Chapter 3

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Optimal Supplies and Synchronous Motors Torque Controls. Design in the d-q Reference Frame ¹

3.1. Introduction: on the controls designed in the Park d-q reference frame

The design and realization of torque control are crucial stages for the control of an electrical motor. In the case of a synchronous motor, it comes down to defining and regulating the stator current references. This explains the importance given to these questions in this book. Chapter 2 was devoted to the controls in the a-b-c reference frame, a very important case, but relatively heavy if we want to discuss it rigorously. This chapter tackles this same question in the d-q reference frame, where it is easier to design controls that are very efficient in terms of precision and dynamics. Indeed, as soon as the digital components facilitated the real time implementations of sophisticated algorithms, the synchronous motor models presented by the Park transformation became excellent tools for designing controls. The first advantage of this model is the fact that the saliency is easily taken into account, as shown by the torque expression as a function of the currents (see for example, equation [3.3] deduced from equation [1.73] in Chapter 1, section 1.5.3). Let us note however that, as educational preliminaries, we will often present the principle of the methods in the case of non-salient pole machines. Moreover, to impose the torque simply consists of regulating “direct” currents.

One of the greatest advantages of the control in the d-q reference frame compared to the controls in the a-b-c reference frame comes from the form of the “control model” deduced from the “dynamic model” of the machine. In particular, in the case of the controls in the a-b-c reference frame, the electromagnetic torque and the back-EMF are given by complicated non-linear expressions. In practice, we bypass this difficulty in Chapter 2 by designing the current and speed regulations, thanks to a simplifying hypothesis, i.e. that the “slow” (speed) and “fast” (currents) dynamics are largely decoupled.

Conversely, the dynamic model of a synchronous machine in the d-q reference frame is much simpler (see [3.1] to [3.3]). Therefore, the control model designed in this reference frame is also simple (see, for example Figures 3.3a and b). Thus, we can consider high dynamics speed controls, without these simplifying hypotheses, forcing us to limit these performances. Thus, we can also consider implementing very fast speed loops and impose on the transfer function poles close to those of the current loop. Indeed, if desired, we can easily take into account the couplings in the design of currents and speed controllers.

Moreover, in the d-q reference frame, the sinusoidal magnitudes are observed as “continuous” magnitudes. It is easy to evaluate their performances and, for example, a current controller with integral effect – simple to design and implant – is enough to erase the stator errors on the torque.

The price to pay is the necessity of carrying out the Park transformation twice (see below Figure 3.1, which will be completed by Figures 3.4 and 3.5) for the “self- control” of the supply voltages and the “reconstruction” of stator currents. But this is well carried out by the modern digital components.

In addition to the problems of pure control (regulations), the Park transformation helps us to determine the optimal currents (usually meaning the minimization of the Joule losses) imposing a given torque. This question has been tackled by analytical means in Chapter 2 for the non-salient pole machine (Chapter 2, sections 2.5 and 2.6). In this chapter, geometrical interpretations will give us very general tools for the case of salient pole machines, similarly to extensions of the Park transformation to the machines with cogging torque and non-sinusoidal field distributions (section 3.6).

3.2. Dynamic model (case of the salient pole machine and constant excitation)

Here we consider the case of constant excitation machines, as permanent magnet machines, and we write the dynamic model of the machine, by clearly emphasizing the fact that the currents are chosen as state variables:

[3.1]

[3.2]

[3.3]

Figure 3.1 gives a functional diagram well suited to the control in the d-q reference frame. It highlights the fact that equations [3.1] to [3.3] represent the “fast part” (“electric” variables of the motor represented on a gray tint background) and we also study the equations of the “slow part” (related to the “mechanical” variables, i.e. the speed and position).

To clarify some control laws, it is practical to rewrite model [3.2] under the following form:

[3.4]

With the vocabulary of automatics, we consider the terms e_{d _ tot} and e_{q _ tot} as “disturbances”.

3.3. First approach to determine of optimal current references (d-q reference frame)

Equation [3.2] shows that the only knowledge of the desired torque (in practice the reference torque) does not help us to determine the desired currents (in practice the reference currents i_{d _ ref} and i_{q _ ref} ), since there is only one equation and two unknowns. There is thus an infinity of solutions. We need an additional equation that can be given by an optimization. We encounter several optimization cases.

The case of simplification of the implementation, consists of taking again the d axis current equal to zero, optimizing the non-salient pole machines control (minimization of the Joule losses), i.e.:

[3.5]

This solution is often used for low power motors.

We also frequently (especially in high power) seek to minimize the Joule losses, which amounts to minimizing the current amplitude or its square root:

[3.6]

It seems to be “friendly” going into polar coordinates, which amounts to introducing the control angle α (see also [1.37], Chapter 1, section 1.4.2). Let us take the opportunity to note that the polar form is often very efficient, which is really common in classical electricity and electrical engineering (amplitude and phase). However, we often forget this possibility when using the Park components, usually defined in Cartesian coordinates.

Consequently we have:

[3.7]

In the following, we will leave out the index “ref” to lighten the formulas. Under these conditions, the torque expression (deduced from [3.3]) becomes:

[3.8]

And the optimization (minimization of the currents amplitude for a given torque) leads to cancelling out the partial derivative, which gives the equation:

[3.9]

From previous results, we deduce that the optimization is described by two equations, [3.8] and [3.9]. One of them, [3.9], teaches us that there is a relation between the optimal control angle and the optimal current amplitude. It has the form of a second degree equation ([3.10]), for which we must choose the physically realistic solution:

[3.10]

We note that, if the machine is with non-salient poles (or if L_d is really close to L_q), we find again the solutionaα_ref = -π/2 . The other equation, [3.8], is the relation between the torque reference and the references of the optimal amplitude and control angle: the resolution of these equations can be a priori digitally done and the results stored in a table. We can also admit that the fluctuation of the optimal control angle is low (see [STU 00a]) and we can admit a constant value for the control angle (its mean value or its value for the nominal torque). We then adjust only the current amplitude (by resolution of the second degree equation given by [3.8]). A possible functional diagram is given in Figure 3.2.

Section 3.6 will be devoted to a global approach of the current supply optimization of the salient poles synchronous motor, while including the cogging torque effect and the non-sinusoidal field distributions. This approach relies on a geometrical representation of the isotorque curves.

3.4. Determination of the current controls designed in the d-q reference frame

3.4.1. Principle of control by model inversion: example of the proportional controller with compensations

The easiest way to determine the current controls, which will carry out the torque control while optimizing the functioning, consists of designing an “input-output linearization” ([LEP 90], [GRE 97]). This linearization consists of imposing on the currents physically feasible dynamics (the currents are described by their Park components). For example, we can start with model [3.2] and impose on the currents 1^st order dynamics for the currents of the d and q axes:

[3.11]

The time constants τ_id andτ_id are chosen by the designers quite often as a function of the digital implementation possibilities (sampling frequency, etc., see Chapter 5 of this book) and of the constraints due to the environment (physical possibilities of the inverter). Model [3.4] immediately gives the voltages to impose in the d-q reference frame. This is the core of the control algorithm:

[3.12]

with:

[3.13]

We observe that this control contains two types of terms:

– the terms e_{d _ est} and e_{q _ est} contain “compensation” terms (by addition or substraction) of the ohmic drop and above all of “decoupling”, to make the two axes independent from one another. Let us note that in control laws [3.12] and [3.13], the values of i_d and of i_q actually used are values rebuilt by calculation, as we will see further (see Figure 3.5) when the indexes “est” (standing for “estimated”) are used in formulas [3.12] and [3.13];

– and a proportional regulation term, whose gains are imposed by the dynamics chosen by the designer:

[3.14]

This control can be illustrated by Figure 3.3a, which is remarkably simple. We will see that this “control model” requires however a quite complex material realization. Let us also note that the control defined by [3.12] must be considered as a “generic” version from which we can imagine all sorts of variants:

– neglect some terms ([BUY 91]), when we observe that they remain very small and negligible compared to the preserved terms;

– or replace in the compensation terms the measured or estimated (by calculation) values by the references ([ROB 95]). These references have two advantages: they are not noisy and they anticipate the future values of the variables, which is favorable to the dynamic performances;

– finally, replace the proportional controller with another type of controller: proportional-integral (PI), or integral-proportional (IP), or with phase lead, “RST”, integral-proportional-derivative (PID) or “with hysteresis” (or “fork”, etc.). A monograph ([MON 11]) largely discusses these questions ([NAA 11], [NAO 11], [PIE 11a], [PIE 11b]).

Design has been discussed in the d-q reference frame, which leads to a simple model, illustrated in Figure 3.3a. Moreover, this diagram shows that if the calculations are done quickly and without any errors (thus if e_{d_est} and e_{q_est} are really equal to e_d and e_q), it can be simplified even more, in order to become the diagram in Figure 3.3b¹.

3.4.2. Self-control

If the “control model” defined by Figures 3.3a and b is that simple, it is thanks to functions usually called “self-control”. We have seen that self-control consists of supplying the stator of the machine as a function of the rotor position p₁ · θ. In our case, the implementation in the d-q reference frame requires the presence - next to proper control algorithms - of operators which move it from the natural a-b-c reference frame to the d-q reference frame. This requires a position sensor² and specific operations:

- in order to go from the control signals to the voltages effectively imposed on the d and q axes, it is necessary to carry out a reverse operation, as it is described by the diagram in Figure 3.4: a rotation (whose angle is given by the measured position) to move in the set α-β reference frame, and then a Concordia transformation to be in the three-phase reference frame. The control signals are given by:

[3.15]

where G_{0 _ est} is the estimated value of the inverter gain. This value can vary as a function of the state of the environment (continuous bus). There is then a “robustness” problem that we will discuss in this chapter, but that must be taken into account for of the control design. The three-phase control signals are supported by the PWM controlling of the inverter, supplying the three-phase voltages, themselves effectively supplying the machine. Then, the machine equations are written in the Park reference frame. Figure 3.4 explicitly shows that the self-control (written with the Park transformation) is a (multiplying) compensation for the mathematical operations implicitly contained in the machine model. We can also say that the machine is described by strong nonlinear operations (the rotation) and that it is necessary to compensate for these operations. This is what is done by implementation of the Park transformation (mathematical formalization of the “self- control” operation);

– to carry out the current regulations in the d-q reference frame, it is necessary to move from the natural reference frame (where the measures are taken) to the Park reference frame, symbolized by the diagram in Figure 3.5.

Control in the Park reference frame (d-q) is thus more complex to carry out than control in the natural a-b-c reference frame, since there are two changes of reference frame to do (algorithms to implant in real time). But we find now on the market, digital components that easily carry out those operations. We then benefit from the good properties of the variables in the Park reference frame. In particular since they are constant in steady state, an integral effect regulator (such as the one we will show), eliminate the static errors.

3.4.3. Some properties of efficient current regulation

We assume in this section that the parameters intervening in the control defined by algorithm [3.12] (and [3.13]) are perfectly known, that the measures are precise and without delay and that the calculations are instantaneous and without errors. These conditions are ideal, but the consideration of all the sources of imperfections is not part of the subject discussed in this chapter (see Chapter 5 of this book). We will only consider a few of them in the following sections.

However, the ideal hypothesis made here allows us to examine the performances that the designers seek to effectively reach.

The operating conditions considered here are as follows: it is a torque control (and thus without speed regulation) obtained thanks to a current control. The current references are chosen under simplicity considerations (and lead to the comparison between cases):

[3.16]

We impose square waves torque references, alternately positive and negative, the motor being a no-load. The mechanical part is thus purely inertial (viscous frictions: f = 0 ). Figure 3.6 gives an example of transients as we can observe them in the d- q reference frame.

The i_d current is maintained exactly at its reference value (equal to zero) and i_q ,after a short transient, reaches its exact steady state. In the example shown in Figure 3.6, the torque is approximately piecewise constant. Consequently, the speed almost describes line segments (in reality exponential segments) and the position almost follows parabolic arcs.

We observe that the durations of the applied square waves enable the speed to change its sign. The dynamic in speed is thus only limited by the authorized current‘s amplitudes. These are imposed by different considerations related to the environment and to the chosen dimensioning for the different components. They are the maximal amplitudes of current and voltage that the motor and the converter(inverter) can tolerate (frequently, the inverter is the one imposing the most restrictive limits). The currents‘ property of being constant in steady state, when we observe them in the d-q reference frame, is obviously an important asset to judge the performances.

In the example of perfect control considered here, the dynamic of the current i_d and of the current i_q (the latter imposes, in practice the dynamics of the electromagnetic torque), is adjusted by the choice of the time constants τ_id and τ_iqof the chosen model in closed-loop [3.11]. In practice, we could not arbitrarily choose a value too small for these time constants. There are limits related to the used PWM frequency of the inverter, to the sampling frequency of the digital control elements and to the fulfilment speed of the calculations (questions discussed in Chapter 5).

Robustness considerations lead to the choice of values comparable to the values observed in open-loop. In the examples shown in Figure 3.6, we have arbitrarily chosen for time constants, the values of the “electrical time constants” of each axis:

[3.17]

We can also examine Figure 3.7a, giving an enlargement of a current transient, where we can also see the reconstitution of two-phase currents i_α and i_β . These are at “variable frequency” and we observe that when the speed is equal to zero, they are continuous (quasi-equal to zero, instantaneous frequency).

Moreover, when the speed is negative, the two-phase currents constitute an inverse system, that becomes direct when the speed is positive. Figure 3.7b shows that the response time at 95% is of about 10.6 ms, which we compare to 3 ·τ_iq= 10.3 ms: the current loop thus has the desired 1^st order behavior.

Figure 3.8 help us to observe the three-phase currents i_a , i_b and i_c during the transient. Once again the “variable frequency” currents are the “continuous type” (almost constant) at the instant of speed crosses through zero. The order of the phases is also reversed. We can also reconstitute (fictive) “reference three-phase currents” deduced from the Park references by:

[3.18]

Also it is interesting to compare the currents by the three-phase position derivatives of the flux, with respect to the position, since:

[3.19]

The flux position derivatives are thus proportional to the back-EMF, but by dividing them by Ω, we erase the speed variation effect (making the signals difficult to visually apprehend). We also have:

[3.20]

This flux position derivative with respect to the position is the pertinent variable, but physically we can only observe the back-EMF.

Figure 3.9 compares for the first phase the current reference, the current and the derivative of flux. We observe that the phase current, besides a transient (easy to interpret on the bottom curves giving the torque transient, thus in practice the current transient of the q axis), is identical to its reference.

Moreover, we verify this property, compatible with the study of the supply in the a-b-c reference frame: when the torque is positive, the current is in phase with the derivative of flux. This is a property due to the fact that we have chosen i_d __ref = 0and/_g __ref >0. When the torque is negative, these two signals are in phaseopposition, because i_q__ref < 0.

We find again the scalar product properties between the current vector and the derivative of flux vector (Chapter 2, section 2.5.6).

Thus, when the current control is very efficient, the motor properties are excellent. However, a problem comes from the control by inversion [3.12], where there is a proportional control and compensations. It is necessary to know the exact parameters so that the compensations are perfect. There is thus a problem of robustness.

3.4.4. Robustness problems of a proportional controller of the currents

It is sufficient to examine the effect of an error on a significant parameter. Figure 3.10 shows the effects of an error of 10% on the estimate of the excitation flux amplitude (such a situation is realistic: an accident leading to an over-intensity can demagnatize some types of magnets).

We then observe errors in the current i_q and in the electromagnetic torque. The trajectory of the speed is also sensibly altered.

Figure 3.11 takes the same signals as Figure 3.9. We observe that, according to the signs of some signals (torque and speed), the current amplitude can be smaller or larger than their reference. Thus, the electromagnetic torque actually obtained cannot be equal to the desired value (the reference).

Such a control is not considered to have good robustness. Several “robustification” strategies are possible. The most conventional, because it is very simple, consist of adding an integral effect to the controllers. We know that on “continuous” signals (constant in steady state), the integrating circuit will compensate for the errors.

We will examine this strategy in the following section.

3.5. New control by model inversion: example of an IP controller with compensations

3.5.1. Principle

In this section, we propose a solution to make the control more robust: an IP (integral-proportional) structure, whose principle for the d and q axes is given by Figure 3.12.

The model of this regulation is written with an extension adapting [3.11].

[3.21]

The model is completed by the equation of the integral controller:

[3.22]

We observe that these controls are naturally “decoupling”: the dynamics of the d axis are chosen independently of the dynamics of the q axis. This is possible thanks to control [3.12] compensation of (“decoupling”) all the disruptive effects, particularly those due to the couplings between the d and q axes.

We can write the transfer functions of the currents i_d and i_q :

[3.23]

This model will be the one helping us to choose the desired dynamics in closed loop and, consequently, to design the controller.

3.5.2. Performances of the IP regulations for current loops

We ignore the problems of implementation, of calculation time in real time (problems discussed in Chapter 5) and of robustness. The control in the d-q reference frame has excellent properties, and the synchronous motor, with its environment (position sensor, self-control, state rebuilder, inverter controlled in PWM) has properties very similar to those of the “reference motor”, the DC motor.

Indeed, if we choose the reference i_{d _ ref} = 0 (and if we assume that the regulation of the d axis is perfect), the model in the q axis (see the first equation of [3.2]) is identical to the equation of the armature of a direct current motor (therefore the indexes “DCM”):

[3.24]

[3.25]

with the equivalence:

[3.26]

We choose controllers by imposing a fast and damped transient, respecting the physical properties of the system (for robustness reasons).

For example, we impose on each transfer function of [3.23] two identical time constants:

[3.27]

[3.28]

Lastly, we choose that each time constant is equal to a fraction (here, half) of the “electric” time constant of its axis. Thus, we accelerate the system in closed-loop, while respecting the order of magnitude of the natural physical properties of the system; the aim being to obtain a compromise between the rapidity and a good robustness:

[3.29]

where:

[3.30]

It is then a specific case of a very conventional method, a “robust poles placement”. The design rules are very simple:

[3.31]

[3.32]

Figures 3.13a and b show an example of torque transient, with the form of currents in the d-q reference frame. We observe that after a transient imposed by the time constants chosen by the designer (in particular τ_iq , and here: τ_iq=τ_eq∕2), the currents join their steady state without static errors. We thus obtain another almost perfect regulation (except during the brief transient) and we would observe the same wave forms as those observed in the example illustrated in Figures 3.6 to 3.9. Therefore, we will not mention them again for this case.

The response time of the current is attainable by the analytical formula, giving the response for a 2^nd order system, having two equal time constants:

[3.33]

This formula gives a response time at 95% of about 8.4 ms (note: τ_iq= ms 1.7ms), which is verified on the enlargement given in Figure 3.13b.

This response time is of the same order of magnitude as the one we imposed on the proportional current loop (section 3.4.3).

3.5.3. Robustness of the IP controllers for the current loops

We will study the robustness of this regulation in two significant examples.

3.5.3.1. First example: effect of an error on the estimate of the excitation flux

We take again the example of an estimate error of 10% on the excitation flux φ_f . Figure 3.14 gives predictable results:

- thanks to the integrating circuit, the currents are exactly on the reference values without static errors;

-  but, as the current reference of the q axis is calculated by [3.16], i.e. I_{q_ref}(t) = C_ref (t)/p₁ .φf__est with a thus incorrect estimate value of the φf

coefficient, the obtained torque is not the desired torque. This error can only be corrected by a higher level regulation; concretely a speed regulation (these are studied in Chapter 4).

3.5.3.2. Second example: controller with compensation terms

We can plan to use a simple control, always an IP controller, but without the compensation terms, decoupling the d and q axes. In the control law, [3.12], we thus set out e_{d _ est} = e_{q _ est} = 0 . Figure 3.15 gives the obtained transients (without error on the coefficient estimate Φ_f , in order not to complicate the analysis): we observe that the d and q axes are no longer decoupled.

The i_d current leaves its reference value (equal to zero) for a moment. The transient observed on the d axis disturbs the torque value, which drops a bit during the transient. But on the whole, the system “functions correctly”, i.e. it remains fast, stable, without static error in steady state.

We can notice the efficiency of the integral term of the IP controller to correct the disturbances. However, we also verify the interest of compensations to obtain very good quality transients.

3.5.4. Conclusion on the control performances in the d-q reference frame

We observe that a conventional control, naturally robust thanks to its integral effect, as the IP regulation (and a PI regulation would have the same properties for these questions), imposes exactly the desired references in steady state. This property is independent of the speed (at least, as long as the requested frequencies are compatible with the used technical means: mainly the PWM frequency of the inverter and the sampling frequency of the digital controllers). There are thus no “dropping” characteristics, like the one presented and discussed in Chapter 2 devoted to controls in the a-b-c reference frame (see section 2.4). This explains the great popularity of the controls in this reference frame and the reason why the manufacturers of digital control devices of electric motors have designed components specially dedicated to this type of control.

3.6. Optimal supply of the salient poles synchronous motors; geometrical approach of the isotorque curves

3.6.1. General information: a general approach with the torque surfaces

We have seen in Chapter 2 a very comprehensive study on the search for optimal currents in the case of non-salient pole machines by an analytical method (sections 2.5 and 2.6). In this chapter (section 3.3), we have given an elementary approach to the search for the optimal supply of a salient poles synchronous motor. We will detail this question by exposing a powerful approach, based on the geometrical representation of the isotorque curves. This method relies on the most general expression of the torque, given in Chapter 1 (section 1.3.5, equation [1.12], see also [ADN 91, CAR 95, STU 99a, STU 99b, STU 99c, REK 91, PET 00, MAR 91, LIP 91, MAR 93, COL 94]). We recall this expression:

[3.34]

In the expression, three terms appear with very different meanings: the first term is due to the rotor saliency, the second is created by field excitation and the last is the cogging torque. Compared to the studies of Chapter 2, the saliency term is the one complicating the study the most. The presented method relies on a specific formalism to represent the matrix of stator inductances. By taking into account the machines symmetries (see [HOL 96]), and in all generalities (including the non- sinusoidal field distributions), we can note it as follows:

[3.35]

The different terms of the matrix (L_ss (p₁·θ)) are reachable by measures at the machine stator. But the torque equation [3.34] involves the derivative of this matrix. Thus, we introduce a matrix (A (p₁ · θ)), [3.36], defined by its three components f, g, h ([3.37]):

[3.36]

with:

[3.37]

In a 3D Cartesian reference frame, where the axes represent the stator currents i_a, i_b and i_c, and for a given position p₁ · θ , it is possible to make a geometrical representation of the quantity represented by the torque of the synchronous machine.

Indeed, as the torque is given by a quadratic form, the set of points M of coordinates i_a, i_b and i_c, producing a given torque C , is an isotorque surface Υ.

In the general case, it is a quadric, but it can be deteriorated if there is no saliency, or for specific values of p₁· θ . For example, we have seen in section 2.5.6 that for a non-salient pole machine, this surface Υ is a plane P (Figure 2.21).

For a salient pole machine, the nature of the surface Υ will be given by the study of the eigenvalues of the matrix The signature of the quadratic form is the torque (p‘, q‘): p‘ is the number of strictly positive eigenvalues of d(L_ss(2p₁ -9))/d9, and q‘ is the number of strictly negative eigenvalues. The order of d (L_ss (2p₁ ·θ))/dθ is then p‘ +q‘.

In the situations studied here, we encounter the case of 2^nd or 3^rd order (signatures (2,0) or (0,2), signatures (2,1) or (1,2)).

As an example (Figure 3.25),we will give the case of a non-excited synchronous machine with variable reluctance, for which we have represented the three eigenvalues.

The quadric surface evolves between a hyperboloid (when two eigenvalues are of the same sign) and a hyperbolic cylinder (when one of the eigenvalues is equal to zero). We summarize in Table 3.1 the different possibilities.

Thereafter, we will only consider the situations for which the motor neutral is not connected, which imposes a zero-sequence equal to zero component current i₀. The three currents i_a, i_b andi_c then verify i_a +i_b +i_c = 0 : this is the equation of the plane H₁ represented in Figure 2.21. The interesting operating points are then located on the intersection of Υ and H₁. This is a family of curves Γ, that we call isotorque curves, whose Cartesian equation is expressed as a function of the currents i_α and i_β. The parameters of this family of curves are defined by (-C_{em_1_des} +C_d (P₁· θ)) and p₁·θ (see [3.38]).

[3.38]

From [3.38] and [3.36], we obtain equation [3.39] of the curve Γ For a fixed position p₁·θ , the equation of the conic section here can be a straight line when is a plane, in the case of non-salient pole machines, or a hyperbola in the other cases (for salient pole machines, excited or not, with or without coupling between phases):

[3.39]

When ( C-C_d ( p₁ · θ)) evolves, we obtain a family of curves that can be interpreted as the level curves of a surface ς , representing the evolution of a function z(i_α,,i_β, p₁·θ). The quantity z(i_α,,i_β, p₁·θ) is homogeneous to a torque and is represented as an altitude in the direction of the zero-sequence component axis o. The plane H₁ is at an altitude equal to zero:

[3.40]

where the current I₀ is a current with an amplitude of 1 ampere.

In a 3D Cartesian reference frame, where the axes represent the currents i_a,i_β andi_o, the surface ς of equation [3.40] is, in the most general case of a salient pole machine, a hyperbolic paraboloid. Figure 3.16 represents an example of this surface for an excited synchronous machine with salient poles and for a position p₁ ·θ = π/3. Figure 3.16 shows the axes a, band c, α, β and o, d and q . The surface ς is represented in “wire mesh”; the level curves of ς form a family of hyperbolas Γ, projected in the plane i_a + i_b + i_c = 0.

There are thus curves corresponding to the positive torques and those corresponding to the negative torques. In the examples presented further, for readability reasons, we will represent the absolute value of the function z (see [3.40]). Thus, the surface will be “straightened” so that the lower part of the surface goes above plane H1 of the equation i_o =0 to be more visible.

If the machine has non-salient poles, the surface ς becomes an equation of a plane [3.41]:

[3.41]

It goes through the origin of the reference frame and is generated by the vectors [3.42]:

[3.42]

The line Γ then has (u_αβo) = (-e_β e_α o) as a direction vector, and is superimposed with the intersection line between P and H₁ of Figure 2.21. However, the planes P and ς do not have anything simple in common. We will show (sections 3.6.5, 3.6.7 and 3.6.10) how the surfaces ς turn as a function of the position p₁ ·θ.

We will examine the preliminaries in the following sections (sections 3.6.2 and 3.6.3) the relatively simple case of the non-salient pole machines (where the ς surfaces are planes), then the more general cases of excited salient pole machines (section 3.6.6) and of non-excited salient pole machines (section 3.6.8).

These properties will help to establish the expressions of the optimal supply currents, giving the desired torque and compensating for the cogging torque, while minimizing the Joule losses.

3.6.2. Preliminaries 1: case of synchronous machines, with magnets, with non- salient poles and with spatial distribution of the sinusoidal field

The approach will first be clarified in a simple case, the case of the non-salient pole machine where the matrix A(p₁·θ) is equal to zero.

There are thus no second degree terms in [3.39]. Hence, Γ is a line of equation [3.43] ([CHO 93]):

[3.43]

with (see also Chapter 1, section 1.5.2 and Chapter 2, section 2.6.10):

[3.44]

[3.45]

This line (Figure 3.17) rotates in the reference frame (α, β). Hence, the idea is to be located in the rotating reference frame (d,q) by Park transformation (see Chapter 1, section 1.5.3, equations [1.66] and [1.67]) defined here by the rotation of the angle p₁ · θ : (i₂)=(i_α i_β)^t=P(p₁ · θ)·(i_dq).

In the Park reference frame (i.e. in the rotor reference frame), the torque verifies equation [3.46]:

[3.46]

When C_em__{1_des} -C_d(p₁·θ) = 0 , the point H is superimposed with the origin.

The set of points M giving a torque equal to zero (due to the currents only, remains the cogging torque) is superimposed with the direct axis, which justifies that the back-EMF (or the derivative of flux) is really in the direction of the quadratic axis.

To produce a torque, it is necessary to have a component of the stator current in quadrature with the component of the rotor flux or its equivalent. It is necessary to have a component of the stator current in phase with the induced back-EMF.

The examination of Figure 3.17 shows that the first optimal operating point (minimization by Joule losses) is point H. Its coordinates are given by [3.47]. If the desired torque increases, the isotorque curve moves parallel to itself and point H moves away from the origin:

[3.47]

From these components, we can write the expression of the corresponding three- phase currents (in a matrix form [3.48] and in a detailed form[3.49]):

[3.48]

[3.49]

We thus find the optimal currents given in sections 2.6.10 and 2.2.1 (where the cogging torque is already compensated).

3.6.3. Preliminaries 2: case of synchronous machines with magnets, with non- salient poles and with spatial distribution of a non-sinusoidal field - first extension of the Park transformation

This geometrical method helps us to find, by generalizing it, (introduction of cogging torque) the extension of Park transformation to the non-salient pole machines with non-sinusoidal field distribution ([GRE 94], [YAL 94], [GRE 95], [GRE 98], [BOD 99], [MAT 93]), presented in Chapter 1 (section 1.6). For this machine, the matrix A(p₁ ·θ) is equal to zero and Γ is a line (Figure 3.18) of the equation [3.50]:

[3.50]

Examination of Figure 3.8 suggests a change of reference frame defined by formula [3.51]:

[3.51]

We can then rewrite the expression of the isotorque curve ([3.52]):

[3.52]

This method helps us to immediately define the optimal point (minimization of the Joule losses):

[3.53]

As in Chapter 1 (section 1.6, [GRE 94]), we introduce a rotation angle (p₁·θ), which is added to the conventional angle μ(p₁·θ), defined by:

[3.54]

With this transformation, we can write the form of the optimal currents ([3.55] and [3.56]) with recognition of the cogging torque, a solution identical to the expression already given by (i_{opt _ 2,3} ) in Chapter 2 (section 2.5.5, equation [2.80], the case where the zero-sequence component of the current was equal to zero):

[3.55]

[3.56]

These results are illustrated by the wave forms in Figure 3.19.

This is the case of the trapezoidal field distribution machine defined in Chapter 1 (section 1.4.3.1).

We observe (Figure 3.19d) that the optimal current is sensibly different from the conventional square wave current.

3.6.4. Remark: analogy with the p-q theory

It is interesting to notice the analogy of this approach with the p-q theory according to Akagi ([AKA 93], [YAL 94]). For a direct three-phase system, the instantaneous powers, active p, and reactive q, are defined by [3.57]:

[3.57]

The inversion of equation [3.58] gives us:

[3.58]

where i_αp is the active instantaneous current on the α, axis, i_αq is the reactive instantaneous current on the α, axis, i_βp is the active instantaneous current on the β axis and i_βq is the reactive instantaneous current on the β axis. For the motor, we observe that the current i_d is then the image of the reactive power, and the current i_q is the image of the active power.

3.6.5. 3D visualization, case of non-salient pole machines

The surface ς is a plane of equation ([3.59]).

[3.59]

∑ goes through the origin and z (i_α, /_β, p₁ ·θ) > 0 when i_q > 0 . The intersection of ∑ and of H₁ is the direct axis, directed in relation to the α, axis of an angle worth p₁ ·θ +μ (p₁ ·θ). This angle p₁ ·θ +μ (p₁ ·θ) is read in true value in the plane H₁. In addition, the angle between ∑ and H₁ is worth:

[3.60]

This angle is constant if the back-EMF module is also constant, which is the case for non-salient pole machines with sinusoidal field distribution. When the rotor rotates, the plane ς also rotates around the axis o (direction of the zero-sequence component) by forming an angle ξ(p₁ ·θ) between the axis and H₁. We can observe this fact in Figures 3.20a and b.

3.6.6. Generalization to the salient pole machines: case of synchronous magnet machines with sinusoidal field distribution

For the salient pole machines, we gave in Chapter 1 (section 1.3.5, equation [1.5]) the matrix expression of the stator inductances (L_ss2 (9)). We gave factorized forms of it (section 1.5.1, equations [1.48]), first with the inductances L_cs and L_s2, then with the inductances L_d and L_q. These factorized forms are well adapted to define the equations in the two-phase “α-β” reference frame. This is in particular the case for the equation relative to the stator fluxes (see the factorized equations [1.57] and [1.58] of Chapter 1, section 1.5.2), for which we give here the detailed form in [3.61]:

[3.61]

We can deduce from this two torque expressions: a matrix form [3.62] and a detailed form [3.63], that we immediately rewrite under an obvious lighten form [3.64] (the comparison of [3.63] and [3.64] immediately gives the meaning of the terms noted a, b, c, d, e). Form [3.64] will be used to write the following results:

[3.62]

[3.63]

[3.64]

We can recognize (Figure 3.21) an equilateral hyperbola whose coordinates of the center O‘ are given by formulas [3.65]:

[3.65]

If we seek to erase the rectangle term (in i_a · i_β), we must locate ourselves in a reference frame (X,Y) rotated compared to the reference frame (“ α, β ”) of an angle Υ so that:

In this new reference frame, the equation related to the torque has as an expression:

[3.67]

We find again the coordinates of the Park reference frame by an additional rotation of π/4, [3.68].

[3.68]

The graphic representation in Figure 3.21 leads to the immediate determination of optimal conventional operating points:

-  the point H, corresponding to the operating point with minimal Joule losses [DIA 93];

-  the point K, corresponding to the operating point so that the torque is proportional to the current i_q.

3.6.7. Visualization: case of an excited synchronous machine with salient poles

The surface ∑ is a hyperbolic paraboloid, containing two lines belonging to the plane H₁. The first is superimposed with the direct axis, the second is parallel to the quadrature axis but shifted compared to the origin of the quantity OO‘ due to the excitation. The positive part of z(i_α,i_β) is located between the d axis and the axis parallel to q going through O‘ (Figure 3.21). When the rotor rotates, the surface Σ. is distorted, by simple rotation if the machine is “sinusoidal” and in a more complex way by combining a rotation and a dilatation if it is not.

The case presented in Figure 3.22 is the case of the “sinusoidal” machine: we see ς rotating around the o axis and ∑ remaining on the whole invariant. The angle p₁ ·θ is read in full-scale in the plane H₁.

3.6.8. Case of a reluctance synchronous machine

3.6.8.1. Case of machines with mutual inductances

In the case of a reluctance synchronous machine, the excitation term is equal to zero. Thus, the hyperbola center is superimposed with O (Figure 3.23). The distance from O to the hyperbola then increases, which means that the necessary current is greater for the same requested torque (which is normal, since there is no excitation). The optimal point is i_d = i_q , [CHI 91].

3.6.8.2. Case of three-phase machines with mutual inductances equal to zero

For specific construction reasons, some machines have all their mutual inductances equal to zero ([NAG 98], [GIR 97], [FLI 95]). Then, the matrix of the stator inductances has the expression [3.69]:

[3.69]

In that case, the isotorque curve Γ remains a hyperbola centered on the origin and its equation is given by [3.70]:

[3.70]

It is necessary to study it in the following section and we obtain optimal currents [3.71]:

[3.71]

By a different approach, we will find a complete study of this type of machine in [NAG 98].

3.6.9. Case of synchronous machines with variable reluctance and non-sinusoidal spatial field distribution: second extension of the Park transformation

We summarily presented in Chapter 1 (section 1.6), with the help of an algebraic method, the first extension of the Park transformation, limited to the non-salient pole machines with cogging torque. The geometrical approach developed here lead us to find again this transformation extended to the non-salient pole machines with cogging torque (section 3.6.3). We will show that this geometrical method can be extended to salient pole machines, with non-sinusoidal field distribution and cogging torque (see [STU 99a, STU 99b, STU 99c, STU 00b]). Our method is based on the geometrical representation of the isotorque curves and we write the equation of the torque to be inverted as a function of the two-phase currents (“ α, β ”reference frame), under the form [3.72]. Let us recall that the matrix (A(p₁ ·θ)) has been defined in [3.36], with the expressions of the functions f, g and h given in [3.37]:

[3.72]

We give a factorized form [3.76] (form similar to a diagonalization, but less restrictive) that uses two functions S(p₁·θ) and D(p₁·θ) defined by [3.73]:

[3.73]

It is easy to verify that these two functions are related to the eigenvalues λ₁ and λ₂ of the matrix A ( p ₁ ·θ) (with:λ₂ > λ) by the properties [3.74]:

[3.74]

Then, we can introduce an angle α (p₁ ·θ) verifying:

[3.75]

indeed, the matrix A(p₁ ·θ) can now be factorized in:

[3.76]

The examination of the expression of the torque [3.72] and of the factorization [3.76] shows us the interest in changing the reference frame that is a new reference frame (here noted “X-Y ”) defined by [3.77]. We are in a reference frame linked to the conic focal axis. Let us note that the Park reference frame is deduced by an additional rotation of π/4:

[3.77]

And then the torque expression is simplified in [3.78]:

[3.78]

Figure 3.24 represents an isotorque curve in the extended Park reference frame (note: in the sinusoidal case, we have α.(p₁ ·θ) + -π/4 = p₁ ·θ ). In this figure, we observe that the optimal operating points are equally points K and K‘ whose coordinates are:

[3.79]

From which we can deduce the optimal currents expressions in the natural a-b-c reference frame ([3.80] and [3.81]).

[3.80]

[3.81]

Figure 3.25 gives an example of the wave forms (see appendix section 3.8.1, [STU 01]).

To conclude, let us note that extensions can be given for saturated machines ([STU 01, STU 03, MAD 03, MAD 04]) from experimental data of isotorque curves.

3.6.10. Visualization: torque surface of a reluctance synchronous machine

The surface Σ is a hyperbolic paraboloid, containing two lines belonging to the plane H₁. The first is superimposed with the d axis and the second is superimposed

with the q axis. The positive part of z(i_α, i_β,p₁ ·θ) is between the d axis and the q

axis.

When the rotor rotates, the surface ? is distorted, by simple rotation if the machine is “sinusoidal” and in a more complex way combining a rotation and a similarity if not.

The case presented in Figure 3.26 is the case of the “sinusoidal” machine. We see Σ rotates around the axis o and on the whole invariant. The angle p₁ ·θ is read full-scale in the plane H₁.

3.7. Conclusion

In this chapter, we presented torque control methods of salient pole synchronous motors. We also exposed general determination methods of their optimal supply for sinusoidal and non-sinusoidal field distribution machines, with salient and non- salient poles. For non-salient pole machines, we again found some results from Chapter 2 and we gave a first extension of the Park transformation via the angle p₁ ·θ +μ (p₁ ·θ). For salient pole machines, the Park transformation is a very powerful tool to study current controls (sections 3.4 and 3.5), but also to determine the optimal supplies (sections 3.3 and 3.6). We have shown that a second extension of this transformation is possible via the angle α (p ₁ ·θ) + π/4 , in particular for non- sinusoidal field distribution cases. In all cases, we have compensated the cogging torque.

Torque control in the d-q Park reference frame authorizes remarkably simple and very powerful design of current loops: the dynamic range is easily adjustable, and for a sinusoidal distribution machine, a simple controller with integral effect eliminates the static errors. For non-sinusoidal distribution machines, extensions are necessary, but resorting to extended Park transformations always facilitates the implementations.

3.8. Appendices

3.8.1. Numerical parameters values

The dynamics examples presented in this chapter concern a motor, whose parameters have been given in appendix section 1.8.1 of Chapter 1.

For the example of section 3.6.6 (Figure 3.23), the inductances are defined by the following expressions:

[3.82]

[3.83]

3.8.2. Nomenclature and notations

We have given in the main notations the appendices of Chapters 1 (section 1.8.2) and 2 (section 2.8.2). We add here notations specific to Chapter 3.

3.8.2.1. Indexes

Some indexes aim to specify the function of some variables:

– “des” and “ref”: “desired” values normally helping to define “references”;

– “mea” and “est”: some important variables are not directly attainable (especially the case of the Park components) and we only have “measures” or “estimates”, that can be wrong (necessary distinction for the robustness studies);

– X, Y: indexes for the references frames linked to the focal axis of the conics.

3.8.2.2. Variables and parameters of the controls in the Park d-q reference frame

The acronyms, parameters and variables used for the controls in d-q:

– P, IP: proportional controller, integral-proportional controller;

– e_{d_tot,} e_q__{_tot}, e_{d_est,}e_{d_est} : total disturbance terms and their estimated values (for the controls) in the electric d and q axes equations;

– τ_ed and τ_eq: “electric” time constants of d and q axes;

– i_{d_ref} and i_{q_ref}; i_{dq_ref} and α_ref: current references, in Cartesian or polar coordinates (modular, argument). When these references are of “continuous” type (constant by interval), we can use capitals: I_{d_ref}and I_{q_ref};I_{dq_ref};

– k_d and k_q: gain of the proportional controllers of the d and q axes regulations (P or IP case); τ_yd and τ_yq time constants of the integrating circuits (IP case);

– τ_id and τ_iq: time constants chosen to impose the currents dynamics for the proportional gain, in the P case; τ_ad and τ_aq: ibid in the IP case and λ_dq: adjustment parameter of the chosen time constants.

3.8.2.3. Specific functions for optimizations

– f, g, h: function of p₁ ·θ for the matrix writing (A (p₁ ·θ)) (section 3.6.1);

– a, b, c, d, e: functions of p₁ · θ for the optimization of section 3.6.5 (these functions are defined by the comparison of [3.63] and [3.64]);

– S and D: functions of p₁ ·θ for the optimization of section 3.6.6;

– μ(p₁·θ), α(p ₁ ·θ), ?(p₁·θ): rotation angles used for various extensions of the Park transformation;

– ∑ : hyperbolic paraboloid whose level curves are Γ isotorques; ? : isotorque surface expressed in the a-b-c reference frame;

– z(i_α,i_β): surface equation Σ .

3.9. Bibliography

Most of the bibliography was given at the end of Chapters 1 and 2. We give here only a few reminders and specific complements to this chapter.

[ADN 91] ADNANES A.K., “Torque analysis of permanent magnet synchronous motors”, Power Electronics Specialist Conference, 22^nd Annual IEEE, Cambridge, Massachusetts, United States, p 695-701, 1991.

[AKA 93] AKAGI H., NABAE A., “The p-q theory in the three-phase systems under non sinusoidal conditions”, ETEP, vol. 3, p. 314-326, January 1993.

[BOD 99] BODIN F., Commande des moteurs synchrones à aimants permanents à force contre électromotrice non sinusoïdale. Minimisation des harmoniques de couple et commande sans capteur mécanique, PhD Thesis of the Nantes University, November 1999.

[BUY 91] BUYSE H., LABRIQUE F., ROBYNS B., DENTE P., “Digital field oriented control of a PPM motor using a simplified strategy for controlling the Park components of the stator surrents”, in J. Robert and R. Le Doeuff (eds), Modelling and Control of Electrical Machines, New Trends, Netherlands, p. 203-207, 1991.

[CAR 95] CARON J.-P., HAUTIER J.-P., “Modélisation et commande de la machine synchrone”, 3EI 95 / SEE, Les moteurs synchrones et leurs applications industrielles, Gif- sur-Yvette, 30-31 March 1995.

[CHI 91] CHIBA A., NAKAMURA F., FUKAO T., RAHMAN A., “Inductances of cageless reluctance-synchronous machines having non-sinusoidal space distributions”, IEEE Trans. on Industry Applications, vol. 27, no. 1, p. 44-51, January-February 1991.

[COL 94] COLAMARTINO F., MARCHAND C., RAZEK A., “Considerations of non-sinusoidal field distribution in a permanent magnet synchronous motor control”, 5^th Conference of Power Electronic Variable Speed Drives, p. 508-513, 1994.

[CON 71] CONCORDIA C., Synchronous Machines, John Wiley and Sons, London, UK, 1971.

[DIA 93] DIALLO D., CHAMPENOIS G., DAFFIX H., “On-line adjustment of the torque angle for the torque optimisation and diagnosis purposes in anisotropics PSMS”, IECON, Hawaii, United States, p. 887-890, 1993.

[FLI 95] FLIELLER D., Etude de systems électrotechniques non linéaires échantillonnés. Application à la modélisation et à la commande de machine à réluctance variable à double saillance, PhD Thesis, Pierre and Marie Curie University, 27 October 1995.

[GIR 97] GIRARDIN M., “Torque control of a three-phase direct reluctance motor”, ISIE‘97, Guimarases, Portugal, 1997.

[GRE 94] GRENIER D. D., Modélisation et stratégies de commande de machines synchrones à aimants permanents montés en surface, PhD Thesis, ENS Cachan, September 1994.

[GRE 95] GRENIER D., LOIUS J.-P., “Modeling for control of a non-sinewave permanent magnet synchronous drives by extending Park‘s transformation”, Mathematics and Computers in Simulation, no. 38, p. 445-452, 1995.

[GRE 97] GRENIER D., DESSAINT L.-A., AKHRIF O., BONNASSIEUX Y, LE PIOUFLE B., “Experimental nonlinear torque control of a permanent-magnet synchronous motor using saliency”, IEEE Trans. On Industrial Electronics, vol. 44, no. 5, p. 680-687, October 1997.

[GRE 98] GRENIER D., YALA S., LOIUS J.-P., “Définitions d‘extensions de la transformation de Park pour la commande de moteurs synchrones à aimants permanents montés en surface”, The European Physical Journal, Applied Physics, p. 233-246, 1998.

[GRE 04] GRENIER D., STURTZER G., FLIELLER D., LOIUS J.-P., “Extension de la transformation de Park aux moteurs synchrones à distribution de champ non sinusoïdales”, in Louis J.-P. (ed.), Modélisation des machines électriques en vue de leur commande, Concepts généraux, Hermès, Paris, 2004.

[HOL 96] HOLTZ J., SPRINGOB L., “Identification and Compensation of Torque Ripple in High-Precision Permanent Magnet Motor Drives”, IEEE Trans. On Industrial Electronics, vol. 43, no. 2, p. 309-320, April 1996.

[JOU 01] JOUVE D., BUI G., “High performance servo Drive Design for distributed Motion Control”, PCIM‘2001 Conference, Nuremberg, Germany, 19-21 June 2001.

[LAM 99] LAM B.H., PANDA S.K., XU J.X., “Torque ripple minimization in PM synchronous motors-An iterative learning control approach”, IEEE Conf. of Power Elect. and Drives, PEDS‘99, Hong Kong, China, vol. 1, p. 141-149, July 1999.

[LAW 64] LAWRENSON P.J., AGU L.A., “Theory and performance of polyphase reluctance machines”, IEE Proc., vol. 111, no. 8, p. 1435-1445, August 1964.

[LEP 90] LE PIOUFLE B., GEORGIOU G., LOIUS J.-P., “Application des Commandes non- linéaires pour la régulation en vitesseou en position de la machine synchrone self- controlée”, Revue de Physique appliquée, no. 25, p. 517-526, June 1990.

[LIP 91] LIPO T.A., “Synchronous reluctance machines – a viable alternative for A.C. drives ?”, Electric Machines and Power Systems, vol. 19, p. 659-671, Hemisphere Publishing Corporation, 1991.

[LOU 99] LOIUS J.-P., BERGMANN C., “Commande numérique des machines synchrones”, Techniques de l‘Ingénieur, D 3644, May 1999.

[LOU 04] LOIUS J.-P. (ed.), Modélisation des machines électriques en vue de leur commande, Concepts généraux, Hermès, Paris, 2004.

[MAD 03] MADEMLIS C., “Compensation of Magnetic Saturation in Maximum Torque to Current Vector Controlled Synchronous”, IEEE Trans. Energy Conversion, vol. 18, no. 3, p. 379-385, September 2003.

[MAD 04] MADEMLIS C., Kioskeridis I., MARGARIS N., “Optimal Efficiency Control Strategy for Interior Permanent-Magnet Synchronous Motor Drives”, IEEE Trans. Energy Conversion, vol. 19, no. 54, p. 715-723, September 2004.

[MAR 91] MARCHAND C., Contribution à l‘étude de l‘optimisation du couple dans les servo moteurs synchrones à aimants permanents, PhD Thesis, Paris 6 University, 1991.

[MAR 93] MARCHAND C., RAZEK A., “Optimal torque operation of digitally controlled permanent magnet synchronous motor drives”, IEEE Proceedings, Pt. B, vol. 140, no. 3, p. 232-240, 1993.

[MAT 93] MATSUI N., MAKINO T., SATOH H., “Auto-compensation of torque ripple of direct drive motor by torque observer”, IEEE Transaction of Industry Applications, vol. 29, p. 187-194, February 1993.

[MON 11] MONMASSON E. (ed.), Power Electronic Converters: PWM Strategies and Current Control Techniques, ISTE, London and John Wiley & Sons, New York, 2011.

[NAA 11] NAASSANI A.-A., NAOUAR W., MONMASSON E, SLAMA-BELKHODJA I., “Sliding mode current control for a synchronous motor”, in Monmasson E. (ed.), Power Electronic Converters: PWM Strategies and Current Control Techniques, ISTE, London and John Wiley & Sons, New York, 2011.

[NAG 98] NAGEL N. J., Complex rotating vector analysis and control of a switched reluctance motor, Thesis, Wisconsin-Madison University, United States, 1998.

[NAO 11] NAOUAR W., MONMASSON E., SLAMA-BELKHODJA I., NAASSANI A.A., “Predictive current control for a synchronous motor”, in Monmasson E. (ed.), Power Electronic Converters: PWM Strategies and Current Control Techniques, ISTE, London and John Wiley & Sons, New York, 2011.

[PET 00] PETROVIC V., ORTEGA R., STANKOVIC A. M., TADMOR A, “Design and implementation of an adaptative controller for torque ripple minimization in PM synchronous motors”, IEEE Transactions of Power Electronics, vol.15, n°5, September 2000.

[PIE 11a] PIERFEDERICI I. S., MEIBODY-TABAR F., “Hybrid current controller with large bandwidth and fixed switching frequency”, in Monmasson E. (ed.), Power Electronic Converters: PWM Strategies and Current Control Techniques, ISTE, London and John Wiley & Sons, New York, 2011.

[PIE 11b] PIERQUIN J., DAVIGNY A., ROBYNS B., “Current and voltage control strategies using resonant correctors: examples of fixed-frequency applications”, in Monmasson E. (ed.), Power Electronic Converters: PWM Strategies and Current Control Techniques, ISTE, London and John Wiley & Sons, New York, 2011.

[REK 91] Rekioua T., Contribution à la modélisation et à la commande vectorielle des machines synchrones à aimants permanents, PhD Thesis, I.N.P Lorraine, 11 July 1991.

[STU 99a] STURTZER G., FLIELLER D., LOIUS J.-P., “Extension de la transformation de Park appliquée aux moteurs synchrones à réluctance variable non sinusoïdale”, Colloque EF‘99, Lille, p. 30-36, 30-31 March 1999.

[STU 99b] STURTZER G., FLIELLER D., LOIUS J.-P., GABSI M., “Inverse modelling of non-sine wave synchronous reluctance motors”, IEEE-IEMDC‘99, p. 107-109, Seattle, United States, 9-12 May 1999.

[STU 99c] STURTZER G., FLIELLER D., LOIUS J.-P., GABSI M., “Inverse modeling for saturated synchronous reluctance motors or permanent magnet synchronous motors”, EPE‘99, CD- ROM, Lausanne, Switzerland, 7-9 September 1999.

[STU 00a] STURTZER G., SMIGEL E., Modélisation et commande des moteurs triphasés – Commande vectorielle des moteurs synchrones – Commande numérique par contrôleur DSP, Ellipses, Paris, 2000.

[STU 00b] STURTZER G., FLIELLER D., LOIUS J.-P., GABSI M., “Extension de la transformation de Park aux moteurs synchrones à entrefer variable non sinusoid aux et saturés”, Revue Internationale de Génie Electrique, Hermès, p. 313-345, vol. 3, no. 3, October 2000.

[STU 01] STURTZER G., MODÈLE inverse et réduction de l‘ondulation de couple pour machines synchrones déduits des courbes isocouples. Extension de la transformation de Park pour moteurs synchrones à poles saillants non sinusoidaux et saturés, PhD Thesis, ENS Cachan, 2001.

[STU 02] STURZER G., FLIELLER D., LOIUS J.-P., “Extension of the Park‘s transformation Applied to Non-Sinusoidal Saturated Synchronous Motors”, EPE Journal, vol. 12, no. 3, p. 16-20, August 2002.

[STU 03] STURZER G., FLIELLER D., LOIUS J.-P., “Mathematical and experimental method to obtain the inverse modelling of non-sinusoidal and saturated synchronous reluctance motors”, IEEE Transactions on Energy Conversion, vol. 18, no. 4, p. 494-500, December 2003.

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1 Chapter written by Damien FLIELLER, Jean-Paul LOIUS, Guy STURTZER and Ngac Ky NGUYEN.

1. This diagram could have been even more idealized by simplifying by the coefficients G₀ and k_i . We have chosen to leave them in evidence, to avoid making this diagram too abstract, by thus recalling the necessary existence of the interfaces between the parts “controls” (signals) and the parts “systems” (power).

2. The controls without position sensor are the subject of Chapters 8 and 9 of this book.