Voltage in Circuit with Two Batteries and Six Resistors

[icesalmon]

Homework Statement

A circuit is constructed with six resistors and two batteries as shown. The battery voltages are V1 = 18 V and V2 = 12 V. The positive terminals are indicated with a + sign, The values for the resistors are: R1 = R5 = 40 Ω, R2 = R6 = 156 Ω R3 = 69 Ω, and R4 = 113 Ω. The positive directions for the currents I1, I2 and I3 are indicated by the directions of the arrows.The First question asks me to find the voltage at R₄

Homework Equations

V = IR

The Attempt at a Solution

I wasn't seeing any other way to find v4 than by making a loop for all three sections of the circuit
For the first loop I chose a clockwise orientation
-i₃R₃ - V₁ +i₁R₁ = 0

For my second loop I chose another clockwise orientation
i₂R₂ - V₂ - i₆R₆ + V₁ + i₃R₃ = 0

and for the final loop I also choose a clockwise orientation
V₂ -i₅R₅ - i₄R₄ = 0

Looking at the junction next to R₂ I get one more equation
i₂ - i₃ = i₁

Looking at junction B I get yet another, fifth, equation
i₄ - i₅ = i₆

my problem is going to arise every time I have to label currents and come up with equations at junctions or loops. I understand how to create a system of equations using these but I will sometimes get solutions to the matrix with all my currents being zero. Also, I'm sometimes creating systems where I have Nx(N+2) rows and columns and I find that creates problems as well, I usually miss an equation. Any advice on how to fix these problems would be greatly appreciated, thank you.

 

[ehild]

Homework Statement

A circuit is constructed with six resistors and two batteries as shown. The battery voltages are V1 = 18 V and V2 = 12 V. The positive terminals are indicated with a + sign, The values for the resistors are: R1 = R5 = 40 Ω, R2 = R6 = 156 Ω R3 = 69 Ω, and R4 = 113 Ω. The positive directions for the currents I1, I2 and I3 are indicated by the directions of the arrows.The First question asks me to find the voltage at R₄

Homework Equations

V = IR

The Attempt at a Solution

I wasn't seeing any other way to find v4 than by making a loop for all three sections of the circuit
For the first loop I chose a clockwise orientation
-i₃R₃ - V₁ +i₁R₁ = 0

For my second loop I chose another clockwise orientation
i₂R₂ - V₂ - i₆R₆ + V₁ + i₃R₃ = 0

Indicate i₆ in the figure.

and for the final loop I also choose a clockwise orientation
V₂ -i₅R₅ - i₄R₄ = 0

Looking at junction the junction next to R₂ I get one more equation
i₂ - i₃ = i₁

Looking at junction B I get yet another, fifth, equation
i₄ - i₅ = i₆

Note that i₅ and i₄ are the same, as r5 and r4 are connected in series.

Also note that the voltage across the series resultant of r₄ and r₅ is equal to V2, so it is very easy to answer the first question.

ehild

 

[icesalmon]

okay so i found R₄₅ and calculated the voltage through it to be 8.86V

 

[ehild]

okay so i found R₄₅ and calculated the voltage through it to be 8.86V

You mean the voltage across R4 is 8.86 V.

ehild

 

[icesalmon]

So I can't use an equivalent circuit using R₄₅ as opposed to R₄ and R₅ for the rest of my questions?

 

[NascentOxygen]

So I can't use an equivalent circuit using R₄₅ as opposed to R₄ and R₅ for the rest of my questions?

Probably can. What are the remaining questions? That 1st one was very easy (a trick question to catch anyone napping).

 

[ehild]

So I can't use an equivalent circuit using R₄₅ as opposed to R₄ and R₅ for the rest of my questions?

You can, but the voltage across R45 (which is R4+R5) is V2.
And you do not need to use R45 for the further calculations. It does not change anything if you omit it.
What you need, is the relation between I1, I3, I6 at node a.

ehild

 

[icesalmon]

okay I'm still confused about labeling currents on my diagram.

what do I call the currents coming out of batteries #1 and #2? It should matter because it will affect the equations at the nodes which will effect my system

 

[icesalmon]

I don't understand why you have I₃ at node a, wouldn't that cancel the current coming out of R₃ how can the battery produce current when the current always cancels over R₃. If those currents do cancel, then I₁ = I₂ at the node to the left of R₂.

 

[ehild]

I don't understand why you have I₃ at node a, wouldn't that cancel the current coming out of R₃ how can the battery produce current when the current always cancels over R₃. If those currents do cancel, then I₁ = I₂ at the node to the left of R₂.

The battery produces emf, that is, voltage. The same current flows through each element between two nodes. So I3 flows through R3, then through the battery, and enters into the node a. Name the currents between the nodes.

ehild

 

[icesalmon]

there's I₃ in between the nodes through V₁
I labeled current going through V₂ I₄

 

[icesalmon]

i₁ + i₆ = i₃ at node a

 

[ehild]

i₁ + i₆ = i₃ at node a

I1 and I3 both flow into the node a. If I6 flows into the node, too, I1+I3 +I6 =0.

ehild