>> Remember, heating is due to power not current.
>
> That's interesting - this is the first I've heard of this.

It's simple physics, any electrical energy lost inside a component must be 
converted to some other form of energy, usually the majority of which is 
heat.  The electrical energy lost inside a component does not depend solely 
on current, but on power.

Example 1: 10 amps going through a thick cable of resistance .001 ohms. 
Power dissipated = 1 W.  Heating effect: very small

Example 2: 10 amps going through a heating element of resistance 10 ohms. 
Power dissipated = 1000 W.  Heating effect: a warm room.

You can compare those examples to the tracks on your PCB and the IC.

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