Rearranging Electric Circuits: Rules and Topological Equivalence

[temaire]

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?

 

[epenguin]

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.

 

[temaire]

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?

 

[gneill]

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.

 

[lorenb]

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

 

[epenguin]

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.