Microstrip filter design: question about a part of a book

Hi

book title: Microstrip Filtersfor RF/Microwave Applications, by JIA-SHENG HONG M. J. LANCASTER

In chapter 5 of the book, the design of Stepped-Impedance, L-C Ladder Type Lowpass Filter is described. A third degree lowpass prototype with Chebyshev response is chosen and LC parameters are calculated.

up to here, it is all OK...

for realization of Microstrip equivalence, it specifies the ε_r and thickness of the substrate and then it says:

"Following the above-mentioned considerations, the characteristic impedances of the high- and low-impedance lines are chosen as Z0L =93 ohms and Z0C = 24 ohms"

and it uses Z0L and Z0C to calculate the initial length of microstrip sections.

my first question is: how Z0L and Z0C have been calculated? as far as i searched, the formulas for calculating impedance of microstrip line depend also on the width of the line. but at this stage of the design the width is not known yet and is to be calculated later.

the pages of the book are attached: Pages from Microstrip-Filters-for-RF-Microwave-Applications.pdf

my second question is regarding equations 5.2 and 5.3 in the attached file. where do they come from?

thanks in advance for your kind help

regarding my first question above:
there is some explanation in the attached file on how to choose Z0L and Z0C, and it has explained that the value of impedance and width of microstrip line should be determined simultaneously, so we choose a value for the impedance first. but the point is that what is best choice and why it has chosen Z0L=93 and Z0C=24. there are plenty of other values. are these values chosen by experience?

Here are my thoughts: 0.2mm wide high impedance section may be manufactured anywhere. When high impedance line is needed I usually use 0.2mm line, or sometimes 0.1524mm (6mil). 4mm wide low impedance section looks "normal" when it's quarterwave length is greater than line width.
I think it must be designed in this way: if with Z0=50 Ohm Z0L is too thin or Z0C is too wide (W>L), then choose Z0 in a such way that Z0L is manufacturable and Z0Cs width W is not too wide. It would be interesting to hear other comments.

Inductors can be considered Short Circuited Transmission Lines which are smaller than lambda/4
Capacitors can similarly be considered Open Circuited Transmission Lines which are smaller than lambda/4.
Under all these circumstances, all lumped elements have an equivalent Transmission Line counterparts.So, a Filter with lumped elements can be transformed into Transmission Line equivalent.
This is valid for filters too..

thanks for replying

I found it ! :)

The limitation on the width comes from the excitation of higher order modes (as was mentioned in the attached file). the formula for the cutoff frequency of the first higher-order mode:

HOM.bmp

increasing the width of the strip, lowers the spurious-free bandwidth of the filter. so i think the width of the strip is chosen so that the desired spurious-free range is satisfied.

and as Georgy.Moshkin said, the limitation on inductive part, comes from the fabrication process.


but my second question still remains: where the formulas 5.2 and 5.3 in the attached file come from?(which are used for estimation of the strip lengths of different sections) and how can i extend it to more complicated circuits with more element in higher order filters?

Pages from Microstrip-Filters-for-RF-Microwave-Applications.pdf

thanks a lot