Calculating value of resistors in parallel - Page 1
Forum section: electronics / beginners

author: soldar
author_rank: Super Contributor
timestamp: January 02, 2019, 03:45:10 pm
content: 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 ohmMeasuring between terminals 1 and 4 the meter reads 181 ohmMeasuring between terminals 2 and 4 the meter reads 90.4 ohmMeasuring between terminals 1 and 3 the meter reads 62.5 ohmWe 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. etcIf 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.

author: Zero999
author_rank: Super Contributor
timestamp: January 02, 2019, 04:29:22 pm
content: https://en.wikipedia.org/wiki/Wheatstone_bridge

author: mbest
author_rank: Contributor
timestamp: January 02, 2019, 05:24:03 pm
content: 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+R490.4 = R3 // R1+R2+R4181 = R4 // R1+R2+R362.5 = R1 // R2+R3+R460.3 = R3 // R461.7 = R1 // R246.9 = R1 // R489.6 = R2 // R330.8 = R1+R2 // R3+R4Write 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.

author: soldar
author_rank: Super Contributor
timestamp: January 02, 2019, 06:23:01 pm
content: Quote from: Zero999 on January 02, 2019, 04:29:22 pmI cannot see how that helps at all. Maybe I am missing something. Maybe you could add an explanation rather than a bare link.

author: soldar
author_rank: Super Contributor
timestamp: January 02, 2019, 06:30:10 pm
content: mbest, that is what I tried to do but got bogged down in the algebra. It seems it is not easy.

author: IanB
author_rank: Super Contributor
timestamp: January 02, 2019, 06:40:04 pm
content: 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?

author: soldar
author_rank: Super Contributor
timestamp: January 02, 2019, 06:51:01 pm
content: Quote from: IanB on January 02, 2019, 06:40:04 pmOK, shows us how. A lot of people are saying it's easy but nobody shows how it is actually done.Quote from: IanB on January 02, 2019, 06:40:04 pmIt 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.

author: IanB
author_rank: Super Contributor
timestamp: January 02, 2019, 06:59:26 pm
content: 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...

author: aneevuser
author_rank: Frequent Contributor
timestamp: January 02, 2019, 10:16:55 pm
content: Quote from: mbest on January 02, 2019, 05:24:03 pmWrite 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.

author: HackedFridgeMagnet
author_rank: Super Contributor
timestamp: January 02, 2019, 10:40:32 pm
content: 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.

author: ArthurDent
author_rank: Super Contributor
timestamp: January 02, 2019, 10:51:24 pm
content: Here.

author: paf
author_rank: Regular Contributor
timestamp: January 02, 2019, 11:02:07 pm
content: 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+G32) Shorting 1 with 3 and 4 with 2, between those two terminals Gy=G2+G43) Shorting 1 with 2 and with 3, between those terminals and 4 Gz=G3+G44) Shorting 1 with 2 and with 4, between those terminals and 3 Gt=G2+G1Measure 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.

author: soldar
author_rank: Super Contributor
timestamp: January 02, 2019, 11:02:30 pm
content: Quote from: ArthurDent on January 02, 2019, 10:51:24 pmThanks! 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.

author: soldar
author_rank: Super Contributor
timestamp: January 02, 2019, 11:06:43 pm
content: Quote from: paf on January 02, 2019, 11:02:07 pmEasy 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.

author: ArthurDent
author_rank: Super Contributor
timestamp: January 02, 2019, 11:25:50 pm
content: 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.

author: kulky64
author_rank: Regular Contributor
timestamp: January 02, 2019, 11:35:39 pm
content: Quote from: ArthurDent on January 02, 2019, 11:25:50 pmExcept you didn't come up with anything new. All numbers what you posted are already in OPs first post.

author: ArthurDent
author_rank: Super Contributor
timestamp: January 02, 2019, 11:59:56 pm
content: 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.

author: kulky64
author_rank: Regular Contributor
timestamp: January 03, 2019, 12:07:32 am
content: 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.

author: soldar
author_rank: Super Contributor
timestamp: January 03, 2019, 12:29:31 am
content: 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.

author: The Electrician
author_rank: Frequent Contributor
timestamp: January 03, 2019, 03:04:12 am
content: soldar, as you said in the first post, it's a problem with 4 equations in 4 unknowns:

author: IanB
author_rank: Super Contributor
timestamp: January 03, 2019, 03:31:14 am
content: 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.

author: paf
author_rank: Regular Contributor
timestamp: January 03, 2019, 09:14:46 am
content: Quote from: soldar on January 02, 2019, 11:06:43 pmHave 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.

author: soldar
author_rank: Super Contributor
timestamp: January 03, 2019, 09:25:38 am
content: Quote from: IanB on January 03, 2019, 03:31:14 amYes, 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.Quote from: IanB on January 03, 2019, 03:31:14 amI 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.Quote from: IanB on January 03, 2019, 03:31:14 amYeah 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.

author: soldar
author_rank: Super Contributor
timestamp: January 03, 2019, 10:01:35 am
content: Quote from: paf on January 03, 2019, 09:14:46 amIf 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.

author: soldar
author_rank: Super Contributor
timestamp: January 03, 2019, 10:17:52 am
content: Quote from: soldar on January 03, 2019, 10:01:35 am

author: paf
author_rank: Regular Contributor
timestamp: January 03, 2019, 10:25:40 am
content: Quote from: soldar on January 03, 2019, 10:01:35 amSorry, I don't have time to help you doing a simple calculation, to give results that are only useful to you. But I have time to help everybody who is facing a similar problem to yours.In order to simplify the equations you can:1) Short circuit some terminals in order to have the parallel of any two resistors in the circuit.2) Write the equations in terms of conductance, not resistance, so the equations will be simpler.Once again, more complete:Calling the conductance G1=1/R1 and so on, you can do the following:1) Shorting 1 with 4 and 3 with 2, measure between those two terminals: Gx=G1+G32) Shorting 1 with 3 and 4 with 2, measure between those two terminals: Gy=G2+G43) Shorting 1 with 2 and with 3, measure between those terminals and 4: Gz=G3+G44) Shorting 1 with 2 and with 4, measure between those terminals and 3: Gt=G2+G15) Shorting 2 with 3 and with 4, measure between those terminals and 1: Gw=G4+G16) Shorting 1 with 3 and with 4, measure between those terminals and 2: Gu=G2+G3

author: soldar
author_rank: Super Contributor
timestamp: January 03, 2019, 12:06:40 pm
content: I have several of these hot plate units and they all have the same three resistor configuration. Originally they had rotary switches which selected the power setting but the switches are gone.I have set up one using four on/off switches as shown but that leaves several combination of switches with no function or duplicate power settings and it requires having a table at hand showing the power for each combination.I have been thinking about the possibility of building or adapting a multi-position switch. The best idea I can come up with would be a rotary shaft with cams which would actuate four micro-switches. Maybe I will give it a try.Maybe another possibility would be a 3 bit counter which could be moved up/down and the 3 bits converted to the necessary 4 bits which would control a relay each. Maybe simpler than building mechanical parts.

author: The Electrician
author_rank: Frequent Contributor
timestamp: January 03, 2019, 02:51:07 pm
content: Quote from: paf on January 02, 2019, 11:02:07 pmThis is a good idea, but the Gt you have chosen is a linear combination of Gx, Gy and Gz, namely: Gt=Gx+Gy-GzIf you short 2 with 3 and with 4, between those terminals and 1, Gt=G1+G4then a solution is possible:

author: ArthurDent
author_rank: Super Contributor
timestamp: January 03, 2019, 04:07:49 pm
content: I had originally written out the 4 equations with 4 variable and thought about matrix reduction but being lazy quickly threw that method aside. I decided to make some assumptions using Occam’s Razor and that, plus a little insight, worked. It is interesting seeing how some still want to do it the long and hard way and are reluctant to accept another simpler way - not that the other ways aren’t correct. It reminded me of this short video.

author: The Electrician
author_rank: Frequent Contributor
timestamp: January 03, 2019, 04:59:47 pm
content: Quote from: ArthurDent on January 03, 2019, 04:07:49 pmThis is the 21st century, and we have computers to do the algebra nowadays. There are only 4 unknowns in soldar's first post, and one would think that the proper 4 equations would yield a solution. In reply #2, the first 4 equations mbest gave are sufficient, and those are the 4 equations I used in reply #19. I could see right away that those 4 equations wouldn't be solvable by hand in the amount of time I was willing to devote. But, I was certain that modern mathematical software probably wouldn't have a problem, and in fact Mathematica found a solution in a fraction of a second.I then thought I would see what a symbolic solution looks like, so I substituted symbolic variables for the numeric values on the right side of the equals signs. Mathematica was still crunching away after 1 minute, so I stopped it because I know from experience that when Mathematica runs that long, the result will be several pages of albegraic expressions.paf's suggestion in reply #11 is a good one, and after fixing his small boo-boo, 4 equations result that can be solved by hand. These equations can also be written as a matrix formulation:Nowadays there's no reason to shy away from a problem involving a lot of algebra when we have all this computer power at our disposal.

author: soldar
author_rank: Super Contributor
timestamp: January 04, 2019, 11:14:45 pm
content: It seems these hot plates often have the problem that the selector switch burns out and is difficult or expensive or impossible to replace.I have replaced it with four switches, as shown. Now I think I could replace the four switches with four relays, solid state or electro-mechanical, and control the relays with a very low current single pole, eight position switch. A bunch of diodes would select which relays would be activated in each position.

author: HackedFridgeMagnet
author_rank: Super Contributor
timestamp: January 05, 2019, 03:29:13 am
content: Quote from: soldar on January 04, 2019, 11:14:45 pmSelector switch burning out, yes. Mine oven has done it twice. It's selector switch is about 10 pole, single throw , with cams turning it off and on for about 8 positions. Both times I have reorganised and simplified the wiring to adapt to the burnt out positions. With a corresponding loss of features.From memory it switches Neutral as well as Active to give different control outcomes.Because I like my oven I might do something similar to your relay idea, it is a double oven and it was a bit of trouble to adjust the cabinet to fit, so a new control solution would probably be simpler.ps. You have to be sure the over temp cutout and high temp cooling fan are fail safe though.

author: soldar
author_rank: Super Contributor
timestamp: January 05, 2019, 08:11:00 am
content: Quote from: HackedFridgeMagnet on January 05, 2019, 03:29:13 amThe switches carry big currents and will fail sooner or later because the contacts burn out. Electro-mechanical relays might have the same problem and it is probably best to install solid state relays well oversized and they should last forever. Just make sure they have good heat sinking.There are quite cheap rotary single circuit switches on ebay but they are plastic and need to be mounted where they will not get too hot.If you install the relays in the same (electrical) place as the switches are then there should be no problem with protections, fans, etc.My original switch had one more position that would require one more relay, 5 instead of 4, but I figure it is not worth it as I have enough choice with the four relays.Of course, another alternative option for power control is to just set it to the max power and use a Triac power control but I prefer not to do that because of the poor power factor and the noise it introduces in the line but you can ameliorate that with a good filter.