Bosch Drawer Microwave causes ARC Faults
Dealing with the problem now:
One of the first things to try is just swapping the breaker with an identical one in the same panel. In my experience, there's a lot of individual variation in the exact behavior of both AFCI breakers and appliances, that sometimes you can get the problem to go away just by using a different combination of individual components. It's harder to swap the microwave (typically), but you can also try exchanging the oven at the store to see if a different one (even if the same make and model) would work okay.
That said, the only real solution to this problem is to have the breaker replaced by the manufacturer. It is almost certain that you have no actual arc fault, and that the breaker is being tripped unnecessarily.
You can also report the problem here: https://www.afcisafety.org/home-owners/unwanted-tripping/
I have been struggling with the same issue, for several years now, in a newly-constructed home where AFCI breakers were required on almost every circuit due to relatively new code requirements. It has not helped that the electrician who did all the work for the construction shows little to no interest in doing anything to investigate, never mind fix the problem. But the one thing that has, for the most part, helped is to replace the breakers. It seems the manufacturer (Square D in my case) updates the AFCI logic in the breaker over time, to reduce the number of false positives.
Since the latest attempt by the electrician to deal with the problem by replacing the most problematic breakers, while they introduced a whole new problem (breakers tripping randomly even when there's not even any load on the circuit), almost all of the appliance-related tripping has been eliminated.
(The one exception is that now we have a breaker for the furnace that trips the instant the furnace blower motor turns on, but only if the utility power is out and we're running on the generator instead. As you can see, it's a bit like playing whack-a-mole.)
You or your electrician need to work with the manufacturer to report the problem, telling them the exact make and model of appliance that is causing the problem, and get updated breakers that will work. If possible, that is…this is hit or miss under the best of circumstances, and it requires that the manufacturer (and any electrician that's involved) are willing to be cooperative and support the breaker and installation.
Why does the microwave trip the breaker?
As far as how the microwave can cause the tripping, it's simply due to the switching of electricity that occurs in the appliance. Motors with brushes are among the worst (fortunately, more and more appliances are being designed with DC brushless motors), but other things like plasma screen TVs (which use tiny arcs to generate the plasma discharge that lights the pixels), higher-end toaster ovens (we have a Breville that modulates the heating element intensity in a way that can trip the AFCI), and yes, microwave ovens, can all have similar problems.
In the case of the microwave oven, it will depend on the individual design. But ovens have motors (which are a possible problem, depending on the motor), for the carousel, ventilation, and the &quot;stirrer fan&quot;, and the switching on and off of the magnetron can also create signals that would fool the breaker.
I can't tell you exactly what component in the microwave is causing the false tripping, but it's not surprising that something is. And because of the chaotic nature of the AFCI behavior, you may or may not get the exact same behavior from a different individual breaker or oven, even if the same make and model, never mind if you tried a different manufacturer for either.
Why bother using the web form to report the problem?
Note that by reporting the problem on the form linked above, you can at least give the manufacturer a chance to investigate and update the design of their breaker for that particular scenario. It doesn't necessarily help your specific issue right now, but it could improve the situation for you and everyone else in the long run.