exact knowledge of the motor phase voltages in the whole speed range. The use of voltage sensors may be eschewed if the reference voltage signals, generated by the control algorithm, can be used instead. To this aim, an accurate compensation of most of inverter non-idealities is essential. This paper presents a novel self-commissioning procedure for the cancellation of inverter non- idealities, based on repetitive control. The main advantage is that the compensation generated by the procedure automatically includes the IGBT parasitic effects during current zero-crossing, whose exact knowledge is one of the major problems in most of the standard dead-time cancellation techniques. Despite its elaborated theoretical background, the method requires few com- putational resources and it is easy to implement. Mathematical developments, design hints and an extensive batch of successful experimental tests are included in the paper.
Index Terms—Induction Motor Drives, Harmonic Distortion, Learning Control Systems, Phase Locked Loops, Pulse Width Modulated Inverters
I. INTRODUCTION
I
T is well known that the dead-times-related voltage distor- tion in inverter-fed electrical drives negatively affects the performances of both open loop control algorithms (like the V/Hz control for induction motor drives) in the low speed region, and sensorless AC drives, which often make use of the voltage references instead of the actual measured ones.The problem of dead-time distortion compensation has been widely discussed in literature, with the purpose of compen- sating not only the well-known step-like ideal distortion, but also by including some second-order non-linearities. In [1] a PI regulator is used to compensate the differences between the reference and actual voltage measured with a dedicated circuit.
Other solutions ([2]-[3]) try to cancel non idealities by means of IGBT and diode models. Assumed a certain accuracy of the models, these approaches can be quite efficient, at the cost of an increasing computational effort and of a rather complicated off-line measurement batch. Some solutions ([4], [5], [6]) use observers that rely on motor parameters, along with their related uncertainties.
Instead of using either models or voltage measurements, further works exploit the harmonic distortion in phase currents to cancel the disturbances in reference voltages. In [7] and [8] two on-line algorithms calculate the voltage compensation trying to minimise the voltage distortion. However, the on-line
S. Bolognani is with the Department of Electrical Engineering, University of Padova, Padova, Italy (e-mail: bolognani@die.unipd.it).
L. Peretti and M. Zigliotto are with the Department of Technique and Management of Industrial Systems, University of Padova, Vicenza, Italy (e- mail: luca.peretti@unipd.it, mauro.zigliotto@unipd.it).
convergence have negative influence on the overall dynamic behaviour. In [9], a repetitive control is used for an induction motor (IM) drive in the angle domain, since in that reference the dead-time distortions are claimed to be constant. The main disadvantage is that the transformation between the time domain and the angle domain needs a measurement of both slip and rotor speed, and therefore the method does not fit for sensorless drives.
Nevertheless, due to the periodical nature of voltage dis- tortion at steady state, the use of a repetitive control for dead-time compensation is intriguing. The rather specialized subject of internal model controls (to which the repetitive control belongs) was first investigated around the beginning of the 80’s. Perhaps boosted by the use of computers in control applications, the ability to store a whole period of the disturbance signal made possible the practical application of these techniques [10]. One of the first examples of repetitive control application was the rejection of periodic disturbances acting on the track-following servo system of optical disk drives [11]. In recent years, repetitive control has been used in electric drive applications, as for example in [12] in which the torque ripple has been profitably reduced.
In this work, the repetitive control is exploited to generate an application-specific look-up table (LUT) that will work at low current levels, where the behaviour of power mod- ules noticeably deviates from the ideal switch. The LUT is excluded for higher current levels, where the conventional step-like compensation works properly. The proposed method dodges both IGBT models and voltage measurement. Even the accurate current zero-crossing detection, bottleneck of most standard strategies, is less stringent. A further advantage over existing solutions is the low computational requirement, due to the LUT-based approach, suitable for low-cost applications.
As well, the procedure can be also regarded as the first step in advanced parameter identification techniques for sensorless drives. Several experimental tests have been carried out to prove the effectiveness of the solution, and its generality as well.
II. DEAD-TIMES EFFECTS
The inverter dead-times generate a phase voltage distortion that can be expressed as:
udist,skn= td
Tc
Udcsgn(isk) (1) whereiskis the current of thek-phase (k = a, b, c), Tcis the PWM switching period, Udc is the dc-link voltage, and td is
Figure 1. IGBT commands and inverter actual voltage output
equal to:
td= tof f− ton− td,s (2) where tof f is the IGBT fall time,ton is the IGBT rise time, and td,s is the safe interval between the commutation edges of the upper and lower devices in the inverter leg (Figure 1).
In a space vector notation, (1) can be expressed by [13]:
udist,s= u∗s− us= −4 3
td
Tc
Udcsgn(is) (3) where:
sgn(is) = sgn(isa) + sgn(isb)ej23π + sgn(isc)ej43π
sgn(isa) + sgn(isb)ej23π + sgn(isc)ej43π
(4)
As can be seen from (3), the distortion induced by the dead- time affects both amplitude and phase of the actual voltage vector, with respect to its reference. The voltage distortion (1) can be expressed in a synchronousdq rotating frame, locked to the phaseϑi of current space vector, as follows (see Appendix B):
udist,sd(ϑi) = 4td
πTc
Udc
"
1 −
+∞X
n=1
(−1)n 2
36n2− 1cos(6nϑi)
#
udist,sq(ϑi) = 4td
πTc
Udc +∞X
n=1
(−1)n 12n
36n2− 1sin(6nϑi) (5) Aside from the constant term inudist,sd, the distortion can be approximated as a sum of6n-th harmonic sinusoidal compo- nents (n = 1, . . . , +∞). The same harmonic behaviour occurs if an angle with constant phase advance (or constant phase lag) is used instead of ϑi. The proposed self-commissioning procedure is performed off-line, at constant load, by a standard V/Hz open-loop control, so that the built-in reference voltage angleϑu∗, which shows a constant phase advance with respect to ϑi, is used instead. In the selected xy reference frame, fixed to the voltage vector u∗, the induced 6n-th harmonics distortion on stator current vector componentsisx andisyare still clearly recognisable. As an example, Figure 2 reportsisx
andisy when the prototype IM 1 (parameters are reported in Appendix A) is operated under V/Hz control at no load, with a reference voltage vector rotating at a frequency of 1 Hz.
Figure 2. Stator currents waveforms: a) isx, b) isy.
It is also worth to note that the distortions expressed in (1) create a homopolar component equal to:
uhom= td
3Tc
Udc(sgn(isa) + sgn(isb) + sgn(isc)) (6)
A. IGBT parasitic effects
The IGBT parasitic effects heavily deteriorate the common step-like dead-time compensation. These effects can be man- aged by a proper look-up table, whose input and output are the phase current amplitude and the voltage correction, respec- tively. Quite often, this method is accomplished by either off- line measurements or complex IGBT models [2]. Conversely, the proposed technique performs the task as a part of its self- commissioning procedure. The parasitic capacitances affect IGBT commutations in proximity of the zero-crossing of each current phase. Since there are six zero crossings during a2π rotation of a synchronous frame, the related distortion effect is a sum of 6n-th order harmonics (n = 1, . . . , +∞), which overlaps the harmonics coming from (5).
III. THE PROPOSED SELF-COMMISSIONING PROCEDURE
The proposed self-commissioning algorithm exploits the presence of periodic components of the disturbance, at steady state, caused by voltage distortion in stator currents. The use of the currents, instead of the voltages, let any voltage measurement be avoided, which is actually one of the main goals. The procedure begins at start-up: an IM is operated by an open-loop V/Hz control with a voltage reference vector rotating at 16.67 Hz, while the phase currents are acquired.
A reasonably constant (or null) load torque is required. In this steady state condition, two main steps are performed in sequence.
In the first step, the feedforward block is activated (FFW block of Figure 3), with the purpose of compensating the constant component of (5).
In the second step, the phase currents are transformed into a rotating xy reference frame, fixed to the reference voltage vector of the V/Hz control. The resultingisx andisy compo- nents are high-pass filtered to get the6n-th components that, changed in sign, constitute the error input to a repetitive block, (Figure 3). The repetitive compensation is tuned on the6n-th harmonics, which in this case are located at6n16.67 = n100 Hz. The details are reported in Section V.
As shown in Figure 3, the proposed voltage compensation is the sum of the feedforward and the repetitive actions. The self- commissioning procedure runs until the input to the repetitive block, which is the 6 − th harmonics ripple superposed to isx andisy, is reduced within a specified peak-to-peak bound.
The bound is defined as a percentage of the nominal motor current. In this work, a value of 4% has been chosen. With this choice, in our tests the whole procedure lasts about 15000 PWM switching periods, i.e.1.5s @ 10kHz.
During the convergence of the repetitive algorithm, the compensation voltage patterns added to the V /Hz voltage reference are continuously stored in a circular buffer. After convergence, the buffer is transferred into three application- specific look-up tables (LUTs), one for each inverter leg, for use during normal drive operations. LUTs will be addressed by the phase currents input, and they will return the corrections to the voltage reference vector as output. Since the DC-link voltageUdcmay differ from the one present during the LUTs creation, the output will be corrected by the ratio of the former to the latter.
As it will be evident in Section V-C, the peculiarity of the generated LUTs is a smooth voltage compensation around the phase currents zero-crossing, which accounts for the behaviour of IGBT commutation at low current levels. As a result, the current zero-crossing detection is less critical, since an error in current acquisition due to superimposed noise will result in a very small voltage compensation error. This behaviour is different compared to classic step-like compensation strategies, in which accurate current zero-crossing detection is mandatory and can even cause instability at low speeds if not properly detected.
A. Implementation issues
The first comment is about the implementation of the FFW block (Figure 3). It is not trivial, since the constant component
4td
πTcUdcis referred to a reference frame fixed toϑi, (see (5)).
The use of any other reference frame would make the constant
respect toϑi, so that a repetitive action based onϑi,P LLwould have reduced performances. The use ofϑu∗, supported by the considerations of Section II and Figure 2, turns out to be a simple and effective solution.
In short, for the reasons specified above, the frame associ- ated toϑi,P LLhas been used for the compensation of the fixed term (FFW block), while thexy frame, linked to ϑu∗, has been used for the compensation of the 6 − th order harmonics, by the repetitive algorithm.
Last, a comment on the choice of the motor speed during the self-commissioning procedure. Dead-time distortion is relevant especially at low voltages (and thus, at low speed). This fact would suggest to run the self-commissioning procedure at low speed. On the other hand, the selection of an extremely low speed implies a great memory consumption by the repetitive algorithm, which needs to store a whole period of the input error. Therefore, the choice of 16.67Hz as voltage vector frequency during the self-commissioning is an attentively considered trade-off. Moreover, since the LUTs are obtained at a fixed and predefined speed (16.67Hz, in this case), there is an implicit acceptance of sub-optimal cancellation of IGBTs parasitic effects, only for their speed-dependent part.
Nevertheless, experimental results reported in Section VI show that it is quite negligible.
IV. THE FEEDFORWARDPLL-BASED COMPENSATION
As mentioned before, the constant component of (5) cannot be fixed by the repetitive control, because it is not included in the set of6n-th harmonics. Therefore, it has to be compensated by a focused feedforward action. A dq-PLL system is used to obtain an estimation ϑi,P LL of the ϑi angle, which is used for theabc − dq and dq − abc coordinates transformations, as reported in Figure 4, which itemizes the structure of the FFW block of Figure 3.
The basic scheme of a dq-PLL system is reported in Figure 5-a. Currentsisdandisqare obtained from theabc − dq trans- formation of the measured currentsisa,isb andisc. The angle ϑi,P LLused in the transformation is the output of a closed loop composed by a PI regulator and an integrator, which forcesisq
to zero. Actually, the locked condition is acknowledged when isq is sufficiently small. The center frequency ωc is useful if an estimation of the current frequency is known a priori.
A. dq-PLL stability analysis
Several tuning methods have been proposed in literature for the dq-PLL PI regulator. Many authors also propose the
Figure 4. Feedforward compensation (F F W ) block details
Figure 5. dq-PLL system: (a) complete scheme, (b) simplified scheme for stability analysis
simplified scheme of Figure 5-b for the stability analysis, as reported in ([14], [15], [16], [17]). The dq-PLL is designed to track only the fundamental componentIsdq of the current vector. The time constant τc = 1.5 Tc approximates the sam- pling delay in the digital implementation (Tc is the sampling period). As ϑi− ϑi,P LL approaches zero (dq-PLL in locked condition), the simplificationsin(ϑi− ϑi,P LL) ≈ ϑi− ϑi,P LL leads to a linear system with the following open loop transfer function:
GHP LL= Kp(1 + sτi) sτi
1 1 + sτc
Isdq
s (7)
The phase margin results:
mϕ= π + [−π + arctan(ωattrτi)+
+ arctan(ωattrτc)]
⇒ τi= 1 ωattr
tan (mϕ+ arctan(ωattrτc)) (8) TheKpvalue can be obtained by imposing a unitary amplitude of (7) at the crossover frequency:
Kp(1 + sτi) sτi
1 1 + sτc
Isdq
s
s=jωattr
= 1 (9)
From (9) the value of KpIsdq is obtained:
KpIsdq =τiωattr2 p
1 + ωattr2 τc2
p1 + ω2attrτi2 (10) When the dq-PLL is locked and isq is forced to zero, isd
represents the amplitude of the fundamental component of the current vectorIsdq. In turn,isd is used for the on-line tuning
Figure 6. PI tuning with fixed-bandwidth feature
Figure 7. Experimental PLL output: (a) isacurrent, (b) ϑi,P LLoutput
ofKp, as shown in Figure 6. As a design hint, it is convenient to set a lower bound to isd, to limit the range ofKp. It has been found that a reasonable choice is isd,min = 1% of the nominal motor current.
B. Experimental results for the dq-PLL algorithm
Since it is requested that the dq-PLL follows only the fundamental harmonic of the stator current (which is located at16.67Hz), the crossover frequency of the open loop transfer function (7) should be designed to damp the higher harmonics.
In this work,ωattr= 2π40 rad/s has given satisfactory results.
From (8) and (10), by imposing mϕ= π3 rad, one calculates τi= 222 and KpIsdq= 29520. Figure 7 reports the outline of isaandϑi,P LL during the first step of the self-commissioning procedure, when only the feedforward contribute is active. The residual current distortion, due to the6n-th harmonics, will be compensated by the repetitive algorithm presented in the next section. The estimated angle ϑi,P LL follows accurately the outline of isa, with a zero-crossing in correspondence to isa
maximum, as required by Park’s transformation.
V. THE REPETITIVE CONTROL-BASED COMPENSATION
As explained earlier, the repetitive control technique is applied to cancel the periodic disturbances due to dead-times at steady state. Mathematically, the repetitive block of Figure 3 can be described by the following transfer function in the z-domain [10]:
REP (z) = KREP zKa
zM − FR(z) (11)
Figure 8. Bode diagram of the repetitive block with FR(z) filter
whereFR(z) = NDF RF R(z)(z) is a filter that limits the repetitive con- trol gain at high frequency [12]. The parameterKaintroduces a leading action on the controller to compensate the delay of the process (in this case, the cascade of the voltage inverter and the IM), while KREP is the controller gain. In the case of FR(z) = 1, the repetitive block shows an infinite gain for the frequencies multiple of1/(M Tc), and M can be selected to match the repetitive control resonant frequency with that of the input disturbance. In presence of a filter FR(z), the parameter M is appropriately modified to take into account the phase delay introduced byFR(z) itself. In the present case of a linear-phase FIR filter, it is sufficient to subtract fromM the filter group delay, which is constant. Obviously, sinceM is an integer, the filter taps have to be designed so as to let the group delay be a multiple of the sampling frequency.
In Section III the choice of 16.67 Hz for the reference voltage vector of the V/Hz control has been highlighted. As a consequence, the distortion-related6 − th harmonic is located at 100 Hz, which also becomes the resonant frequency of the repetitive control. A sample frequencyFc = 10kHz (control periodTc= 100 µs) leads to the setting M = 100.
The filter FR(z) is a simple moving average filter with three taps. Its corresponding group delay is Tc, perfectly compensated by choosing M = 99. The Bode diagram of the repetitive block of Figure 3, comprehensive of theFR(z) filter, is shown in Figure 8.
A. Current FIR filters
Although not strictly necessary, the abc phase currents are filtered (only during the self-commissioning) with another moving average FIR filter with three taps, to get smoother look-up tables. To avoid that the filter phase delay affects the look-up tables, a simple compensation technique is used. Let a vector gf = gf,x+ jgf,y, rotating at a constant speed equal toω, be the output of a filter, delayed by ϑ with respect to the filter input. A new vector g, with a phase advance of ϑ with respect to gf is sought. Mathematically, one can write:
gf = ge−jϑ= gej(ωt−ϑ)= g[cos(ωt − ϑ) + j sin(ωt − ϑ)]
(12)
g=
1 − jϑ =
1 + ϑ2gf (14)
The denominator of (14) can be expressed by a Taylor expan- sion:
g≈ (1 + jϑ)(1 − ϑ2)gf (15) whose xy components are
gx= (1 − ϑ2) (gf,x− ϑgf,y)
gy = (1 − ϑ2) (gf,x+ ϑgf,y) (16) Since ϑ is the phase delay of a FIR filter, which is inherently linear, the (16) are of immediate application.
B. Stability analysis of the repetitive algorithm
The repetitive control is activated at steady state in a precise and predetermined working condition, which in the present work was characterised by a reference voltage vector frequency of 16.67 Hz, at no load. The following stability analysis has anyway general validity.
The IM state-space model, expressed in a generic frame rotating atωg, is given by
d dt
is
λr
= A11 A12
A21 A22
is
λr
+B1
0
us
y = [ 1 0 ] is
λr
(17) where vectors and matrices assume the following form:
is= [ isx isy]T stator currents λr= [ λrx λry]T rotor fluxes us= [ usx usy]T stator voltages
A11= − 1 Lt
Rs+ Rr
L2m L2r
I − ωgJ A12= Rr
Lt
Lm
L2rI − ωme 1 Lt
Lm
LrJ A21= RrLm
Lr
I A22= −Rr
LrI − (ωg− ωme)J B1= 1
Lt
, I =h 1 0
0 1 i
, h 0 −1 1 0
i
Rs, Rr, Ls, Lr are the stator and rotor resistances and self- inductances respectively, Lm andLtare the magnetising and
Figure 9. Block scheme for the stability analysis of the repetitive control
Figure 10. Subsystem for the stability analysis of the y axis loop
transient inductances andωme is the electromechanical motor speed. In this work, ωg is set equal to the reference voltage vector speed, to obtain the IM model in the xy reference frame. It is worth to note that in the xy frame there is a cross-coupling between the axes that cannot be neglected a- priori. From (17) and its specifications, the transfer functions between the inputsusx,usyand the outputsisx,isyare readily obtained. Consequently, the block scheme for the stability analysis can be rearranged as depicted in Figure 9.
For eachxy axis, the control input is the difference between the actual current ripple and its reference, which is obviously null (Figure 9). At a glance, the y axis can be viewed from pointA to point B as a separate subsystem, described by the block scheme of Figure 10, in which it is
F5= KREPNF
DF
NP A
DP AzKa
zM −
NF R
DF R − KREP
NF
DF
NP A
DP A
N1
DENzKa
| {z }
H2(z)
F6= F22F5 (18)
As reported in [12], a sufficient condition for the y axis stability is:
|H2(z)|z=ejωTc < 1 (19) in which ω sweeps from zero to the Nyquist frequency FN = Fc/2. Figure 11 shows the H2(z) plot in the Nyquist coordinates at no-load, forω ∈ [0, FN]. It has been obtained with the parameters of the IM 1 (see Appendix A) and by setting KREP = 0.03 and Ka = 3. The condition (19) is satisfied and thus F5 is stable. Other simulations have been performed by varyingKREP from 0.1 to 10 times the nominal values, and by varying the slip speed from 0 to 5% ofωg. It
Figure 11. H2(z) as function of z = ejωTc
Figure 12. Block scheme for the stability analysis of the x axis loop
has been experienced that H2(z) is scarcely affected by such variations, and it always remains within the unity circle.
SinceF2is stable, from (18) the stability ofF5implies that of F6, too. With reference to the block scheme of Figure 12, the same procedure can now be applied to the stability analysis of thex axis. The transfer function eisx/ei∗sx is calculated by expanding theREP block, obtaining:
eisx
ei∗sx
= KREPNF
DF
NP A
DP A(F1+ F6)zKa
zM −
NF R
DF R − KREP
NF
DF
NP A
DP A
(F1+ F6)zKa
| {z }
H1(z)
(20) As before, the system is stable if:
|H1(z)|z=ejωTc < 1 (21) withω ranging from zero to FN. Using the same parameters as for theH2(z) analysis, and by performing the same robustness analysis, it has been found that the diagram of H1(z) in the Nyquist coordinates always resides within the unity circle, proving the overall stability of the proposed repetitive control.
C. Look-up table creation and elaboration
An output example of the self-commissioning procedure is reported in Figure 13, which shows the look-up table for the a phase of IM 1. The profile of Figure 13 is directly obtained by adding the repetitive control output and the FFW contribute (Figure 3), but it still needs some further post- processing to be effective. As depicted in Figure 13, the multiple step-like behaviour is due to to the lack of information about the homopolar component (6), which is lost in the xy reference frame. Analysing the contribute of the homopolar
Figure 13. Experimental acquisition of a look-up table
Figure 14. Look-up table post-processing
component around each current zero-crossing, with the aid of (6), it is soon demonstrated that the compensation voltage without homopolar component is 2/3 of the full one. Therefore, around the zero-crossing, the homopolar component can be recovered by multiplying the values in Figure 13 by the term 3/2, while the extremes of the look-up table are substituted by the conventional compensation (1). This makes sense because the outer side values correspond to higher phase currents, in which the IGBT behaviour is less critical. The resulting post- processeda phase current LUT is shown in Figure 14.
The table has a resolution of about 7 mA. As expected, for low current levels the ideal step of (1) is smoothed by IGBT and diode parasitic effects. Moreover, and according to the common understanding of the commutation phenomena, the small hysteresis shows that there is a slightly different behaviour for negative and positive derivative of the phase currents. Consequently, a precise compensation could make use of two different look-up tables for the two cases.
VI. EXPERIMENTAL RESULTS
The proposed procedure has been evaluated on three dif- ferent induction motors, whose parameters are reported in Table I, making use of a fast control prototyping laboratory setup. The filter and repetitive block parameters are the same as those used in the stability analysis of Section V-B for all IMs, except forKREP that has been experimentally increased from 0.03 to 0.5 in the specific operating point, reducing the execution time while preserving the system stability. Figure 15,
Figure 15. a) phase voltage and b) current: no compensation, 60 rpm
Figure 16. a) phase voltage and b) current: step-like compensation, 60 rpm
Figure 17. a) phase voltage and b) current: proposed compensation, 60 rpm
Table II
COMPARISON OF VOLTAGETHD (EXPERIMENTAL RESULTS, @60RPM)
THD THD THD
no compensation conventional proposed
IM 1 0.23 0.17 0.09
IM 2 0.30 0.21 0.05
IM 3 0.37 0.17 0.05
16 and 17 report a comparison between voltage and current measurements for IM 1 at no load with a rotating frequency of 1 Hz (60 rpm), in case of no dead-time compensation, conven- tional step-like compensation and compensation based on LUT of Figure 14 respectively. Very similar results are obtained with all the other IMs. As a measure of the effectiveness of the proposed solution, Table II e III collect the voltage and current THD obtained from experimental measurements on the different prototypes. The advantages of the proposed solution are clear.
Table III
COMPARISON OF CURRENTTHD (EXPERIMENTAL RESULTS, @60RPM)
THD THD THD
no compensation conventional proposed
IM 1 0.09 0.08 0.05
IM 2 0.11 0.05 0.02
IM 3 0.12 0.09 0.03
VII. CONCLUSIONS
In this paper, a self-commissioning procedure for an accu- rate compensation of inverter non-idealities have been pre- sented. The proposed approach uses repetitive control to create application-specific look-up tables, which can be used instead of a conventional step-like compensation in an IM electric drive, while maintaining a low-cost profile and low computational effort.
A complete stability analysis has been developed, and experimental results on three IMs have been presented. The comparison between the voltage and current THD in the three cases further confirms the effectiveness of the proposed solution, which is going to be implemented by an industrial partner.
APPENDIXA
IM 1ELECTRICAL PARAMETERS Rs 2.74Ω Rr 1.70Ω Ls 0.161 H Lr 0.161 H
Lm 0.157 H p 1
APPENDIXB
DEAD TIME DISTORTION IN A REFERENCE FRAME ROTATING WITHϑi
In order to obtain the expressions (5), let us consider a balanced three phase current system:
isa= I cos(ϑi), isb= I cos
ϑi−2π
3
, isc= I cos
ϑi−4π
3
(22) and the following transformation matrices (from theabc frame to the stationaryαβ frame and from the αβ frame to the dq frame, which rotates with a generic angleϑg):
Tabc/αβ=2 3
1 −1
2 −1 2 0
√3
2 −
√3 2
, Tαβ/dq= cos(ϑg) sin(ϑg)
− sin(ϑg) cos(ϑg)
(23) From (1) and (22), the sgn(isa) function is described by:
sgn(isa) =
1, for 0 ≤ ϑi<π 2
−1, forπ
2< ϑi<3π 2 1, for 3π
2 < ϑi< 2π
(24)
Each sgn(isk) function can be easily expressed as a Fourier series. For that purpose, the following definition of Fourier
series of a genericf (ϑi) function has been used:
f (ϑi) =1 2a0+
+∞X
n=1
[ancos(nϑi) + bnsin(nϑi)] (25) where:
an=1 π
Z π
−π
f (ϑi) cos(nϑi)dϑ, bn=1 π
Z π
−π
f (ϑi) sin(nϑi)dϑ (26) Since sgn(isa) is an even function, its Fourier coefficients are immediately obtained as follows:
an= 1 π
Z π
−π
sgn(ia) cos(nϑi)dϑi= 4
nπsin nπ 2
bn= 1 π
Z π
−π
sgn(ia) sin(nϑi)dϑi= 0 (27) Considering that sgn(isa) leads sgn(isb) and sgn(isc) by 2π/3 and4π/3 respectively, it is:
sgn(isa) = X+∞
n=1
4
nπsin nπ 2
cos (nϑi)
sgn(isb) = X+∞
n=1
4
nπsin nπ 2
cos
n
ϑi−2π
3
sgn(isc) = X+∞
n=1
4
nπsin nπ 2
cos
n
ϑi−4π
3
(28) Since the term sin (nπ/2) is equal to ±1 for odd values of n and is null for even values of n, expressions (28) can be rewritten as follows:
sgn(isa)=
+∞X
n=0
(−1)n 4
(2n+1)πcos ((2n+1)ϑi) sgn(isb)=
+∞X
n=0
(−1)n 4 (2n+1)πcos
(2n+1)
ϑi−2π
3
sgn(isc)=
+∞X
n=0
(−1)n 4 (2n+1)πcos
(2n+1)
ϑi−4π
3
(29) Substituting (29) into (1) and transforming in theαβ stationary reference frame, one can obtain:
udist,sα=
+∞X
n=0
uαMcos ((2n + 1)ϑi)
1−cos
(2n + 1)2π 3
udist,sβ=
+∞X
n=0
uβMsin ((2n + 1)ϑi) sin
(2n + 1)2π 3
(30) where:
uαM = (−1)n 8 3π(2n + 1)
td
TcUdc, uβM =√
3uαM (31) As can be easily demonstrated, (30) are null for even harmon- ics and for multiples of the third harmonic. As a consequence, (30) can be rewritten as:
udist,sα= 4 π
td
Tc
Udc cos(ϑi) −X
α
!
udist,sβ= 4 π
td
TcUdc
sin(ϑi) +X
β
(32)
udist,sd= 4 π
td
Tc
Udc1 −X
α
cos(ϑi) +X
β
sin(ϑi)
udist,sq= 4 π
td
Tc
Udc
X
α
sin(ϑi) +X
β
cos(ϑi)
(34) which are those reported in (5).
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Silverio Bolognani (M’76) is a native of the Trento province, in the North of Italy. He received the Laurea degree in Electrical Engineering from the University of Padova, Italy, in 1976. In the same year, he joined the Department of Electrical Engi- neering at that University, where he was involved in the analysis and design of thyristor converters and synchronous motor drives. After that, he started the Electrical Drives Laboratory where a variety of researches on brushless and induction motor drives are carried out in the frame of European and Na- tional research projects. He is presently engaged in researches on advanced control techniques for motor drives and motion control and on design of ac electrical motors for variable speed applications. He is author of 3 patents and more than 200 papers on electrical machines and drives. He has been serving International Conferences as member of the Steering or Technical Committees, as well as invited speaker. At present he is Chairman of the IEEE North Italy IA/IE/PEL Joint Chapter. His teaching activity was first devoted to Electrical Circuit Analysis and Electromagnetic Field Theory and, later, to Electrical Drives and Electrical Machine Design. He is now Full Professor of Electrical Converters, Machines and Drives and Head of the Department of Electrical Engineering at the University of Padova.
Luca Peretti was born in Udine, Italy. He received the M. S. degree (with honors) in electronic en- gineering from the University of Udine, Italy, in 2005. During 2005, he was in the Electric Drives Laboratory at the Department of Electrical, Manage- ment, and Mechanical Engineering of the University of Udine, Italy. In 2006, he joined the University of Padova as a Ph.D. student in Mechatronics and Industrial Systems at the Department of Technique and Management of Industrial Systems, Vicenza, Italy. His main research activity concerns sensorless control and parameter estimation techniques for electrical motor drives and magnetorheological fluids in industrial and civil applications.
Mauro Zigliotto (M’88) is a native of Vicenza, Italy.
After the degree, he worked in industry, as R&D manager, developing microcontroller-based control systems for electric drives. From 1992 to 1999, he was a Senior Research Assistant in the Elec- tric Drives Laboratory, University of Padova. Since 2000, for five years, he worked at the Department of Electrical, Management and Mechanical Engineer- ing of the University of Udine, as Associate Profes- sor of Electric Drives. In November, 2005 he joined the Department of Technique and Management of Industrial Systems, University of Padova, where he started the Electric Drives Laboratory. Advanced control strategies for ac motors are Prof. Zigliotto’s main research interest, and he has published about extensively in this area.
He has been serving International Conferences as member of the Steering or Technical Committees. At present he is the secretary of the IEEE IAS-IES- PELS North Italy Joint Chapter.
