ATC corrects for the Nerstian response for a membrane potential (i.e. pH electrode). Temperature is a factor in the Nernst equation. At 25 oC a pH electrode theoretically generates 0 mV at pH 7.01 and +177.48 mV in pH 4.01. The pH 4.01 value represent 59.16 mV/pH unit and since pH 4.01 is 3 pH units away the theoretical voltage is 59.16 x 3 @ 25 oC.
pH 7 is known as the isopotential (iso = same) point in which Theoretically it should always be 0 mV. For the pH 4.01 value (or slope) the mV/pH will change with temperature. For example, 54 mV/pH unit at 0 oC and 64 mV/pH at 50 oC. Again, these values are calculated by changing the temperature value in the Nernst equation.
pH meters with ATC use the Nernst equation to correct for temperature variations as predicted by the Nernst equation.
It is critical to note that the values on the bottles/packets of pH buffer solutions are the pH of the solution at that specific temperature. It is the reason why some meters will display a different pH value after calibration. The meter is programmed to display the solution pH buffer value for for the temperature being measured. The pH at any given temperature for a NIST traceable buffer is 100% known while the pH of wort at 25 oC versus 50 oC is not known. The actual wort pH can be different at different temperatures. It cannot be predicted with 100% certainty since the composition is never 100% exactly the same. So for a pH meter we program what we know. This is not the same as ATC. ATC does not correct for solution pH but for the Nernstian behavior.
Temperature does impact the resistance of the glass membrane. As temperature goes up the resistance goes down. It is one of the main concerns when measuring at very high temperatures. Ohm's law stated that V = IR. (V=voltage, I = impedance, R = resistance). The voltage is predicted by Nernst and R is the resistance of the glass membrane (around 100 megaohms for general purpose glass at 25 oC). With voltage staying the same and resistance decreasing then the impedance is affected, which has to be within an ideal range for a pH meter to work correctly. It is recommended to use pH electrodes with high temperature (HT) glass for measuring above 150 oF.
Bottom line:
1) ATC will correct the slope based on the temperature of solutions used for calibration.
2) With ATC, most meters will display the solution pH (the value printed on the bottle)
3) For Manual Temp Compensation (MTC), measure the temperature of the buffer and adjust meter pH value to match the pH value for given temperature from the table printed on the bottle.