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I am reading Electronic Principles by P.E. Gray and C.L. Searle and I came across the following statement.

If there are n capacitors and no capacitor loops, the highest power of s in the characteristic equation is n

Is there a proof of the following statement? and also what is meant in a capacitor loop?

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  • \$\begingroup\$ This seems to be a round-about way of saying that an n-th order circuit has a characteristic equation of order n, no? \$\endgroup\$ – Hearth Nov 6 '19 at 0:09
  • \$\begingroup\$ @Hearth I was looking to understand more why a circuit with n capacitors (provided there are no capacitor loops) will have a characteristic equation with degree n? \$\endgroup\$ – dilinex Nov 6 '19 at 0:12
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    \$\begingroup\$ What level of proof do you need? It's easy enough to demonstrate by induction. There's an unstated assumption that the circuit contains only resistors and capacitors, no inductors. With zero capacitors, there's no frequency dependence, so s^0, with one there's s^1 and so on, it's almost a tautology, every time you add one capacitor, it's easy to find a configuration which increases the power of s by one, and no more. \$\endgroup\$ – Neil_UK Nov 6 '19 at 3:39
  • \$\begingroup\$ I'm not seeing something here - how do you get a 2nd order circuit (I assume that means a 2nd order differential equation), with 2 capacitors and nothing else? Perhaps it's something to do with 'capacitor loops' (what are they?). \$\endgroup\$ – Chu Nov 6 '19 at 19:58
  • \$\begingroup\$ @Chu the statement is not mentioning other components, but neither any (wired) connection. Since the latter obviously needs to be assumed (in order to form loops), I think the first needs to be assumed as well. \$\endgroup\$ – Huisman Nov 6 '19 at 20:01
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The book can be found on: https://mirror.thelifeofkenneth.com/lib/electronics_archive/GraySearle-ElectronicPrinciples_text.pdf

On page 556 in PDF / page 553 in book, the section starts with:

To find the general relationship between a1/a0 and the network elements, we consider a linear active network that contains n capacitors and no other energy storage.
For simplicity, we develop the relation for a1/a0 for the three- capacitor network of Fig. 15.3a, recognizing that the PROOF can be readily generalized.

This generalization of the proof can be demonstrate by induction, as @Neil_UK suggests, by adding another capacitor in Figure 15.3 and appending the equation for the next capacitor to eq. 15.23a and following equations.

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