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I have a fairly simple situation: N-Channel MOSFET with base connected to Raspberry Pi's 3.3V source connected to ground and Drain connected this way: +12V ----> Vacuum Cleaner (60W) -----> Drain. Guy in the electronics shop told me, this transistor's (1D33AA BUZ11) gonna carry big current no problem, but as I stated in the question it is getting very hot, and the vacuum cleaner reaches its full power slowly (but the hotter the transistor the more power provided)

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Short answer: You're not giving it enough gate voltage for it to turn on completely. It needs 10V for the guaranteed Rds(on) of 40m\$\Omega\$.

The BUZ11 is a fine MOSFET, but it is not designed to operate from a 3.3V drive voltage. With 3.3V on the gate (relative to source) it will only turn on partially, and will begin to heat rapidly with a heavy load. The threshold voltage has a negative temperature coefficient, so the hotter it gets, the more it tends to turn on. Needless to say, this is not a good way to operate the MOSFET.

You could either get a MOSFET that has a guaranteed on-resistance with a 3.3V or lower drive voltage, or boost the gate drive voltage to allow you to use a part like the BUZ11.

There may not be any MOSFETs that are as good as the BUZ11 with 3.3V drive (especially the 50V voltage rating), at any price. If you can reduce the voltage rating to 20V, there may be a number of them available- but in TO-220, I see only the IRL3103PBF, which is probably okay but not guaranteed at 3.3V drive.

You could simply use a transistor to drive the MOSFET gate as so:

schematic

This would also tend to protect your Raspberry pi from some types of failure and bad connections.. a gate to drain short on a direct-driven MOSFET would otherwise release the smoke.

Note that "high" = "off" since the BJT acts as an inverter. If this is undesirable, you can add another BJT inverter.

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Is it possible to provide said 10V with a bipolar transistor? So I could turn mosfet with bipolar which is turned on with 3.3V. Will it work? –  Michał May 4 at 14:34
    
@Michał See my edit. –  Spehro Pefhany May 4 at 14:34
    
I've done exactly the same thing in Every Circuit circuits simulator and the MOSFET seems to be turned on by the little voltage that is between the NPN and 10k resistor (like 23mV). Will this really happen? I don't want to wire it before I'm sure it's ok. Raspberry pi is easily burnt, unfortunately. –  Michał May 4 at 15:07
2  
No, the BUZ11 is only partially on at 3V and is guaranteed to pass less than 1mA at 2.1V (the \$V_{TH}\$ specification). Your simulator is broken if it thinks the MOSFET will conduct more than uA with 23mV on the gate. Make sure the grounds are well connected and don't run the 12V power ground through the RPi. If you want to be completely safe, replace the BJT with an optoisolator and keep the 12V power circuit separate. –  Spehro Pefhany May 4 at 15:20

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