# Unknown resistors - Kirchhoff laws

#### Andrei Monsanto Boysen

Joined Nov 30, 2014
24
Hello all,

In the attached image you find the circuit for which I'm supposed to figure out the values for the resistors R2, R3 and R6. I can have the switch S1 in ON and OFF positions. I'm supposed to take only the voltage between A and B. All other resistors have their resistance values showed in the circuit's figure. I don't have any information about current.

I couldn't progress much here, I tried several approaches, like coming up with the mesh equations and so on. But I couldn't isolate the unknown resistors. I always end up having more variables than equations.

I'm not asking for the equations, just some hint on how to proceed here, so I can try myself.

Any help is welcome. Thanks a lot!

#### WBahn

Joined Mar 31, 2012
29,883
So, if I understand you correctly, you have two voltage measurements, one with the switch open and one with it closed.

When the switch is open, what constraints can you come up with for the values of R2 and R3?

#### Andrei Monsanto Boysen

Joined Nov 30, 2014
24
I think I have solved this. Here goes an abbreviated version of my solution.

First of all, I just analyzed the circuit when the switch is open. With this solution I found:

R2 = 150 ohm
R3 = 160k ohm

I started taking the mesh equations (looking at the circuit diagram, consider M1 being the left mesh and M2 the right one, both considering clockwise currents). Current I1 for M1 and I2 for M2.

M1:
[1] 202*I1 + R2*I1 - 180*I2 = 12

M2:
[2] 269*I2 + R3*I2 = 180*I1

Considering that we know the voltage between A and B, I could write
[3] I1 = 5.96 / (22 + R2)

Doing node analysis at node A, I came up with this:

I1 = I2 + Ir5

Where Ir5 is the current through R5 resistor. Well, since R5 is in parallel with the resistor association of R3-R4-R7, so the voltage drop on R5 equals the voltage between A and B, thus I know the current through R5. Btw, the voltage between A and B is 6.04 V .
So, the current thru R5 is 3.36*10^-2 A .
This allowed me to find

[4] I2 = 6.04 / (89 + R3)

Now, replacing I1 in equation [1] with the result from [3], I found:

[5] 180*I2*(22 + R2) = 939.92 - 6.04*R2

Now rewriting the node equation with the values known, I got:

[6] 5.96 / (22 + R2) = 3.36*10⁻2 + 6.04 / (89 + R3)

Applying [4] to [5]:

[7] R2 = (9890.16 + 155*R3) / (269 + R3)

And applying [7] to [6] and solving it to R3 gives:

[8] R3 = 167k Ohm

Replacing [8] in [7] gives:

[9] R2 = 154.8 ohm

Do you agree with this solution? Let me know if I did anything wrong here or if you have found a better and more elegant solution.

Apparently, the values look ok.

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#### drc_567

Joined Dec 29, 2008
1,156
Please clarify ... How did you come up with Vab?

#12

#### Andrei Monsanto Boysen

Joined Nov 30, 2014
24
It was the only value that could be measured. It was manually measured. Sorry for not stating this before.

#### #12

Joined Nov 30, 2010
18,224
I'm as stuck as drc. Your drawing (and the whole first post) has only one voltage and no currents labeled. You can't do Ohm's Law or Watt's Law without 2 known numbers.

So, now that you've measured from A to B, is that with the switch open or closed?
Seems to me you need to measure once for each condition of the switch.

#### Andrei Monsanto Boysen

Joined Nov 30, 2014
24
The Vab = 6.04 V is for the S1 switch at open state.
When it is closed, then Vab = 4.03 V .

As I said in the beginning of my second post, I did all analysis considering the S1 switch to be open.

Again, I apologize for the lack of information of OP. Thanks for pointing this out to me! ;-)

#### WBahn

Joined Mar 31, 2012
29,883
I think I have solved this. Here goes an abbreviated version of my solution.

First of all, I just analyzed the circuit when the switch is open. With this solution I found:

R2 = 150 ohm
R3 = 160k ohm
The Vab = 6.04 V is for the S1 switch at open state.
When it is closed, then Vab = 4.03 V .
How did you get these values for R2 and R3?

Are you sure that these are the ONLY values for R2 and R3 that could yield that voltage with the switch open?

#### Andrei Monsanto Boysen

Joined Nov 30, 2014
24
The way I found those values is in my second post, my solution is there.
No, I'm not sure of this answer yet. This is why I posted it here.

#### WBahn

Joined Mar 31, 2012
29,883
The way I found those values is in my second post, my solution is there.
No, I'm not sure of this answer yet. This is why I posted it here.
Ah. I thought you were just giving the values that you got for them and then proceeding to find the rest. Since there's an issue with those two, I didn't look further.

So with that in mind...

I think I have solved this. Here goes an abbreviated version of my solution.

First of all, I just analyzed the circuit when the switch is open. With this solution I found:

R2 = 150 ohm
R3 = 160k ohm

I started taking the mesh equations (looking at the circuit diagram, consider M1 being the left mesh and M2 the right one, both considering clockwise currents). Current I1 for M1 and I2 for M2.

M1:
[1] 202*I1 + R2*I1 - 180*I2 = 12

M2:
[2] 269*I2 + R3*I2 = 180*I1
These are fine, but you need to properly track your units:

M1:
[1] (202 Ω)·I1 + R2·I1 - (180 Ω)·I2 = 12 V

M2:
[2] (269 Ω)·I2 + R3·I2 = (180 Ω)·I1

Considering that we know the voltage between A and B, I could write
[3] I1 = 5.96 / (22 + R2)
[3] I1 = 5.96 V / (22 Ω + R2)
Doing node analysis at node A, I came up with this:

I1 = I2 + Ir5

Where Ir5 is the current through R5 resistor.
That's not good enough -- what direction is Ir5 going through R5? Like voltages, currents have polarity.

Don't make people reverse engineer your work to come to the guess that Ir5 is defined to be flowing downward. If nothing else, what if you MEANT to have it flowing upward and messed up your node equation? The folks would be on a different page than you and chaos would ensue.

Well, since R5 is in parallel with the resistor association of R3-R4-R7, so the voltage drop on R5 equals the voltage between A and B, thus I know the current through R5. Btw, the voltage between A and B is 6.04 V .
So, the current thru R5 is 3.36*10^-2 A .
So far, so good.

This allowed me to find

[4] I2 = 6.04 / (89 + R3)
[4] I2 = (6.04 V) / (89 Ω + R3)

Now, replacing I1 in equation [2] with the result from [3], I found:

[5] 180*I2*(22 + R2) = 939.92 - 6.04*R2
How did you get this?

[2] (269 Ω)·I2 + R3·I2 = (180 Ω)·I1
[3] I1 = 5.96 V / (22 Ω + R2)

(269 Ω)·I2 + R3·I2 = (180 Ω)·(5.96 V / (22 Ω + R2))

[(269 Ω)·I2 + R3·I2](22 Ω + R2) = (180 Ω)·(5.96 V) = 1072.8 V·Ω

If nothing else, what happened to R3?

#### Andrei Monsanto Boysen

Joined Nov 30, 2014
24
R6 is easily found since we know all other resistors now.
I'll post corrections based on your appointments, thanks a lot for spending time looking into this.

I ran a quick simulation using the values I got and it matched what I observed.

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#### WBahn

Joined Mar 31, 2012
29,883
What if R2 = 70 Ω and R3 = 91 Ω ?

#### RBR1317

Joined Nov 13, 2010
712
With two conditions measured for Vab, there are two node equations for node A. However, there are three unknown resistor values. Two equations, three unknowns. How is it possible to solve that?

#### The Electrician

Joined Oct 9, 2007
2,971
In post #3, you say:
Now, replacing I1 in equation [2] with the result from [3], I found:

[5] 180*I2*(22 + R2) = 939.92 - 6.04*R2
But [2] is: 269*I2 + R3*I2 = 180*I1

It contains R3; what happened to R3 when you made the substitution?

Edit: I just noticed that WBahn already asked you this question. But he failed to ask you if you had substituted the values you got back into the circuit to see if they gave 6.04 volts at A-B. He went easy on you and only said that there is an issue with the values you got, instead of harping on the necessity of checking your results.

Your simulation didn't give 6.04 volts. Doesn't this indicate a problem?

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#### WBahn

Joined Mar 31, 2012
29,883
In post #3, you say:

But [2] is: 269*I2 + R3*I2 = 180*I1

It contains R3; what happened to R3 when you made the substitution?

Edit: I just noticed that WBahn already asked you this question. But he failed to ask you if you had substituted the values you got back into the circuit to see if they gave 6.04 volts at A-B. He went easy on you and only said that there is an issue with the values you got, instead of harping on the necessity of checking your results.

Your simulation didn't give 6.04 volts. Doesn't this indicate a problem?
I stopped at the first issue and ignored everything beyond that.

His sim result was 6.051 V, which is close enough that round off in an intermediate result in the hand calculation could account for the difference.

I also just realized that (I think) I was using 5.96 V for Vab when I came up with the alternate resistor values.

Yep, I was -- so the values I should have suggested would be R2 = 67 Ω and R3 = 91 Ω.

#### The Electrician

Joined Oct 9, 2007
2,971
I couldn't progress much here, I tried several approaches, like coming up with the mesh equations and so on. But I couldn't isolate the unknown resistors. I always end up having more variables than equations.

I'm not asking for the equations, just some hint on how to proceed here, so I can try myself.

Any help is welcome. Thanks a lot!
When this happens it means there is not a unique solution. You get to arbitrarily choose one variable. After you take care of the problem WBahn and I pointed out, you could choose an arbitrary value for R2 and then solve for R3 (and R6 for the case where S1 is closed).

But, beware, it may be that even though you can choose R2 arbitrarily, the range of values for the choice may be constrained if you don't want to end up with negative values for R3 and R6. Also, the sensitivity of the choice may vary over the range of possible values.

I would suggest that starting with the case where S1 is open, derive an expression for the voltage at A-B leaving R2 as a symbolic variable, but using the numerical values for R1, R4, R5 and R7. Then you can explore the properties of that expression.

#### Andrei Monsanto Boysen

Joined Nov 30, 2014
24
In post #3, you say:

But [2] is: 269*I2 + R3*I2 = 180*I1

It contains R3; what happened to R3 when you made the substitution?

Edit: I just noticed that WBahn already asked you this question. But he failed to ask you if you had substituted the values you got back into the circuit to see if they gave 6.04 volts at A-B. He went easy on you and only said that there is an issue with the values you got, instead of harping on the necessity of checking your results.

Your simulation didn't give 6.04 volts. Doesn't this indicate a problem?
I have fixed a typo there. It should be:

"Now, replacing I1 in equation [1] with the result from [3], I found:"

Thanks for pointing this out.

#### Andrei Monsanto Boysen

Joined Nov 30, 2014
24
I stopped at the first issue and ignored everything beyond that.

His sim result was 6.051 V, which is close enough that round off in an intermediate result in the hand calculation could account for the difference.

I also just realized that (I think) I was using 5.96 V for Vab when I came up with the alternate resistor values.

Yep, I was -- so the values I should have suggested would be R2 = 67 Ω and R3 = 91 Ω.
Thanks a lot. I managed to get the expected right answer for this and your values are much closer to it:

R2 = 68 Ω
R3 = 100 Ω
R6 = 33 Ω .

I'm reviewing my calculations now.
Thanks a lot for all the help.

#### The Electrician

Joined Oct 9, 2007
2,971
I have fixed a typo there. It should be:

"Now, replacing I1 in equation [1] with the result from [3], I found:"

Thanks for pointing this out.
Try solving for the voltage across A-B using nodal analysis. What works for me to check my results is to solve a circuit with both mesh and nodal analysis. As has been pointed out, and as you have noticed, you have more variables than equations, so if you get a unique result as a result of all your algebra, you have made a mistake.

#### The Electrician

Joined Oct 9, 2007
2,971
Thanks a lot. I managed to get the expected right answer for this and your values are much closer to it:

R2 = 68 Ω
R3 = 100 Ω
R6 = 33 Ω .

I'm reviewing my calculations now.
Thanks a lot for all the help.
What do you mean by "expected right answer"? Do you have a result given by your instructor?