Determining poles and zeros from those of interconnected two-port networks

Your question pertains to a particular network, but a general approach to answering the question applies to all networks. Middlebrook published an early paper on the subject.

Middlebrook, R.D., 1989. Null double injection and the extra element theorem. IEEE Transactions on Education, 32(3), pp.167-180.

Later he generalized this approach somewhat.

Middlebrook, R.D., Vorpérian, V. and Lindal, J., 1998. The N extra element theorem. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 45(9), pp.919-935.

Vorpérian wrote an entire book addressing the subject.

Vorpérian, V., 2002. Fast analytical techniques for electrical and electronic circuits. Cambridge University Press.

I believe this book by Basso also addresses the technique.

Basso, C.P., 2016. Linear circuit transfer functions: An introduction to fast analytical techniques. John Wiley & Sons.

There's a lot of material in these articles and books, more than can be replicated in a brief StackExchange response, but all of them describe how to analyze these networks. That is different and, I submit, much better than answering the question you have asked: how to analyze a particular network.

@Andyaka I'm interested in two cases: One where the L's and C's are the same value, and another when the L's are all the same but the C's alternate between C1 and C2

For the case when L and C are the same for all stages I think this transmission line derivation for characteristic impedance may provide some pointers. The thing is this; when the Ls and Cs are equal value (even though their positions have swapped), the output impedance of one stage tends can be the input impedance of the next stage. You can imagine that if your actual lumped circuit was symmetrical (by doubling up on input and output capacitances), then the same sort of math is involved.

Here is the standard transmission line solution for references and maybe you can use this method for your lumped sections joined together

So, consider a "lump" of transmission line connected to the continuation of that transmission line (\$Z_0\$): -

• R is series resistance of cable for a given length
• L is series inductance of cable for a given length
• G is parallel conductance of cable for a given length
• C is parallel capacitance of cable for a given length
• \$Z_0\$ to the right is the continuation of the cable

Therefore the impedance looking into the left is: -

$$Z_0 = R + j\omega L + Z_0||\dfrac{1}{G + j\omega C}$$

$$= R + j\omega L + \dfrac{\frac{Z_0}{G+j\omega C}}{Z_0 + \frac{1}{G+j\omega C}}$$

$$= R + j\omega L + \dfrac{Z_0}{1 + Z_0(G+j\omega C)}$$

$$Z_0[1 + Z_0(G+j\omega C)] = [R+j\omega L][1 + Z_0(G+j\omega C)]+Z_0$$

$$Z_0 + Z_0^2(G+j\omega C) = R+j\omega L + Z_0[(R+j\omega L)(G+j\omega C)]+Z_0$$

$$Z_0^2(G+j\omega C) = R+j\omega L + Z_0[(R+j\omega L)(G+j\omega C)]$$

The important thing next is to recognize that \$(R+j\omega L)(G+j\omega C)\$ is insignificant as the "lump" approaches zero length and we are left with: -

$$Z_0^2(G+j\omega C) = R+j\omega L$$

hence $$Z_0 = \sqrt{\dfrac{R+j\omega L}{G+j\omega C}}$$