ΦAbstract – Optimal waveforms for maximum torque per ampere control of Permanent Magnet Synchronous Machines were explored via numerical techniques in the past. This paper introduces an analytical approach for computation of the optimal stator excitation in Permanent Magnet Synchronous Machines including surface mount and interior permanent magnet configurations. Furthermore, a unified solution for optimal current excitation of Permanent Magnet Synchronous Machines and Synchronous Reluctance machines are presented. The developed analytical solutions enable the online calculation of optimal excitation for minimum copper loss. It also provides guideline for multi-objective optimization when efficiency is of interest. Simulation results are provided for validation.

Index Terms— Maximum Torque Per Ampere, Minimum Copper Loss, Optimal Excitation, Permanent Magnet Synchronous Machine, Interior Permanent Magnet Machine, Synchronous Reluctance Machine

I. INTRODUCTION PTIMAL excitation of Permanent Magnet Synchronous Machine (PMSM) for maximum efficiency has been accomplished via tuning of d and

q axis currents considering parameter variation caused by saturation and ambient conditions [1]. Current waveform optimization has received less attention in the literature. The optimal waveform excitations for permanent magnet synchronous machines (PMSM) have been mostly computed for the torque ripple mitigation, associated with maximum efficiency [2]-[9]. These problems have been usually solved using iterative numerical optimization due to the complexity of multiple optimization objectives and various constraints. As the Maximum Torque Per Ampere (MTPA) and minimum torque ripple are generally popular objectives for non-ideal machines, a separate investigation into each problem is necessary to reveal the optimal waveform solution.

The relationship between back-EMF, current harmonics and the corresponding torque harmonic contents in PMSM has been studied in [2]-[5]. A closed form solution for minimum torque ripple was provided in [6]. It is shown that stator currents should contain the same harmonic contents as the respective back-EMF in order to generate more torque. Nevertheless, the relative ratio between each harmonic magnitude is not specified explicitly. This problem is rather solved numerically in [4],[7], and [8]. Although [9] provides an analytical method for minimum torque ripple and maximum efficiency

Wei Wang is with the Department of Electrical Engineering,

University of Texas at Dallas, Dallas, TX 75080 USA (e-mail: wxw108020@utdallas.edu).

Babak Fahimi is with the Department of Electrical Engineering, University of Texas at Dallas, Dallas, TX 75080 USA (e-mail: fahimi@utdallas.edu).

Morgan Kiani is with the Department of Engineering, Texas Christian University, Fort Worth, TX, 76129, USA ( e-mail:m.kiani@tcu.edu).

excitation of PMSM using d-q reference frame model, a theoretical validation of the claim on the aspect of maximum torque per unit copper loss is absent. Reference [10] presents an analytical expression in time domain for minimum ripple and maximum efficiency control of PMSM. This paper solves the maximum average torque problem under RMS current constraint via frequency domain approach since it provides better appreciation of the relation between harmonics and the torque production.

This paper aims at providing an expression for optimal current waveform as a function of each harmonics’ magnitude and frequency for better understanding the torque production in PM machines. Torque ripple is not listed as a constraint in this paper. The MPTA control is of great importance for applications in which high torque density is required and operation under harsh conditions such as continuous operation under partial failure of the machine is of interest.

In this paper, a unified approach for MTPA control of a PMSM drives is developed. It starts with the derivation of optimal current waveforms for the reaction torque in PMSM drives. Subsequently, the torque equations for PMSM with magnetic saliency is developed accounting for both reaction and reluctance torque components. Next, the optimal waveform for reluctance torque is derived. Finally, a general unified solution for MTPA problem of PMSM, BLDC, IPM and synchronous reluctance machines is presented. Analytical examination of solution via case study and simulation are provided for validation.

II. MTPA PROBLEM DEFINITION OF PMSM

In non-salient PMSM, the interaction between different stator phases does not produce motional torque. Also, the magnetization is provided by PM which is mounted on the rotor. Thus the torque produced by each phase is independent of the other phases. In this study, only single phase voltage and current are analyzed as the other phases are assumed to have the same waveforms with proper phase shift. This assumption is valid for symmetrical PMSM. For the sake of simplicity, only a three phase two pole machine is analyzed here. However, it can be generalized to include PMSM with higher number of poles and phases.

A. Torque Per Ampere Definition for PMSM The back-EMF of the PMSM machine can be

represented by a Fourier series cos cos 120 1

Maximum Torque Per Ampere Control of Permanent Magnet Synchronous Machines

Wei Wang, Babak Fahimi, Morgan Kiani

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978-1-4673-0142-8/12/$26.00 ©2012 IEEE 1013

cos 120 The corresponding three phase currents can be defined

as cos cos 120 2

cos 120

It is well acknowledged, the current should be in phase with voltage to produce maximum power. Thus is set to zero hereafter (2). Excluding the iron losses the electromagnetic power of the machine is then approximated by 3

The average power over one electrical cycle is given by: 12

32 4

Under constant rotor speed , the average torque is 32 5

It is shown in (5) that the average torque is contributed to by all the harmonics which are common between both back-EMF and current. This is also mentioned in [7], and [9]. Notably, those current harmonics which do not have a counterpart in back-EMF may still affect the torque ripple but do not result in any gain in average torque.

The total rms current is given by 32 6 Assuming that the phase resistance is R and neglecting

resistance variation due to the skin effect and temperature variation, the total copper loss is given by 32 7

Torque Per Ampere (TPA) is then expressed as 1 / 32 8

The Maximum TPA problem is then defined as finding optimal current such that

Subject to the following constant current limit

32

III. MTPA PROBLEM SOLUTION AND ANALYSIS FOR PMSM

In the above section, the MTPA problem of a surface mount PMSM is defined. In this section, the optimization problem will be solved first without any limitation. Then the optimization problem under the constraints will be introduced.

A. Solution for MTPA The problem can be solved using Lagrange optimization

method. Transforming the problem into standard optimization problem with a multiplier ,

, , … , 32 9

The optimal solution exists when the partial derivative for each variable is set to zero, 3…3 0…0

32 0 10

Solving the equation set (10), the following results can be obtained: 13 11 ∑√6 12

Substituting (11), and (12) into (2) and (8) one yields: 13 13 13 13 32 / 14

B. MTPA under Harmonics Constraint In an actual drive system, there might be practical

reasons which can limit harmonic spectrum of the drive system. For example, a limited bandwidth of controller will cut off frequencies which are out of band. A star connected winding will eliminate specific current harmonic content. In these situations, a slight modification will satisfy both MTPA and harmonics requirement. If the machine is a star connected with phases (phases are 360/m degrees electrical angle apart) with a drive which only offers frequencies up to harmonics, simply adding the

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following boundary condition to equations(11), (12), (14) will suffice. , 2 , 3 , 4 , 5 , … (15)

Notably, with the additional limits on current harmonics, the resulting MTPA solution is degraded from its original value.

C. Analysis As can be seen from (11), without applying any

constraint on the current waveform, the current harmonics are proportional to the back-EMF harmonic contents. If the harmonics are normalized with respect to the first order harmonic, the normalized optimal current harmonic spectrum that is obtained should be the same as the normalized back-EMF harmonics spectrum. In other words, in time domain, the instantaneous current value should be proportional to its phase back-EMF to ensure minimum copper loss. This is expressed by (13). For instantaneous torque regulation, [10] shows a similar result. This implies that for MTPA control, the optimal current is sinusoidal for a surface mount PMSM designed with pure sinusoidal back-EMF whereas the optimal current waveform is trapezoidal for BLDC with a trapezoidal back-EMF.

Another observation is that under optimal current excitation, MTPA is only a function of back-EMF harmonics which are dictated by the magnetization of the PM and machine design. This also shows the linearity of current and torque relationship in PMSM. The magnetic saturation is not considered here as the surface mount PMSM has relatively large air gap and can be assumed to be linear over a large range of operation without losing accuracy in modeling. It can also be noted that the machine design sets the boundary for the optimal performance of the machine. This, in turn provides guidance for machine design. It is preferable to allow the existence of back-EMF harmonics for most effective usage of the expensive Permanent Magnet material. When certain harmonics in currents are absent, for example 3rd harmonics current in three phase star connected PMSM, the PMSM magnetic field needs to be configured to eliminate the generation of the corresponding back-EMF harmonics.

IV. MTPA PROBLEM DEFINITION AND MAXIMUM RELUCTANCE TORQUE PRODUCTION FOR IPM

For Interior Permanent Magnet (IPM) machine, there are two sources of torque. One source is the reaction torque between the stator field and rotor filed. The other source is reluctance torque due to the magnetic saliency and stator excitation. The optimization excitation needs to properly balance the reaction torque and reluctance torque. In addition, optimal current waveform solution which maximizes the reaction torque and reluctance torque needs to be calculated.

In this study, the machine phase back-EMF and inductance profiles of each phase are assumed to have the same waveform except for an adequate phase shift (120 degree). For the sake of simplicity, only a three phase two pole machine is analyzed here. However, the developed result can be modified to accommodate IPM machines with higher number of pole pair and phases.

A. Maximum Torque Per Ampere Definition for IPM The IPM is assumed to have similar back-EMF due to PM on rotor as in previous sections. Assuming that, the rotor starts at a position for which the q axis coincides with phase-a axis. The inductance profiles can be defined as follow [13], cos 2 cos 2 120 16 cos 2 120 12 cos 2 12 cos 2 60 17 12 cos 2 60 In which, , , refer to self inductances and , , , denote mutual inductances.

The current can be represent by

cos sin 18

cos 120 sin 120 cos 240 sin 240

The total torque is made up of two components, out of which the average reaction torque is calculated in section II. 32 19

In which the refers to the nth harmonic magnitude of flux linkage created by permanent magnets on the rotor.

The reluctance torque [11] is expressed by 12 12 12 20

Substituting (16), (17) into (20), the torque equation is given by: 2 sin 2 2 sin 2 120 21 2 sin 2 120

Due to the limitation of space, the derivation process of the torque equation is not included here. The average reluctance torque due to variation in self inductance is given by:

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1 sin 2 sin 2 120 sin 2 120 34 – – 22

The average reluctance torque due to mutual inductance variation is

2 , 23

3, 3 1, 0, 1, 2, …32 ,

3, 3 2, 0, 1, 2, …32 ,

The total reluctance torque is given as follows:

for positive torque production, n=3k+1 94 ,

92 , 24

for negative torque production, n=3k+2 94 ,

92 , 25

The rms current is defined as 32 26

The optimization problem is then defined by arg _

Subject to 32

B. Maximum Reluctance Torque Per Ampere For better understanding of the impact of current

harmonic on torque production, the optimal waveform for each sources of torque is solved first. Optimal current waveform for reaction torque production is already addressed in section III. This sub section introduces the optimal current waveform for reluctance torque.

Equation (24) and (25) show that only certain combination of current harmonics produce nonzero net torque. It is necessary to identify this harmonics and carefully select the current magnitude to obtain maximum torque. For those harmonic contents which do not contribute to average torque production, the corresponding harmonic magnitude should be zero. Thus the 0, 3 1 , 0, 1, 2, … 3 2, 0, 1, 2, … (27)

The optimal reluctance torque production problem can be solved in two steps. First, the optimal current magnitude ratio within each pair of torque production harmonics is determined. Then unit current is optimally distributed among different harmonic pairs such that torque production is maximized. Since the equations for positive and negative torque production in (24), (25) are almost the same; only the positive torque production situation is analyzed here.

The solution for maximum reluctance torque per unit current is straight forward. Within each pair of frequencies, the two harmonics contribute equally to the torque production. It can also be solved using Lagrange method. The solution is , 3 1 (28)

Therefore, the torque equation reduces to 184 , 29

From (29), the contribution for each harmonic pair is equal to each other. Calculating the TPA2, it turns into a function irrelevant to the current magnitude of each harmonic pair rms currents as shown in (30). 32 30

Eq. (29) implies that there are infinite numbers of solution for maximum reluctance torque production as long as (28) is satisfied.

In this section, the torque equation is defined for average torque. Thus, (28) only guarantees that the average torque is maximum. In simulation, it is shown that when n=1(k=0), the torque is ripple free whereas the other harmonic pairs create significant torque ripple for same average torque production. Nevertheless, it still provides the flexibility to excite the machine with certain amount of higher harmonics without losing the optimality in torque per unit current. For example, when signal injection is used for position estimation, selecting frequency based on (28) would relax the limitation on the injected signal magnitude and frequency.

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In a practical drive system, the bandwidth of the current is preferred to be constrained for obtaining a relatively smooth current, torque profile and the possibility of real time implementation. For same average torque production, the lower the frequency of excitation current is, the lower the copper and core loss will be. Thus the optimal solution should only contain the first harmonic.

V. MAXIMUM TORQUE PER AMPERE SOLUTION FOR IPM

A. Maximum TPA Solution of IPM Under the current band limitation, the total torque is expressed by 32 3 31

0 3 1, 1, 2, 3 … …

In which is the band limit of the drive system. The Lagrange optimization function is defined as

,…, , ,…, ,32 332 32

The optimization current fulfills the following equation sets, 0, 33

0 34

32 35

For n 3k+1,

3 , 0 36

For n = 3k +1,

2 32 0 37 32 0 38

Solving the (36), (37), and (38) results in: 24 9 39 34 9 40

For n 3k+1, 3 , 0 41

By solving the above equation sets, the optimal value is among one of the solutions.

Defining the following parameters,

, 3 1

, 3 1

The solution for is , , , 42

In which, , , , is one of the standard solutions for cubic equation, 0 43

In which 1283 16 16 192 72 36 216 81 44

The general solution for the standard cubic equation is listed below

, , , 3 2 3 1 √33 21 √36 2 45

2 9 27

4 3 2 9 27 /

Substituting into (39), (40), the optimal magnitude for each harmonics can be obtained. Although it appears that there is imaginary part in the solution at first glance, the imaginary part is cancelled eventually when all parameters are brought in.

B. A Case Study on Basic Harmonics The optimal current waveform properly dispatches the

torque into the reaction and reluctance torque sources. The expression for the optimal solution is relatively complex. A detailed check at the basic harmonic would help with the examination the effectiveness of the optimal solution. It provides insight into the optimization problem. Moreover, in practical implementation, usually only the first harmonic is of concern. Examination of the optimal currents for only the basic frequency provides guidance for drive control algorithm.

Define a sub-problem for only the first harmonic

, , 1

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Subjects to + ) = C}.

In the above equation, , , 3 corresponds to , and ( ) in 0synchrnous reference frame

model[13] respectively.

This sub-problem can be easily solved by replacing the appropriate parameters in the (44).The result is shown below, 24 9 34 9 46

3 14 3 1443 When the demand torque is high or the reference current

is high, the optimal value of is very close to negative 1.5 which makes almost equal to . When reference current is low, the optimal value of and the ratio between and are very sensitive to the machine parameters. If the current magnitudes are chosen according to (46), loss will be effectively minimized when machine operates at high speed region or light torque, light load condition.

C. The Generality of the MTPA solution One might notice that the solution for MTPA control of

IPM can be extended to other machines by properly configuring the values of , and .

For example, a PM machine with zero value represents a machine without structural magnetic saliency. It is either a BLDC or surface mount PMSM depending on the back-EMF harmonic contents. Under this situation, the optimal solution of (39)-(44) reduces to solution (13)-(14). When , are equal to zero which means no PM or excitation on rotor, it is a Synchronous Reluctance Machine. In this case, the solution is the same as in (27) which is optimal current waveform for Synchronous Reluctance Machine. Thus the presented solution (39)-(44) is a general solution, under linear assumption, for MTPA control of synchronous type machines including surface mount PMSM, BLDC, Synchronous Reluctance Machine and IPM.

VI. SIMULATION RESULTS AND DISCUSSIONS In this section, lumped parameter simulations and

analysis are conducted to validate the proposed solution for MPTA control of PMSM and IPM with random combination of back-EMF harmonics. Optimal waveforms are calculated and plotted to prove the effectiveness of the proposed MTPA solution.

A. Surface Mount Permanent Magnet Machine Assuming the flux linkage harmonics contents of a two

pole three phases PMSM machine is

1st=1, 2nd=0.0, 3rd=0.33, 4th= 0.0, 5th = 0.2, 7th = 0.15

For a RMS phase current of 1A, fig.1 shows the optimal current waveform, the sinusoidal current waveform and the back-EMF. It is apparent that the optimal current for MTPA control is proportional to the real time back-EMF value. The torques under sinusoidal and optimal current excitation is plotted in fig.2. As shown in this figure, the optimal current produces more torque than sinusoidal excitation. Although there is an increase in torque ripple, it is still good for certain applications which is tolerant to torque ripple requirement. Table I shows the average torque comparison between the optimal and conventional waveform which is sinusoidal with zero Id current at various load levels. The optimal current waveform shows about 8.2% improvement in average torque production under the same RMS current.

Fig.1 Optimal Current waveform for PMSM

Fig.2 Torque result for Optimal Excitation and Sinusoidal Excitation

B. Interior Permanent Magnet Machine Assuming the phase flux linkage harmonics of a two pole three phases IPM machine is

1st=0.1, 2nd=0.01, 3rd=0.02, 4th= 0.01, 5th = 0.01, 7th = 0.01

And 2 .

In this optimization problem, the bandwidth of the current profile is limited to 7th harmonic. For a RMS phase current of 1A, fig.3 shows the normalized optimal current waveform, the sinusoidal current waveform with Id=0, and back-EMF. Fig.4 shows the simulated torque result under optimal current waveform and the conventional Iq=Id sinusoidal current waveform excitation at 16A rms. From this figure, the torque generated is bigger than the conventional method with an increased torque ripple. Table II shows the average torque under different control strategy

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at various current values. The results show that there is more than 15% improvement at 8A (low current level) and 2% at 32A (high current level). This matches the qualitative analysis in section V.

As stated earlier, the optimal waveform varies at different RMS current level. Fig.5 shows the waveform transition at different current levels. With the increase of current, the phase angle of current tends to lead the back-EMF. This leading angle eventually converges at around 45 degree electrical angle.

Fig.3 Optimal current waveform for IPM at 1A phase rms current

Fig.4 torque result for MTPA control and Id=Iq, sinusoidal excitation of

IPM at 16A rms.

Fig.5 variation of optimal current waveform at different current level

TABLE I

RESULT COMPARISON FOR MTPA CONTROL OF PMSM

PMSM RMS current (A)

Average Te (N.M)

TPA (N.M/A)

Sinusoidal Id=0 1 2.12 1.224 Optimal 1 2.29 1.322

Sinusoidal 5 10.6 1.224 Optimal 5 11.45 1.322

TABLE II

RESULTS COMPARISON FOR MTPA CONTROL OF IPM

IPM RMS Current (A) Average Te (N.M) TPA

(N.M/A) Sinusoidal Id=0 8 1.703 0.142 Sinusoidal Id=Iq 8 1.77 0.147

Optimal 8 2.03 0.168 Sinusoidal Id=0 32 6.93 0.144 Sinusoidal Id=Iq 32 13.975 0.291

Optimal 32 14.395 0.299

VII. CONCLUSION A unified approach for Maximum Torque Per Ampere control of Permanent Magnet Synchronous Machines are developed. A general solution for surface mount PMSM, BLDC, IPM, and Synchronous Reluctance Machine is provided and validated through theoretical analysis and simulation.

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[3] Bolton, H.R.; Ashen, R.A.; , "Influence of motor design and feed-current waveform on torque ripple in brushless DC drives," Electric Power Applications, IEE Proceedings B , vol.131, no.3, pp.82-90, May 1984

[4] Le-Huy, Hoang; Perret, Robert; Feuillet, Rene; , "Minimization of Torque Ripple in Brushless DC Motor Drives," Industry Applications, IEEE Transactions on , vol.IA-22, no.4, pp.748-755, July 1986

[5] De La Ree, J.; Boules, N.;, "Torque production in permanent-magnet synchronous motors," Industry Applications, IEEE Transactions on , vol.25, no.1, pp.107-112, Jan/Feb 1989.

[6] Hung, J.Y.; Ding, Z.; , "Minimization of torque ripple in permanent magnet motors: a closed form solution," Industrial Electronics, Control, Instrumentation, and Automation, 1992. Power Electronics and Motion Control., Proceedings of the 1992 International Conference on , vol., no., pp.459-463 vol.1, 9-13 Nov 1992

[7] Hanselman, D.C.;, "Minimum torque ripple, maximum efficiency excitation of brushless permanent magnet motors," Industrial Electronics, IEEE Transactions on , vol.41, no.3, pp.292-300, Jun 1994.

[8] Chapman, P.L.; Sudhoff, S.D.;Whitcomb, C.A.;, "Optimal current strategies for surface-mounted permanent-magnet synchronous machine drives," Energy Conversion, IEEE Transactions on , vol.14, no.4, pp. 1043- 1050, December, 1999.

[9] Sung Jun Park; Han Woong Park; Man Hyung Lee; Harashima, F.; , "A new approach for minimum-torque-ripple maximum-efficiency control of BLDC motor," Industrial Electronics, IEEE Transactions on , vol.47, no.1, pp.109-114, Feb 2000.

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[11] Jie Fang; Heising, C.; Staudt, V.; Steimel, A.; , "Modelling of anisotropic synchronous machine in stator-reference frame including torque calculation," Electrical Systems for Aircraft, Railway and Ship Propulsion (ESARS), 2010 , vol., no., pp.1-6, 19-21 Oct. 2010

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[13] P.C. Krause, O. Wasynczuk, S.D. Sudhoff. , Analysis of electric machinery and drive systems, 2nd ed., John wiley and sons, New York, 2002.

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[14] Han-Woong Park; Sung-Jun Park; Yang-Woo Lee; Soon-Il Song; Cheul-U Kim; , "Reference frame approach for torque ripple minimization of BLDCM over wide speed range including cogging torque," Industrial Electronics, 2001. Proceedings. ISIE 2001. IEEE International Symposium on , vol.1, no., pp.637-642 vol.1, 2001.

IX. BIOGRAPHIES

Wei Wang (S’09 )was born in Hubei, China. He received B.S.c degrees in electrical engineering from Huazhong University of Science and Technology (HUST), Wuhan, China in 2008.

Currently, he is Ph.D. student at Renewable Energy and Vehicular Technology Laboratory, EE department, University of Texas at Dallas, Dallas, TX. His research of interests includes design, analysis, and control of electric machines and power electronics converters.

Babak Fahimi (S’96---M’00---SM’02) was born in Tehran, Iran. He received the B.S. and M.S. in electrical engineering from the University of Tehran with highest distinction in 1991 and 1993 and Ph.D. degree in electrical engineering from Texas A&M University, College Station, in 1999.

During 1999-2002, he was a Research Scientist at Electro standards Laboratories, Inc., Cranston, RI. Currently he is a professor of electrical engineering at University of Texas-Dallas. He is the author or coauthor of more than 200 technical papers and 15 book chapter including a chapter in John Wiley Encyclopedia on electrical and electronics engineering. He is the holder of six U.S. patents and has seven more pending. His areas of interest include analysis of electromechanical energy conversion, digital control of adjustable speed motor drives, and design and development of power electronic converters.

Dr. Fahimi has been the general Chair of the IEEE applied power electronics conference and expo (APEC) in 2010. He has been the chairman of the electric machines committee in IEEE Industrial Electronics Society.

Morgan Kiani has been a faculty member in the Department of Engineering at Texas Christian University since spring of 2011. She received her Ph.D . degree in Electrical Engineering from the University of Texas at Arlington in 2009. She was an adjunct faculty member at the University of Texas at Arlington prior to joining TCU. Her research interests include modeling and analysis of energy converters, numerical methods in power system simulation, integration of distributed generators in smart grid , fault detection in wind farms, energy harvesting, energy diversification and power electronics. Her teaching interests include electric circuits, power systems, power electronic systems and electromechanical energy conversion. Dr. Kiani has authored several scientific articles in peer reviewed journals and conference proceedings. She has one pending patent on active control of Doubly Fed Induction Generators (DFIG) used in wind energy harvesters.

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