I was wondering if a simple way to measure the temperature of a thermocouple is to use an inductor, and it looks like a viable approach, perhaps even providing precision comparable to or better than the standard op-amp approach, though it requires much thicker sensor leads.
Thermocouples can generate substantial power — those used for safety cutoffs in gas appliances typically power solenoids directly — but very low voltages — typically 20–40 μV/°, with 41 μV/° for the common type-K thermocouple (which is the one commonly used in gas appliances). Being made of solid metal, they have very low output impedances, but the low voltages are not particularly friendly to precise sensing with essentially capacitive CMOS.
So what if we use an inductor and a switch to generate a higher-voltage waveform from the low-voltage thermocouple signal, then measure the waveform with a cheap microcontroller?
The conventional approach to solving this problem is to use an op-amp (maybe a chopper to mask out flicker noise) to amplify the output voltage to a level where you can conveniently digitize it, but it occurred to me that perhaps you could put a thermocouple in series with an inductor, then open the circuit so that the inductor is obliged to discharge through a higher, known resistance.
To be more specific, consider the problem of trying to measure a temperature between 600° and 1200° (against a reference temperature we will assume is 0°) using a type-K thermocouple in series with a 100-μH, 1-Ω inductor in parallel with a 100-Ω resistor. At ΔT = 1200°, the thermocouple generates 49.2 mV, thus pushing 49.2 mA through the inductor (and 0.492 mA through the resistor). When the thermocouple is disconnected, the 49.2 mA starts to flow through the resistor instead, raising the voltage across it to 4.92 V, which is easily measured without further amplification. The current is then declining at 49.2 A per microsecond, so we only have a few microseconds (L/R = 1 μs) to measure it in, which is challenging but not infeasible with an off-the-shelf STM32.
A larger inductor would give us more time, but to avoid annoyingly high voltages, would require a correspondingly smaller resistor, and so parasitic resistances would start to play a larger role.
A different approach is to use a capacitor instead of the resistor for the inductor to discharge through, so that the current can oscillate back and forth several times. With the same 100-μH, 1-Ω inductor, our inductor energy ½LI² is ½·100μH·(49.2 mA)² = 121 nJ; if we want our capacitor to charge up to ±2.5 V or less (so we can bias the whole oscillation into the +0–+5 VDC range), then C ≥ 2 · 121 nJ/(2.5 V)² = 39 nF. Using a conventional 0.047 μF capacitor, then, we get up to ±2.27 V, and the oscillation frequency is 1/(2π√(LC)) ≈ 73 kHz, so we need to digitize the result at at least 150 ksps to avoid aliasing, which is straightforward.
Q = sqrt(L/C)/R, which works out to about 46 — about 200 μs per factor of e decay (τ), so the signal should remain detectable for up to a millisecond or two. I think this is probably a higher Q than we’ll actually get in practice, but maybe not much higher.
So the procedure is to open the switch (which needs to operate in less than about 10 μs) and measure the ringing-down signal in order to get a reasonably precise indication of its amplitude over time, and in particular to extrapolate what its initial amplitude was when the circuit opened.
The sources of error I can see here are:
Error estimating the time-domain start of oscillation; an error of 1τ results in an error of e ≈ 2.7, so to get 10°C of error at 1200° we need about τ/120 = 1.7 μs. Measuring this directly would mean we’d need to sample at hundreds of kHz, but that isn’t necessary — we know the precise phase of the oscillation start — the current was at near-max and the capacitor was charged only according to the resistance of the inductor (49.2 mV in the above example). So we only need the accurate phase of the signal (to within 0.78 radians in this case) and to successfully detect its first oscillation. (In theory we don’t need to estimate this because we’re controlling it, but there might be delay in the switch.) Roughly half of our samples give us phase information whose precision is comparable to the precision of the amplitude information in (roughly) the other half; if we have 1000 useful samples (1 ms) then we have a few hundred samples each telling us the phase with something like 12 bits of precision, so we have maybe 16 bits of precision on the phase: something like an error of 15 microradians, which gives us a time-domain error for the cycle start on the order of 30 ps, translating to an amplitude error of 0.15 ppm, and thus totally overkill precision of 180 μK.
Error estimating the amplitude of the signal, which is potentially a sort of weighted average of contributions from all the samples, maybe with some nonlinear filtering to reject out-of-band noise. Our 2.27 V initial amplitude will decrease proportionally by some 2 mV/°, so even 10 bits of precision (in the weighted sum, not each individual measurement) should be adequate. Offsets are inconsequential. If we figure we have about a millisecond of useful 12-bit samples at a megasample per second, that’s 1000 samples, which gives us a precision of about 17 bits, a precision of about 17 microvolts on the 2.27 V initial measurement, or 375 nV on the original low-impedance thermocouple voltage, working out to a totally overkill precision of 9 mK. (Of course, the precision of your analog voltage reference is going to be a limiting factor here, too; if it’s not accurate to 17 μV, you’re not going to hit that.)
Error in estimating the voltage amplification factor of the circuit — the 49.2 mV to 2.27 V factor described above, which comes from ½L(V₀/R)² = ½CV₁², and thus V₁² = (L/(CR²))V₀², so V₁ = (1/R) √(L/C) V₀ = Q V₀, precisely the Q factor of the resonator. This is most sensitive to the precise value of the inductor’s resistance, which will vary with the inductor’s temperature, since the inductor is not a precision resistor. However, the fact that this is precisely the Q factor of the resonator is wonderful — it means that we can measure the rate of ringdown and get a very precise measurement of this very amplification factor, accurate as of that moment. If we let the 2.27 V ring down to 0.5 mV (the limit of a 12-bit ADC) that’s about 8.4τ ≈ 1.6 ms, and getting a 10°/1200° ≈ 0.8% error in the time domain would require an error of about 13 μs — not plausible, since we’re talking about a systematic timing error over many measurements. In the voltage domain, an error in estimating the Q factor of 0.8% would amount to a (differential!) voltage error of 1 - (1.008)⁸·⁴ ≈ 7%, which is even less plausible.
The only remaining source of error in this voltage amplification factor, then, will be parasitic resistance in the thermocouple, its leads, and the switch, which are outside the resonating loop, and the capacitor, which is in the resonating loop but not part of the series resistance that sets the initial inductor current. As an example, a 2N7002’s on-resistance is 7Ω (see My attempt to learn about jellybean electronic components). This could be improved by adding resistance inside the resonating loop, in series with the inductor, thus reducing the Q factor and thus increasing the earlier-mentioned errors; this would probably be a good tradeoff.
Switch slowness error: we’re indirectly measuring a discontinuous step function in current through the thermocouple by way of measuring its 73-kHz component. But discontinuous step functions of current cannot exist in the real world of nonzero parasitic inductances; any real switch, whether solid-state, thermionic, or electromechanical, will take some finite time to open and close, and electromechanical switches have the even more alarming behavior of bouncing when they close. If these errors attenuate or strengthen the 73-kHz component of the step function, you could get potentially totally incorrect measurements or even a blown input circuit.
This is somewhat exacerbated by the fact that MOSFETs with low on-resistance are big power MOSFETs like the IRF540N (44 mΩ), and so they have much larger Qg (71 nC in that case, compared to 2 for the 2N7000 — see My attempt to learn about jellybean electronic components.) Driving 71 nC into the gate in under 100 ns, which I guess would keep the switch-slowness error under about 1% and thus the temperature error under 12° at 1200°, requires a large transient current of 710 mA; the IRF540N datasheet does claim its turn-off delay time is 39 ns and its fall time is 35 ns.
This problem, too, can be ameliorated with a Q-spoiling resistor in series with the inductor, reducing the error due to the switch’s on-resistance and thus allowing the use of smaller transistors. ON Semiconductor’s 2N7000 datasheet claims a 10-ns turnoff time and 1.2–5Ω on-resistance at room temperature.
It could also be ameliorated by using an exotic GaN power transistor like the 39¢ EPC2036 with its 65 mΩ and Qg = 0.91 nC, or possibly an electromechanical relay.
Thermocouple error: the metals in the thermocouple may not be pure, and their purity may change over time, especially at higher temperatures. Eventually it will probably burn out. It should be easy to make this the dominant form of error, but presumably it’s a form of error we can calibrate out, since at any given time it depends only on the temperature we’re measuring.
Cold-junction compensation error: thermocouples really measure a difference in temperature between two points, one of which is typically inside the measuring instrument. This is probably not a major problem for the ±10° precision I’m looking for, since a calibration at 18° is good from 8° to 28° even without any temperature compensation.
Here I’ve talked about the possibility of using an extra resistor in series with the inductor — which could take the form of just winding your inductor with thinner wire, and therefore perhaps just buying a cheaper inductor — but another possibility is using a precision resistor of 10–100 Ω outside the resonant loop. This would avoid reducing the Q (though it’s probably excessive anyway) and reduce the error introduced by resistance in the thermocouple’s leads, but by dropping most of the thermocouple’s voltage, it would also attenuate the signal being measured and make the results sensitive to the unknown drifting parasitic resistance of the inductor.
Since the dominant sources of error here are parasitic resistances (which can be measured), the unknown thermocouple characteristics (which can be calibrated), and switch slowness, you can probably get better measurements by using an electromechanical relay.
A thing to consider is that this approach can quite reasonably measure 100 samples per second, but the thermocouple’s temperature can’t vary under non-disastrous circumstances by more than a hundred degrees per second or so, and in many circumstances, not more than a degree per second. So errors that pertain to only a single measurement, like phase error, can be averaged out over hundreds of measurements to obtain another 4–5 bits of precision.
(XXX I am not sure my calculations of how many bits of precision you get from averaging over some number of samples are right; I think I may have dropped a squaring or a square-root somewhere. The above calculations are based on the idea that quadrupling your number of samples gets you an extra bit of precision, which is 6 dB.)
(This design might also be adaptable for use as an energy-harvesting frontend.)
Horowitz & Hill (2015 edition) talks a bit about the problem of amplifying thermocouples. On p. 936 it explains that you can use a US$16 Cirrus CS5532 24-bit delta-sigma converter to digitize the signal “directly” — by way of its integrated chopper-stabilized programmable-gain amplifier, which you can set to 64× gain to get a ±40 mV input range, or 32× for twice that. They mention that they balanced their thermocouple lines around ground to reduce EMI pickup.
On p. 1033, they also mention the Maxim MAX6675, which is a “thermocouple-to-digital-converter chip” with SPI output, and on p. 1083 they mention the MAX31855 SPI thermocouple ADC with 0.25° resolution from -270° to +1372° and built-in cold-junction compensation. The 1989 edition has a good deal more information on the subject of thermocouples on pp. 989–992, in fact recommending 10kΩ or more of input impedance to avoid errors from lead resistance.
But basically you just rig up an op-amp to amplify the small low input voltage to a larger output voltage, in, for example, the simplest non-inverting configuration: connect the thermocouple to the + input and run a voltage divider across the - input between the op-amp output and ground, with, say, a precision 47kΩ resistor and a precision 1kΩ resistor. This circuit makes no difficult demands of bias current, input range, bandwidth, slew rate, output power, power consumption, stability, gain, or broadband (as opposed to flicker) noise, but it does require a low input offset voltage (and precision resistors). Horowitz & Hill suggest using a chopper-stabilized op-amp, which typically requires a couple of external capacitors, like the US$1.60 MAX9617 — typical offset voltage 0.8 μV + 5 nV/°, with a 10–140 pA bias current. 0.8 μV in this context is a temperature error of 0.02°. This is fine for my intended application of controlling a pottery kiln, but it seems large compared to the much smaller errors I was talking about above. Worse, its worst-case offset voltage is specified as 10 μV, which is a temperature error of 0.2°. Other choppers are comparable — some a little better, others a lot worse.
Horowitz & Hill (1989) recommend rigging up the op-amp as a differencing amplifier in order to get good common-mode rejection of EMI, which I think is in large part a function of the high input impedance they’ve chosen due to their fine wires; but they emphasize that low offset voltage, a microvolt or better, is essential.
In the land of garden-variety op-amps, things are much worse. The US$0.24 Microchip MCP6001T-I/OT I declared in Jellybean ICs 2016 to be “the world’s cheapest op amp” is specified as ±4.5 mV input offset voltage, ±2μV/°. ±4.5 mV would be an error of ±110°. The LM324 is ±2–7 mV, the LM741 is ±1–5 mV, and the CMOS TLC272 is ±1.1–10 mV, though its precision-trimmed TLC277 variant is ±500μV. And that’s before we even get into flicker noise and drift! So if you were in the unfortunate position of having to improvise the thermocouple amplifier with whatever op-amps you could scrounge (see Ghettobotics: making robots out of trash) the straightforward circuit wouldn’t work; the resonant tank circuit this note is about would be a lot better. You might be able to get the straightforward circuit, or an instrumentation-amplifier variant of it, to work with a trimming resistor, since you only need about an order of magnitude improvement to get to the 10° (400μV) precision I wanted.
But can the inductor hack compete with the precision of a chopper-stabilized op-amp for thermocouple sensing? The chopper here is giving us 10 μV of error on a 49.2-mV signal, an error of about one part in 5000, 200 ppm. The inductor hack above is estimated to get an error of about 0.15 ppm from the time-domain start error, 7 ppm from the amplitude estimation error, some small error I didn’t actually estimate from estimating the Q factor (probably dominated by the same voltage-measurement error as the amplitude estimation, and so probably of similar magnitude), and some potentially large errors from parasitic resistances (≈4000 ppm if we assume that we’re using the IRF540N and its on-resistance varies unpredictably by 4 mΩ) and switching softness (say, the IRF540N’s 35-ns fall time reducing the measured voltage at 73 kHz by 0.25%, 2500 ppm — although I may be estimating that wrong). So it might be possible to get the inductor hack below 100 ppm of error, especially by using exotic components like electromechanical relays or GaN FETs.
(But then Horowitz & Hill get serious about precision design and explain the design of HP multimeters with 0.1 ppm error.)
The problem with the switch on-resistance and the resistance of the thermistor and leads in the above design is that it drops some unknown amount of voltage before getting into our resonating tank circuit. Parasitic resistances in the tank itself introduce no error, as long as they’re on the inductor side, because they show up in our Q measurement of the ringdown.
The problem is that we’re faced with a Thevenin voltage-resistance pair, the voltage coming from the thermocouple and the resistance coming mostly from its leads, and we’re trying to estimate the voltage from a single measurement from outside of it. This is clearly going to introduce significant error, and is the reason Horowitz & Hill suggest using a high input impedance, even though the thermocouple itself is very low impedance.
A different approach is to add some more switches with different known, precision, low-drift series resistances. Then we can measure how much the voltage across our tank changes with these various resistances, and thus the internal resistance of our voltage source. In the absence of measurement error, two switches would be enough.
However, if we’re talking about MOSFET switches, we have the problem that the resistance of each switch is significant, significantly different from other switches, and variable over time and with temperature. So it might be worthwhile to instead seek a way of taking multiple measurements to calibrate out the source resistance that lets us keep using the same switch.
Specifically, we can leave the switch closed for less time than is needed for the current through the inductor to reach its steady state, which we’ve calculated above is about 1.6 ms. If we leave the switch open for only 100 μs or 200 μs, for example, the current through the inductor won’t have ramped up to its max, and so we’ll get a smaller pulse when we open the switch again. These shorter pulses will depend less on the parasitic lead resistances, because until the current gets going, those resistances aren’t dropping any voltage, so the initial ramp-up of current in the inductor is independent of lead resistance and indeed of any resistance at all. By taking several different such measurements, it should be possible to calculate the outside-the-loop parasitic resistances with good precision — not just the resistance of the leads but also of the switching MOSFET.