Degenerate circuit concept and its theoretical and practical implications

Question

I came across with the term "degenerate circuit" when I was studying solving linear differential equations with Laplace transform. I have learned that a system is called degenerate when the determinant of coefficient matrix (characteristic polynomial) is zero.

I've searched a lot to try to understand the concept of "degenerate circuit" but I could not fully understand it. Some fragments of what I have searched:

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Network Theory and Filter Design (Vasudev K. Aatre)

         

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Electric Circuits And Networks (For Gtu) (Kumar K. S. Suresh)

         

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Linear Circuit Transfer Functions: An Introduction to Fast Analytical Techniques (Christophe P. Basso)

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My questions are:

• What is the definition of "degenerate circuit"? (I think Basso's book 1st fragment answered it, but I didn't understand)
• What are the theoretical implications of a circuit being degenerate, in terms of stability?
• In "real life", what does it mean to be degenerate? A degenerate circuit, if built, would not work?

Answer 1

The phrase degenerate circuit does not have a strict meaning out of a specific context. It means that in a certain case, something about the circuit differs in a significant way from the equations that apply to the more general case. Usually this means that, though it may still be possible to use the equations you have developed, they may no longer be the best way to analyze the circuit.

This meaning of degenerate is discussed in more detail at: Degeneracy (mathematics) on Wikipedia.

There is another related meaning of degenerate that you may encounter, which is a special case of the previous definition -- a degenerate matrix is one which is not invertible (Invertible matrix, Wikipedia). This can come up when solving a system of linear equations (KCL or KVL) if you mistakenly use the same equation twice, for example.

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Here's a (somewhat contrived) example of how a degenerate case can still use more complicated equations from a general case, though they are not necessarily the best tool. Consider the case of a 2-stage RC filter:

^{simulate this circuit – Schematic created using CircuitLab}

The transfer function, given in hryghr's answer, is

$$ H(s)=\frac{1}{s^2R_1R_2C_1C_2+s(R_1C_1+R_1C_2+R_2C_2)+1}$$

Now, a single-stage RC filter can be considered a degenerate case of the above circuit, with \$R_2 = 0\$. You can plug that into the above equation and get

$$ H(s)=\frac{1}{1+sR_1(C_1+C_2)}$$

but the simpler method would be to just analyze the 1-stage RC circuit directly.