So I implemented the bundled connections on the sensor in a heat-shrinked tube in the water between the heatsinks.
Then I've ran a test to have it "settle" on 30 deg C. That takes about 30 minutes. I measured the difference between the target voltage and the output of the LM335 (10 mV / °C) during 2.5 hours.
But did it stabilize?
As can be seen not really, it oscillated with an amplitude of about 6 mV (0.6° C) with a period of 24 min.
I don't think it will stabilize within a day, so I may have to tune the PID control
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The regulation however will never be fast using this setup. For a practical solution I would have left this path of the water bucket. But then I would also miss the opportunity to learn more about PID control.
One of the main reasons it can't seem to find a stable power output is the very slow feedback. It takes so long that the U2A won't regulate proportionally at the turning points. The charge of C3 goes from min to max and from max to min.
Adding a parallel resistor to limit the amplification might be an solution. But this will also mean that there will always be a small difference between the set temperature and the measured one.
If for example the amplification is 1A/°C and it needs a settling current of 0.1A, then there will be a difference of 0.1°C between the set and the actual temp.
The data shows it will probably need a smaller amplification like 100mA/°C, which would result in a 1.0°C difference. That's not acceptable, especially because it depends on how much current is needed to keep it at the target temp and which is dependent on the environment temp.
Any suggestions on what may stabilize the control?
(I agree with that the water and other poor thermal conductors attribute to the oscillations)