- #1

Davidllerenav

- 424

- 14

- Homework Statement
- For the circuit in Figure 1, find the current and voltage at resistor R1 considering that the values of the voltage sources

they are V1 [V] and V2 [V], and the resistance values are R1 [Ω], R2 [Ω] and R3 [Ω] respectively.

- Relevant Equations
- Kirchhoff's Laws:

##\sum_i I_i=0##

##\sum_i V_i=0##

I tried to solve it by loop currents. So on the left mesh the loop current ##I_1## goes clockwise and on the right mesh the loop current ##I_2## goes counterclockwise.

I ended up with the following equations:

1) ##V_1-R_1(I_1+I_2)-R_2I_1=0##;

2) ##V_2-R_3I_2-R_1(I_1+I_2)=0##.

To find the current on ##R_1##, I tried to find ##I_1## and ##I_2##. I first solved 1) for ##I_1##: ##I_1=\frac{V_1-R_1I_2}{R_1+R_2}##. Then I solved 2) for ##I_2## replacing the expression that I found for ##I_1## :

##V_2-R_3I_2-R_1(I_1+I_2)=V_2-I_2(R_3-R_1)-R_1I_1##

##=V_2-I_2(R_3-R_1)-R_1\left(\frac{V_1-R_1I_2}{R_1+R_2}\right)=V_2-I_2(R_3-R_1)-R_1\left(\frac{V_1-R_1I_2}{R_1+R_2}\right)##

##=V_2(R_1+R_2)-I_2(R_3-R_1)(R_1+R_2)-R_1(V_1-R_1I_2)##

##=V_2(R_1+R_2)-I_2[(R_3-R_1)(R_1+R_2)-R_1^2]-R_1V_1##

##I_2=\frac{V_2(R_1+R_2)-R_1V_1}{(R_1+R_2)(R_3-R_1)-R_1^2}##

Finally, I replaced ##I_2## on ##I_1##: ##I_1=\frac{V_1-R_1\left(\frac{V_2(R_1+R_2)-R_1V_1}{(R_1+R_2)(R_3-R_1)-R_1^2}\right)}{R_1+R_2}##. Am I correct?##=V_2-I_2(R_3-R_1)-R_1\left(\frac{V_1-R_1I_2}{R_1+R_2}\right)=V_2-I_2(R_3-R_1)-R_1\left(\frac{V_1-R_1I_2}{R_1+R_2}\right)##

##=V_2(R_1+R_2)-I_2(R_3-R_1)(R_1+R_2)-R_1(V_1-R_1I_2)##

##=V_2(R_1+R_2)-I_2[(R_3-R_1)(R_1+R_2)-R_1^2]-R_1V_1##

##I_2=\frac{V_2(R_1+R_2)-R_1V_1}{(R_1+R_2)(R_3-R_1)-R_1^2}##

I'm not sure how to find the voltage on ##R_1##, but I think that I need to find ##R_1(I_1+I_2)##, right?