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Why is a capacitor used in a single phase motor and not in three phase motor ?
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Point 1: A rotating magnetic field (RMF) is produced in a 3-phase induction motor when a 3-phase supply displaced from each other by 120 deg in time is applied to a 3-ph winding displaced from each other by 120 deg in space, naturally. This RMF is the reason for the motor to self-start and rotate to a speed just below the synchronous speed of the RMF.

In a 1-phase induction motor there is no RMF and to create one another winding displaced from the main winding by 90 deg in space is provided making it a 2-phase winding. A phase lead of less than around 90 deg is created in time by connecting a capacitor in series with this winding. This RMF (not all that uniform as in 3-phase) is sufficient to self start the motor. An additional capacitor may also be connected across the motor to improve its power factor.

**Point 2: In a single phase motor the magnetic field generated is pulsating in nature. we required rotating magnetic field for generation of torque that's the reason capacitors is used in single phase motor to create balance rotating field.
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A 3 phase motor is a self starting motor, i.e there is no requirement of any external means for the motor to start rotating. This is because a 3 phase induction motor has 3 phase stator winding which generates a magnetic field which changes direction constantly but has a constant magnitude in every direction. (This can be proven using simple vectors). This magnetic field is called a rotating magnetic field (rmf). The rmf interacts with the rotor and hence a torque is produced. So basically, you need a rmf to produce a starting torque. And to generate a rmf you need at least 2 phases.

Now, coming to a single phase motor, it is unable to generate a rmf and thus cannot start on its own. We could start the motor by giving it a starting torque using mechanical means, but that's not a practical solution. (Imagine having to physically rotate a fan by hand to get it started)

So, our aim is to derive a two phase motor from a single phase motor. Hence, we add a capacitor along with an auxiliary winding to the single phase motor. We know current leads voltage by 90 degrees in a capacitor thus now a phase difference is created. Due to this phase difference, a rmf will be generated and the single phase motor will now be able to start on its own.
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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.
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In 1 phase we need split the phases.. That's why use capacitor.. In 3 phase induction already rotating magnetic field produce so there is no need of capacitor
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In order to make single phase motor self starting we need two phases in motor which is achieved with help of capacitor where as three phase motor is self starting and that's why the capacitor is not used in it....
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In 1-phase induction motor capacitor work is generate the auxiliary winding current than excite the main winding to rotor starts rotate.

After rotor start rotating about 75% of their speed than auxiliary windinga is cut off .

Thats why in 1-phase induction motor need capacitor for starting the rotor rotation.hence,single phase induction motor is not self start.

In 3-phase induction motor 3 phases are made angle 120 degree and it helpd to start the generated emf to start the motor.so,in 3-phase induction motor is self start induction motor.
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