### Synchronous Motor

1. Introduction
If a three phase supply is given to the stator of a three phase alternator, it can work as a motor. As is is driven at synchronous speed, it is called synchronous generator. So if alternator is run as a motor. It will rotate at a synchronous speed. Such a device which converts an electrical energy into a mechanical energy running at synchronous speed is called synchronous motor. Synchronous motor works only at synchronous speed and can not work at a speed other than the synchronous speed. Its speed is constant irrespective of load, no doubt, its speed changes for an instant at the time of loading.
2. Types
The two types of synchronous motor are,
1. Three phase synchronous motors
2. Single phase synchronous motor
The single phase synchronous motor are further classified as reluctance motor and hysteresis motor.
The three phase synchronous motor works on the concept of rotating magnetic field. The field produced by stationary three phase winding, which rotates in space is called rotating magnetic field. Its speed is always synchronous and given by,
Ns = 120f/P
Where         P = Number of poles for which winding is wound
f = Frequency of the supply.
3. Rotating magnetic field (R.M.F.)
The rotating magnetic field can be defined as the field or flux having constant amplitude but whose axis rotates in a plane at a certain speed.e.g. permanent magnet rotating in a space produces a rotating magnetic field. Similarly if an arrangement is made to rotate the poles, with constant excitation supplied, the resulting field is rotating magnetic field. So a field produced in an air gap of a rotating field type alternator is of rotating type. But this is all about production of R.M.F. by physically rotating poles or magnet. In practice such a rotating magnetic field can be produced by exciting a set of stationary coils or wi9nding with the help of polyphase a.c. supply. The resultant flux produced in such a case has constant magnitude and its axis rotates in space without physically rotating the winding. Let us study how it happens.

3.1 Production of rotating magnetic field
Consider a three phase windings displaced in space by 120o, supplied by a three phase a.c. supply. The three phase currents are also displaced from each other by 120o. the flux produced by each phase current is also sinusoidal in nature and all three fluxes are separated from each other by120o. If the phase sequence of the windings is 1-2-3, then the mathematical equation for the instantaneous values of the fluxes Φ1, Φ2 and Φ3 can be given as,
Φ1 = Φm sin(ωt) = Φm sin θ                                    ...........(1)
Φ2 = sin (ωt - 120o) = Φm sin (θ - 120o)                ............(2)
Φ3 = Φm sin (ωt - 240o) = Φm sin (θ - 240o)          .............(3)

As windings are identical and supply is balanced the amplitude of each flux is same i.e.Φm . The waveform of three fluxes are shown in the Fig.1(a) while the assumed positive directions of these fluxes in space are shown in the Fig.1(b). Assumed positive direction means whenever the instantaneous value of flux is positive, in vector diagram it must be represented along its assumed positive direction. And if flux has negative instantaneous value then must be represented in opposite direction to assumed positive direction, in the vector diagram. Fig. 1

Let Φ1, Φ2 and Φ3 be the instantaneous values of the fluxes. The resultant flux ΦT at any instant is given by phasor combination of Φ1, Φ2 and Φ3 at that instant. Let us find out at four different instant 1, 2, 3 and 4 as shown in the Fig. 1(a) i.e. respectively at θ = 0o, 60o, 120o and 180o.
Case i)         θ = 0o
Substituting in equations (1), (2) and (3) we get,
Φ1 = Φm sin 0o = 0
Φ2 = Φm sin(-120o ) = -0.866 Φm
Φ= Φm sin (-240o) = + 0.866 Φ
Show positive values in assumed positive directions and negative in opposite directions to assumed positive directions.
Hence vector diagram looks like as shown in The Fig. 2(a).
BD is perpendicular drawn from B on ' ΦT '. Fig. 2 a and b
...                    OD = DA =  ΦT/2
In triangle        ∟OBD   = 30o
...                    cos 30o = OD/OB = (ΦT/2)/(0.866 Φm )
...                    ΦT = 2 x 0.866 Φm x cos 30o
= 1.5 Φm

So magnitude of resultant flux is 1.5 times the maximum value of an individual flux.
Case ii)  θ = 60o
Substituting in equations (1), (2) and (3) we get,
Φ1 = Φm sin 60o = +0.866 Φm
Φ2 = Φm sin (-60o) = -0866 Φm
Φ3 = Φm sin (-180o) = 0
So Φ1 is positive and  Φ2 is negative so vector diagram looks like as shown in the Fig. 2(b).
It can be seen that from the Fig. 2(b), that,
ΦT = 1.5 Φ
So magnitude of the resultant is same as before but is is rotated in space by 60o in clockwise direction, from its previous position.
Case iii)   θ = 120o
Substituting in equations (1), (2) and (3), we get,
Φ1 = Φm sin 120o = +0.866 Φm
Φ2 =  Φm sin 0o = 0
Φ3 = Φm sin (-120o ) = -0.866 Φm
So Φ1 is positive, Φ2 is zero and Φ3 is negative. So vector diagram looks like as shown in the Fig. 2(c). From the Fig. 2(c), it can be proved easily that,
ΦT = 1.5 Φ Fig. 2  c and d

So magnitude of the resultant is once again 1.5 Φm, same as before. While it is further rotated in space by from its previous position at θ = 60o
Case iv)             θ = 180o
Substituting in equations (1), (2) and (3), we get,
Φ1 = Φm sin (180o) = 0
Φ2 = Φm sin (60o) = +0.866 Φm
Φ3 = Φm sin (-60o)
= -0.866 Φm
So Φ1 = 0, Φ2 is positive and  Φ3 is negative. The vector diagram is as shown in the Fig. 2(d).
From the vector diagram, it can be proved that,
ΦT = 1.5 Φm
So magnitude of resultant flux is once again 1.5 Φm but is further rotated by 60o in clockwise direction from its position for θ  = 120o
So for a half cycle of the fluxes, the resultant has rotated through180o. This is applicable for 2 pole winding. From this discussion we can have following conclusions :
a) The resultant of the three alternating fluxes, separated from each other by 120o, has a constant amplitude of 1.5 Φm where Φm is maximum amplitude of an individual flux due to any phase.
b) The resultant always keeps on rotating with a certain speed in space.
This is nothing but satisfying the definition of a rotating magnetic field. Hence we can conclude that the three phase stationary winding when connected to a three phase a.c. supply produces a rotating magnetic field.
The speed of the resultant is in space, for electrical of the fluxes for a 2 pole winding as discussed above.
Key Point : This is nothing but,
omechanical = oelectrical for 2 pole case.
If winding is wound for P poles, then resultant will complete 2/P revolution for 360oelectrical of the fluxes. The relation is exactly similar to what we have discussed earlier in case of alternator. So resultant flux bears a fixed relation between speed of rotation, supply frequency and number of poles for which winding is wound. The relation is derived while studying an alternator. So for a standard supply frequency of f Hz of a three phase a.c. supply and 'P' poles of the three windings, the speed of the rotating magnetic field is Ns r.p.m.
Key Point : So for a rotating magnetic field,
Ns = 120f/P r.p.m.
3.2 Direction of rotating magnetic field
The direction of the rotating magnetic field is always from the axis of the leading phase of the three phase winding towards the lagging phase of the winding. In the example above the phase sequence is 1-2-3 i.e. phase 1 leads 2 by 120o and phase 2 leads 3 by 120o. So rotating magnetic field rotates from axis of 1 to axis of 2 and then to axis of e i.e. in the clockwise direction as seen above. This direction can be reversed by changing any two terminals of three phase winding while connecting them to the three phase supply. So in practice for a phase sequence of R-Y-B, the rotating magnetic field is rotating in clockwise direction, then by changing any two terminals of the winding it can be changed to anticlockwise, as shown in the Fig. 3(a) and (b). Fig.3  Reversal of direction R.M.F.