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I recently thought that it would be fun to build an arduino quadcopter.

As I did some research I realized that the common L293D motor drivers aren't ever used.

Why is this? Could I still use them in my project?

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Bipolar bridge drivers like the L293D, L298, etc have extremely high losses. They are long out of date and a poor choice compared to modern FET chips even for surface vehicles, let alone something that needs a good power to weight ratio for flight. Typically if you made a small robot with one and some AA cells, you'd have to have an extra cell in the pack just to cover the losses of the L293D. An aircraft (and especially a vertical lift one) cannot afford that waste. Also the wasted energy shows up as heat (though I suppose you could position the chip in the propwash).

Actual quadcopters use MOSFET drivers, typically discrete ones carefully selected for a low RDSon at a low enough threshold voltage to fully turn on when driven by an MCU output. Either just one N FET per motor for unidirectional control of brushed motors, or H bridges made of an N FET and either a P FET or else an N FETs with special high side driver circuitry for bidirectional control of brushed motors, or similarly as a half-bridge for each of the three leads of a brushless motor. The MOSFETs may be found either on the main board in a single board quadcopter, or else they may be packaged in modular speed controllers separated from it.

But providing current to drive the motors is the easy part - sequencing brushless motor commutation, and stabilizing a flight platform are the hard parts.

In that regard, an Arduino is a very poor choice as a flight controller. A literal one means carrying around a lot of extra mass and connectors that aren't really fitting. A work-alike on a custom board is possible, but most of the open source projects that started that way have long since moved on to other MCUs with more capability. You do still see ATmega8's within brushless motor drivers though.

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