fully understand, what that means and how to compensate for a filter with arbitrary length of elements.

I have attached the presentation.  On page 3 down, the two equations with tangents, underlined red. Do I understand it right I should put there a sum of every other (L versus C) elements?

So for example, If I have a 7th order filter, I have 4 capacitors and 3 inductors.  To compensate a length of any "L" transmission line in the filter, I have to put there a sum of tangents of all four capacitors in the filter, and do similar with the C length compensation?

Here's a link to the presentation: http://ntuemc.tw/upload/file/20110321102525847f2.pdf (Sorry, it didn't allow me to attach it)

Thank you for help,

So this is the way I have understood how to calculate the lengths. Taking the formula from page 3 of the PDF above,  the length of L element should be this way. The K constant is the correcting part of the bottom formula, the sum of the tangents.

Is that right?


//EDIT: Sorry! A mistake! In the L length formula, there should be lambdaGL / 2pi not the other way round!

Maybe I should make an example so we can better understand each other:

My design goal is a 0.5dB Chebyshev fc 1GHz, 7th order. 50ohm in, 50ohm out.

Using a 0.8mm (32mil) thick FR4 PCB (epsilon_r of 4.6). These impedances/TMLs were chosen.

Zo = 50ohm, 60mil wide, guided wavelength 162.5mm (effective epsilon_r 3.406)
Zc = 10ohm, 500mil wide, guided wavelength 146.8mm
Zl = 110ohm, 8mil wide, guided wavelength 175.5mm

The normalized filter coefficients are: 1.737, 1.258, 2.638, 1.344, 2.638, 1.258, 1.737
From the coefficients, I get this set of lumped element values: 5.5pF, 10nH, 8.4pF, 10.7nH, 8.4pF, 10nH, 5.5pF
So far so good. Now to calculate the uncompensated lengths:
L2 = L6 = 10nH = 17mm
L4 = 10.7nH = 18.4mm
C1 = C7 = 5.5pF = 8.3mm
C3=C5 = 8.4pF = 13mm

How should I compensate the lengths now? For each inductor (L2, L4, L6) should I use a sum of tangents of all capacitive elements from the filter?
Meaning the K in this case should be equal:  K = Zc * ( 2*tan(pi*8.3mm/10) +  2*tan(pi*13mm/10) )
Am I right?


The compensating coefficient K for all capacitor lengths in the filter is 9.3025 ohms or thereabout (hope I calculated those tangents right).
After trying to compensate L2(L6) - the 10nH inductor, the new length is 14.2mm instead of 17mm.  At least it seems a legit number, not some strange bullshit.


Is this just a paper exercise or are you trying to actually build a filter?


Doesn't really matter, so I don't understand such question.

PS: Will build the filter afterwards and measure the performance.


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