Unsure why I am getting this mesh analysis problem wrong

[influx]

I considered loop I₁ and got the correct equation for it. Then I applied the mesh analysis method to loop I₂ but I'm getting one term wrong (its sign).

Starting in the top left corner of loop I₂, 30I₂ - 90(I₂ -I₃) - 5(I₁ - I₂) = 0

Everything is correct apart from the -90, its supposed to be +90 according to the answer provided to me. However, if you start in the top left corner of loop I₂ and travel clockwise, you enter the 90 ohm resistor at the negative end. So surely its -90 rather than +90? I've been using this method successfully so far but I am unsure why its not working here..Thanks

PS: I assumed I₁ > I₂ > I₃

 

[SteamKing]

When you come to a resistor which is shared by two loops, what matters is that you write the proper equation for each loop, using the assumed direction of current flow in each loop you made initially. As long as you write the proper equations for each loop, the solution of the unknown currents should give you the proper values. If they don't, you have made a mistake in writing the equation. If the solution gives you a negative current, it just means that the actual current flows in the direction opposite to your initial assumption.

For the problem above, you can't tell for certain that you have made a mistake until you write out and solve all three loop equations.

 

[gneill]

I considered loop I₁ and got the correct equation for it. Then I applied the mesh analysis method to loop I₂ but I'm getting one term wrong (its sign).

Starting in the top left corner of loop I₂, 30I₂ - 90(I₂ -I₃) - 5(I₁ - I₂) = 0

Everything is correct apart from the -90, its supposed to be +90 according to the answer provided to me. However, if you start in the top left corner of loop I₂ and travel clockwise, you enter the 90 ohm resistor at the negative end. So surely its -90 rather than +90? I've been using this method successfully so far but I am unsure why its not working here..


Thanks

PS: I assumed I₁ > I₂ > I₃

Assuming relationships between current magnitudes doesn't get you anywhere in mesh problems. What matters is making an initial assumption about current directions and then sticking to it.

Usually the choice is to make all the mesh currents either clockwise or counterclockwise. This means that components that are shared by two loops will have the two currents passing through it moving in opposite directions, and so their individual potential changes will also oppose (have opposite sign) in a given equation. So labeling a single "+ -" potential change across each component only makes sense for components where all the currents passing through it do so in the same direction.

In your circuit diagram you should sketch in the mesh currents so that you can visualize the potential changes caused by each current as you do your "KVL walk" around each loop.