DC Networks and Kirchoff's Laws

[rikiki]

Homework Statement

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

Homework Equations

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.

 

[gneill]

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?

 

[rikiki]

gneill, thanks for your help.

We've been shown the branch current method.

 

[gneill]

gneill, thanks for your help.

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), V_A, then you could calculate the current through each of the branches. For the leftmost branch, for example, you 'd have I = (V_A - 13V)/R1. So how to determine V_A? That's where Kirchhoff's Law comes in.

Assume that there's a voltage V_A 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 V_A. Once you know that voltage you can determine the current in each of the branches (including the one you're interested in).

 

[rikiki]

Thanks again for your reply.

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]

Thanks again for your reply.

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.

 

[rikiki]

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]

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:

 

[rikiki]

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]

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.

 

[rikiki]

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]

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

 

[rikiki]

Thanks for all your help.

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]

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.

 

[rikiki]

Brilliant thanks for all your help.

 

[charger9198]

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?

 

[oxon88]

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

 

[gneill]

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.

 

[oxon88]

ok so if i use P=I²R

with the folowing values for I:

I1 = -0.0875
I2 = 0.45
I3 = 0.4

i get:

P_R1 = -0.0875² * 1 = -0.00766 Watts

P_R2 = 0.45² * 2 = 0.405 Watts

P_R3 = 0.4² * 3 = 0.48 watts


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

i get:

P_R1 = 0.0875 * 13 = 1.1375 watts

P_R2 = 0.45 * 14 = 6.3 watts

P_R3 = 0.4 * 12 = 4.8 watts

which is correct?

 

[gneill]

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:

P_R1 = -0.0875² * 1 = -0.00766 Watts

P_R2 = 0.45² * 2 = 0.405 Watts

P_R3 = 0.4² * 3 = 0.48 watts


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

i get:

P_R1 = 0.0875 * 13 = 1.1375 watts

P_R2 = 0.45 * 14 = 6.3 watts

P_R3 = 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.

 

[oxon88]

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]

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.

 

[oxon88]

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]

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

 

[oxon88]

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]

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

 

[mm52]

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]

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.

 

[mm52]

Thanks gneill, makes sense.

 

[biggdogg9]

i understand up to this point...

Thanks for all your help.

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 don't 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.

 

[gneill]

I have follewed up this point quite clearly, however I don't 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.

It's a technique for solving the two simultaneous equations. The idea is to make the coefficients of one of the variables the same in both equations so that when the difference of the two equations is taken that variable drops away leaving a single equation in one unknown (the other variable).

In this case the poster chose to eliminate the I1 variable. To that end he looked at the coefficients of I1 in each equation: 4 and 3. A number that has both as factors is 12, so he multiplied the two equations by suitable values to yield 12I1 in each.

 

[biggdogg9]

Cheers gneil

 

[waltarno]

Hi All,

I too have this question and am a little confused. For the value of L2 I get -0.4545 but if I use this figure to get L1 I get 0.590875 which is nowhere near what you guys are saying.

If I use 0.4545 instead I get somehwere near the answers suggested on here.

Could someone please explain what happens to the minus (is it disregarded because it is indicative of current direction?) or have I done something school boy style?

Thanks

 

[waltarno]

maths used to get to this point :-

(12L₁+9L₂)-(12L₁+20L₂)=3-8

-11L₂=-5

L₂=-5/11

L₂=-0.4545

4L₁+3L₂=1

4L₁+(3*-0.4545)=1

4L₁-1.3635=1

4L₁=1+1.3635

L₁=2.3635/4

L₁=0.590875

 

[gneill]

maths used to get to this point :-

(12L₁+9L₂)-(12L₁+20L₂)=3-8

-11L₂=-5 ← negative on both sides[/color]

L₂=-5/11 ← Incorrect sign[/color]

L₂=-0.4545

4L₁+3L₂=1

4L₁+(3*-0.4545)=1

4L₁-1.3635=1

4L₁=1+1.3635

L₁=2.3635/4

L₁=0.590875

 

[waltarno]

Thanks for the quick reply, obviously didn't spot that!