Three-phase Synchronous Machine: Operation

Let's consider a three-phase alternator with salient pole inductor set in the rotor and with connection of the star stator phases.
The section of a machine of such a type with p=4 polar couples is shown in Figure 1.

 
AIR GAP INDUCTION DOE TO THE INDUCTOR
AIR GAP INDUCTION IN THE SALIENT POLE MACHINES

The outline of the pole shoes is shaped in such a way that the vectorial lines have such lengths to provide an almost sinusoidal induction run, along the corner covered by a polar couple, and with p periods along the whole air gap circumference.

We can notice, in Figure 1 where the distribution of the component is shown along the induction radius at the air gap, that the maximum value, being in correspondence of the symmetry axis of each pole, called polar axis, owhere the air gap has a minimum thickness. 

The induction nullifies in correspondence of the median axis between two poles, called interpolar axis.


Symbols of e.m.f. and currents: + = positive incoming,  · =  positive outgoing
Fig. 1

F =  armature e.m.f.    Ai =  interpolar axis
C =  excitation currents    Ap =  polar axis
AIR GAP INDUCTION IN THE SMOOTH ROTOR MACHINES
In the smooth rotor machines the air gap thickness is constant along the arc of each pole. 

Figure 2, illustrating the rectified sector covered at a polar couple, shows that the vectorial lines due to the excitation m.m.f. develop in the iron by crossing the air gap twice. 

Always assuming the iron permeability very big, the distribution of the rotor slots and of the excitation conductors and of the excitation conductors give rise to a sinusoidal run of the induction. 


Symbols of e.m.f. and currents: + = positive incoming,  · =  positive outgoing
Fig. 2

F =  armature e.m.f.    Ai =  interpolar axis
C =  excitation currents    Ap =  polar axis

 
INDUCED E.M.F.
CONDUCTOR INDUCED E.M.F.
In the normal operation the rotor has constant angular speed and drags in its movement the m.m.f. of the poles and the induction produced by them, turning synchronoulsy with it, i.e. with the same angular speed.
To evaluate the electromagnetical interactions between rotor and stator we can think that the first one is still and the second one turns in the opposite direction. With a good approximation we can think that the armature conductors, set in the stator, are at radial distance from the axis equal to the air gap radius rtr. 
In each one of them the motional e.m.f. is induced, called conductor e.m.f.

The e.m.f. of the armature conductors occupying different angular positions are different among them. 
Their spatial distribution is, as a matter of fact, similar to the induction one, hit has therefore a sinusoidal run, and it is fixed as to the rotor, i.e. it is synchronous with it; it presents a complete sinusoidal oscillation at each polar step 2τ and p complete oscillations along the air gap. 
The maximum values and the minimum ones are in correspondence of the polar axes while the e.m.f. are null on the interpolar axes.
By considering a reference system in agreement with the stator we have that each conductor e.m.f. is therefore sinusoidal in time.

EFFECTIVE VALUE OF THE CONDUCTOR E.M.F.
Of the conductor e.m.f. we are interested, rather than the maximum value, the effective value.
This effective value of the e.m.f. is equal to:

Eq. 1

where:

- f=ω/2π   is the frequency of the e.m.f. itself;
- Φ  is polar flux, i.e. the indiction flux flowing though an inductior pole;
- kf is the shape factor that for a sine wave assumes the value kf =- π /2 1,11;

 The previous formula is also valid if the distribution of the e.m.f. is not sinusoidal.
FREQUENCY AND ROTATION SPEED
We are used to expressing the rotation speed, instead of in radiants per second, in revolutions per minute n. 
The stiff bond existing between mechanical rotation speed of the rotor and angular pulsation of the sine waves of the armature e.m.f. implies that between frequency of the conductor e.m.f. and speed n there is the bond:

Eq. 2
as a matter of fact, in a machine having a single polar couple, the e.m.f. perform a complete oscillation at each turn and therefore so many oscillations per second as many turns per second. 
If the polar couples are more than one the oscillations of the sine waves are p for each turn.
If the operation frequency is imposed, for example f=50 Hz, the rotation speed is fixed by the number of polar couples, n=3000/p, being as lower as these are numerous, as Table 1 shows, the machines with a lower number of poles present the greater speeds and therefore they have rotors subject to the most intense centrifugal forces: it is especially for this reason that they are of smooth type, in massive steel and having a relatively small diameter.
Tab. 1
Rotation speed of the synchronous machines as a function of the number of polar couples, in the operation at f=50 Hz
p 1 2 3 4 5
n 3000 1500 1000  750  600
E.M.F. OF THE ARMATURE WINDINGS
The armature windings are connected in such a way to carry out three equal windings, having a triad of symmetrical e.m.f. (the armature e.m.f.).
The common effective value of the three e.m.f. are equal to:

Eq. 3
where:
- Nm it is the number of windings series connected in each winding,
- ka it is the winding factor, in practice an nondimensional geometrical coefficient lower than 1,
- Ni=2·Nm·Nc   it is the total number of armature conductors series connected in each armature winding.

The three windings make up the internal phases of the machine; their induced e.m.f. can be expressed as:

Eq. 4

 
NO-LOAD OPERATION
The no-load operation is carried out when the machine is excited (Ie¹0) and the armature currents are null. 
Therefore the polar flux only depends on Ie; as a consequence the effective value of the sinusoidal e.m.f. can be expressed as Ei=2·ka·kf·Ni·Φ0·f.
By considering the operation of the synchronous machine at imposed frequency f, the e.m.f. Ei0 can vary only when the polar flux Φ0 vary, and therefore of the excitation current Ie.
NO-LOAD CHARACTERISTIC
Having assumed the armature windings star connected, their e.m.f. coincide with the starry voltages, with effective value E0=Ei0, the linked, present between the couples of armature terminals, have effective value U0=Ö3·E0=Ö3·Ei0, as a function of the single Ie  too:

Eq. 5


T = Air gap characteristic,  V = No-load characteristic
Fig. 3 - Excitation characteristic

This function is called excitation characteristic or no-load characteristic of the synchronous machine and it presents the typical run shown in Figure 3. 
To obtain modest no-load voltages U0, small values of polar flux and of induction are enough; therefore the rotor and stator iron present a high permeability, so the reluctance of the magnetic circuit is essentially determined by the air gap and it is linear (air gap characteristic). 
On the contrary to obtain greater no-load voltages, we need greater polar flux and induction, that imply saturation conditions in the iron, by reducing its permeability; the contribution of the iron sections becomes therefore relevant to the total reluctance of the magnetic circuit, that becomes strongly non linear: increments of  Ie are necessary more than proportional as to the increments of U0. 
The machines generally present a rated voltage Un, corresponding to a condition of modest saturation, with deviation of the 15- 30% as to the air gap characteristic.
If the machine has operated at least once, the iron presents, with a null excitation current, a residual magnetism due to its hysteretical behaviour. Therefore for Ie=0 è there is a residual flux enough to produce a small no-load voltage U0. 
For this reason the characteristic doesn't start for the axes origin.
 
LOAD OPERATION
ARMATURE CURRENTS
The load operation is obtained by connecting the armature terminals to a mains in three-phase sinusoidal speed, so that at the terminal themselves there are sinusoidal currents; having assumed the star connected armature windings, their currents coincide with the ones at the terminals, whose effective value is I=Ii. 
The triad of the winding currents, or armature currents, can be expressed as:

Eq. 6
where φ, shows the phase displacement of each current as to the corresponding no-load e.m.f..
ARMATURE REACTION
The previous currents apply to the magnetic circuit a triad of m.m.f. that is added to the inductor m.m.f. and that is called armature reaction.
To show very synthetically its effect, it is better to remind that at the distribution of synchronous rotating conductor e.m.f. with the rotor corresponds the symmetrical triad of armature e.m.f.; in a similar way the symmetrical triad of armature currents gives rise, in the armature conductors, to a current distribution that moves in the armature conductors by keeping itself synchronous with the rotor and therefore even with the excitation m.m.f.

The same as the distribution of the conductor e.m.f. also the one of the armature currents has an alternating periodical run along the air gap, with periodicity equal to the polar step 2τ and it is turned late as to the distribution of the conductor e.m.f.
The rotor, the distribution of the armature currents, the induction distribution totally produced by excitation and armature currents and the distribution of the conductor e.m.f. are reciprocally fixed, i.e. all synchronous (from which the machine name): they turn at the same speed set by the frequency of the e.m.f. and of the sinusoidal currents at the armature terminals.
To evaluate the deformation of the load air gap induction, it is better to remind that the no-load conductor e.m.f. build up a distribution similar to the one of the air gap induction (Figure 1), having the maximum module in the polar axis and null on the interpolar axis; such a situation is shown in the two-pole machine schematized in Figure 4.

Fig. 4 - No-load operation Ic¹0, Ii=0

A = polar axis
B = slot e.m.f.
C = interpolar axis

Symbols of e.m.f. and currents:
+ = positive incoming
 · = positive outgoing

We can therefore consider the following cases.

Armature currents in phase with the e.m.f.: φo=0.

The armature currents are incoming in the upper pole and outgoing in the lower one  (side Figure 5.a): their distribution is therefore similar to the one of the e.m.f. and alone it would produce the induction Bi schematized in the figure, with the symmetry axes turned of an angle φo as to the ones of the no-load induction Bo, (Figura 4).
Therefore the two m.m.f. by summing give rise to a distribution of load induction B increased in the half poles II and IV and reduced in the I and in the III .
Armature currents in phase opposition with the e.m.f.: φo=π.

The armature currents are outgoing in the upper pole and incoming in the lower one (side Figure 5.b): their distribution is therefore opposite to the one of the conductor e.m.f. and alone it would produce the induction Bi with the symmetry axes turned of an angle φo as to the ones of the no-load induction Bo.
Therefore the two m.m.f. by adding give rise to a distribution of load induction B increased in the half poles I and III and reduced in the II and in the IV.
Armature currents late in quadrature on the e.m.f.: φo=π/2.

The armature currents are incoming in the two left half poles and outgoing in the right ones (side Figure 5.c): their distribution is late in  quadrature as to the conductor one and alone it would produce the induction Bi opposite in every half pole as to the no-load induction Bo.
Therefore the two m.m.f. by adding give rise to a distribution of load induction B decreased wherever there is the so-called demagnetizing effect.
Armature currents late in quadrature on the f.e.m.: φo= -π/2.

The armature currents are outgoing in the two left half polesand incoming in the right ones (side Figure 5.d): their distribution is early in  quadrature as to the one of the conductor e.m.f. and alone it would produce the induction Bi equiverse in each half pole with the no-load induction Bo.
Therefore the two m.m.f. by adding give rise to a distribution of load induction B increased wherever there is a magnetizing effect.
 
SHORT CIRCUIT OPERATION
The short circuit operation carries out when, with excited machine (Ie¹0), we short circuit the armature terminals, so to nullify the chained voltages and therefore the starry ones (E =0): therefore at these terminals we have the short circuit currents that are equal to:

Eq. 7
Where Zs is called synchronous impedance reactance, while Xs is called synchronous reactance. 
Being R<<Xs we can neglect its contribution in the impedance.
If in this case we consider the operation with a fixed frequency, since the no-load e.m.f. Eio, appearing at the numerator is a function of the excitation current Ie only, the same occurs for the short circuit current Icc. 
The run of its effective value Icc as a function of Ie builds up the short circuit characteristic of the synchronous machine (Figure 6).


Fig. 6 - Short circuit characteristic

Eq.7 shows that, other conditions being equal, the short circuit current is in practice late in quadrature as to the no-load e.m.f. (i.e. we have φocc=π/2) and in any case it gives rise to an armature reaction with  demagnetizing effect. 
For this reason, the excitation current being equal, the short circuited polar flux Φ is lower than the no-load one and the magnetic circuit behaviour keeps linear.

From Eq.7 we deduce that the module of the synchronous impedance is equal to Zs=Eio/Icc=Uo/ Icc and its run as a function of Ie we obtain by relating the abscissa of the curves of Uo and Icc (Figure 6): it is variable, owing to the magnetic circuit saturation; anyway in normal operation conditions and in a first approximation it can be considered as constant.