Microstrip High- and Low-Impedance Short-Line Sections

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Expression 4.50 can be obtained by equating the ABCD parameters of the two circuits ?

Could anyone advise ?

Note: Screenshots below are taken from Microstrip Filters for RF Microwave Applications, 2nd Edition

[TheUnnamedNewbie]

Expression 4.50 can be obtained by equating the ABCD parameters of the two circuits ?

Could anyone advise ?

It literally says that they can be obtained from the ABCD parameters.

What you can try is look up is the following:

• How to go from a difference in impedance to ABCD parameters
• How to go from lumped components to ABCD parameters
• How to deal with electrical length in ABCD parameters

And from what you learn about this, you should try and set up the ABCD parameters for both the situations and show equivalence.

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Now the question is how to derive expression 4.50 given the following ABCD matrix ?

I am still trying to derive the ABCD matrix for the following circuit configuration.

Any help ?







Edit : The derivation process could be found at page 25 of the pdf attachment

 Microwave_Network_Parameter.pdf (387.32 kB - downloaded 79 times.)

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The following is the derivation process for expression 4.50

ABCD matrix for transmission line at Figure 4.7(a)



ABCD matrix for π-circuit network at Figure 4.7(b)




Comparing the two ABCD matrices above gives :






Applying trigonometry double angle formula :

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Ok, now to more detailed part:

How to derive expressions (5.3) ?
I suppose that this involves eliminating the assumption : Z_C << Z₀ << Z_L

Note: In the post above, I have derived expression (5.2) for inductor, which is equivalent to expression (4.50)

[promach]

The following is the derivation process for expression 4.52

ABCD matrix for transmission line at Figure 4.8(a)



ABCD matrix for T-circuit network at Figure 4.8(b)




Comparing the two ABCD matrices above gives :






Applying trigonometry double angle formula :

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Summarizing the derivation result from reply #3 and reply #5 , we could derive equation (5.3) easily.

However, what I do not understand is why the extra term for the inductor equation (5.3) is labeled as "shunt susceptance of the high-impedance line" ?

If I am not wrong, the book might have a typo mistake and the correct version of the sentence should be as follows :

The results of Eq. (5.2) do not take into account the series reactance of the HIGH-impedance line and shunt susceptance of the LOW-impedance lines.