Why don’t electric cars use gearboxes?

Simon’s answer is not very accurate so I would like to add some more.

This diagram shows the relationship between the traction force (Zugkraft) and the car’s speed for various gear ratios. The ideal maximum is indicated by the black hyperbolic curve. The car can be driven at a certain speed by applying the maximum traction force. The orange curves indicate the ACTUAL traction force available for different gear ratios. The orange curve at its highest corresponds to the first gear. The one below corresponds to second.

So for any given gear, you can only operate the car only within the corresponding curve’s limits. Note the red areas below the black which are not covered by the orange ones – the points in these areas are not covered by the powertrain’s performance and hence you can’t fully exploit all possible torque scenarios. When shifting from one gear to the adjacent one, all possible states of torque in the red area is practically ‘lost’.

An electric motor is able to do this. The beauty of an electric motor is its ability to continuously change the torque available at any torque below it power curve. There are no red areas. In the above diagram, the black curve would look almost identical to the orange curves. This diagram will help you better understand the concept.

(Source: Renault)

Your electric motor could be operated at any point below the dark grey curve. This eliminates the need to have an additional transmission. Or does it?

Technically, an electric motor could be built to meet the requirements of the vehicle. If you need more power or torque for a larger car, then you build a larger motor. However, you will have to carry more weight for the powertrain. This is not economically feasible. It’s not as cost-effective as fitting a gearbox with one (or sometimes two) ratios. Engineers may have different design ideas, but the transmission of gears will still play an important role in future powertrain designs.

EDIT: Just realized that a multiple-speed transmission would require you to break the engine’s mechanical link in order to change gears. This is similar to what we do with clutches. This is a disadvantage to having an electric motor to drive your car. There are however, solutions. For example, dual-clutch transmissions. Other innovative solutions are being found all over the globe. Here’s an example:

Oerlikon Graziano’s new prototype eDCT transmission for EVs

 

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