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Some rough examples simulating the most common motors in personal computers. Information might not be scientifically accurate, it's only from what I've learned from pulling crap apart and poking it and googling about.
From left to right:
Brushed DC Motor: a standard Faraday motor, two (or three) coils are wrapped around poles on the armature. When the motor turns, the brushes swap contacts to swap the polarity through the coils. These motors are used where raw guts is required and speed or precision and noise is not important; eg ejecting and returning a motorised CD/DVD drive tray. Power is controlled by current flow and direction can be controlled by polarity. Brushed motors are usually quite thick and chunky and take up a lot of space. And there's a spark every time the brushes reconnect to the coils which makes a lot of EMI. The weight of the coils around the armature also works as a flywheel, so they're quite hard to position accurately.
Brushless DC motor: every dc fan I've found is one of these. Two stationary coils with a common tap are would around a single round stator, or four stator poles. A four or six pole ring magnet sits in the armature surrounding the stator. Putting straight dc through either coil does nothing. The secret to making it spin is in the Hall-Effect sensor, which is placed on the stator and detects which pole is closest. The hall sensor then switches current through the opposite coil and blocks current through the nearby coil. These motors can only turn in one direction and stall very easily. This makes them useful as cooling fans. The position of the armature cannot be determined, but the hall sensor switch frequency can be read directly to determine the fan's rpm. Simple designs will have a single IC containing the hall sensor and a transistor; more elaborate fans will be packed with ICs, filter caps, thermistors and even little PWM controllers onto a pcb inside the fan.
Three Phase Spindle Motor: these motors have three stationary coils and three or four wires, and get driven by three AC phases 120° apart. Most coils are wound onto multiple poles. All coils are tapped together to a common node. Current must flow through two coils to cause the motor to spin. The speed of the motor is controlled by the AC frequency and the direction is controlled by the order of the phases. These motors are used in floppy drives, older hard disk drives, cd and DVD drives to spin the disk platter, laser printers to reflect the laser across the page. Some of these motors have three hall sensors which are used in the same way as the brushless DC motor. Three phase motors need some pretty fancy circuits to operate from DC and most of them have dedicated ICs to drive them. They're built very flat, with lots of bearings for smooth operation, and can spin at crazy speeds, eg 35k rpm for 50x CDROM. They're very good for purposes that require constant consistent accurate speed, but they're not well suited to driving gears or belts.
Bipolar Stepper Motor: these motors have four coils arranged in pairs. To drive the motor, two AC phases 90° apart are used. Each phase will cause the motor to step to the next polarity Each coil can be energised in steps for precise positioning, and can also hold a position very tightly. I haven't found any containing any hall sensors. These motors are in CD/DVD drives to position the lens, usually with a worm gear. They're thick and chunky and won't freewheel much in your hands from having a great deal of magnetic resistance.
Unipolar Stepper Motor: similar to Bipolar except the coil are connected to a common ground, so they're easy to drive using pulses of DC in steps. They have five or six wires, one for each coil and one or two for ground. They're driven with four phases 90° apart, and the pulse length can be 25% to 50% of the duty cycle to control speed vs torque. These are most common in printers and scanners, they have heaps of torque and they're usually geared up for heavy duty operation for stuff that needs precise position control. They don't have hall sensors that I've seen, but sometimes they have an extra pair of wires going to a thermistor inside to signal if they're running hot.
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