The purpose of a motor is to deliver power to a load. Given a certain voltage u, the delivered power is equal to the torque times the speed:
The maximum delivered power Pout,max for a given voltage can be found by differentiating this function and setting it to zero. The result will be a specific torque and speed. If this process is repeated for other voltages, a line of points where maximum power is delivered will result.
The efficiency h of a motor is defined as the net output power divided by the input power. For a given voltage, the maximum efficiency can be found just like the maximum power and will lead to a certain torque and speed. By repeating this procedure for other voltages, a line of points where maximum efficiency is obtained will result.
The resulting lines of maximum power and maximum efficiency can be shown in the torque speed plot. Choosing a good motor is always a compromise between efficiency and output power. A good choice is to choose a motor that has the major part of the load curve inside the lines of maximum power and maximum efficiency. This area is called the Desired Operating Area.