Averaging thermistor readings

Question

I have 9 thermistors arranged in a grid, and I want to average the readings from these thermistors without code (that is, average them in a circuit rather than reading all the values into a microcontroller and averaging within code) - for speed. The 'readings' are voltages measured via ADC on an Arduino in a typical thermistor set up with the ADC line straight after the first resistor but right before the thermistor, such as shown below.

I was originally intending to use a voltage averaging circuit based on Millman's theorem- until I suddenly realised that this won't work because no current flows from the thermistor circuit to the ADC.

Does anyone know of a way to wire up 9 thermistors such that the readings I can obtain is the average of the 9, or if there's a version of a voltage averaging circuit that works on open loops?

Edit with more info:

Thermistors are equispaced in a 3x3 grid with a 150mm pitch. Ranging from 40 degrees Celsius to -40 degrees Celsius. All resistors should be very close to each other, they all measure the same thing, but 9 is used instead of one to account for uneven temperature distributions. The change in temperature should be gradual but I'm limited by the Arduino's built in analogueRead function which is about 0.001 second sample time.

Answer 1

There is a method to produce pseudo-linear NTC response (reasonably linear over a 50C range) via the addition of a parallel resistor.

^{simulate this circuit – Schematic created using CircuitLab}

The additional parallel resistor, R2, can assist in linearising aspects of the NTC

Between 25C and 90C it is ~linear, with the exact variation dependent on tolerance and the BETA of the NTC. The region that linearisation can be achieved is also influenced by the parallel resistor (ie if it required for low-temp, then the characteristics can be shifted)

The buffered output can be fed into an OPAMP summing circuit with a 1/9 gain to provide the averaging result

^{simulate this circuit}

Answer 2

Thermistors are very nonlinear. If you actually want an equally-weighted (in temperature) average, I suggest digitizing and linearizing the temperature readings and then averaging them digitally.

Consider, for example, simply connecting the thermistors in series. The lowest temperature one would dominate by far. Connecting them in parallel would have the opposite result- the highest temperature one would dominate.

Alternately, you could consider using a different kind of sensor.

There are ways to get a more linear output from a thermistor with additional resistors, but I think it's more trouble than it's worth and it reduces the sensitivity so would require more signal conditioning.

Edit: Ran some optimization routines to see roughly what the typical Wheatstone bridge linearization method would yield, using least squares Broyden–Fletcher–Goldfarb–Shanno optimization, with a targeted output of 10mV/°C. As I recalled, the WB method is quite good for narrow ranges- at 25°C range (eg. 15°C to 40°C) with a typical thermistor the theoretical error could be kept to almost +/-0.1°C, however over a range such as 0°C to 100°C the theoretical deviation from linearity was more like +/-5°C.

The method requires reading a differential voltage per thermistor and, of course, the 4 additional resistor tolerances affect the accuracy in a real application.

Answer 3

As mentioned, NTC thermistors are non-linear and thus difficult to average or sum in hardware.

There are linear alternatives though:

• Thermocouples are fully linear.
• PT100 sensors are almost linear in 0-100°C area.
• Integrated chip sensors such as LM35 have linear output.

Any of these would be possible to average using simple passive circuit. But NTC thermistors are usually the cheapest type so averaging in software is probably the preferable solution.

Answer 4

This is a typical plot of the Steinhart-hart equation. (wiki link)

ContourPlot[{1/t == 
   1.4 10^-3 + 2.37 10^-4 Log[r] + 9.9 10^-8 (Log[r])^3
  },
  {t, 273 - 40, 273 + 40}, {r, 0, 35^3},
 ContourStyle -> Red,
 GridLines -> Automatic,
 FrameLabel -> {"Temperature (K)", "Resistance (\[CapitalOmega])"},
 Epilog -> {Style[
    Text["Typical plot of the\nSteinhart-hart Equation", {290, 
      30000}], 14, Bold]}
 ]

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If (let's say) the thermistors at the right side of the grid are at 40C, and those at the left side are at -40C, then your voltage readings will vary widely as the thermistor resistance would range between few hundred and several tens of thousands of ohms. A simple average of such voltages wouldn't mean much. I cannot think of a non-linear sum circuitry that would find the 'average' and send that to Arduino. My next question would have been: How far is the Arduino from this arrangement? I think you get the point.

My advice would be to use eight independent temperature sensors (current-based) placed according to the geometry of the frame/housing of the device under test along with an 8-input ADC interface and do numeric processing in code.