# DC Networks and Kirchoff's Laws

## Homework Statement

Calculate the value of the current through the 12V battery shown in figure (attached).

V = IR
I1 = I2 + I3

## The Attempt at a Solution

I've calculated the current for the left hand loop, see below:

Loop 1:
E_1- E_2- V_1- V_2=0
E_1- E_2= V_1+ V_2
E_1- E_2=I(R_1+ R_2)
13-12= I (3+1)
1=4I
1/4=I
I=0.25A

I understand it's a process of doing the same to the right hand side and then subtracting the two sides? I'm just unsure as to when i should be adding the voltages and resistors and when i should be subtracting them.Is it related to the direction of the current and voltage potential? If anyone can help here it would be greatly appreciated. Thanks.

## The Attempt at a Solution

#### Attachments

• attempt.jpg
22.4 KB · Views: 1,524
• figure.bmp
200.8 KB · Views: 1,080

gneill
Mentor
You have two loops which will influence each other in terms of voltage and current. So a single equation for one loop isn't going to capture the interplay.

What circuit analysis methods have you learned?

We've been shown the branch current method.

gneill
Mentor

We've been shown the branch current method.

Okay. It looked as though you were attempting to apply Kirchhoff's Voltage Law (KVL) around a loop, but branch currents are good too! That boils down to Kirchhoff's Current Law (KCL), which states that the sum of the currents entering a given node of the circuit must equal the sum of the currents leaving that node.

Your circuit has three branches and two nodes. For the nodes, label the bottom horizontal conductor as "ground". Label the top horizontal conductor "Node A".

If you happened to known the voltage at node A (with respect to ground), VA, then you could calculate the current through each of the branches. For the leftmost branch, for example, you 'd have I = (VA - 13V)/R1. So how to determine VA? That's where Kirchhoff's Law comes in.

Assume that there's a voltage VA at node A. Write the equation for the sum of the currents leaving Node A for each branch connected to that node. I gave you the example for the first branch. Solve the resulting equation for VA. Once you know that voltage you can determine the current in each of the branches (including the one you're interested in).

#### Attachments

• Fig1.jpg
12.8 KB · Views: 3,774

For the left hand loop I have:

+E1 - I3R3-E3-I1R1 = 0
19 - 3I3 - 12 - I1 = 0

-3I3 - I1 = 7
I2 = I1 - I3

I1 - I2 = 7

For the right hand loop I have:

+E2 - I3R3 - E3 - I2R2 = 0
14 - 3I3 - 12 - 2I2 = 0

-3I3 - 2I2 = 2
I1 = I2 + I3

I1 - 3I2 = 2

Multiply left hand side by four and add to right hand side ...

4I1 - 4I2 = 28

Left hand loop plus right hand loop ...

5I1 - 7I2 = 30

How can I know calculate the value of I1 and I2?

gneill
Mentor

For the left hand loop I have:

+E1 - I3R3-E3-I1R1 = 0
19 - 3I3 - 12 - I1 = 0
Where did the 19 come from? Doesn't E1 = 13V?
-3I3 - I1 = 7
I2 = I1 - I3

I1 - I2 = 7

For the right hand loop I have:

+E2 - I3R3 - E3 - I2R2 = 0
14 - 3I3 - 12 - 2I2 = 0

-3I3 - 2I2 = 2
I1 = I2 + I3

I1 - 3I2 = 2

Multiply left hand side by four and add to right hand side ...

4I1 - 4I2 = 28

Left hand loop plus right hand loop ...

5I1 - 7I2 = 30

How can I know calculate the value of I1 and I2?

After you straighten out the voltage of E1, the first two equations that you wrote in bold constitute two equations in two unknowns. You can solve for both I1 and I2 with them. For example, if you subtract one equation from the other you should eliminate I1 and be left with an equation for I2.

Thanks for that, didn't even spot that.
So ...

I1 - I2 = 1

(I1 - I2) - (I1 - 3I2) = 1 - 2

2I2 = -1

I2 = -1 / 2 = -0.5A

Therefore if I1 - I2 = 1, I1 - -0.5 = 1

I1 = 0.5

If I1 = I2 + I3, 0.5 = -0.5 + 1

So current at E3 would be 1Amp

gneill
Mentor
One problem I just spotted (should have caught it sooner) is that, according to your assumed current directions you should have I3 = I1 + I2. This is going to change your result, since you've written I1 = I2 + I3.

Here's the figure again with the assumed currents:

#### Attachments

• Fig1.jpg
8.7 KB · Views: 3,337
Ok thanks for seeing that. So,

If I2 = -0.5A

I1 - I2 = 1

Therefore I1 = 0.5

I3 = I1 + I2

I3 = 0.5 + -0.5

I3 = 0A

Current at E3 would be 0A?

gneill
Mentor
You'll have to go back and rework the loop equations where you've used the current sum relationship. The current through E3 is not zero.

Thanks for all your help to date. I've come up with a final figure of 0.95A for I3. Does this seem right?

gneill
Mentor
Thanks for all your help to date. I've come up with a final figure of 0.95A for I3. Does this seem right?

Unfortunately, no Something must have gone awry in the math.

How about one more attempt, from the beginning? I'll get things started, using the equations that you've written before:

KCL for top node: I3 = I1 + I2 ............(1)

KVL for left-hand loop:

13 - 3*I3 -12 - 1*I1 = 0

Replacing I3 using (1):
4*I1 + 3*I2 = 1 ............(2)

KVL for right-hand loop:

14 - 3*I3 - 12 - 2*I2 = 0

and using (1):
3*I1 + 5*I2 = 2 ............(3)

You should be able to solve (2) and (3) for I1 and I2, then use (1) to find I3.

I

So ...

4I1 + 3I2 = 1 1
3I1 + 5I2 = 2 2

Multiply 1 by 3
12I1 + 9I2 = 3

Multiply 2 by 4
12I1 + 20I2 = 8

(12I1 + 9I2) - (12I1 + 20I2) = 3 - 8
-11I2 = -5
I2 = -5 / 11
I2 = 0.45

4I1 + 3I2 = 1
4I1 + 3x0.45 = 1
4I1 + 1.35 = 1
4I1 = 1 - 1.35
= -0.35
I1 = -0.35 / 4
I1 = -0.0875

I3 = I1 + I2
= -0.0875 + 0.45
= 0.3625
= 0.4A?

gneill
Mentor
Your method is good. You might want to keep a few more decimal places in intermediate values to prevent rounding errors from creeping into the final result.

Bravo.

Brilliant thanks for all your help.

I think your I2 should be
-0.4545 which changes your I1 to to 0.8409 giving I3 = 0.39 A.
This would affect your answer when working out the power dissipated on each resistor wouldn't it?

If i wanted to work out the power dissipated in each resistor. would i use P=I^2*R or P=I*V ??

gneill
Mentor
If i wanted to work out the power dissipated in each resistor. would i use P=I^2*R or P=I*V ??

Yes

Both are applicable depending upon what quantities you have to work with. There's also V2/R that'll work if you happen to have the voltage across the resistor.

ok so if i use P=I2R

with the folowing values for I:

I1 = -0.0875
I2 = 0.45
I3 = 0.4

i get:

PR1 = -0.08752 * 1 = -0.00766 Watts

PR2 = 0.452 * 2 = 0.405 Watts

PR3 = 0.42 * 3 = 0.48 watts

but if i use the other formula P=I*V

i get:

PR1 = 0.0875 * 13 = 1.1375 watts

PR2 = 0.45 * 14 = 6.3 watts

PR3 = 0.4 * 12 = 4.8 watts

which is correct?

gneill
Mentor
ok so if i use P=I2R

with the folowing values for I:

I1 = -0.0875
I2 = 0.45
I3 = 0.4
As was pointed out by charger9198 in a previous post, there seems to be some issue with the calculated values for the currents. I've just taken a look at the circuit and I can see that the values are not those given above

You should recalculate the currents before moving forward.
i get:

PR1 = -0.08752 * 1 = -0.00766 Watts

PR2 = 0.452 * 2 = 0.405 Watts

PR3 = 0.42 * 3 = 0.48 watts

but if i use the other formula P=I*V

i get:

PR1 = 0.0875 * 13 = 1.1375 watts

PR2 = 0.45 * 14 = 6.3 watts

PR3 = 0.4 * 12 = 4.8 watts

which is correct?
Well the second set is definitely incorrect because the voltage you need to use is the potential that appears across each resistance. This will NOT be the values of the individual cells in the branches; it will be the voltage drop due to the current passing through a given resistor.

ok so i re calculated the current as follows:

I1 = 0.09091 A

I2 = 0.45455 A

I3 = 0.36364 A

do these values look more accurate?

gneill
Mentor
ok so i re calculated the current as follows:

I1 = 0.09091 A

I2 = 0.45455 A

I3 = 0.36364 A

do these values look more accurate?

Yes, their magnitudes look okay.

Yes, their magnitudes look okay.

i now get:

PR1 = -0.090912 * 1 = -0.008265 Watts

PR2 = 0.454552 * 2 = 0.4132 Watts

PR3 = 0.363642 * 3 = 0.3967 watts

gneill
Mentor
They should all be positive values. I2 will be positive no matter the sign of I. Energy being dissipated as heat in a resistor doesn't care which way the current flows

yes ok I see you point,

PR1 = -0.090912 * 1 = 0.008265 Watts

PR2 = 0.454552 * 2 = 0.4132 Watts

PR3 = 0.363642 * 3 = 0.3967 watts

they seem very small values to me

gneill
Mentor
Small currents and small resistances yield small power dissipations. In a large system they can still add up!

How do we get a negative current in I1? does this mean the direction of current is opposite to that shown on the diagram and does the law now change to I3 = I1 - I2?

gneill
Mentor
How do we get a negative current in I1? does this mean the direction of current is opposite to that shown on the diagram and does the law now change to I3 = I1 - I2?

Nothing changes. The equations to solve the circuit were based upon assumed current directions that may or may not turn out to be correct assumptions. No matter, the math takes care of such things. If a current that was thought to flow one way but turns out to flow in the opposite direction, it will end up with a negative value. No big deal. All the equations still apply exactly as written, just use the values obtained.

Thanks gneill, makes sense.

i understand up to this point....

So ...

4I1 + 3I2 = 1 1
3I1 + 5I2 = 2 2

Multiply 1 by 3
12I1 + 9I2 = 3

Multiply 2 by 4
12I1 + 20I2 = 8

(12I1 + 9I2) - (12I1 + 20I2) = 3 - 8
-11I2 = -5
I2 = -5 / 11
I2 = 0.45

4I1 + 3I2 = 1
4I1 + 3x0.45 = 1
4I1 + 1.35 = 1
4I1 = 1 - 1.35
= -0.35
I1 = -0.35 / 4
I1 = -0.0875

I3 = I1 + I2
= -0.0875 + 0.45
= 0.3625
= 0.4A?

I have follewed up this point quite clearly, however I dont understand the reason why we have to multiply equation 1 by 3 ? If somebody could explain this too me or even send me a link where I can understand why.