Yes, the auxiliary winding is smaller and has lesser number of turns than the primary winding.
There are basically two ways in which you can use this capacitor with the motor
Capacitor start Induction Motor
Capacitor run Induction Motor
In the first one, we use a capacitor just to start the motor, i.e to provide a starting torque. The capacitor is connected long enough to bring the motor to about 75% of the full speed and then the Capacitor is disconnected from the motor using a centrifugal switch. The motor will continue to run on its own since a single phase induction motor is self sustainable once started.
However, in a capacitor run motor, we have two capacitors- start capacitor and a run capacitor. The start capacitor is same as the one described above. It helps to increase the starting torque. However, a run capacitor is used to create a rmf while the motor is running. They are designed for continuous duty so we must use low loss capacitors such as polymer capacitors.
The rating of a capacitor is an important factor especially in run capacitor because if a wrong capacitance value is installed, it will create an uneven magnetic field. This will cause an uneven rotation, loss in efficiency, noise and overheating. The rating of the capacitor can be calculated using the following forumla:
Xc=1/2πfc
Where Xc is the reactance which can be adjusted according to the current.
Usually, start capacitors are generally rated above 70uF with different voltage ratings depending on the application. The capacitance value of run capacitors are lesser, generally ranging from 1.5uF - 100uF.