Help concering understanding Circuits (Linear Algebra Related)

[MMhawk607]

In our Linear Algebra class we barely touched on (and I mean barely touched up on) circuits. Thing is, it will be on the exam, but just 1 question.

My problem is understanding how to get the formula to use Cramer's Rule

Of course I know how to set these up into Matrices and do all of that. But my issue is how do they get these formulas?

Why is E1 positive and E2 negative
Why is R2 (Ia - Ib) but on the other it's (Ib - Ia)

Any help is appreciated. I'm not asking for a problem to be solved. Just need sort of a lesson on how to read these sorts of images.

 

[tiny-tim]

Hi MMhawk607!

(try using the X₂ button just above the Reply box )

Why is E1 positive and E2 negative
Why is R2 (Ia - Ib) but on the other it's (Ib - Ia)

because you go round the same way as that round arrow is pointing

so the left arrow goes through E₁ from + to -, so that's +E₁,

while the right arrow goes through E₂ from - to +, so that's -E₂,

similarly, for the "left equation", R₂ is positive, so that's R₂I_a, but the left and right arrows go opposite ways through R₂, so R₂I_b must be negative

(you could draw the diagram with one or both of the round arrows the other way round …

then a lot of things would be minus instead of plus, but it would all give the same result in the end)

 

[mynick]

http://imageshack.us/a/img850/1476/kclcapture.jpg

KCL=the sum of the currents entering a node is equal to the sum of the currents leaving that node. For each loop following direction of the arrow around the loop you have drawn we get:
Loop 1: Ia=I2+Ib => I2=Ia-Ib
Loop 2: Ib=Ia+I2 => I2=Ib-Ia

Examine each loop individually(hide the part of the circuit that does not matter) so that you won't scratch your head wondering why I2 is positive in both cases...I only applied what KCL says.
i have re -drawn your image to make the above equations clear.For Loop 1 ignore red arrows. For loop 2 ignore black arrows.

 

[MMhawk607]

Ohhhh I see it now! Thank you guys. The professor just rushed through this part and didn't really give us time to ask questions about it. I appreciate it!

 

[Studiot]

Hello MMHAwk, What are you studying, Physics, Maths or Elec Eng?

I ask this because you should be careful analysing the circuit as presented.

In your linear algebra class you have no doubt learned that you can transform a set of linear equations into one with a new basis. This is equivalent to directly manipulating the equations, or using matrix operations to solve them.

In linear electrical circuit theory there are several ways to analyse the system and solve a set of linear equations. They should all lead to the same solution set.
The different ways are equivalent to change of basis.

The method you have presented is Maxwells Mesh Method. This is a change of variable method, rather than a change of basis method.

Be careful not to confuse it with Kirchoff's methods, one of which mynick showed. Mesh analysis uses differnt variables and needs one les equation to solve the system compared to Kirchoff's branch currents or nodal analysis. Further it is impossible to solve a system by branch currents alone, as they do not involve voltages.

If you really wish to use Kirchoff then