Quote: 
Here is the in loop temperature for various loop settings.
Legend: The first number is proportional gain, the F is integral gain ("fast" as called by the 3040) To note:
 I don't know why the loop oscillates for lower gain settings
 The loop seems most stable for the highest possible gain settings ("300 Fast"), so I will use this.
 This is actually Kelvin/rtHz

Making sense of this  I spent a couple minutes sitting down and writing down the right equation for the complete open loop transfer function. Assuming an integral gain of "fast", we have:
OLTF = 3.68e3*Kp*(s+8.5/Kp)/s/(s+0.0167)
Where Kp is the proportional gain. When we make Kp 50, we put the zero from the PI loop at 0.17, where the pole is at 0.0167. This makes a nice long chunk of f^2 in the loop, murdering the phase. As we crank the gain up to 300, we push this zero down, and shorten the stretch of f^2.
Ideally with an integral gain of "fast" we would have a proportional gain of ~500.
If we go to "slow", then we have for our complete open loop transfer function:
OLTF = 3.68e3*Kp*(s+1.6/Kp)/s/(s+0.0167)
For a gain of "1 slow", as in the plot, we find the zero is 2 orders higher than the pole, which is consistent with this setting ringing way more than the others. A gain of "100 slow" would work for balancing the zero and pole. Our highest proportional gain available with a "slow" setting is 50, which is similar to the situation above, where the zero is about a factor of 2 before the pole, thought with a lower UGF than above.
