I'm kind of working through Grob's Basic Electricity (9th edition) and just finished Chapter 9, on Kirchhoff's Laws. Most of the material I am ok with, but in the problems for the chapter, I'm having some difficulty with numbers 10, 11, and 12. Number 10 requires using the method of branch currents to solve for the current and voltages across a number of resistors. Here's a copy of the illustration provided:
I've included my work with the answers I got, below. I'm probably not following the method to a T, but I'm not terribly worried about that. The answers aren't in the back of the book, but I modeled the circuit on a simulator and its results agreed with mine.
V1-V(R1)-V(R3)=0
V2-V(R3)-V(R2)=0
V2-V(R3)-V(R2)=0
15-15*I3-10*I2=0
15-15*(I1+I2)-10*I2=0
15-15*I1-15*I2-10*I2=0
15-15*I1-25*I2=0
15*I1+25*I2=15
I2=(15-15*I1)/25=3/5-3/5*I1
V1-V(R1)-V(R3)=0
10-10*I1-15*I3=0
10-10*I1-15*(I1+I2)=0
10-10*I1-15*I1-15*I2=0
10-25*I1-15*I2=0
25*I1+15*I2=10 -----------------------------> 25*1/16+15*I2=10
25*I1+15*(3/5-3/5*I1)=10..................... 10-25/16=15*I2
25*I1+45/5-45/5*I1)=10........................ 160/16-25/16=15*I2
25*I1+9-9*I1=10................................... 135/16=15*I2
16*I1=1................................................. I2=9/16 A
I1=1/16 A
V1-V(R1)-V(R3)=0
10-10*1/16-15*I3=0
10-10/16=15*I3
160/16-10/16=15*I3
(150/16)/15=I3
I3=10/16 A
V(R1)=10*1/16=10/16 V=0.625 V
V(R2)=10*9/16=90/16=5.625 V
V(R3)=15*10/16=150/16=9.375 V
Number 11 says to use KVL to prove that the sum of the voltages is zero in each of the three closed loops. For the left loop, going counter-clockwise from the negative terminal of V1, it says the answer is -10 V + 16.875 V - 15 V + 8.125 V = 0; for the right loop, counter-clockwise from the positive terminal of V2, it says the answer is 15 V - 16.875 V + 1.875 V = 0; for the outside loop, going counter-clockwise from the negative terminal of V1, it gives -10 V + 1.875 V + 8.125 V = 0.
Honestly, I have no idea what to do with the left loop, with its two voltage sources, but for the right loop, I would think 15 V - 15*10/16 - 10*9/16 = 0 (or 15 V - 9.375 V - 5.625 V = 0) should be the answer. For the outside loop, I would think - 10 V + 10*1/16 + 15*10/16 = 0 (or 10 V + 0.625 V + 9.375 V = 0) should be the answer. As I mentioned, these are not the answers the book gives. Could someone please help me understand where I am going wrong?
Number 12 involves analyzing the same circuit using the method of mesh currents, but I'll try to work through that one again before asking.
Thank you!
I've included my work with the answers I got, below. I'm probably not following the method to a T, but I'm not terribly worried about that. The answers aren't in the back of the book, but I modeled the circuit on a simulator and its results agreed with mine.
V1-V(R1)-V(R3)=0
V2-V(R3)-V(R2)=0
V2-V(R3)-V(R2)=0
15-15*I3-10*I2=0
15-15*(I1+I2)-10*I2=0
15-15*I1-15*I2-10*I2=0
15-15*I1-25*I2=0
15*I1+25*I2=15
I2=(15-15*I1)/25=3/5-3/5*I1
V1-V(R1)-V(R3)=0
10-10*I1-15*I3=0
10-10*I1-15*(I1+I2)=0
10-10*I1-15*I1-15*I2=0
10-25*I1-15*I2=0
25*I1+15*I2=10 -----------------------------> 25*1/16+15*I2=10
25*I1+15*(3/5-3/5*I1)=10..................... 10-25/16=15*I2
25*I1+45/5-45/5*I1)=10........................ 160/16-25/16=15*I2
25*I1+9-9*I1=10................................... 135/16=15*I2
16*I1=1................................................. I2=9/16 A
I1=1/16 A
V1-V(R1)-V(R3)=0
10-10*1/16-15*I3=0
10-10/16=15*I3
160/16-10/16=15*I3
(150/16)/15=I3
I3=10/16 A
V(R1)=10*1/16=10/16 V=0.625 V
V(R2)=10*9/16=90/16=5.625 V
V(R3)=15*10/16=150/16=9.375 V
Number 11 says to use KVL to prove that the sum of the voltages is zero in each of the three closed loops. For the left loop, going counter-clockwise from the negative terminal of V1, it says the answer is -10 V + 16.875 V - 15 V + 8.125 V = 0; for the right loop, counter-clockwise from the positive terminal of V2, it says the answer is 15 V - 16.875 V + 1.875 V = 0; for the outside loop, going counter-clockwise from the negative terminal of V1, it gives -10 V + 1.875 V + 8.125 V = 0.
Honestly, I have no idea what to do with the left loop, with its two voltage sources, but for the right loop, I would think 15 V - 15*10/16 - 10*9/16 = 0 (or 15 V - 9.375 V - 5.625 V = 0) should be the answer. For the outside loop, I would think - 10 V + 10*1/16 + 15*10/16 = 0 (or 10 V + 0.625 V + 9.375 V = 0) should be the answer. As I mentioned, these are not the answers the book gives. Could someone please help me understand where I am going wrong?
Number 12 involves analyzing the same circuit using the method of mesh currents, but I'll try to work through that one again before asking.
Thank you!