Tricks in finding the transfer functions of passive quadripoles?

Question

Electronics

In short: we are interested in methods for calculating the voltage transfer coefficient for a passive four-terminal network that do not require the compilation of a complete system of equations for the circuit. Under the cut is an example of a circuit when a typical simplified calculation method cannot be applied, and the question is what other methods of simplifying circuits are (besides the usual convolution of parallel and series resistances and applying the delta-star transformation ).
Of course, the interest here is completely idle, since in practice the system of equations describing a linear circuit can always be symbolically solved using some mathcad, and then ask mathcad to divide the expression for the voltage at the load by the expression for the input voltage, and get the exact formula for the transfer coefficient.

The problem, but rather a general question, arose when a student friend contacted me with a request to tell me how to find the voltage transfer coefficient for some passive four-terminal network.
Theory; as they think in general terms Anyone who has not in vain listened to lectures on linear circuits at a university will immediately (in any case, rather quickly) remember what exactly they asked me about: there is a certain bunch of points i, let from 1..N, connected in some order by complex resistances; the voltage transfer coefficient is the ratio Ů _ij / Ů _km , for a four-terminal network, in which the points k, m (preferably not the same) receive the input signal, and the output voltage is removed from the points i, j.
It is quite obvious that the ratio of signals Ů _outand Ů _input for a passive four-terminal network (that is, consisting only of complex resistances, read - resistors, capacitances and inductances) is determined solely by the circuit structure and the values ​​\u200b\u200bof the elements. However, any person who passed the TOE knows all this without me.
... just like the usual method of finding the voltage transfer coefficient. Indeed, the Kirchhoff system of equations in any case gives us the answer (K̊ = f(Z ₁₂ ,Z ₁₃ ,...Z _N-1,N ), but at what cost, yeah. it is crooked.
Naturally, people rarely do this in practical classes. The thing is that there are certain situations in which the expression of the transfer function can be obtained more simply without compiling the entire system of equations. It is these situations that I would like to talk about.
So, there is some circuit, the transfer function of which (let it be the ratio of voltages for definiteness) we would like to obtain.
Yes, you can make a system of equations and derive an expression for the transfer function we need. But sometimes you can reduce the amount of calculations: for example, if a quadripole can be (by convolution) brought to the form
, then its transfer function is the ratio Z ₂ / (Z ₁ + Z ₂), which is usually much simpler than a system of equations. This, by the way, was the method that was proposed to be used by a student I knew.
There is an artificial technique that simplifies the calculations. Unfortunately, this technique can not always be applied, and it was impossible to apply this method to the scheme that corresponded to his version of the task (I no longer remember whether the teacher invented the scheme on the fly, or took it from the problem book).
Question: what other methods of circuit simplification do you know? Are there other ways to obtain the transfer function of a circuit that do not require the compilation of a complete system of equations?

Answer 1

I don’t know any tricky ways, but in the diagram above, you can apply the triangle-star transformation to the 3-5-6 contour ...