Author Topic: Calculating value of resistors in parallel  (Read 5884 times)

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Offline soldarTopic starter

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Calculating value of resistors in parallel
« on: January 02, 2019, 03:45:10 pm »
This is a purely mathematical & Kirchoffian problem.

I have the heating element of a hot plate and it has four terminals. I am assuming internally it has four resistors (heating elements) connected as the sides of a square or diamond. I suppose it could have two more elements connected in diagonal but I think this is very unlikely.  I have access only to the four corner terminals and not to the individual elements.

If I had the values of the four resistors I could easily calculate the value of the different combinations but here I have the opposite problem: measuring only from the four corners determine the value of the individual resistors.  I start writing equations and they soon get so complex it is hopeless.

Measuring between terminals 2 and 3 the meter reads 327 ohm
Measuring between terminals 1 and 4 the meter reads 181 ohm
Measuring between terminals 2 and 4 the meter reads 90.4 ohm
Measuring between terminals 1 and 3 the meter reads 62.5 ohm

We have four unknowns and we would need four equations so the above should be enough to resolve the problem.

From the above it seems that R1 would be the lowest value, then R3, then R4 and R2 the highest.

If I connect terminals 1 & 2 together and measure resistance to 4 I get 60.3 Ohm. etc

If I connect 1&2 and measure to 3&4 then I get 30.8 ohms.

Can anyone help me calculate the resistance of the four individual elements?

If using equations directly is too complicated I wonder if it could be solved using recursive iterations with a spreadsheet.
« Last Edit: January 02, 2019, 03:48:22 pm by soldar »
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Offline Zero999

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Offline mbest

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Re: Calculating value of resistors in parallel
« Reply #2 on: January 02, 2019, 05:24:03 pm »
I'm still learning, when I see a textbook problem like this, I immediately start thinking algebraic substitution or linear equation solving.

First step I would take it to collect the measurements in a more useable form, and/or create circuit diagrams for all the measurements.

Measurement 1 of 327 ohm is from a circuit where R2 is in parallel with R1,R3,R4 in series, etc...
327 = R2 // R1+R3+R4
90.4 = R3 // R1+R2+R4
181 = R4 // R1+R2+R3
62.5 = R1 // R2+R3+R4
60.3 = R3 // R4
61.7 = R1 // R2
46.9 = R1 // R4
89.6 = R2 // R3
30.8 = R1+R2 // R3+R4

Write the calculation for each circuit and then solve for R1, by substitution, until only a single variable remains?

Does this seem like the correct approach?  I tried to solve it this way, but came up a bit short on the algebra, probably need more coffee.
 

Offline soldarTopic starter

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Re: Calculating value of resistors in parallel
« Reply #3 on: January 02, 2019, 06:23:01 pm »
https://en.wikipedia.org/wiki/Wheatstone_bridge
I cannot see how that helps at all. Maybe I am missing something. Maybe you could add an explanation rather than a bare link.
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Offline soldarTopic starter

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Re: Calculating value of resistors in parallel
« Reply #4 on: January 02, 2019, 06:30:10 pm »
mbest, that is what I tried to do but got bogged down in the algebra. It seems it is not easy.
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Online IanB

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Re: Calculating value of resistors in parallel
« Reply #5 on: January 02, 2019, 06:40:04 pm »
Assuming it is a diamond I think the algebra would be straightforward. But why do you think it is a diamond? Isn't it more likely that there are graduated heat settings available by switching in different heating elements? For example, do you get 90 ohms between 1 and 2?
 

Offline soldarTopic starter

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Re: Calculating value of resistors in parallel
« Reply #6 on: January 02, 2019, 06:51:01 pm »
Assuming it is a diamond I think the algebra would be straightforward.
OK, shows us how. A lot of people are saying it's easy but nobody shows how it is actually done.

But why do you think it is a diamond? Isn't it more likely that there are graduated heat settings available by switching in different heating elements? For example, do you get 90 ohms between 1 and 2?
It is most likely a square or diamond as shown for several reasons like, that is what I have found in other cases and because it is the simplest, but if you have any other explanation which fits the data then let's see it.
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Online IanB

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Re: Calculating value of resistors in parallel
« Reply #7 on: January 02, 2019, 06:59:26 pm »
I'm at work now and don't have time to look at it. Maybe this evening if nobody else has solved it by then...
 

Offline aneevuser

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Re: Calculating value of resistors in parallel
« Reply #8 on: January 02, 2019, 10:16:55 pm »

60.3 = R3 // R4
61.7 = R1 // R2
46.9 = R1 // R4
89.6 = R2 // R3


Write these four in terms of conductances, and they would be soluble, but appear to be inconsistent:

1/60.3 = G3 + G4 (1)
1/61.7 = G1 + G2 (2)
1/46.9 = G1 + G4 (3)
1/89.6 = G2 + G3 (4)

where Gk = 1/Rk.

(4)-(1) seems to give a different value for G2-G4 to that given by (2)-(3) but I have a headache right now so can't think about this further.
« Last Edit: January 02, 2019, 10:35:16 pm by aneevuser »
 

Offline HackedFridgeMagnet

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Re: Calculating value of resistors in parallel
« Reply #9 on: January 02, 2019, 10:40:32 pm »
can you apply a small voltage to the terminals, say 12v and then measure the voltage on the other terminals.
this could help you confirm the topology.
then reverse the polarity just in case there is a semiconductor inline.
Maybe something isn't making good contact which may effect the resistance results.
 

Offline ArthurDent

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Re: Calculating value of resistors in parallel
« Reply #10 on: January 02, 2019, 10:51:24 pm »
Here.

 
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Offline paf

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Re: Calculating value of resistors in parallel
« Reply #11 on: January 02, 2019, 11:02:07 pm »

Well, if you are not good at algebra, cheat a little with some short circuits. 

Calling the conductance G1=1/R1 and so on, you can do the following:

1)  Shorting 1 with 4 and 3 with 2, between those two terminals   Gx=G1+G3
2)  Shorting 1 with 3 and 4 with 2, between those two terminals   Gy=G2+G4
3)  Shorting 1 with 2 and with 3, between those  terminals and 4  Gz=G3+G4
4)  Shorting 1 with 2 and with 4, between those  terminals and 3  Gt=G2+G1


Measure Rx, Ry, Rz and Rt. Invert those and you got  Gx, Gy, Gz and Gt.

Solve the equations, and invert the found values to have the resistors.


 
 
 

Offline soldarTopic starter

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Re: Calculating value of resistors in parallel
« Reply #12 on: January 02, 2019, 11:02:30 pm »
Here.
Thanks! That fits the data pretty nicely. It makes the value of one resistor infinitely large and it makes sense that there are only three resistor elements.  I am really curious as to how you did it.
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Offline soldarTopic starter

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Re: Calculating value of resistors in parallel
« Reply #13 on: January 02, 2019, 11:06:43 pm »
Solve the equations, and invert the found values to have the resistors.
Easy to say isn't it? The problem is that several people come in and suggest a way but they don't do it. And I have thought of those ways and I always get stuck. So I would like anyone who thinks they can solve it to actually do the work of solving it and showing us how it is done. Because I just get stuck along the way.
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Offline ArthurDent

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Re: Calculating value of resistors in parallel
« Reply #14 on: January 02, 2019, 11:25:50 pm »
I first assumed no real hotplate would have a diamond shaped resistor configuration because it would probably make the switching too complex. Next I rounded the values so they were more common and easier to visualize. I then noticed that 60+90+180=330 so I guessed there might be just 3 resistors, not 4. Then I tried different series combinations to see if I could get a combination that fit the data given and found one. On the first go around I copied the last equation wrong and that slowed me down until I realized that it was 3*4-1*2 which would put the 3 resistors in parallel to give 30 ohms where I originally got 90 because I had 3*4-1 so I was missing one jumper.

It actually took a while to see what to do and there were a few dead ends before I got it right. No higher math was used, only the parallel resistor equation. It is simple once you see how it's done.
« Last Edit: January 03, 2019, 01:32:02 am by ArthurDent »
 

Offline kulky64

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Re: Calculating value of resistors in parallel
« Reply #15 on: January 02, 2019, 11:35:39 pm »
I first assumed no real hotplate would have a diamond shaped resistor configuration because it would probably make the switching too complex. Next I rounded the values so they were more common and easier to visualize. I then noticed that 60+90+180=330 so I guessed there might be just 3 resistors, not 4. Then I tried different series combinations to see if I could get a combination that fit the data given and found one. On the first go around I copied the last equation wrong and that slowed me down until I realized that it was 2*4-1*2 which would put the 3 resistors in parallel to give 30 ohms where I originally got 90 because I had 2*4-1 so I was missing one jumper.

It actually took a while to see what to do and there were a few dead ends before I got it right. No higher math was used, only the parallel resistor equation. It is simple once you see how it's done.

Except you didn't come up with anything new. All numbers what you posted are already in OPs first post.
« Last Edit: January 02, 2019, 11:40:00 pm by kulky64 »
 

Offline ArthurDent

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Re: Calculating value of resistors in parallel
« Reply #16 on: January 02, 2019, 11:59:56 pm »
kulky64 - "Except you didn't come up with anything new. All numbers what you posted is already in OPs first post."

You may have missed it but the question was what was the values of the four individual resistance elements, not the measured terminal resistance. I showed that the original assumption of the diamond shape was wrong and showed the correct wiring of the resistors. I also determined that the values of the "four" individual resistance elements were 60, 90, 180, and no fourth resistor, in case you missed that point. These values were not initially known because it was assumed that there was some configuration that put resistor elements in parallel and the values of individual elements were unknown. I didn't see your solution.
 

Offline kulky64

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Re: Calculating value of resistors in parallel
« Reply #17 on: January 03, 2019, 12:07:32 am »
I thik OPs assumption on the configuration of the resistors is correct, because he provided redundant (more than necessary) measurements that are supporting each other.
 

Offline soldarTopic starter

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Re: Calculating value of resistors in parallel
« Reply #18 on: January 03, 2019, 12:29:31 am »
ArthurDent, that was very ingenious of you and I thank you!  I sheepishly admit that I should have thought of that because I have seen similar heaters but instead of the elements being enclosed they are open, of the spiral wire that gets red hot type, and they have the configuration you discovered. I don't know why I didn't think of it. Sometimes intuition and ingeniousness beats raw math and algebra.

Doing it by algebra brute force should have yielded an extremely high or infinite value for the assumed fourth element.

The problem with the values as they are is that the different combinations of switching sometimes results in repeated values of resistance and power so of 16 possible combinations of the four switches, 6 result in zero power, six result in unique values and three result in repeated values.

Two things that come to my mind now is that it would be interesting to use a thermal camera to see the distribution of heat on the hot plate as the different elements are used and also that for combinations that result in the same power it may be preferable to use the one that uses more resistors so that the heat is more distributed and it would prolong the life of the heater.

All very interesting. Thank you!

E.T.A. I did a quick calculation of the resistance and power for each combination of switches but chances are there might be some mistake because my head hurts. I will review it again tomorrow.

By the way, this was/is a hotplate that was controlled by a rotary switch that failed and I could not find a replacement so I fitted four switches as shown and I have a card with a table showing the switch combinations and corresponding power. The most common combinations I know by heart now. :)
« Last Edit: January 03, 2019, 12:44:51 am by soldar »
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Offline The Electrician

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Re: Calculating value of resistors in parallel
« Reply #19 on: January 03, 2019, 03:04:12 am »
soldar, as you said in the first post, it's a problem with 4 equations in 4 unknowns:

« Last Edit: January 03, 2019, 03:07:05 am by The Electrician »
 

Online IanB

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Re: Calculating value of resistors in parallel
« Reply #20 on: January 03, 2019, 03:31:14 am »
OK, now I'm home from work, I've looked at the problem as promised.

I see that you have a solution, and the problem is basically solved.

I found that an algebraic solution was essentially intractable, and was indeed much harder than I suspected it to be.

Instead I have attached an Excel workbook that shows how to tackle the problem using the solver add-in. Since you provided all possible measurements it is possible to look at this as an error minimization problem and find the most likely values of the resistors based on your measurements. Once it became clear that R2 was a very large value I eliminated it from the network and re-solved for the others.

As you will see the results are that R1 = 61.7 Ω, R3 = 89.6 Ω and R4 = 180.9 Ω.

It seems likely that your resistance measurements were not as accurate as they might be, since some residual errors remained after solving.
 

Offline paf

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Re: Calculating value of resistors in parallel
« Reply #21 on: January 03, 2019, 09:14:46 am »
Solve the equations, and invert the found values to have the resistors.
Easy to say isn't it? The problem is that several people come in and suggest a way but they don't do it. And I have thought of those ways and I always get stuck. So I would like anyone who thinks they can solve it to actually do the work of solving it and showing us how it is done. Because I just get stuck along the way.

Have you read all my message? 

I have said that short circuiting some terminals, and measuring the resistance you can have simpler equations.
You can measure the parallel of any set of two resistors on the circuit.

After doing that, the equations are simpler.

I cannot measure the resistance in your circuit, from my home. But, will gladly do it if you show me how. 
 
 

Offline soldarTopic starter

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Re: Calculating value of resistors in parallel
« Reply #22 on: January 03, 2019, 09:25:38 am »
I found that an algebraic solution was essentially intractable, and was indeed much harder than I suspected it to be.
Yes, I have tried several times over the years and I always got stuck. It was just an academic exercise so I just dropped it each time. It never occurred to me to see if it might be only three elements. Once you think of that possibility it is easy to find out which element is missing because the reading between those two terminals becomes the sum of the other three individually. At that point the problem is trivial.

I just had a look at another heater I got from the trash but this one has the elements exposed as the whole thing was covered with glass. You can see I have marked the three resistor elements. It has four connectors and the only problem and the reason it was discarded and replaced is that the connectors are failing due to the heat. The rest works fine and repairing the connectors is trivial. What I have found fails quite frequently is the rotary selector switch and those are expensive and difficult or impossible to find.

In my case I fitted four switches as shown and it works well enough for me but I suppose I could fit three-position switches which would allow connecting each terminal to high/off/low rail. That would give more combinations for power.

  Instead I have attached an Excel workbook that shows how to tackle the problem using the solver add-in. Since you provided all possible measurements it is possible to look at this as an error minimization problem and find the most likely values of the resistors based on your measurements. Once it became clear that R2 was a very large value I eliminated it from the network and re-solved for the others.
I know how to use Excel to solve for one unknown by doing successive iterations - approximations but with four unknowns I do not know how to do it. Thanks for that sheet. I will study it and learn. I always find Excel to be very useful but my knowledge is limited.


As you will see the results are that R1 = 61.7 Ω, R3 = 89.6 Ω and R4 = 180.9 Ω.

It seems likely that your resistance measurements were not as accurate as they might be, since some residual errors remained after solving.
Yeah my readings / data was taken from a quick reading of power used in each position and then converted to ohms assuming a certain voltage. Just a rough approximation. But once you realize there is no fourth resistor then the value of the others are taken directly between terminals. The values you arrived at are the values I mention in my OP. The solution was right there staring at me and I just wasn't seeing it.

Thank you all for your input.
« Last Edit: January 03, 2019, 09:48:50 am by soldar »
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Offline soldarTopic starter

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Re: Calculating value of resistors in parallel
« Reply #23 on: January 03, 2019, 10:01:35 am »
Have you read all my message? 

I have said that short circuiting some terminals, and measuring the resistance you can have simpler equations.
You can measure the parallel of any set of two resistors on the circuit.

After doing that, the equations are simpler.

I cannot measure the resistance in your circuit, from my home. But, will gladly do it if you show me how. 
If you think you have equations that will resolve the problem you do not need the actual values; just give them names. Like "short here and there and we have this and that in parallel, let us call that value K", etc. then with the givens K,L,M,N you can go on to resolve and give us the final result.

But, furthermore, if you look carefully at the OP you can see I already short circuited some terminals and gave the values I obtained which, indeed, simplified things but not enough to make it easy. The values were right there all along.
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Offline soldarTopic starter

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Re: Calculating value of resistors in parallel
« Reply #24 on: January 03, 2019, 10:17:52 am »
Have you read all my message? 

I have said that short circuiting some terminals, and measuring the resistance you can have simpler equations.
You can measure the parallel of any set of two resistors on the circuit.

After doing that, the equations are simpler.

I cannot measure the resistance in your circuit, from my home. But, will gladly do it if you show me how. 
If you think you have equations that will resolve the problem you do not need the actual values; just give them names. Like "short here and there and we have this and that in parallel, let us call that value K", etc. then with the givens K,L,M,N you can go on to resolve and give us the final result.

But, furthermore, if you look carefully at the OP you can see I already short circuited some terminals and gave the values I obtained which, indeed, simplified things but not enough to make it easy. The values were right there for you all along. I know that with four unknowns you only need four equations to solve but I gave more redundant, information so that readers could use which data to use and so that it could be used to confirm any results obtained.
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