Chapter 1

Self-controlled Synchronous Motor: Principles of Function and Simplified Control Model ¹

1.1. Introduction

Every synchronous machine supplied by an electronic power converter and functioning with variable speed needs to ensure control of the converter that supplies it, in the sense that the rotational speed of its rotor imposes the frequency of the voltages and of the currents the converter provides to the stator windings. However the name “self-synchronous machine” usually refers to the case where the electronic power converter that supplies it is a thyristor bridge whose commutations are ensured by the voltages the machine develops at its terminals [BON 97, BUH 77, CHA 88, KLE 80, PAL 99]. This bridge is supplied with direct current (DC) via a smoothing inductance by an input converter that is itself usually a thyristor bridge connected to the alternating current (AC) mains. We thus obtain the principle diagram in Figure 1.1.

The synchronous machine can run as a motor or as a generator:

– to run as a motor, the input bridge works as a rectifier and supplies the intermediate DC circuit with electric power that is transferred to the synchronous machine by the bridge that supplies it, which works as an inverter.

– to run as a generator (braking operation) the bridge connected to the synchronous machine works as a rectifier and transfers the electric power generated by the synchronous machine to the network via the input bridge, which works as an inverter.

Operation with a positive or negative speed is possible since a change in rotation sense at the level of the synchronous machine corresponds to a permutation of the order of phase succession and to an inversion of the polarity of their voltages. This means that we only have to adapt the control sequence of the thyristors of the bridge that supplies it accordingly.

The assembly in Figure 1.1 therefore corresponds to a drive system that can work in the four quadrants of the torque-speed plane: running as a motor or a brake is possible in both rotational directions [BON 97, PAL 99].

Taking into account the remark made on the effect of the change in direction of rotation, for the sake of simplicity, all of the examples presented in this chapter will be done for a positive rotational speed and we will point out at the end of the study how to transpose the results in the case of a negative rotational speed.

1.2. Design aspects specific to the self-controlled synchronous machine

The self-controlled synchronous machine is used for strong to very strong power applications and systematically uses machines with field winding [BON 01]. This type of machine is characterized by the high value of cyclical inductance LCS from the stator windings, whose impedance at the nominal working frequency ranges from 1 to 1.5 per unit1 [CHA 83].

Such a cyclical inductance value is not compatible with supply by a thyristor bridge that is functioning as a current commutator, as it will attempt to impose pulsed currents in the windings. Using appropriate constructive measures, it is therefore advisable to decrease the apparent inductance of the windings towards the abrupt variations of the currents flowing into them. We obtain this result by providing the rotor with large dimensioned damper windings [BON 00, KLE 80].

1.3. Simplified model for the study of steady state operation

To establish this model we are going to consider that the machine has a cylindrical rotor, that there is no saturation effect, that the stator is three-phase with a Y connection of its windings and that the machine rotor circuits consist of a field winding and two damper windings in short-circuit: one (d-axis damper winding) aligned with the field winding; and the other (q-axis damper winding) in quadrature with the former. The field winding is assumed to be supplied by a current source If (see Figure 1.2). We also assume that all the mutual inductances between the windings of the stator and rotor sinusoidally vary according to the electric angular position of the rotor, em, which is equal to its mechanical angular position m multiplied by the number of pole pairs p of the machine. The modeling principles have been presented in [LOU 04, LOU 10].

In steady state, the electromotive forces (emfs) generated by the field winding in the phases of the armature are sinusoidal functions whose period T is set by the electric speed of the machine rotor :

If Mf represents the maximal value of the mutual inductance between the field winding and a phase of the stator, the magnitude E0 of these emfs is equal to:

[1.1] 

In steady state, the currents in the stator windings are periodic time functions with the same period T as the emfs.

We can represent these currents as a development in Fourier series, which leads us to distinguish the following in each:

– a fundamental component of angular frequency ;

– harmonic components of angular frequency, kpm, with k being an integer >1.

The fundamental components of the currents, with angular frequency, , generate an armature reaction rotating field whose rotational speed is synchronous with that of the rotor. For these components, the inductance of the stator windings is equal to the cyclical inductance, LCS.

The harmonic components with angular frequency, kpm, generate armature reaction rotating fields whose rotational speed with respect to the rotor is equal to (k±1)m according to whether these harmonics form inverse or direct systems. For these components, if we neglect the resistance of the damper windings, the apparent inductance of the stator windings is equal to:

[1.2] 

where:

– LR is the inductance of a damper winding; and

– M0 is equal to times the maximal value M of the mutual inductance between a damper winding and a stator winding.

Inductance LC has a value of around 15% of the cyclical inductance and constitutes what we call the transient inductance of the machine [BON 00, CHA 83].

The equivalent circuit of one phase of the machine, towards the fundamental component of the current flowing into it, is made up of the cyclical inductance and the resistance of the winding in series with the emf, which is generated therein by direct current If that flows in the field winding (see Figure 1.3).

In this diagram, e0x is the electromotive force induced by the field winding in phase x, x ∈ [ a, b, c]; i1x is the fundamental component of the current that flows into it; and u1x is the fundamental component of the voltage at its terminals.

If is the phasor representative of emfs e0x, and and are the phasors representative of currents and voltages i1x and u1x, we have:

[1.3] 

The equivalent circuit of a phase towards the harmonic of rank k of the current flowing into it is reduced to resistance RS in series with an inductance equal to LC, (see Figure 1.4).

We can combine the equivalent circuit, which is valid for the fundamental components of phase currents and voltages, with those valid for the harmonic components of these variables by rewriting equation [1.3] as follows:

[1.4] 

From this equation, we can in fact substitute a new equivalent circuit for the circuit in Figure 1.3, where an inductance LC and a resistance RS are put in series with an emf ecx with an angular frequency pm, whose representative phasor corresponds to the first two terms on the right-hand side of equation [1.4] (see Figure 1.5):

[1.5] 

With the voltage and current components of angular frequency kpm, the voltage source ecx of angular frequency pm appears to be a short-circuit due to the superposition principle, which is applicable since we assume that the machine is unsaturated. For current harmonics, the circuit in Figure 1.5 thus amounts to that in Figure 1.4. We can therefore apply the whole current ix (fundamental + harmonics) to it to find the total voltage ux at the phase terminals.

It is from this equivalent circuit − known as “a commutation electromotive force behind the commutation inductance” [BON 00, CHA 88], where we will disregard resistance RS − that we are going to study the functioning of the rectifier bridge that supplies the machine and from it how the machine functions.

1.4. Study of steady-state operation

To perform this study (see Figure 1.6), we are going to suppose, as indicated in the introduction, that the rotational speed m is positive, that the current I at the input of the bridge that supplies the machine is perfectly smooth and that the thyristor bridge works in binary commutation. We can refer to the classical study of rectifiers [SEG 92] but consider that the reference directions of the voltage U and phase currents ix have been inverted with respect to those normally used for the study of rectifier circuits. This is done in order to take into account the fact that, in the case of a self-controlled synchronous motor, normal operation corresponds to electric power supplied to the machine, i.e. going from the DC to the AC circuit, and hence to the operation as an inverter of the bridge that supplies the machine.

In these conditions, with the adopted reference directions, to ensure commutations we need to turn on each thyristor with a sufficient advance firing angle with respect to the point where:

– the commutation emf of the phase to which it is linked stops being smaller than that of the preceding phase if it belongs to thyristors with common anodes (+ input terminal);

– the commutation emf of the phase to which it is linked stops being greater than that of the preceding phase if it belongs to one of the thyristors with common cathodes (- input terminal).

The advance firing angle is sufficient if the current in the phase, whose thyristor is turned on, reaches the value I before its commutation emf is crossed with that of the preceding phase. The angle , between the point where the current in the phase takes the value I and the point of crossing of the emf, is called the extinction angle. We generally give this angle a value between /9 and /6.

According to the rotor electrical position angle em = pmt, the commutation emfs can be written as:

[1.6] 

where EC and C are two parameters linked to the field winding current If, to the rotational speed m of the rotor, to the extinction angle and to the current I imposed at the input of the bridge supplying the machine. We will later see how EC and C are linked to current If, m, and I.

If we consider the commutation of current I from phase c to phase a at the + input terminal as an example, this commutation starts at:

when thyristor Ta1 is turned on (see Figure 1.6).

From this point until:

where it ends, we can write:

[1.7] 

with:

From this we can infer the equation that rules the evolution of ia during the commutation; as dI/dt = 0 (since we assume that current I is constant), we obtain:

[1.8] 

By calculating the integral of this equation, we get the evolution of ia during the commutation.

Taking into account that:

we have:

[1.9] 

When:

current ia becomes equal to I and current ic, which is equal to I − ia, cancels out leading to the blocking of Tc1 and to the end of the transfer of current I from phase c to phase a.

We have:

[1.10] 

or, by introducing an overlap angle equal to − , which corresponds to the interval of simultaneous conduction of Ta1 and Tc1:

[1.11] 

Relationship [1.10] according to the values of Ec, m and I, allows us to fix the value we need to give to the advance firing angle α to have a given extinction angle :

[1.12] 

Voltage U, at the input of the thyristor bridge that supplies the machine, consists of, at each output terminal, three sinusoidal arches with a notch due to commutation at the beginning of each arch. The voltage at the – terminal is symmetrical to the voltage at the + terminal, with a shift of /3. The mean value of the voltage at the + terminal with respect to the neutral point of the stator windings (taken as a reference point for drawing the voltages in Figure 1.6) is half the mean value of voltage U. We obtain this mean value by considering the third of the period, for instance, which begins with the firing of thyristor Ta1:

[1.13] 

where the second term reflects the notch effect due to commutation and corresponds to the drop in voltage on Lc due to current ia during its growth from zero to I.

We obtain:

[1.14] 

since:

and:

By using relationship [1.10], equation [1.14] can be written as:

[1.15] 

or:

[1.16] 

By introducing angle , equation [1.16] becomes:

[1.17] 

If, as is the case in Figure 1.6, the conditions of functioning correspond to a low value, we can admit that:

– we will not make a significant error by assuming that cos/2 = 1, which leads to:

[1.18] 

– the amplitude of the peak value I1 of the fundamental components of currents ia, ib and ic is almost equal to that of pulsed currents of amplitude I and width 2/3:

[1.19] 

– phase shift (leading) to the fundamental components of currents ia, ib and ic with respect to voltages eca, ecb and ecc is almost equal to:

[1.20] 

– the fundamental components of voltages ua, ub and uc at the terminals of the phases of the machine are almost equal to the commutation emfs eca, ecb and ecc since the voltages at the terminals of the machine only depart from these emfs by the commutation notches whose width, equal to , is small.

1.5. Operation at nominal speed, voltage and current

Admitting that, for nominal conditions of operation, the overlap angle is small (we will check this a posteriori by taking into account the order of magnitude of the commutation inductance), on the basis of the approximations introduced at the end of the previous section, we can consider that the nominal operating point will be reached when the machine is running at its nominal speed mN if:

– we set direct current I to a value IN equal to , where I1N is the normal amplitude (i.e. the peak value) of the fundamental component of phase currents (see equation [1.19]). This occurs when we supply the input rectifier with a control circuit to control the current it delivers;

– we set the commutation emfs to a value ECN equal to U1N, where U1N is the nominal amplitude (i.e. the peak value) of phase voltages. This occurs via the effect of the field current If on the value E0 of emfs induced by the field winding.

By substituting in relationship [1.12] for I and U1N for Ec, we have:

[1.21] 

As LCpmN is roughly equal to 0.15 p.u. (see section 1.3), the second term in square brackets is about equal to 0.16. For an extinction angle of /9 (20°), relationship [1.21] gives a value of roughly /9 + /10 (38°) for the advance firing angle of the thyristors, N =N + N, and hence a N angle of roughly /10 (18°). This value is small, as initially assumed.

To find value E0N of E0, which will lead to the desired amplitude of Ec, we use relationship [1.5] and the vectorial diagram that is associated with it (see Figure 1.7) drawn by using this approximation:

We have:

[1.22] 

[1.23] 

Equation [1.22] sets amplitude E0N to give to emfs e0x and hence that to be given to current If.

Equation [1.23] gives the leading phase shift angle N of the commutation emfs on emfs e0x induced by the field winding. Since the emf e0a of phase a is in advance of /2 on the electric position em of the rotor, if we select a position for zero for which the flux induced by the field winding in phase a is maximum, the value of angle C that appears in equations [1.6] is equal to:

[1.24] 

at the nominal operating point.

We can then link the firing instants of the thyristors to the electrical position of the rotor. Thus, thyristor Ta1 must be fired for the following positions of the rotor:

The positions of the rotor corresponding to the firing of the other thyristors can easily be inferred from that of Ta1, considering that:

– the firing of Tb1 and Tc1 are respectively shifted by 2/3 and 4/3 with respect to those of Ta1;

– the firing of Ta2 is shifted by a- with respect to those of Ta1;

– the firing of Tb2 and Tc2 are respectively shifted by 2/3 and 4/3 with respect to those of Ta2.

The mean value of the electromagnetic torque produced by the machine is obtained by expressing that the mechanical power produced Cemm is equal to the active power received by sources e0x. Thus (see Figure 1.7), we have:

[1.25] 

Taking into account that:

equation [1.25] leads to:

[1.26] 

The mean value of DC voltage at the input of the bridge that supplies the motor Uav (see equation [1.18]) is equal to:

It can be written:

[1.27] 

as shown in Figure 1.7, or:

[1.28] 

By multiplying this voltage by the current I to find the power at the input of the bridge that supplies the machine, we again find the value of the power received by sources e0x. This is normal, since we have neglected all of the sources of loss.

The instantaneous torque, like the instantaneous voltage at the input of the bridge that supplies the machine, shows a ripple at six times the frequency of the phase voltage and currents.

1.6. Operation with a torque smaller than the nominal torque

If, at nominal speed mN and nominal field current current IfN, we decrease the value of current I without modifying the positions of the firing angles of the thyristors with respect to the position of the rotor, the amplitude of fundamental components of phase currents is decreased proportionally to the decrease in I. The phase shift (equal to N + N − /2) increases slightly with respect to the emfs produced by the field winding, insofar as the overlap angle slightly decreases. If we ignore this variation, which is small, we obtain a decrease in torque that is proportional to the decrease in current I, just like in a DC machine.

1.7. Operation with a speed below the nominal speed

Insofar as we can disregard the voltage drops on the winding resistances with respect to those on the winding inductances, a decrease in the speed of rotation at constant currents I and If and at constant advance firing angle leads to a decrease in voltages at the machine terminals proportional to the decrease in speed. Indeed, both the emfs induced by the field winding and the voltage drops on the stator winding inductances are proportional to the electric speed of the rotor; whereas the value of angle is unaffected by the variation in frequency.

However, when the rotational speed drops below a small proportion of the nominal speed (in practice from 5 to 10%, depending on the machine parameters) we can no longer disregard the effects on resistances. The emfs at the machine terminals become insufficient to properly ensure the commutations of current I from one machine phase to the other.

We then resort to the following artifact: each time we need to commute current I from one phase to the next (at the + or – terminal), we cancel out this current by altering the control of the input bridge. Canceling out current I leads to all the thyristors of the bridge that supplies the motor being blocked. We then control the firing of the thyristors through which current I has to circulate after commutation and, thanks to the input bridge current, we can restore I. To speed up this process, we generally provide smoothing inductance with a free wheeling thyristor that is triggered each time we want to cancel out the circulation of current I in the stator windings of the motor, see Figure 1.8 [CHA 88].

1.8. Running as a generator

When the synchronous machine is run as a generator there is an inversion of the direction of energy transfer between the machine and the DC circuit.

The power received by the machine becomes negative and corresponds to an energy transfer from the machine to the DC circuit. Since current I has to be positive due to the irreversibility of current in thyristors, the transition of the synchronous machine from being used as a motor to being used as a generator involves the mean value of voltage U at the input of the bridge that supplies the machine being inverted by action on the thyristors’ value of advance firing angle . At set values of I, If and m, by giving a value equal to − , we invert the value of the DC voltage. At the level of the windings, the transition of from a value below /2 to a value close to modifies accordingly the phase shift of the fundamental components of the currents with respect to the emfs induced by the field winding. The phase shift goes from a value below /2 to a value above /2, and thus changes the sign of cos(N + N − N/2).

1.9. Equivalence of a machine with a commutator and brushes

The connection that the thyristor bridge − which supplies the machine − establishes between the DC circuit and the statoric windings is equivalent to that established by a commutator with three segments 2/3 wide equipped with a pair of brushes of wide whose position s follows electric position θem of the rotor (see Figure 1.9). We will notice the presence of a diode in series with the pair of brushes on this figure. This takes the irreversibility of the current of the thyristor bridge into account.

If S is the position of the center of the brush linked to the + terminal and we match S = 0 with the center of the commutator segment linked to phase a, this brush links:

– phase a to the + terminal from S = −/3 −/2 to S = +/3 +/2;

– phase b to the + terminal from S = +/3 −/2 to S = +/2;

– phase c to the + terminal from S = − /2 to S = + 2/3 +/2.

From then on, it is enough for S to be linked to the electric position of the rotor by the following relationship:

[1.29] 

So the intervals during which each stator winding is linked to the + terminal by the commutator-brush system correspond to intervals during which the common anode thyristor connected to the winding is ON.

Similarly, since the brush linked to the – input terminal is shifted by π with respect to the brush linked to the + terminal, it connects each stator winding to the – terminal during the intervals corresponding to conduction of the common cathode thyristor to which the winding is connected.

If we neglect width of the overlap intervals, the brushes become punctual and the circuit seen from the access terminals of the brushes (armature circuit) contains at every instant two stator windings in series; on one turn of the brushes, we see six successive configurations, each corresponding to a commutation interval (the interval separating two successive changes in configuration, see Figure 1.10):

– from S = −/3 to S = 0, the armature is made of windings a and b in series;

– from S = 0 to S = /3, the armature is made of windings a and c in series;

– from S = /3 to S = 2/3, the armature is made of windings b and c in series;

– from S = 2/3 to , the armature is made of windings b and a in series;

– from S = to S = + /3, the armature is made of windings c and a in series;

– from S = + /3 to S = + 2/3, the armature is made of windings c and b in series.

Since the position of the brushes is linked to the electric position of the rotor by relationship [1.29], and since we assume that the overlap angle has a negligible value, the armature circuit is made of the series connection:

– of phases a and b for −/3 − ( + + /2) < em < −( + + /2);

– of phases a and c for − ( + + /2) < em < /3 − ( + + /2);

– of phases b and c for /3 − ( + + /2) < em < 2/3 − ( + + /2);

– of phases b and a for 2/3 − ( + + /2) < em < − ( + + /2);

– of phases c and a for − ( + + /2) < em < + /3 − ( + + /2);

– of phases c and b for + /3 − ( + + /2) < em < + 2/3 − ( + + /2).

This circuit possesses a resistance RA equal to 2RS and a self-inductance LA equal to 2LCS.

On the commutation interval corresponding to:

where the armature circuit is made of phases a and b in series, the mutual inductance MAf between this circuit and the field winding is equal to:

[1.30] 

namely:

[1.31] 

This mutual inductance varies between:

and:

Its mean value < MAf > on the given interval is equal to:

[1.32] 

On interval −( + + /2) < em < −(( + + /2) + /3, the mutual inductance MAf is equal to:

[1.33] 

namely:

[1.34] 

Mutual inductance MAf varies again between:

and:

with a mean value equal to:

and so on.

The calculation of the inductance value MAf shows that on a commutation interval, position A of the armature with respect to the field winding varies with speed from a position equal to − ( + + /2) − /6 at the beginning of the interval to − ( + + /2) + /6 at the end of the interval (see Figure 1.11).

At this moment, a change in configuration and hence armature circuit is imposed by the transition of one of the brushes from a commutator segment to the following (see Figure 1.10) and leads to the armature circuit returning to a position with respect to the field winding that is equal to − ( + + /2) − /6. This configuration change causes a transfer of current I from the armature circuit, which stops being in service, to the armature circuit that is activated. Insofar as we neglect duration of commutations, this transfer is assumed to take place instantaneously.

Similarly to what has been said on its position towards the field winding, during each commutation interval the position of the armature towards the d-axis damper winding varies with speed from − ( + + /2) − /6 to − ( + + /2) + /6 and its position towards q-axis damper winding changes from − ( + ) − /6 to − ( + ) + /6, since axis q is shifted by /2 with respect to axis d.

In a referential linked to the electric position of the rotor, we can consider the synchronous machine and the diode bridge that supplies it as a machine with a commutator and brushes. The circuit seen from the brushes (armature circuit) possesses a resistance RA equal to 2RS. It has a self-inductance LA equal to 2LCS and mutual inductances MAf, MAd and MAq, with the field and damper windings whose values depend on position A of the armature with respect to the field winding (axis d of the rotor). This position varies repeatedly, with speed , from − ( + + /2) − /6 to − ( + + /2) + /6.

1.10. Equations inferred from the theory of circuits with sliding contacts

By referring to the works of M. Garrido on the general equations of electrical machines [GAR 68, GAR 71, GAR 72, MAT 04], the description self-controlled synchronous machine function (see section 1.9) involves considering the behaviour of this machine to be that of a non-lineic circuit with sliding contacts and discontinuous commutation.

Without proving them, we are going to use the results of the studies presented in [GAR 72] to establish a macroscopic model of the behaviour of the self-controlled synchronous machine. This model, as indicated in the reference, is based on the transition to mean values of voltages and currents, taken on a commutation interval, but also the mean values of the machine parameters, which vary during the interval.

Thus, at the level of machine parameters we replace:

– the instantaneous position A of the armature, which varies from − ( + + /2) – /6 to − ( + + /2) + / 6 during a commutation interval, with its mean position during this interval:

[1.35] 

– the value of mutual inductance MAf between the armature and the field winding, which varies from to , with its mean value during this interval:

[1.36] 

– the value of mutual inductance MAd between the armature and the d-axis damper winding, which varies from to , with its mean value:

[1.37] 

– the value of mutual inductance MAq between the armature and the q-axis damper winding, which varies from to , with its mean value:

[1.38] 

At the level of the different voltages and currents, we replace their instantaneous values with a sliding mean taken during a time interval ΔTC corresponding to the duration of a commutation interval:

We thus obtain the following values:

[1.39] 

[1.40] 

[1.41] 

[1.42] 

[1.43] 

According to the average parameters and the macroscopic variables that have just been defined, by neglecting the terms of second order we obtain the following macroscopic electrical equations:

[1.44] 

[1.45] 

[1.46] 

[1.47] 

In these equations:

– Rf and Lf are the resistance and self-inductance of the field winding;

– RR and LR are the resistance and self-inductance of a damper winding;

– Mdf is the mutual inductance between the d-axis damper winding and the field winding.

The equivalent diagram in Figure 1.12 corresponds to equations [1.44] to [1.47].

In steady state <A>= −( + + /2), and . The equations become:

[1.48] 

[1.49] 

[1.50] 

[1.51] 

The power converted from electrical into mechanical energy Cemm is equal to the product of current <I> by the voltage produced by sliding (the terms in ) in the equation of the armature circuit.

By comparing equation [1.48], after having neglected the voltage drop RA < I >, with equation [1.28] of the “classic” study, where /2 = 0 to take into account the hypothesis of instantaneous commutation, we can notice the equivalence between the results given by the macroscopic model inferred from the theory of circuits with sliding contacts and the “classic” study based on the theory of rectification.

1.11. Evaluation of alternating currents circulating in steady state in the damper windings

The macroscopic model shows that, in steady state, the mean values of currents in the damper windings are zero. These currents therefore do not have a DC component. The variation in the position of the armature with respect to the field winding, and hence with respect to the damper windings within each commutation interval, is however going to induce AC currents in the damper windings.

If during each commutation interval we consider a time t' whose origin matches the beginning of the interval, we can write:

[1.52] 

[1.53] 

Taking into account these variations in mutual inductances and the fact that in steady state I and If are constant (see the hypotheses made on these currents in sections 1.3 and 1.4), the equations of the damper windings become [BUY 82]:

[1.54] 

[1.55] 

If we disregard the resistance RR of damper windings, we obtain:

[1.56] 

[1.57] 

The values of constants K and K' are set by the fact that the mean values of id and iq are zero:

[1.58] 

[1.59] 

Figure 1.13 gives the evolution as a function of t′ of and on a commutation interval. The abrupt variations these currents undergo at the changes in commutation intervals come from the transfers of current I from one armature circuit to the following, which take place at these times.

The AC currents that circulate in the damper windings are induced by the harmonic components of the currents circulating in the stator phases of the machine2 .

The calculation of these currents, which as we have just seen is easy when we use the commutator machine model, is mainly used to ensure dimensioning of the damper windings that takes into account the conditions of functioning peculiar to the supply of the machine with a thyristor bridge.

1.12. Transposition of the study to the case of a negative rotational speed

In the case where the rotational speed is negative, we need only invert the direction of reference adopted for position m of the rotor, so that the rotational speed becomes positive once again.

We only need to take into account that this change in direction of reference reverses the polarity of the emfs induced by the field winding (which corresponds to a phase shift of ) and swaps the role of phases b and c (see Figure 1.14). All of the results of the analysis we have performed in the case of a positive rotational speed are applicable.

1.13. Variant of the base assembly

For very high power applications, the base assembly that has just been studied is adapted using a machine equipped with two three-phase stator windings, mechanically shifted by 30 electrical degrees, one in relation to the other. Each stator winding system is supplied with an input rectifier via a DC smoothing inductance (see Figure 1.15) [BON 01].

With this solution, we decrease the torque ripple during normal operation but reduce the problem of canceling out torque during low-speed commutations (see section 1.7). Indeed, due to the 30° shift between the two winding systems, the commutations of the two systems are shifted by 1/12th of an electrical period. When one cancels out the current in a winding system, the current carries on circulating in the other and we retain 50% of the torque.

1.14. Conclusion

In this chapter, we have studied how the self-controlled synchronous motor functions. We started with a classical study based on a “transient emf in series with a commutation inductance”-type stator model. We have then shown the similarities existing between the functioning of the rectifier supplying the machine and that of a three-segment commutator equipped with a pair of brushes. We have ended up with a model of “non-lineic circuit with sliding contacts and discontinuous commutation”-type for stator windings and the bridge that supplies them. Using Garrido’s work, we have then built an equivalent model of the self-controlled synchronous machine that is similar to that of a DC machine.

The results of the classical study, as much as the model of the equivalent DC machine, can be used to design the control system and regulation of the self-controlled synchronous machine.

1.15. List of the main symbols used

A list of the main symbols used in this chapter and in Chapter 2 is given at the end of Chapter 2.

1.16. Bibliography

[BON 97] BONAL J., Entraînements Électriques à Vitesse Variable, Technique et Documentation, Paris, France, 1997.

[BON 00] BONAL J., Entraînements Électriques à Vitesse Variable: Interactions Convertisseur-réseau et Convertisseur-moteur-charge, Technique et Documentation, Paris, France, 2000.

[BON 01] BONAL J., Utilisation Inductrielle des Moteurs à Courant Alternatif, Technique et Documentation, Paris, France, 2001.

[BUH 77] BUHLER H., Einführung in die Theorie Geregelten Drehstromantriebe, Birkhauser, Bale, Switzerland, 1977.

[BUY 81] BUYSE H., THIRY J.M., GARRIDO M., “Une nouvelle méthode de commutation du courant d'induit d'une machine synchrone autopilotée”, Actes du Colloque Européen du Carrefour de la Force Motrice, pp. 127-133, Paris, France, December 10-11, 1981.

[BUY 82] BUYSE H., THIRY J.M., “Contribution to the transient analysis of the self controlled synchronous machine”, Proceedings of the International Conference on Electrical Machines ICEM'82, pp. 546-549, Budapest, Hungary, September 5-9, 1982.

[CHA 83] CHATELAIN J., Machine Électriques, Presses Polytechniques Romandes, Lausanne, Switzerland, 1983.

[CHA 88] CHAUPRADE R., MILSANT F., Electronique de Puissance. 2. Commande Moteurs à Courant Alternatif, Eyrolles, Paris, France, 1988.

[GAR 68] GARRIDO M.S., Contribution à la théorie dynamique des systèmes électromécaniques, Thesis, Institut National Polytechnique de Lorraine, Nancy, France, 1968.

[GAR 71] GARRIDO M.S., “Les équations générales des machines électriques déduites de l'électromagnétisme”, Revue E, vol. VI, no. 9, pp. 269-273, 1971.

[GAR 72] GARRIDO M.S., GUDEFIN E., “Equations des circuits à commutation non linéiques”, Revue E, vol. VII, no. 3, pp. 53-60, 1972.

[KLE 80] KLEINRATH H., Stromrichtegespeiste Drehfeldmaschine, Springer, Vienna, Austria, 1980.

[LOU 04] LOUIS J.-P., Modélisation des Machines Électriques en Vue de Leur Commande: Concepts Généraux, Hermès-Lavoisier (Traité EGEM, série Génie électrique), Paris, France, 2004.

[LOU 10] LOUIS J.-P., Commandes Classiques et Avancées des Actionneurs Synchrones, Hermès-Lavoisier (Traité EGEM, série Génie électrique), Paris, France, 2010.

[MAT 04] MATAGNE E., GARRIDO M., “Conversion électronique d'énergie: du phénomène physique à la modélisation dynamique” in : Modélisation des Machines Électriques en Vue de Leur Commande, LOUIS J-P (ed.), Hermès-Lavoisier (Traité EGEM), Paris, France, 2004.

[PAL 99] PALMA J., Accionamentos Electricos de Velocedade Variavel, Fundacao Colousta Gulbenkian, Lisbon, Portugal, 1999.

[SEG 92] SEGUIER G., Les Convertisseurs de l'Électronique de Puissance. 1. La Conversion Alternatif Continu, Technique et Documentation, Paris, France, 1992.

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1 Chapter written by Francis LABRIQUE and François BAUDART.

1 In the ‘per unit’ system, the values of statoric impedances at the nominal frequency are divided by the value of a base impedance (equal to the phase nominal voltage divided by the nominal current) in order to obtain dimensionless values.

2 If we disregard the duration of the commutations, the currents in the machine phases are almost rectangular and have harmonics of rank 6k+1, k being an integer, [SEG 92] and the resulting currents in the damper windings have angular frequencies equal to 6kpωm. The currents in the damper windings thus have a frequency equal to six times that of statoric currents, as shown in Figure 1.13.