Rearranging Electric Circuits: Rules and Topological Equivalence

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Discussion Overview

The discussion revolves around the rules and principles governing the rearrangement of electric circuits, specifically focusing on topological equivalence and the conditions under which two circuit diagrams can be considered equivalent. Participants explore the implications of circuit layout changes on electrical properties.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether there are specific rules for rearranging circuits and seeks clarification on the equivalence of two given circuit diagrams.
  • Another participant points out a potential short circuit in the original drawing, suggesting that it would affect the circuit's behavior and questioning the equivalence without further clarification.
  • A correction is made regarding the presence of a short circuit, leading to a discussion on whether the corrected circuit is equivalent to the original.
  • One participant asserts that as long as the connections remain unchanged, the rearrangement of components does not affect the circuit's equivalence.
  • Further clarification is provided on the concept of nodes, emphasizing that the arrangement of components is irrelevant as long as the connections to nodes are maintained.
  • Another participant elaborates on the idea of topological equivalence, explaining that the physical layout can change without altering the electrical connections, thus preserving equivalence.

Areas of Agreement / Disagreement

Participants generally agree on the principle of topological equivalence in circuit rearrangement, but there are differing views on the implications of specific circuit configurations, such as the presence of a short circuit. The discussion remains somewhat unresolved regarding the nuances of certain configurations.

Contextual Notes

There is an assumption that participants have a shared understanding of circuit components and topology, but specific definitions of terms like "node" and "topological equivalence" are not explicitly agreed upon. The discussion also does not resolve the implications of the short circuit in the context of equivalence.

temaire
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I just have a general question on rearranging circuits. Is there are certain rule that dictates how and when you can rearrange circuits? For example, in the diagram below, is the left circuit equal to the right circuit? And how would I know that?

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I don't know whether you need to produce more formal arguments but it seems to me obvious that in the second diag. you have a short-circuit. You seem to have a constant-current generator, the voltage across the circuit would be zero as would be the current through the resistors. In the case this short circuit is a mis-drawing and should not be there it is a bit less obvious but still fairly trivial I'd say.
 
Sorry, I made a mistake in drawing the right circuit. There shoudn't be a short circuit there. But without the short circuit, the right circuit is equivalent to the left circuit?
 
Yes. R1 is in parallel with the current supply in both diagrams. You can rearrange the component locations on the page to your heart's content so long as all the connections remain the same.
 
as long as the elements are connected to each "node" the same way. It does not matter where you put them. Make sure you have a good understanding of what a node is. In bother figures there are only two nodes
 
Yes - in case it is not obvious or similar problems aren't, just remember this is topological, that is as long as you keep the original connections, no more no less, you haven't changed anything electrically and you have an equivalent circuit. The lines represent perfect conductors.

So on your left fig. take top and bottom left corners and place them inside the right rectangle, keeping all connections. So you've obtained inside the right rectangle a branch that goes top L corner - R1 - bottom L corner. What were previously T junctions are still points from which 3 wires radiate - looks different but no connections have changed - topological equivalence. Now you could slide your middle branch connected to R1 along the top side, similarly at the bottom, till it looks like your second figure (corrected as you said) and you are still not changing anything because those lines are just perfect electrical connections.

These things are easier done than said.
 

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