PID autotune fails 'Temp too high' with 12 V heater cartridge but works with 24 V?

Question

So, I have created my own heater block for my 3D printer and mistakenly was using a 12 V, 40 W heater cartridge thinking that was the standard for my Ender 3, when in fact it is supplied with a 24 V, 40 W.

You may have seen my earlier post about when using autotune the temperature would overshoot by a large margin and returned the following error.

PID Autotune failed! Temperature too high

Graph of Overshoot:

I was unable to remove this overshoot even through manual tuning.

Now I have switched from the 12 V cartridge to the 24 V this problem is resolved, and I can now run the autotune. It follows a much more gradual curve when heating up so doesn't trigger the same error.

However, I am unsure why this is the case? Can anyone explain why the 12 V heater cartridge results in too high of a temperature increase?

It seems counterintuitive to me as I would have thought the higher voltage cartridge would heat up faster as opposed to vice versa?

Answer 1

It is all about resistance.

This requires some formulae:

$U = Voltage$
$I = Current$
$R = Resistance$
$P = Power$

$U = I \times R$

$P = U \times R$

$ R = \dfrac{P}{I^2} = \dfrac{U^2}{P} $

The 12 V, 40 W cartridge has a resistance of about 3.6 Ω.

If you use this cartridge at 24 V, this caculates to a power of 160 W!

This means that there is an enormous influx of heat that is hard to control, hence the overshoot.