Abstract: The current waveforms for optimal excitation of surface- sinusoidal hack emf motors are excited with sinusoidal current mounted permanent-magnet ...
IEEE Transactions on Energy Conversion, Vol. 14, No. 4, December 1999
1043
Optimal Current Control Strategies for Surface-Mounted Permanent-Magnet Synchronous Machine Drives P.L. Chapman, Member
S.D. Sudhoff, Member
Purdue University West Lafayette, IN 47907-1285 Abstract: The current waveforms for optimal excitation of surfacemounted permanent-magnet synchronous machines are Set forth. Four different modes are considered, involving varying degrees of minimization of rms c ~ ~ and ~ ntorque t ripple. m e optimized waveforms are markedly different than the traditional sinusoidal or rectangular excitation schemes. Inclusion of cogging torque and arbitrary degree of torque ripple minimization generalizethis work over that of previous authors. An experimental drive and a detailed computer simulation verify the proposed control schemes. Keywords: permanent-magnet motors, optimization, torque ripple,
hmshless dc
I.
INTRODUCTION
Permanent-magnet synchronous machine (PMSM) drives are arguably the most efficient drive available due to minimal rotor losses and the absence of magnetizing CUrrent. These machines can be constructed with either sinusoidal or nonsinusoidal hack emf waveforms. These latter machines have been shown to offer greater power density, however they are normally subject to IoW-freqUency torque ripple unless it is actively mitigated [l]. In designing PMSM drives, two factors Of interest are efficiency,' and, for some applications, minimization of torque ripple. There are two general methods by which both of these areas can he addressed. The first involves the design of the machine itself in order to maximize efficiency and minimize torque ripple (note, however, that these are not the same objective and a compromise must he made for a given application). The second method relies on control Of the inverter to establish the optimal operation of a given machine [l]. Again, the maximization of overall efficiency must be weighed against minimization of torque ripple in SitUatiOns where torque ripple is objectionable. This paper focuses on the second method of minimizing losses and torque ripple through appropriate choice of the excitation waveform. Although it is often the case that PE-227-EC-0-01-1999 A paper recommended and approved by the IEEE Electric Machinery Committee of the IEEE Power Engineering Society for publication in the IEEE Transactions on Energy Conversion. submitted May 18. 1998; made available for ... . Manuscript .^^^ printing January LU, .IYYY. ^^
C.A. Whitcomb, Member Massachusetts Institute of Technology Cambridge, MA 02139-4307 sinusoidal hack emf motors are excited with sinusoidal current [2] and trapezoidal back emf motors are excited with rectangular currents [3], neither of these ideal hack emf's exist. Furthermore, it has heen shown that rectangular curre,,t excitation of trapezoidal is not only to achieve, hut also requires higher rms current than is actually necessary. There has been considerable work in the area of choosing optimal current excitation of PMSM drives to address these issues, For example, in [4-51 a method is set forth wherein selected current harmonics are injected to cancel a few of the lower frequency torque harmonics. A method for canceling all torque harmonics and simultaneously minimizing the rms stator current was set forth in [6]. A similar result was obtained in [7] although the mathematical analysis is quite different. These ideas were extended in [8] where constrained inverter voltage was considered and numerical optimization was used. Unfortunately, all of these papers neglect cogging torque. Although cogging torque is included in [9], that paper does not address current minimization. Further, [9] utilized an iterative method to calculate the appropriate current harmonics that would he difficult to implement in a real-time control in which the torque command is varying. Two further disadvantages of all of these approaches are that the optimized solutions require a number of current harmonics that depends directly on the hack emf waveform, which detracts from the designers freedom, and that [4-81 lack experimental verification. This paper, which is loosely based on the analysis in [lo], demonstrates methods whereby rms stator current and torque ripple may he minimized to an arbitrary degree with the effects of cogging torque included. This allows tradeoff of rms stator cnrrent for torque ripple as the application demands, effectively tying together the functionality of [4-81 while providing several advantages. These advantages include the fact that the number of current harmonics in the proposed optimal solution is not determined uniquely by the hack emf allowing the designer additional discretion in balancing torque ripple, rms stator current, and complexity of the control. In addition, unlike Other methods that address coaeinr toraue . 191. the method uresented is non-iterative and therefore amenable to real-time opimization as the command exDerimental
---
-
confirmation of the urouosed control schemes using a commercial PMSM is p r h d e d . 11. MACHINE MODEL
The permanent-magnet synchronous machine model is set forth in this section. In general, the voltage equation of a three-phase machine may he expressed as
0885-8969/99/$10.00 0 1999 IEEE
1044
where rs is the stator resistance, kabc is the flux linking the such that kobc represents the normalized back emf with stator windings, and io,, is the vector of stator currents. In (I) k: =mkGy and P is the number of machine poles, The and throughout this work, vectors are in the form cogging torque, Tcog, may be expressed fahc
=ka
fb
fc1T
and may designate voltage, flux linkage, or current. Furthermore, any vector fahc is represented trigonometric Fourier series fashion as
(2)
T~~~= ~ ~ ~ c o s ( z d ~ n(z8,) , ) + ~ ~ s i in
(7)
Z€. 7
Z = { z = 3 y : y ~ N , T #0] i~
where T: and Tiare constants associated with the Fourier series of the cogging torque. The set Z is defined as such since three-phase machines with integer or half-integer numbers of slots per pole per phase can be shown to have only triplen cogging torque harmonics in 8,. In case the machine under study has an integer number of slots per pole per phase, which is more commonly the case, the restriction may be narrowed to cover only z=6y in Z. However, the analysis herein is not f&"' are the appropriate Fourier series coefficients written in substantially simplified by this restriction. terms of q- and d-axis components [lo], and will be written 111. TORQUE RIPPLE ANALYSIS = bF Furthermore, I is the more compactly as Since the optimization process to follow involves set of integers, and N is the set of natural numbers. The set M is elimination or minimization of torque ripple, it is necessary to defined such that each harmonic m exists only as a positive, express the components of the ripple in closed form. This negative, or zero sequence, since only balanced, symmetrical begins by expanding (6) by substituting the series representation machines are considered. For example, a 5" harmonic only (3) in for both current and normalized back emf. In the exhibits a negative phase sequence, and therefore is represented following, the sets M and N correspond to the back emf and with m=-5. Triplen harmonics are always zero sequence current harmonics, respectively. The first term of this quantities, and it is redundant to define zero sequence for m expansion corresponds to the product of normalized a-phase both positive and negative, therefore, the restriction that y E N back emf and a-phase current: is on M. Under the assumptions that the machine has surface T ~=-.a', P~ ~ , mounted magnets (and is therefore non-salient), incurs no eddy mMnsN currents in either the rotor or stator structures, and exhibits no magnetic saturation, the flux linkage equations may be written k r i r (cos((m-n)B,) + cos((m+ n)8,))+
fz
fr]'.
CC[
/Zabc = L s i ~ h c +akkLbc
(4)
where A! defines the peak strength of the fundamental d-axis component of stator flux linkage due to the magnet, and L, is an inductance matrix of the form
+ krir (sin((m + a)@,)+ sin(@ - n ) ~ ? , )+) kFiT (sin((m+n)B,) k"'i"' d
-sin((m -n)B,))
(cos((m-n)~,)-cos((m+n)B,))I
(8)
The total expression for torque is then
where the (b) and (c) components of (9) are obtained by substituting (8, and (8, ) ;+ into (8), respectively. where L. and L, are the stator self and mutual inductances, Analysis of (8-9) yields that the summations of cosine and sine respectively. In (4), kbhc is a trigonometric shape function in are zero except for m f n E ( 3 x :x E I] Therefore, there are torque ripple components only at frequencies that are integer the form of (3), normalized such that k i r = I , B~ multiples of three times the electrical rotor speed. By letting coenergy techniques, the electromagnetic torque is expressed as
-2)
..I
.
1045
h = Im fnl , one may formulate two separate equations for each First, this reduces the current sourced by the inverter and possible harmonic, h; one each for cosine and sine in the forms therefore decreases inverter conduction losses. Second, it reduces stator losses in the machine, though strictly speaking, T&,pie = R,h cos(he,) (10) stator losses do not exactly scale with the rms current due to skin effect, hysteresis, and eddy currents. However, it shall he assumed that the frequency independent effects at the Tdh,rippipp* = Rdh sin(h6,) (11) frequencies of interest dominate stator loss. Therefore. for minimization of system loss herein, it will he considered where the coefficients are given by sufficient to minimize rms current. It is necessary to define two vectors of interest. The first is ~h = T h + 3 P a / [,,;th #n-h)q &h * k;b-/i))t;'](12) the vector 1 1 2 2 m of currents to he optimized. #EN +
R: = T:
+?Ea! 2 2
+
C [ ( X d n + h F b *db - h ) ) i " '1+ ( _ X " + 1h + , * t 64- h ) ) i ~
]
(13)
nE N
...r
iy
t/aiEN
(18)
Similarly, a vector of hack emf constants may he defined.
In (IZ-l3), further simplification has been achieved by realizing that each harmonic exhibits a positive, negative, or zero phase
Although (10-13) represent the general expression for torque ripple harmonics, there is some maximum value of h, and therefore some finite number of ripple equations, that need he considered since N, M, and Z are finite SetS. In Pa'tiCUh, h,,,
i = l qa lr i qaz r ...
= max{lm - nl,lm + nl, z )
Ear P
kpz' P
...
k?'
..] V p i E M
kda,.
(19)
Note that this only applies if M=N to preserve dimensionality. However, in case this requires N to he undesirably large, M may he truncated at the designer's discretion. Given the constraint that I:, is to be minimized the minimum solution [I 11 may be used to find the optimal current vector which is
(13
VrnEM,neN,zeZ,hm,E3N The harmonics that must then be considered are given by h E H = ( 3 x :x 5
k=
-
i,,
Tf1 = k T?c$,, kkT
(21)
2 2
(16) where MC stands for 'Minimum Current' mode. Note that the current harmonics in this case are directly proportional to the It is also convenient to specify the rms torque ripple for hack emf harmonics so that the current waveform should have later analysis. the exact same shape as the back emf. Often, PMSM's with nearly sinusoidal hack emf are (17) excited with only fundamental q-axis current (i.e., purely T A= (R,h + he X sinusoidal). This case, designated as MCS (Minimum Current,
N]
E[ Y kdh Y]
Equation (17) is the same for
ir and ';i
,except the sum is taken over Sinusoidal) mode, is a special case of (21) where k =
-
.
In the analysis to follow, it is assumed that the hack emf of the motor is given and fixed (i.e. k % and Mare given) and the current vector, i::, , and set N, are to he determined.
IV.MINIMUM CURRENT EXCITATION With the torque ripple equations defined, the stage is set to derive the current vector for various modes of operation. It is of interest to minimize the rms current for two principle reasons.
iMCS=,,=
Te*
E,] q
E;]
= 1. (74
?$in
Here, it is assumed that the rotor position sensor of the drive has been set such that all of the fundamental back emf is in the qaxis. This case will act as a comparative basis for the other optimization routines.
1046
V. MINIMUM CURRENT, CONSTANT TORQUE EXCITATION
harmonics is presented. This will be designated MCMR (Minimum Current, Minimum RiDDk) mode. The relative degree to which each quantity IS minimized is defined by the designer and may easily be changed to suit a given application. The minimization procedure in the previous section utilized solution, however now that quantitiesare to be minimized, the more general optimization method of LaGrange multipliers [121 is used. The function to be minimized is given as -* f=ar&++T;,+n(+T, - T ~ ) (26)
In many applications, it is necessary to achieve as close to constant torque as possible. Therefore, the current harmonics will be selected to cancel all torque harmonics, including those due to cogging torque, such that a given average torque is maintained, This case shall be designated as MCNR (Minimum Current, No Ripple) mode. The current command to achieve constant torque can be obtained by setting (12-13) to zero and setting (14) equal to the value' This set Of equations may be where the positive constant a may be specified to control the represented in the form relative amount of minimization to be placed on therms current. Ki=t (23) The only equality constraint is that the average torque be met. where the matrix is formed (12-14) such that one row This constraint is multiplied by the Lagrangian multiplier, A . contains the of (14) and each additional row A linear system of equations with a unique solution is formed corresponds to one value of h (a torque harmonic) in (12-13). by setting to zero each partial derivative offwith respect to each Each column corresponds to a value o f n (a current harmonic) in current and 2 . Utilizing (l2-14,17), these partial derivatives the current vector i (18). The t vector contains the average may he written as torque command, T,' , and the negatives of the cogging torque
af = 0 = 2 a i r - 12 2p n ; q -
harmonics, T," and T j , wbicb correspond to the TOWS of K. a i 4p r Once this matrix is formed, there must be at least as many ,(kTh columns as rows to ensure a non-singular system. To make sure this is the case, the set N must be defined as discussed below. + Solving (23) using the minimum norm solution [ 111 yields
hsHnsN +
kt(p-l)
4
)(k;b
[(lrr"+
i,,
+&,tip E
+
k;(n-h))
+
@+h
k;(P-h))(k;h
k;(*-h)
)It,.nr
-
)]ir]
+ k : ( P - h ) ) + T j ( k $ + h Tk:b-h))]
(27)
k i ( P - h ) ) ( t y h-t k i ( " - h ) ) + ( k T h 7 k i ( p - h ) ) ( k i ( " - h ) k r h
=KT(KKT)-lt
This procedure minimizes the rms current subject to the constraint of eliminating electromagnetic torque ripple. a f = 0 = 2aiF - + f n ; n F + 2 However, for the MCNR case, the set N has yet to be specified. heHnc;N In order to eusnre that all ripple components are exactly k ~ - l E l l ( k *ky-ll)+(kyh ~ ~ Tk~-hl)(k;~-hl -k ~ ' ] ] i ~ cancelled, the number of columns of K must meet or exceed the [(k;h number of rows. Therefore, the set N should be built up until +[pi+**kf'-hJ)(k;+'l ik;("-bJ) + ( k * (4 n-b' - k;+hl(k;(l-*l k;+'*l]id"') this is the case. For some sets of back emf and cogging torque harmonics, a square matrix can be produced for a particular set + ~ : 2 ~ $ ( , ] [ ~*k~"-'") , ( k ~+ + ~ j (~k : ( ~ - ' " - k4p+'')] (28) ~ C H N. While this K would result in currents that exactly cancel the torque harmonics and produce the required average torque, the minimum current constraint is ignored. Therefore, forming a = 0 = T," -?pa/ 2 2 m x k " rq' E" 'q + k r Q ' (29) an square K does not necessarily result in an optimal solution. "EN
air
+
af
VI. MINIMUM CURRENT, MINIMUM TORQUE RIPPLE EXCITATION
where p E N . These equations, when written for all p in the set N , compose a non-singular linear set that may be easily solved for the desired currents. It is interesting to note that in this case, no restriction is placed on N. This allows the designer considerable flexibility in both the selections of harmonics to be used and in the relative minimization of torque ripple versus TmS current.
In the previous section, it was required to produce zero torque ripple with the smallest rms current excitation. This required a given number of current harmonics for a certain motor configuration. However, in many cases it is only uractical to consider a uarticular order of current harmonics due to limitations on the precision of the rotor position sensor, VII. NUMERICAL EXAMPLE bandwidth of the inverter controller. meed and memorv TO validate the methods presented in the previous sections, constraints of the control electronics, and the relatively low levels of current that are sometimes given by (21) or (25). In the optimization routines were implemented for two motors, one this section, a method of simultaneously minimizing torque Of which is a test motor with moderately non-sinusoidal back ripple and rms current with a given number of current emf and the other is a hypothetical trapezoidal motor. Each
1.0
0.6
-
.
ka 0.0
.
.
-0.6 -1.0
15
...
. . . I
-
8 .
3E: 8 1 0 .
$3 4 .$ .s 8
p"
'
42
L L
.., ......;.'
0-
...I
2.0
sinusoidal .trapezoidal
. 1.6 .
..'' ' ' ' ' ''
1.8
-
g g
1.4
..+.
60
.
$ 2
3
.s
22 1.0 .
'
.n
eP
0.6
0.4 . 0.2 0.0,
.
V,(W
40
[ ;0.8 .. ....." ......
..
sq--wave
Fig. 6 Percent decrease in I'R losses as a function of bsck emf wave shape; MCNR mode VS. MCS made
I
3 1.2 .
--+ Back emf waveform
.
~
1049
-1.5 2
1
T,(Nm) (sirn.)
1
01
zr
0 Fig. 8 Simuloted, measured signals for MCS mode
1 Fig. 9 Simulated, measured signals for MC mode
1.5 I
reached. Discussion of this regulator is beyond the scope of this paper; however, it will be presented in detail in a future paper. In Figs. 8-10, since a suitable torque transducer was not available, the measured torque waveform was obtained by digitally capturing the abc currents and applying (6-7) off-line. Although this method relies on precise knowledge of the back emf harmonics, this method was validated in [13] by comparing it to a high bandwidth torque observer that utilized a very highresolution position resolver. In Fig. 8, the current vector for MCS mode is simply i' = 1A , which has an rms value of 0.7071 A. For MCNR P mode in Fig. 10, the optimal currents are ib = 0.9784A, i 43 = -0.1153A, and
ii5 = -0.0198A1which results in an rms
value of 0.6968 A. This reduction in rms current comes in spite of the increased harmonic content since the fundamental is reduced significantly. From Fig. 10 it can be seen that almost no low kquency torque ripple (essentially zero) is present in MCNR mode. Fig. 8 shows there is also very little torque ripple present in MCS mode, since this machine is only somewhat non-sinusoidal. In particular, this is because the third harmonic, which contributes nothing to the average or ripple torque with sinusoidal excitation, is the dominant harmonic. The only ripple present in Fig. 8 is due to the fundamental current interacting with the small 51h harmonic
^. LI
0
Fig. 10 Simulated, measured signals for MCNR mode
1
1050
back emf, k;5
,
This interaction gives a ripple component (12) [7l
at 68, with a magnitude of 0.0084 Nm, or 0.6% of the average torque. Despite the modest improvement in torque ripple, the MCNR mode does require lower nns current than MCS mode, as stated above. Larger improvements in torque ripple occur for more non-sinusoidal motors such as the hypothetical motor presented in the previous section. Fig. 9 shows that MC mode produces the most obvious degree of torque ripple, however, it has the lowest current (0,6965 A), in this case, MC mode rms current is not significantly lower than MCNR mode, which suggests that for this motor is advantageous to use MCNR mode since it produces much less torque ripple. The high frequency torque ripple evident in ~ i 8-10 ~ is ~due , to the hysteresis switching of the inverter.
IX. CONCLUSIONS of current control for arbitrary back emf Several PMSM drives have been presented and compared. The control strategies minimize rms current and torque ripple due to both cogging and interaction of the Current and airugap flux. One mode allows the designer considerable flexibility in the relative minimization of rms Current versus torque ripple and in the selection of current harmonics that are most feasible to implement in hardware for the given application. These of both a strategies were illustrated numerically in the hypothetical trapezoidal motor and a test motor. All modes, regardless of the degree of torque ripple considered, were shown to produce reduced rms current, particularly as the back emf wavefonn becomes more non-sinusoidal. An experimental drive coupled with a detailed computer simulation further demonstrated the effectiveness of the proposed strategy. X. ACKNOWLEDGEMENTS The authors wish to acknowledge the Office of Naval Research for financial support and Todd Walls of Emerson Electric Company for donating the test motor.
XI. REFERENCES T.M. Jahns, W.L. Soong, "Pulsating Torque Minimization Techniques for Permanent Magnet AC Drives-A Review," IEEE Transactions on Industrial Electronics, Vol. 43, No. 2.1996, pp. 321-330. P.C. Krause, 0.Wasynczuk, S.D. Sndhoff,Analysis of Electric Machinery, IEEEPress, 1995. T.M. Jahns, 'Toque Pmduction in Permanent-Magnet Synchronous Motor Drives with Rectanaular Current Excitation," IEEE Transactionson Industry Applications, VocIA-20, No. 4,1984, pp. 803-813. H. Le-Huy, R. Perret, R. Feuillet, "Minimization of Torque Ripple in Brushless DC Motor Drives," IEEE Transactions on Industry Applications, Vol. IA-22, No. 4, 1986, pp. 748-755. F. Piriou, A. R m k . R. Perret, H. Le-Huy, 'Toque Characteristics of Brushless DC Motors with Imposed Current Waveforms," Conference Record: /AS Annual Meeting (Industrial Applications Sociely). 1986, pp, 176-181. J.Y. Hung, 2. Ding, "Minimizntion of Toque Ripple in PermanentMagnet Motors: A Closed Form Solution," Proceedings of the I t hIEEE Industrial Electronics Conference, 1992, pp. 459-463.
c.M g , 1. Ha, "An Efficient Torque COnhol Algorithm for BLDCM with
a General Shape Back EMF," PESC Record- Power Electronics $ecialists conference, 1993. pp. 451-457. [81 D.C. Hanselman, "Minimum Toque Ripple, Maximum Efficiency Excitation of Brushless Permanent Magnet Motors," IEEE ~ m n r a ~ f i o ~ on Industrial Electronics. Vol. 41, No. 3,1994, pp, 292.300.
[91
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1131
koLTi~o~p~~~~%;e~p~:''~,
~ ~ " &Y?:"i
m E Tranractions on industryApplicationr, vol. 29, N ~ 6,. 1993, pp. 1141.1149, P.L. Chapman. S.D. Sudhoff, C.A. whitcomb, "Multiple Reference Frame Analysis Of Non-sinusoidal Bmshless DC Drives," accepted for publication in IEEE Transactions on Energy Conversion. 1998. W.L. Bmgm, Modem control prentice H ~ I1991, , D. Zwillinger (Ed.), Standard Mathematical Tables and ~ ~CRC Press, 19%. s. b g z s. SUI. "Direct Torque COntd Of Brushless DC Motor With Nonideal Trapezoidal Back EMF," IEEE Transactions on Power Electronics, Val. IO, No. 6, 1995, pp. 796.802,
PatrickL. Chapman 1974- (S '94, M '96) is native to Cenhlia. Missouri. He received the degrees of Bachelor of Science and Master of Science in Electrical Engineering in 1996 and 1997, respectively, from the University of MissouriRalla. currently. he is pursuing a Ph.D. in Elechical Engineering at Purdue University. During his education, he has conducted research in the areas of power electronics, elecvic machinery. and solid-state power systems, scotL D. Sudhoff (M .88) received the BSEE, MSEE, and ph,D, degrees from Purdue University in 1988,1989, and 1991, respectively. prom 1991.1993 he served as a half-time v ng faculty and half-time consultant for P.C. Krause and Associates. From 1993-1997 he served as an assistant professor at the University of Missouri-Rolla and became an associate professor at U M R in 1997. later in 1997, he joined the faculty at Pudue University as an associate professor. His interests include the analysis, simulation, and design of elechic machinery, drive systems, and finite ineha power systems. He has authoned or CO-authomJ overtwentyJoumal Papers in these
LCDR
whlteomb, USN (M ,97) received a BS in Nuclear Engineering from the university of Washingtan, seanle,WA in 1984, an SM in ElecvicalEngineering and Computer Science from MIT in 1992, a Naval Engineer degree from MlT in 1992, and a Fh.D. in Mechanical Engineering from the University of Maryland in 1998. His current position is as an Associate Pmfessor of Naval ArchitecNre and Marine Engineering at MIT in Cmhridge MA. Until recently he was a program Officer in the Ship Sttuc~resand Systems Science and Technology Division (ONR 334) at the Office of Naval Research in Arlington Virginia overseeing basic research in elechical distribution systems, as well as applied research for surface ship and submarine machinery and electrical systems. He was also the Systems Engineering Manager for the Power Electronic Building Block (PEBB) project team at ONR. His 21 years of naval service has included tours onhoard the USS SCAMP (SSN 588). shipwork coordinator for consttuction work on nuclear attack submarines at the Supelvisor of Shipbuilding, Conversion, and Repair, Croton. Connecticut. and as a ship research design engineer for design of submarine systems at the Naval Slirface Warfat Center, Carderock, Maryland. His main research interests are in multidisciplinary design optimization of complex systems and electric power technology applications to naval combatant ships.
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