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The dual use of positive & negative feedback in an op amp ("Analog Linearization of resistance temperature detectors")
Per the Analog Applications Journal (4Q 2011), Bruce Trump wrote an article entitled, "Analog Linearization of resistance temperature detectors".
I'm trying to understand two things:
1) It is normally assumed that positive feedback to used with an op amp to ensure an oscillation (e. g. op amp oscillator) or at least "snap" level change (Schmitt trigger). This is the first configuration I've seen that remains steadily fixed over time; why is that? Is it because the negative feedback is greater than the positive feedback?
2) Looking at the attached Excel spreadsheet (RTD_Linearization_v7.xls) -- these are by far the largest set of equations I've seen being compiled into so few cells -- how does one derive the equations for R5 in the spreadsheet? What I'm looking for is the starting set of equations along with a brief overview of how the result was derived...even better, would be the actual derivations from start to finish. In any case, I presume the starting equation are three nodal equations each with an initial condition, and somehow we end up with a root-mean-square being applied to massage the data. But that's only a guess.
Even those of you who are reading this and don't really care -- take a gander at the R5 equation in RTD_Linearization_v7.xls, I promise you it will make you sweat.
Any information this forum provides would be greatly appreciated.
This is in regards to possible application to a precision temperature sensor used in a weather station application.
Thanks for reading the article.
You are correct, the amount of positive feedback is very small compared to negative feedback. As a result, the circuit is quite stable. With most circuits, adding some positive feedback merely changes the closed loop gain of the circuit. There is no reason to do this with most circuits. With this circuit, the amount of positive feedback changes with temperature as the resistance of the RTD changes.
The basic equation governing the circuit is shown at the bottom of page 22 in the journal. This is a relatively simple nodal equation written purely by inspection. Why does it become so complicated? The key is that the resistance of the RTD varies with temperature. Three versions of the equation are written. All three look basically the same except different variables are assigned to represent the high, mid and low values of the RTD resistance at the three temperatures of zero error. Variables are assigned to the high, mid and low values representing the desired output voltages at the three temperatures. The three equations are then solved simultaneously by substitution of variables, then solved for R5.
I used Mathcad to do the algebra as it gets far more messy than I could possibly attempt by hand. The concept is simple but the math is very long and messy. The equation was then split into pieces because Excel cannot accept the whole equation in one cell.
I hope this is clear.
I reread your question this morning and decided that I should add some more detail:
The math solution does not involve any RMS of errors. It relies on the fact that error is very nearly minimized by finding the R5 that creates zero error at the endpoints and midpoint temperatures. The math only deals with these three points. The resulting solution very nearly minimizes error at 1/4 and 3/4 scale where the error will be nearly equal and opposite polarity.
I've attached a Powerpoint file, copying and pasting the steps in Mathcad. The notes were for my own documentation purposes so that I could retrace my steps but I think you can follow the procedure.
2677.Annotated math solution.ppt
I really appreciate you taking the time to answer the question and provide the followup information. The PowerPoint attachment is exactly what I was looking for. Just enough information to guide me to understanding the solution. It will also give me a chance to use to torture my MATLAB's symbolic tools for the first time. Since I have limited time to work on this, I'm probably being pinging on you about this time to time.
Thanks again for you help, Mr. Trump. Really made my day.
I'll be curious whether you can retrace my same steps in MATLAB. I am an amateur in using Mathcad so I'm not sure that I took the most direct approach. I crashed the program a few times in the process. I'm very confident in the final result, however, as it gives correct answers. Please give me an update.
I've had some additional insights with regard to the effect of series connection resistance. If you come to the point of circuit implementation you may want to come back for some further information.
Good luck and regards, Bruce.
Your problems are my problems. I, too, am an amateur with MATLAB. I'm using a student version, BTW, and don't know if there maybe a limit of the number of variable entries.
I should have said this earlier: Be patient, this is going to take several weeks to a couple of months...not because the your linearization equations are difficult to figure out (per your data), but computers being computers, GIGO applies here (G being your's truly). Namely: I will have to iterate through and figure out how what MATLAB really wants as opposed to what I think it wants.
Presuming success, I'll be more than happy to pass on my notes.
Again, thanks for your efforts, Bruce.
Bump. Sorry for the slow response, other events took precedence and took a while to write the basic equations correctly. Tedious work. Initially, when I first ran the solution, it argued it couldn't find an explicit solution; that took several hours to find that error. In another case, a wrong sign (- for a +, or vice versa, ended up generating at least 2.5 pages of the solution in Word; again, it took a few hours to find the error; wash, rinse, repeat...
Now I can confirm, tentatively, that R5's solution in Mat Lab, matched the PowerPoint slide (i. e., generated 3 solutions: 0, positive, and negative, and a visual inspection indicated that the variables "looked right"), but did not check for a one-for-one correspondence yet.
I'll be working on find the solution to Vout the next few days when I get a chance.
When I wrap this up, I'll post a PDF and some other files of my adventures in Pt RTD linearity.
Sorry I missed your posting. You are in a corner of our forums that I don't normally watch closely.
If you can get the math expression for R5 into a text editor you have a chance of getting it correctly entered into Excel. With the text editor, you can do search and replace of variable names for Excel cell names and make other edits systematically to get an error-free Excel formula. It does need to be broken into several cells as it is far too long for one cell.
Good luck, Bruce.
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