Mesh Analysis / Sign Convention - Conceptual Problem

[Zatman]

Homework Statement

I am having a little trouble using mesh analysis. I'm trying to stick rigorously to the passive sign convention, as I'm told that this is sensible.

Consider the circuit diagram attached, where I have added the mesh currents in orange. In red are the polarities for the components where it is obvious - of course, using the passive sign convention.

How do I assign the polarities for the other components, i.e. the 6V voltage source and the 5 and 1 ohm resistors? For example, the 6V source cannot be assigned a polarity that satisfies the passive sign convention for both mesh 1 and mesh 3.

As it turns out, assigning positive to the top and negative to the bottom gives the correct answer (after the mesh analysis), but I'm not sure how you can tell.

 

[gneill]

Potential drops in the direction of the current flow. Since mesh currents are treated as separate currents in shared components (like the 5 and 1 ohm resistors in your example), there's a separate potential drop associated with each current.

You need to account for both changes in potential when you write the equation for a loop.

 

[Zatman]

Thank you gneill, that makes sense. However, I still get the incorrect equations when I try to analyse my circuit, so I guess I'm still missing something.

I get, for mesh 1, 2 and 3 respectively,

2i_1+5(i_1-i_2)+6=0
\Rightarrow7i_1-5i_2=-6

12+(i_2-i_3)+5(i_2-i_1)=0
\Rightarrow 5i_1-6i_2+i_3=12

4i_3+6+(i_3-i_2)=0
\Rightarrow -i_2+5i_3=-6

Not sure what I'm doing wrong. :(

 

[SteamKing]

Well, for one thing, in loops 1 and 3, the 6V source has different polarities simultaneously. This can only lead to confusion.

http://en.wikipedia.org/wiki/Mesh_analysis

It's best to pick one polarity and adjust the sign of the voltage according to the assumed direction of the current when writing your mesh equations. There is no rule that says voltages must always be positive.

 

[Zatman]

Okay, but that raises the question: how do I know which way to assign the polarity to the 6V source? I could either assign it to match up with the mesh 1 current (which gives the correct answer) or the mesh 3 current (which gives the incorrect answer).

 

[gneill]

Okay, but that raises the question: how do I know which way to assign the polarity to the 6V source? I could either assign it to match up with the mesh 1 current (which gives the correct answer) or the mesh 3 current (which gives the incorrect answer).

Voltage sources are not passive components like resistors. They have a fixed polarity which you must use. A voltage source always produces the same potential difference across its terminals regardless of the current flowing though it.

 

[Zatman]

Voltage sources are not passive components like resistors. They have a fixed polarity which you must use. A voltage source always produces the same potential difference across its terminals regardless of the current flowing though it.

Got it, thanks!

For the record, the book I'm using doesn't use the passive sign convention. So I've got into the habit of ignoring their labelling and redoing it. So in this example I changed the labelling of the voltage sources!