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All air conditioning units, heat pumps, and furnaces come with capacitors for the fan and compressor unless it is a three phase system. Three phase systems are generally commercial.
A capacitor helps the motor run or start depending on what type of capacitor it is. In brief, it provides an energy boost to get the motor started. A start capacitor performs the function of a centrifugal switch. There is a brief pulse to start the winding taht couses the motor to rotate. The run capacitor are designed to provide a phase shift for the motor.
If a capacitor has aged from normal usage it becomes too weak causing the motor not to run. This is not an uncommon; many A/C units require new capacitors after a few years.
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To understand the differences between three different types of capacitors, refer to overview diagrams below from Packard, one of the world's leading manufacturers of capacitors.