Chapter 5

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Digital Implementation of Vector Control of Synchronous Motors ¹

5.1. Introduction

Vector control of AC motors, supplied by a PWM voltage inverter helps to carry out electric actuating systems; these systems have the performances required by various scopes, such as haptics, robotics, precision machining or the spatial field [GUS 03, KHA 06].

In order to achieve this, the torque control of synchronous motors aims to control the electromagnetic torque of the machine through the regulation of its Park currents of the d and q axes [LOU 99]. This control, whose principle has been described in the previous chapters, needs, among other things:

– the inverse transformation of the line currents;

– the estimate (ê_d ,ê_q) of the coupling terms (e_d ,e_q) due to the transformations;

– and the estimate ê₀ of the EMF, induced in the machine, noted e₀ .

In this chapter, we take particular interest in the digital implementation of the vector control of synchronous motors. This is why these different stages are not described in the following. The estimate of the coupling terms and of the

electromotive force (emf) is assumed to be perfect and thus these terms are perfectly compensated. The motor model is thus simplified and its electric part consists only of the equations of the d and q axes voltages, leading to the following first order transfer functions:

[5.1]

The model and the vector pulse width modulation (vector PWM) of the voltage inverter are found again in a very detailed manner in the literature [LOU 96, KHA 06, SIM 99]. In this chapter, the whole is described at first by a simple gain K_inv , to which a delay T_inv is added later on. The simplified block diagram of the ideal analog vector control of a synchronous motor can then be represented in Figure 5.1.

A calculation method of the analog regulators and simulation results is presented in the following section and will be used as the starting point of the digital implementation of the vector control of the studied synchronous motor.

However, the transfer from the analog to the discrete domain model comes with several problems deteriorating the performances of the carried out feedback:

– problem of the choice of the sampling periodT_e ;

– problem of the delays introduced by the calculations in the microprocessor and by the vector PWM;

– quantization problem due to the analog to digital converter in the measurement of the line currents;

– resolution problem of the incremental encoder necessary to measure the position of the motor and from which the speed calculation is carried out by digital differentiation;

– delay problem between the angle used in the Park and the inverse Park transformations and its instantaneous value.

These are the main elements discussed in this chapter.

5.2. Classical, analog and ideal torque control of a synchronous motor

5.2.1. Calculation of the current regulators

The torque control and thus in current of a synchronous motor is usually done with the help of Proportional Integral (PI) controllers, whose transfer function is written:

[5.2]

The transfer functions of the d and q axes, modeling the synchronous motor, are almost identical, the L_d and L_q inductance parameters being the only difference. This is why, the calculation of the controllers gains will only be presented for the q axis, which is the axis representative of the electromagnetic torque.

First, the delay T_ond of the voltage inverter is neglected. The inverter/motor system can thus be described by the following transfer function H(s):

[5.3]

The regulation must lead to a first order system H_elec (s), so that the bandwidth is finally equal to f_elec :

[5.4]

As a general rule, the zero of the controller is chosen to be equal to the pole of the system to be regulated and in this case, τ_i = L_q/R_s Once the controller is inserted into the system, the transfer functions in open loop OL (s), and in closed-loop CL (s), become:

[5.5]

By analogy with system [5.4], the calculation of the proportional gain leads to:

[5.6]

5.2.2. Determination of the current references

In order to optimize the torque supplied by the PMSM for a given current, and thus to minimize the losses of the electric system, a good control strategy is to take the d axis current reference such as i_d^* = 0 . In this case, the obtained electromagnetic torque is proportional to the i_q current, hence the choice of the q axis current reference from the desired torque reference such as:

[5.7]

The 3 / 2 coefficient appearing in the torque expression is due to the application of the Clarke transformation rather than the Concordia transformation on the three- phase currents. Indeed, the Clarke transformation preserves the component amplitude and helps to avoid the format overruns in the framework of a digital implementation.

For applications where overspeed is necessary, a defluxing strategy can be applied. In this case, the speed range is larger, but in return, the current reference i_d^* is no longer equal to zero and must take into account the following limits:

– torque limit so that we always respect relation [5.8], where i_max is the maximal current supported by the motor windings:

[5.8]

– voltage limit due to the emf [KHA 03]. Indeed, in steady state and if the stator resistances are neglected, the voltages of the d and q axes are written:

[5.9]

If i_d^* = 0 , the stator voltage is given by the following relation:

[5.10]

This voltage is acceptable as long as it is lower than the voltage limit V_lim set by the motor manufacturer (|V| <V_lim ) . If not, the reference following the d axis of the current is given by [5.11] and helps to deflux the PMSM. In this way, the voltage limit is not exceeded.

[5.11]

5.2.3. Parameters of the studied synchronous motor

In this chapter, the chosen synchronous motor is a permanent magnet synchronous machine with non-salient poles (L_d =L_q =L) . This machine is designed for robotic applications.

Its main characteristics are presented in Table 5.1.

Table 5.1. Parameters of the studied motor

Nominal power	Pn	120 W
Nominal voltage	Vn	10 V
Rated current	In	2.6 A
Nominal torque	Cn	108 mN.m
Rated speed	Nn	5,000 rpm
Number of poles pairs	p1	1
d and/or q axes stator winding inductance	L	0.17 mH
Stator resistance	Rs	0.65 Ω
Inertia moment	J	85 g/cm2

5.2.4. Simulation results of the ideal analog vector control of synchronous motors

In order to validate the controller and the calculated gains - proportional K_p and integral τ_i – the system is simulated under Matlab-Simulink The current reference of the q axis is taken to be equal to the rated current of the PMSM. Figure 5.2 shows the currents (i_d,i_q) and the current references (i ^*_d, i ^*_q) for specifications defining a response time corresponding to a bandwidth f_elec of 1 kHz.

The obtained simulation results show that the motor is equivalent on the electric level to a first order system, whose response time is about 0.48 ms. This corresponds to the desired bandwidth in the specifications. These ideal results are the basis for all that follows and are the objective to reach after the digital implementation of the vector control of the studied synchronous motor.

5.3. Digital implementation problem of the synchronous motor vector control

From a practical point of view and considering the complexity of AC motor control, the natural tendency is to go towards the digital implementation of the vector control. This requires fast microprocessors, with high calculation powers such as DSP or FPGA-type controllers [MON 07], [TOL 04].

The constraints inherent to the numerical digital domain are however a world in itself, where different notions appear. These notions, such as the periods of sampling, chopping and regulation, the analog digital converters (ADC), the quantization, the PWM and the calculation delays, are not found in the analog domain. These different elements are described and studied in the following sections.

5.3.1. The interfaces, sources of restrictions

Figure 5.3 schematically shows the different elements or interfaces necessary for the implementation of a digital vector control. It will list, first intuitively, the sources of the digital problems, to which the designer is confronted.

The indices k and k′ represent the sampling of the measured variables at the instants kT_e and k′T_Ω where T_e and T_Ω are respectively the regulation periods of the currents and speed loops.

Figure 5.3 shows the necessity for new interfaces for the transfer from the analog to the digital domain. These interfaces are the main sources of digital implementation problems and the subject of this chapter. Let us quote among them:

– the analog to digital converters allowing the transformation of analog signals of phase current images, into digital data;

– the incremental encoder of resolution R_c leading to the position measure from two binary signals in A and B in phase quadrature sent towards a DSP counter;

– and the PWM.

These three interfaces introduce the discretization notions and thus the notions of the sampling period T_e and of quantization discussed below.

Another important characteristic of the digital domain is its calculation time T_c , necessary for the different data processing carried out in the microprocessor:

– calculation of the Park components of the stator currents;

– position calculation;

– speed estimate from the position;

– digital current regulation;

– reference voltage rotation;

– vector pulses width modulation;

– estimate and compensation of the coupling terms and of the emf;

– and external speed and/or position regulation loop according to the application.

The calculation time can generally be neglected in the FPGA case, but it is a non-negligible magnitude in the case of the DSP-type controller‘s processors.

5.3.2. Time diagram

To target the delay sources of a digital control, it is necessary to carry out a time diagram in order to visualize all the temporal data of the problem.

As a general rule, the regulation of a synchronous motor has three main loops:

–two internal regulation loops of period T_e for the d and q axes currents;

–and an external speed regulation loop of a higher T_Ω period, a multiple of T_e.

The vector PWM is largely discussed in the literature. In this chapter, we will only keep the different commutation instants ti_h - i going from 0 to 7 – of the switches of the inverter, defined by the application times T¹ _h and T_h⁴ of the null vectors V₀ and V₇ and by the application times T_h ² and T_h ³ of the voltage vectors adjacent to the reference voltage.

A time diagram gathering all the temporal data relative to the notions we have already mentioned, is shown in Figure 5.4; T_h being the chopping period of the PWM carrier. For educational and didactic reasons, we have chosen to present the case where sampling and chopping periods are equal (T_e =T_h), but in general these periods are chosen so that T_e=T_h/2, which reduces the number of commutations of the inverter switches by half.

5.3.3. Digital implementation constraints of the vector control of a synchronous motor

The time diagram emphasizes several important elements that will be discussed one by one in the rest of this chapter and that we briefly present below:

- the analog measures of the phase currents are sent towards an analog to digital converter, in order to discretize them. The latter needs the choice of a suitable sampling period T_e (see section 5.4.1) and introduces a quantization error studied later on in section 5.6.1. This quantization problem is also found in the position measure of the motor shaft. This measure is made from the incremental encoder of finished R_c resolution and the consequences pass on the speed calculation that is done by digital differentiation (see section 5.6.3);

–the application times (r¹_hk,T²_hk,T³_hk,T⁴_kk) are calculated from the data measured and sampled at the instant t_k−1. They are only available after a non-negligible T_c calculation time. Thus, in practice, this data is only sent to the registers of the vector PWM of the DSP at the next chopping period, i.e. at the t_k instant. This leads to an

important delay, higher than T_c, on the level of the system response and whose consequences are studied in section 5.5.1;

–the vector PWM, by its commutation principle, introduces a delay generally represented in the inverter transfer function, in our case under the name T_inv. This delay is quantified in section 5.5.3, which will then lead to its correction in the current regulators‘ calculation;

–transformations from reference voltages in the Park reference frame, to those in the Clarke reference frame necessary for the vector PWM are done after a generally non-negligible calculation delay. The measured position at the instant t_k will thus evolve and it is necessary to find methods, in order to compensate for the introduced error (see section 5.7).

5.4. Discretization of the control system

The previous sections have shown several critical durations in the system:

–the chopping period of the voltage inverter T_h;

–the sampling periods of the regulations of current T_e and speed T_Ω;

–the calculation delays of the reference voltages T_c;

–and the vector PWM delay T_inv.

These different periods and delays must be taken into account in the choice of the sampling period T_e of the system.

5.4.1. Choice of the sampling period

The digital implementation of the vector control of a synchronous motor necessarily goes through the choice of the sampling period T_e , suitable for the measure of the currents and position data, and consequently for the speed calculation. This period‘s higher value is limited by the period defined from Shannon‘s theorem, and its lower value is limited by different technological limits of the elements present in the loop, such as the delay necessary for the different control calculations T_c .

Shannon‘s theorem states that the sampling frequency of a signal must be equal to or higher than the double of the maximal frequency contained in this signal, in order to convert this signal from an analog to a digital form. The bandwidth of the current loop being adjusted to a cut-off frequency of about f_elec , the sampling period of the currents must thus verify the following relation:

[5.12]

In practice, a value clearly lower than the Shannon period is necessary, for the simple reason that no physical signal really has a limited band. It is thus generally common to take a sampling period at least ten times smaller than the Shannon period, hence the new relation:

[5.13]

The smaller the sampling period is, the more the information contained in the analog signals is preserved. However, a too low sampling period leads to digital conditioning difficulties for the regulators (quantization problem). A compromise must thus be found. In this chapter, the chosen value for the sampling period is T_e = 40 μs. It is then practical to define all the other critical durations at the same value [Gus 03]. In that case, the chopping period T_h = T_e corresponds to a switching frequency of the inverter switches of 25 kHz, which is in the norms of the MOSFET switches of our application.

5.4.2. Choice of the sampling instant

The choice of the sampling instant is necessarily added to the choice of the sampling period. Indeed, the vector PWM implies the application of a series of rectangular pulses of a constant period T_h and of an amplitude that is null or equal to the continuous voltage at the inverter input. The width of these pulses is such that on each commutation period, the mean value of the voltages supplied by the inverter is equal to the value of the reference voltages [GUS 03]. The consequence of the application of the spike trains at the terminals of the motor is that the ripples are induced in the stator currents of the latter, as shown in Figure 5.5.

The amplitude of the current ripples mainly depends on the stator winding inductance, on the amplitude of the voltages applied to the terminals of these windings and on the chopping period T_h . The shorter the latter is, the more the ripples are reduced.

The choice of the sampling instant in the chopping period influences the error induced in the currents measure. Considering that the electric time-constant is generally much longer than the chopping period, we can generally assume that the evolution of the current during this period is linear between two successive commutations. In the case of a symmetric modulation, the currents reach at the beginning and the middle of the period, the same values they would have reached if the applied voltages were constant. Taking measures at these instants thus ensures a theoretically null measure error. Moreover, it has the advantage of being carried out at an instant with no occurring commutation. This helps to avoid electromagnetic disturbances associated with the commutations‘ influence on them [GUS 03]. In addition, in this case, the measured value is then almost equal to the mean value of the current in the windings.

5.4.3. Implementation of the digital control

Once the sampling period and instant are chosen, the digital regulation of the currents of a synchronous motor can be carried out by two different categories of methods. The most frequently used is a regulator synthesis by transposition from the analog to the digital domain: in a first phase, it consists of determining an analog controller and then transforming it into a digital controller, with equivalent performances. This type of method is very frequently used in the industrial sector and will be discussed in the following sections of this chapter.

The second type of synthesis method is called direct synthesis in digital. This is less frequently used in the industrial sector than synthesis by transposition. It however leads to more delicate adjustments when the models of the processes to override are well known. Let us quote for example RST regulators [LAR 96], the predictive control [MOR 09], the adaptive control, etc.

The transposition objective is thus to find a digital regulator for which the behavior of the digital feedback is the closest possible to the behavior of the analog feedback. Given the sampling and quantization effect, we can see that the analog feedback coming from the transposition will be equivalent at best to the analog feedback, but never better.

Several transposition techniques are frequently used. The Euler transposition methods are based on the digital approximation of the derivation, according to the following relations:

– approximation by front discretization:

[5.14]

– approximation by back discretization:

[5.15]

The Tustin approximation method, also known as bilinear transformation, results from the digital integration approximation by the trapezoids method and is written:

[5.16]

Finally, transposition by conservation of the poles and of the zeros (CPZ) aims to preserve the regulator behavior by individually transposing its poles and zeros and by ensuring compatibility between the final values of the continuous and discrete controllers.

Bode diagrams of the discretized regulators using these different methods, are drawn in Figure 5.6. In our case, T_e being very low compared to the dynamics of the looped system, all the discretization methods give good results.

We will proceed in the following study with the bilinear transformation that better preserves the harmonic richness of the regulator. In that case, the digital PI regulator is written:

[5.17]

hence:

[5.18]

The block diagram of the regulator is represented in Figure 5.7, where the variables x_k correspond to x_k=x(t_k) = x(kT_e). The gain K_pd=r₀ is the proportional gain of the discrete regulator and t_{i d} =r₀+r₁ is the integral gain.

A K_awu gain is added, after saturation of the reference voltage in order to avoid, in case of saturation, the integral term diverging. This gain, called anti-windup [TEX 98], is defined as:

[5.19]

5.4.4. Simulation of the control with discrete regulators

The digital control is simulated under Matlab-Simulink. Figure 5.8 shows the currents (i_d ,i_q) measured and sampled at the period T_e , compared to the current references (i_d^* ,i_q^*) . The obtained results show the validity of the PI discretization and of the choice of the sampling period for the digital implementation of the vector control of the synchronous motor.

The discretization of the regulators of the first order systems modeling the motor, is thus validated. In the following, we will introduce a second problem of the digital implementation: the delays due to the calculation times and to the vector PWM of the inverter.

5.5. Study of the delays introduced by the digital implementation of the vector control of the synchronous motor

Two main delays are in the system to be regulated: T_c representing the delays in the analog to digital converters and the calculation times in the microprocessor, and T_inv the delay introduced by the vector PWM of the voltage inverter [BUH 93], [BOC 09]. These delays are gathered later on in a single term noted T_Σd :

[5.20]

5.5.1. Simulation results after introduction of the delays in the system

In order to observe the delays introduced in the system and to study the robustness of the chosen regulatorwith respect to these delays, the global delay T_Σd is introduced into the model of the initial analog system between the reference voltages and the real voltages obtained at the output of the voltage inverter. Figure 5.9 shows, in that case, the q axis currents obtained for different values of T_Σd .

The system response has a 2^nd order form with a pure delay and has an overrun that becomes all the more important when the delay is long. Moreover, ripples appear, which is generally not acceptable in the frame of the precision applications

to which this chapter is devoted. This result can easily be demonstrated. Indeed, the system of electric equations modeling the motor, to which we can add a delay T_Σd , introduced between the reference voltages and the real voltages, is written according to the transfer functions:

[5.21]

The results presented later on can be transposed from one axis to another. We can proceed with the study on the q axis only. The delay T_Σd, described by the Laplace delay operator e-^TΣd.s., can be approximated by a pole in the system so that the new system to be regulated, H_q,r (s), is written:

[5.22]

The system with the PI regulator initially calculated thus compensates for only one pole of the system and the new transfer function of the corrected system in closed-loop is written:

[5.23]

with:

[5.24]

The new values of the angular frequency and the obtained damping factor depend on the delay introduced. The more the delay increases, the more the system oscillates with an increasingly important overrun. It is thus necessary to review the regulator calculation, by introducing the delay T_Σd in the system, in order to reduce its effects.

5.5.2. Calculation of the new regulators after taking into account the delays

Let us take again a PI controller in order to regulate the currents:

[5.25]

The zero of the aforementioned regulator is taken to be equal to the slowest pole of system [5.22], so that t_iΣ =L /R_s. The transfer functions in open-loop OL_Σ (s), and in closed-loop CL_Σ(S) then become:

[5.26]

The choice of the proportional gain of the PI regulator must thus be done by a compromise between the system speed and its stability in terms of overruns and oscillations. The optimal adjustment, when a slight overrun is acceptable, is obtained for a damping factor £_Σ equal to √2/2. When no overrun is desired, it is then sufficient to take £_Σ equal to 1. In those two cases, the gain expression K_pΣ is written:

[5.27]

Figure 5.10 shows in these two cases the response in current of the system regulated for different calculation delays. The obtained results indeed show a rapidity decrease and an overrun increase for an increase in the damping factor. In the first case, £_Σ = 2√2, the system has an overrun lower than 5% with better response times than when the delays were neglected. In the case where £_Σ = 1, there is no overrun, but the obtained response times are more important and are increasingly different from the defined specifications.

5.5.3. Simulation after delays correction and system discretization

In the following the chosen regulator is the one leading to a response the closest possible to that of the ideal case (analog without delays), while preserving an overrun lower than 5%. Concerning the studied delays, let us recall that they are the sum of two main terms:

–T_c being the calculation and sampling delays;

–and T_inv the response delays of the inverter.

The sum of these two terms can be approximated in the case of the implementation on a microcontroller by T_Σd ≈1.5T_e, where T_e is the sampling and regulation period of the currents of the discretized system [BOC 09]. Indeed, let us recall that for this type of implementation, T_c and T_e are very close. As for the PWM, it can be seen as a sliding averaging filter, inducing a delay equal to T_h/2 = T_e/2. Thus, in this chapter, the chosen sampling period is 40µs and the studied delays will thus later on be considered, so that T_Σd = 60µs.

Once this data is defined, the calculated analog regulator can be discretized by following the same reasoning as in section 5.4.3. Figure 5.11 shows the d and q axes currents obtained after correction of the delays introduced into the system and after discretization of the regulator.

The simulation results obtained validate the regulation after the correction of the delays introduced into the system by the digital implementation of the vector control. The response is aperiodic with, at the origin, a pure delay equal to the calculation delays T_Σd . The current response time is about 0.4 ms, which corresponds to the specifications defined for this application.

5.6. Quantization problems

The question of the digital implementation delays having now been discussed, we can study the quantization problems, introduced in the system by current and position sensors, by digital calculation of the speed and by vector PWM of the voltage inverter.

5.6.1. Quantization affecting the current measures

Regulation of the currents of the d and q axes requires phase current measures i₁ and i₂ , to which a Park transformation is applied, that itself requires the rotor position measure θ . Figure 5.12 shows an example of the acquisition principle of the phase current i₁ , the current acquisition i₂ being done in the same way.

Two resistive shunts lead to two voltages, images of the phase currents. An optocoupler amplifies these voltages and ensures the isolation of the power variables compared to the control magnitudes of the electronic board. Two stages with operational amplifiers adapt the measures amplitude to correspond to the characteristics of the Q bits analog to digital converter (ADC) of the DSP controller, chosen for the digital implementation of the vector control. The ADC then carries out the analog to digital conversion and the sampling at the period T_e of the voltages, images of the phase currents.

Given the fact that the measured values of the stator currents intervene in the control process, we must avoid the ripples due to the PWM supply leading to frequency aliasing [GUS 03]. A solution to this problem is to filter the currents before measuring them, but in that case, delays will be introduced by the filtering circuits. This leads to the reduction of the control performances. Another solution is to eliminate the frequency aliasing at the level of the measures, by choosing a sampling period T_e multiple integer of the chopping period T_h and by ensuring a synchronization of these two periods. This is indeed the case in our application.

The acquisition of the phase currents finally comes down to a sampling of period T_e and to a gain K_ADC (Figure 5.13), to which we need to add the quantization effect due to the limited number of bits of the analog to digital converter. By setting out Q as the resolution of the ADC of the DSP, we can then calculate the smallest variation of the digitized signal δI , also called the quantization step, by setting out ΔI the maximal amplitude of the measured signal.

[5.28]

The quantization step and thus the quantization error are smaller when the number of bits of the ADC is high. The choice of the ADC is thus done by a compromise between the precision of the desired measures and the admissible cost for the purchase of a DSP containing a high resolution ADC.

In order to study the quantization effect on the system, the latter is simulated by adding an analog to digital converter with various resolutions. Figure 5.14 shows the current responses of the system obtained for a number of ADC bits going from 4 to 8 bits.

Figure 5.14 shows that the system response is deteriorating for an ADC of 4 bits resolution, whereas for ADC with resolutions higher than 8 bits, the obtained responses are satisfactory. Nowadays, the converters available on the microprocessors market have a high resolution (Q > 8). The problem introduced by quantization is thus controllable in the frame of this type of application.

This brings us to the problem of the fixed point and of the floating point in the microprocessors. Indeed and up until very recently, the microprocessors worked only with integer numbers. It was thus necessary to scale all the variables in the interval [−1;1[, and then to multiply them by a power of 2, according to the calculation format of the microprocessor. This often leads to format overruns, which must be prevented at the expense of an often important resolution loss in the system. Nowadays, microprocessors have new calculation capacities, with fixed or floating points, leading to a better use of the converters.

In this domain, the FPGA show a better flexibility than the DSP controllers. Despite that fact, the latter are the most frequently used for the torque control or in speed of the electric machines; nowadays the Texas Instruments company has the most frequently used DSP controllers [TEX 98].

5.6.2. Quantization at the level of the position measure

To calculate the Park currents i_d (k) and i _q(k) from the phase currents i₁ (k) and i₂ (k), it is necessary to know the rotor position θ(k), also sampled at the T_e period.

The rotor position measure of the PMSM is carried out via an incremental encoder with R_c resolution. This encoder sends two binary signals, A and B in phase quadrature towards a DSP counter, counting the ascending and descending pulse edges of the received signals, in order to find the motor position and its rotation direction. The counter is reset at each top encoder signal received from the third Z signal of the encoder, as seen in Figure 5.15 [TEX 98].

The position measurement is thus equivalent to a sampling of period T_s, followed by a gain K_encoder equal to the quantization step δθ, corresponding to the chosen incremental encoder of resolution R_c:

[5.29]

The quantization problem is thus also raised on the level of the measure of the motor rotor position. The position is all the more precise when the resolution of the chosen incremental encoder is high. Currently, the coders resolution on the market is quite important and generally higher than 250 points per round, which highly minimizes the incidence of this quantization error on the measure of the motor position. The choice of the encoder will thus also be a compromise between the precision of the measure and the price of the desired encoder.

5.6.3. Calculation of the speed by digital differentiation

Calculation of the motor rotation speed is necessary in the presented vector control for:

– the calculation of the coupling terms and of the emf, must be compensated for in the current regulation loops;

– the speed regulation loop which is found in any variable speed application , but is not presented in this chapter.

The two previous sections have shown that the quantization errors, because of the technological advance on the converters and the incremental coders level, do not influence, or just a little, the dynamics of the regulated systems. However, this is not the case for the speed calculation done by digital differentiation at the period T_Ω, from position measures, sampled at the T_e period.

The literature shows that three main methods are used in the digital calculation of a motor speed [GAL 96], [BHA 97], [KHA 06a]:

–by direct pulse counting method (DPCM);

–by single-pulse time measurement method (SPTMM);

–and by constant elapsed time method (CETM).

In all these cases, the calculation of the motor speed is carried out by simple digital differentiation of the position measures and so that:

[5.30]

where ω_n is the calculated speed at the instant t_n , θ_n the position measured by the incremental encoder and T the time elapsed between the two position measures θ_n and θ_n-1.

According to the chosen method for the speed calculation, the quantization error will depend either on the encoder resolution or on the period of the internal clock of the DSP chosen for the application.

Considering that the speed regulation loop is not discussed in this chapter, we will not develop the quantization notion on the motor speed calculation level, but we refer the readers interested in this problem to [KHA 06].

5.6.4. Quantization in the vector PWM of the voltage inverter

The last interface to introduce a quantization problem is the DSP controller peripheral, carrying out the vector PWM of the voltage inverter supplying the motor.

Let us recall that the current regulation loop supplies, at the output, voltage references v* and v*, leading to the calculation of the application times of the voltage vectors, corresponding to the desired reference. These application times (T_h¹, T_h², T_h³, T_h⁴ defined in section 5.3.2, are compared to a carrier signal, in order to obtain logical control signals of the MOSFET switches in the framework of this application.

However, the carrier signal generated by the DSP is not continuous. It depends on the DSP internal clock and thus varies with temporal period steps equal to the clock period T_H = 25 ns.

This phenomenon, presented in Figure 5.16, shows that in the end, the vector PWM and the voltage inverter can be represented by a zero order hold (ZOH) block of period T_h , since the voltage references are refreshed only once per current regulation period, by a gain K_inv and a delay T_inv , and by a quantifier with a quantization step δ_PWM defined by the relation:

[5.31]

The obtained quantization step gives the equivalent resolution of:

[5.32]

i.e. in our case, an 8 bit resolution.

Finally, let us note that it is possible to refresh the application times every half- period instead of every period.

5.7. Delays in the reverse Park transformation

The time diagram of the operations carried out for the digital implementation of the vector control (see section 5.3.2), shows that all the calculations carried out in the DSP lead to a non-negligible delay T_c before the sending of the voltage references towards the vector modulation unit of the DSP. It is thus important to take into account the fluctuation of the rotor position between the sampling instant and the moment when the reverse Park transformation is applied to the voltage references v_d* and v_q*.

In that case, the schematic diagram of the regulation is presented in Figure 5.17, Q'(k) being the position to be defined for the reverse Park transformation.

The rotor displacement during this time interval, because of the discrete characteristic of the reverse Park transformation, leads to a recoupling of the d and q axes of the motor [ROB 91] and deteriorates the monitoring of the references trajectory. These effects are more annoying for high speed machines [FU 96]. A technique for reducing these phenomena consists of introducing a compensation term θ_c(k) into the reverse Park transformation angle, such as:

[5.33]

The choice of this compensation angle value helps to minimize the trajectory error and to almost cancel the recoupling between the two axes [GRE 97, YAL 98].

5.8. Conclusion

We have presented in this chapter the digital implementation of the vector control of a synchronous motor with permanent magnets and non-salient poles. We have shown the different critical points due to the digitization of the control, their consequences and the main solutions proposed in the literature.

We have thus discussed the choice of the sampling period; the discretization of the analog regulator synthesized from the Park model of the machine; the delays introduced by the voltage inverter and the calculation times; the quantization problems, at the level of the measures of current, of position or of the vector PWM; and finally a compensation technique for the calculation delay in the Park rotation of the voltage references.

This presentation only gives a general overview of the problems that need to be solved during the digital implementation of the vector control of a synchronous motor. The proposed solutions must be adapted on a case-by-case basis and reconfigured depending on the studied application. In practice, despite all the precautions taken by the user in the calculation of the regulators, it will always be necessary to make some adjustment modifications to the final system.

5.9. Bibliography

[BHA 97] BHATTI P, HANNAFORD B., “Single-chip velocity measurement system for incremental optical encoders”, IEEE Transactions on Control Systems Technology, vol. 5, no. 6, p. 654-661, November 1997.

[BÖC 09] BÖCKER J., BEINEKE S., BÄHR A., “On the Control Bandwidth of Servo Drives”, EPE2009, Spain, 2009.

[BUH 93] BUHLER H., Conception de systèmes automatiques, Presses polytechniques et universitaires romandes, Switzerland, 1993.

[FU 96] FU Y., LABRIQUE F., BUYSE H., “Sensitivity of various synchronous motors field oriented control structures to the discretization effects related to their digital implementation”, Proceedings of the CESA‘96 IMACS Multiconference, France, vol. 1, p. 616-621, July 1996.

[GAL 96] GALVAN E., TORRALBA A., Franquelo L.G., “ASIC implementation of a digital tachometer with high precision in a wide speed range”, IEEE Transactions on Industrial Electronics, vol. 43, no. 6, p. 655-660, December 1996.

[GRE 97] GRENIER D., LABRIQUE F., MATAGNE E., BUYSE H., “Discretization effects on the control of VSI fed PM synchronous motor drives”, Electromotion International Journal, vol. 4, no. 4, p. 155-163, 1997.

[GUS 03] GUSIA S., LABRIQUE F., GRENIER D., BUYSE H., SENTE P., “Réflexions sur l‘implantation numérique et l‘analyse en temps discret de la commande vectorielle des machines courant alternatif: une synthèse”, Electrotechnique du futur, Gif-sur Yvette, France, 9-10 December 2003.

[KHA 03] KHATOUNIAN F., MONMASSON E., BERTHEREAU F., DELALEAU E., LOUIS J.P., “Control of a doubly fed induction generator for aircraft applications”, Proceedings of IEEE IECON 2003, p. 2711-2716, 2003.

[KHA 06] KHATOUNIAN F., Contribution à la modélisation, l‘identification et à la commande d‘une interface haptique à un degré de liberté entraînée par une machine synchrone à aimants permanents, PhD Thesis, école normale supérieure de Cachan, 2006.

[KHA 06-a] KHATOUNIAN F., MOREAU S., MONMASSON E., LOUVEAU F., “Speed estimation improvement after decreasing the encoder resolution for a haptic interface”, IEEE ISIE 2006, Montreal, Canada, 9-13 July 2006.

[LAR 96] DE LARMINAT P., Automatique: commande des systèmes linéaires, Hermès, Paris, 1996.

[LOU 96] LOUIS J.P., BERGMANN C., “Commande numérique, systèmes triphasés: régime permanent”, Techniques de l‘ingénieur, no. D3642, November 1996.

[LOU 99] LOUIS J.P., BERGMANN C., “Commande numérique des machines synchrones”, Techniques de l‘Ingénieur, no. D3644, 1999.

[MON 07] MONMASSON E., CIRSTEA M., “FPGA design methodology for industrial control systems – A review”, IEEE Transactions on Industrial Electronics, vol. 54, no. 4, p. 1824-1842, August 2007.

[MOR 09] MOREL F., XUEFANG L.-S., RETIF J.-M., ALLARD B., BUTAY C., “A comparative study of predictive current control schemes for a permanent-magnet synchronous machine drive”, IEEE Transactions on Industrial Electronics, vol. 56, no. 7, p. 2715-2728, July 2009.

[ROB 91] ROBYNS B., LABRIQUE F., BUYSE H., “Digital control with decoupling state feedback of AC motors”, Proceedings of the Third International Workshop on Microcomputer Control of Electric Drives, IEEE, p. E1-E9, July 1991.

[TEX 98] Texas Instruments, Field Orientated Control of 3-Phase AC-Motors, application report Bpra073, February 1998.

[SIM 99] SIMON E., Implementation of a Speed Field Orientation Control of 3-phase PMSM Motor using TMS320F240, Texas Instruments, application report SPRA588, September 1999.

[TOL 04] TOLIYAT H. A., CAMPBELL S. G., DSP-Based Electromechanical Motion Control, CRC Press, Texas A&M University, United States, 2004.

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1 Chapter written by Flavia KHATOUNIAN and Eric MONMASSON