The two most common types of hub motors used in electric bicycles are brushed and brushless. Many configurations are available, varying in cost and complexity; direct-drive and geared motor units are both used. An electric power-assist system may be added to almost any pedal cycle using chain drive, belt drive, hub motors or friction drive. BLDC hub motors are a common modern design. The motor is built into the wheel hub itself, and the stator fixed solidly to the axle, and the magnets attached to and rotating with the wheel. The bicycle wheel hub is the motor. The power levels of motors used are influenced by available legal categories and are often, but not always limited to under 750 watts.
However, each country is free to allow faster s-pedelecs if they go through collective or individual type approval. Typically, an S-Pedelec can be associated with "mofas" or "mopeds" or "light scooters". Germany has such a solution: S-pedelecs have max. 500W motors, max 45km/h, max 300% support, can drive up to 20Km/h without pedalling, plus other constraints regarding security, and are associated with Mopeds. In the City you can't use bicycles paths, but you can in the country side.

In the theoretical electric bike we considered up above, we had the dynamo/motor driving the back wheel directly, simply by pressing on the tire. Most electric bikes work a different way. They have compact electric motors built into the hub of the back or front wheel (or mounted in the center of the bike and connected to the pedal sprocket). Take a look at the hub of an electric bike and probably you'll see it's much fatter and bulkier than on a normal bike. You can read more about how these motors work in our main article about hub motors.
On the other hand, the battery doesn’t lock into the bike, which means you can’t just leave it on your bike when you park at the bike rack. Anyone could walk by and simply remove your battery. That seems like an oversight to me, though perhaps the designers assumed that such a small battery would just be easy to take with you. And it is. My wife could probably lose this battery in her purse.
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