Understanding KVL Equations for Circuit Analysis

[Tiziano]

Homework Statement

The following circuit is given.

I intend to calculate the current in every resistor (every quantity except i1, i2, i3, is known).
My textbook states that ℰ3-ℰ1 = (R1+R2+R3+R4+2r)*i, but I think it should be -ℰ3+ℰ1 on the left-hand side, since the current enters the negative and leaves the positive terminal of ℰ1 and does the opposite with ℰ3.

Homework Equations

Ohm's generalized law for a closed circuit (the algebric sum of the emf's equals R*i, where R is the total resistance)

The Attempt at a Solution

I've simply computed the same equation of the book, but I don't understand why my signs are wrong.

 

[Dr.D]

Since you did not show any work, it is difficult to help you.

It is also not clear what is indicated by the curved arrow across the middle leg just below C. What are we to suppose this is?

 

[Simon Bridge]

My textbook states that ℰ3-ℰ1 = (R1+R2+R3+R4+2r)*i, but I think it should be -ℰ3+ℰ1 on the left-hand side, since the current enters the negative and leaves the positive terminal of ℰ1 and does the opposite with ℰ3.

Isn't that what they did? (change sides change sign).

The rule is: the sum of the voltages is zero. So, for the big loop:
$\mathscr{E}_1 -R_1i_1 -R_2i_3 -\mathscr{E}_3 +r i_3 +R_4i_3 +R_3i_1 +ri_1 = 0$ ... tidy it up, and express as emf = other stuff.

I don't think that ℰ3-ℰ1 = (R1+R2+R3+R4+2r)*i is correct as written though.

 

[gneill]

Hi Tiziano,

To me it looks like you're writing KVL equations for loop analysis of the circuit using loops as follows:

Is this correct?

If so, I agree with you that the book's equation ℰ3-ℰ1 = (R1+R2+R3+R4+2r)*i does not handle the voltage sources correctly for the given loop with current i (shown in blue above), and that your own version is correct.