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I have a question that is a bit abstract. Assuming I have two proper linear circuits, A and B, which have:

  • The same ideal components (for example, voltage source, inductor, capacitor, resistor)
  • The same number for each type of components (for example, 1 voltage source, 3 capacitors, 1 resistor, and 1 inductor)
  • The same value for each component (for example Vs=1V, L=1uH,C1=3uF, C2=5uF, C3=7uF, R=1kΩ)
  • Any KVL (Kirchhoff's Voltage Law) equation that exists in A also exists in B, regardless of the order (for example, \$V_L + V_C + V_S = 0\$ is the same as \$V_L + V_S + V_C = 0\$ or \$-V_L - V_S - V_C = 0\$)

Can we conclude that:

  • The voltage across each element in circuit A will be the same as the voltage across the corresponding element in circuit B?
  • The current through each element in circuit A will be the same as the current through the corresponding element in circuit B?

If this is not true, could you provide an example?

By proper circuit, I mean a normal circuit without contradictions, such as not having two ideal voltage sources with different values in parallel.

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  • \$\begingroup\$ Have you read anything about the limitations of KVL? Wikipedia does an OK job of explaining \$\endgroup\$
    – MrGerber
    Commented Jul 17 at 7:01
  • \$\begingroup\$ Also relevant physics.stackexchange.com/questions/685833/… \$\endgroup\$
    – MrGerber
    Commented Jul 17 at 7:06
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    \$\begingroup\$ Imagine yourself running around on a hillside. You start and stop at the same point. Can you create a running track around that hillside where you don't end up at the same height? \$\endgroup\$
    – MrGerber
    Commented Jul 17 at 7:10
  • \$\begingroup\$ Are the circuit topologies the same? If yes, then the 'same number' condition seems redundant, which suggests the topologies are different. Different arrangements of the same number and same value components can trivially have different voltages and currents (think voltage divider one or the other way up). If the circuits are completely identical, ie cloned, same topology, component values, then KVL is irrelevant, the circuits will obviously be and behave identically. \$\endgroup\$
    – Neil_UK
    Commented Jul 17 at 7:11
  • \$\begingroup\$ @MrGerber I just checked Wikipedia, and I think it's not related because it discusses lumped circuits and transmission lines, whereas this is specifically for lumped circuits. \$\endgroup\$
    – hana
    Commented Jul 17 at 7:18

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The question can be answered trivially, once it's properly defined how alike the two circuits A and B are. The OP answered a question about the topologies of the circuits in a comment ...

Are the circuit topologies the same? Assume that we don't know anything beyond the given assumptions. If they are the same, then there is nothing more to discuss.

If A and B are identical, as in clones of each other, so copy one and paste to the other, they are trivially identical in every way. No need to invoke KVL, they are totally identical. As the OP said, nothing more to discuss. So they must be different in some way.

If A and B are not identical , but have the same component values, then the simplest counter-example that comes to mind is this. No need to invoke KVL, they are different by inspection. Consider the voltage at the junction of the resistors, with respect to either of the supply terminals.

schematic

simulate this circuit – Schematic created using CircuitLab

There is a simpler pair of circuits with only two resistors in a voltage divider configuration, flipping the top and bottom resistors, but that could be argued to be 'the same' with only a sign change in the power supply voltage.

(edit)
However, my two circuits above are disallowed by the OP's 4th bullet, all KVL loops are the same. I think this implies that the two circuits are copy/paste clones, but I've been wrong before on 'obvious' identities in topology and graph theory. I'm therefore reluctant to claim that two graphs (circuits) that have the same set of loop sub-graphs (KVL loops) are necessarily identical. I think a proof or refutation of this particular question is better addressed on MathOverflow.
(/edit)

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  • \$\begingroup\$ This is what I was getting at in my latest comments. \$\endgroup\$
    – MrGerber
    Commented Jul 17 at 11:20
  • \$\begingroup\$ The 4th bullet point says that any KVL equation that exists in circuit A also exists in circuit B. However, your two circuits don't satisfy this condition. For example, the loop with the 10kΩ and 100kΩ resistors in the first circuit doesn't exist in the second circuit. \$\endgroup\$
    – hana
    Commented Jul 18 at 5:05
  • \$\begingroup\$ @hana In that case I have missed the point. I think the 4th bullet implies the circuits are clones, copy-pasted, and so identical, but I'm not sure. If you think so, I suggest you write an answer to your own question. It might be worth asking the question on Mathoverflow, it may be a better fit there, as there may be some graph theory result about whether if two graphs (circuits) have the same cyclic sub-graphs (voltage loops for KVL) they are necessarily identical or not. This topological argument was too deep for me, and will probably be for most electronic engineers. \$\endgroup\$
    – Neil_UK
    Commented Jul 18 at 7:39
  • \$\begingroup\$ How could they be different then - it would be the same as asking whether KVL holds no matter where you start or which way you go around the circuit. As long as you don't change direction suddenly, and keep your polarities correct, it's correct. \$\endgroup\$
    – MrGerber
    Commented Jul 18 at 8:06
  • \$\begingroup\$ How could we use the normal arithmetic operators if we suddenly decided that the axioms of algebra weren't valid? If we weren't able to order the summed terms around as long as we keep the polarity - how would the additive axiom and Rearrangement Property of Addition still work? \$\endgroup\$
    – MrGerber
    Commented Jul 18 at 8:12

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