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`US00632675UB1
`
`(12; United States Patent
`US 6,326,750 B1
`
`Marcinkiewicz
`Dec. 4, 2001
`(45) Date of Patent:
`
`(10) Patent No.:
`
`(54) ACTIVE REDUCTION OF TORQUE
`IRREGULARITIES IN ROTATING
`MACHINES
`
`(75)
`
`Inventor:
`
`‘
`'
`_
`Joseph G. Man:|nk1ewIez,St.(3har1es.
`MO (US)
`
`(56)
`
`(73) Assigneez Emerson Electric Co.,St. Louis, MO
`(US)
`
`References Cited
`.
`U.S. PATEN] DOCUMENTS
`4,236,202 *
`8/1981 Clancy et al.
`4,533,224 4-
`1/1937 G,-me,
`4,312,591 *
`341989 Bertram E1 al.
`5,223,775 * H1993 Mnngeau
`5,444,341 ‘
`3/1995 Kneifel, Halal --
`5,616,999 *
`4/1997 Malsumura etat. .
`5,744,921 "
`4,v’I998 Makatan
`
`.
`
`(*) Notice:
`
`Subject to any disclaimer, the term ofthis
`Pam“, is amended or adjusted under 35
`U.S.C- 15403) by 0 day5-
`
`5937-741 * 3-‘W0 "Yamadfi et a1-
`6,046,534 *
`4/ZCUO Becerra
`* cited by examiner
`
`--
`
`318.7696
`313,954
`.. 310541
`318/432
`318/432
`318/632
`318.054
`
`-- 313-"721
`318,-'254
`
`
`
`4
`_:
`_
`2
`( 1) App‘ N0 M} 59543
`(22) Filed:
`Dec. 14, 1999
`
`(6-0)
`
`Related U.S. Application Data
`Provisional application No. 60/139,703, filed on Jun. 17,
`1999'
`7
`(5l_)
`Int. Cl.
`(52) U.S. Cl.
`
`........................................................ HOZP 7/00
`313.’432; 318/437; 318/434;
`318/802
`(58) Field of Search ..................................... 3183432, 437,
`318.1609, 610, 434, 254, 809, 807, 802.
`803, 822, 439, I38, 700, 720-724; 323./220-354
`
`Cogging Torque
`Rejection Feed
`
`Primary Examx'ner—Robe\:t E. Nappi
`Assistant Examiner—Edgardo San Martin
`(74) Attorney, Agent, or Ft'rm—Howrey Simon Amold &
`White, LLP
`
`,
`ABSTRACT
`(57)
`A method and apparatus for actively controlling the energi-
`zation of the stator windings of a rotating electromagnetic
`machine to actively overcome any tendency of the rotor of
`the machine to prefer certain angular positions with respect
`to the stator over other angular positions.
`
`24 Claims, 28 Drawing Sheets
`
`153
`
`BOM Exhibit 1040
`|PR2014-01121
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`
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`U.S. Patent
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`Dec. 4, 2001
`
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`1
`ACTIVE REDUCTION OF TORQUE
`IRREGULARITIES IN ROTATING
`MACHINES
`
`The present application claims priority to Provisional
`Application Ser. No. 60,139,703 filed Jttn. 17, 1999, for an
`Active Reduction of Torque Irregularities in Rotating
`Machines.
`
`BACKGROUND OF 'I‘[II:‘. INVI:}N'I'lON
`
`Many electromagnetic machines in general. and penna-
`nent magnet electric motors in particular, exhibit torque
`irregularities as the rotor rotates with respect to the stator.
`Such irregularities produce non-uniform torque output and,
`thus. non-uniform rotation of the rotor. These torque irregu-
`larities may be caused by the physical construction of a
`given machine. They can result from, for example, a bearing
`that sticks in a given rotor position or the fact that, because
`of the electromagnetic characteristics of the machine, the
`rotor tends to prefer certain angular positions with respect to
`the stator. Torque irregularities resulting from the electro-
`magnetic characteristics of an electromagnetic machine are
`commonly known as “oogging" irregularities and the result-
`ant non-uniform rotation of the rotor or non-uniform torque
`output is known as “cogging.”
`In permanent magnet machines, cogging most often
`results from the physical construction of the machine.
`Specifically, the utilization of rotors having discrete north
`and south outer poles results in a circumferential distribution
`of magnetic flux about the rotor circumference that is not
`smooth, but choppy. Additionally,
`the stators commonly
`used with such machines are formed in such a way that the
`magnetic fluxes generated by the stator windings provide a
`flux distribution about the stator circumference that is not
`smooth. The combination of such rotors and stators, and the
`accompanying non-smooth flux distributions. produces
`undesired cogging irregularities. Rotor output non-
`uniformities may also be produced by back emf harmonics
`produced in certain machines.
`For many motor applications the slight non-uniformity in
`the rotation of the rotor caused by torque irregularities is of
`little or no consequence. For example,
`in large motors
`driving large loads, slight variations in the output torque will
`not significantly atfect the rotor speed and any slight varia-
`tions in rotor speed will not significantly alfcct the system
`being driven by the machine. In other applications, where
`the rotation of the rotor or the torque output of the motor
`must be precisely controlled or uniform, such non-
`uniformity is not acceptable. For example, in servomotors
`used in electric power steering systems and in disk drives,
`the rotational output of the rotor or the torque output of the
`motor must be smooth and without significant variation. In
`many such applications the maximum acceptable peak-to-
`peak torque ripple or rotational speed ripple as a percentage
`of full load mean is on the order ofonly 1% to 2%. Because
`such machines typically have small, relatively low mass
`rotors,
`torque irregularities of a small magnitude can
`adversely impact the output of the machine.
`Prior art approaches to reducing the undesirable conse-
`quences of torque irregularities in electromagnetic machines
`have focused on relatively complex rotor or stator construc-
`tions designed to eliminate the physical characteristics of the
`machines that would otherwise give rise to the irregularities.
`While the prior art machine construction approaches can
`result in reduction of torque irregularities, the approaches
`require the design and construction of complex rotor and
`
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`stator components, such complex components are typically
`diflicult to design, ditlicult to manufacturer, and much more
`costly to produce than are conventionally constructed com-
`ponents. Moreover, many of the physical changes required
`by such prior art solutions result in a significant reduction in
`the efficiency or other performance parameters of the result-
`ing machines over that expected of comparable conventional
`machines. Thus, many of the prior art attempts to reduce
`torque irregularities do so at the cost of machine perfor-
`IIIHIJCC.
`
`invention to provide an
`It is an object of the present
`improved method and apparatus for reducing the negative
`consequences of torque irregularities that do not suffer from
`the described and other limitations associated with the prior
`art.
`
`BRIEF SUMMARY OF THE INVENTION
`
`The present invention generally relates to a method and
`apparatus for controlling the energization of the stator wind-
`ings of a rotating electromagnetic machine to actively over-
`come any tendency of the rotor to prefer certain angular
`positions with respect
`to the stator over other angular
`positions. In accordance with one exemplary embodiment
`the present invention relates to a method comprising the acts
`of: generating an electrical representation of the angular
`position of the rotor with respect to the stator; controlling the
`energization of the stator windings in response to the elec-
`trical representation of the rotor with respect to the stator to
`actively counteract the tendency of the rotor to prefer ccnain
`angular positions with respect
`to the stator over other
`angular positions.
`In accordance with yet another embodiment, a controller
`is provided for energizing the stator windings of a rotating
`electromagnetic machine to provide smooth output torque,
`the machine including a rotor that prefers certain angular
`positions with respect to the stator over other positions, a
`torque rejection circuit that receives as an input a represen-
`tation of the angular position of the rotor with respect to the
`stator and generates at an output an energization command
`corresponding to any tendency of the rotor to prefer the
`received angular position and an energization circuit respon-
`sive to the output of the torque rejection circuit that ener-
`gizes to the stator windings of the rotating electromagnetic
`machine so as to provide smooth output torque.
`BRIEF DESCRIPTION OF TIIE FIGURES
`
`FIG. 1 illustrates a system It] constructed according to
`certain teachings of the present disclosure.
`FIG. 2 provides a representation of measured Bemf data
`(line to neutral) one phase of an exemplary three-phase
`twelve slot, eight-pole permanent magnet motor (12s8p PM
`machine).
`FIG. 3 provides a representation of measured cogging
`torque for the 12s8p PM machine used to generate the data
`reflected in FIG. 2.
`FIG. 4 illustrates Bemf data from FIG. 2 and an exem-
`plary fit divided by 100 along with a fit error.
`FIG. 5 provides an exemplary plot of the magnitude of the
`FFI‘ of the cogging torque data represented by FIG. 3.
`FIG. 6 provides a representative illustration of the cog-
`ging data of FIG. 3/10, an exemplary fitfl0 and a lit error.
`FIG. 7 provides an exemplary plot of appropriate control
`currents for the exemplary motor rellected by FIGS. 2 and
`3 as a
`function of the angular position of the rotor
`(mechanical).
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`3
`FIG. 8 provides an exemplary plot of shaft torque using
`active cogging rejection and the cogging torque for an
`exemplary system constructed in accordance with certain
`teachings found in this disclosure.
`FIG. 9 provides an exemplary plot of appropriate control
`currents and output torque multiplied lU for a 3N"‘m output
`for the machine represented by FIGS. 2 and 3.
`FIG. 10 provides exemplary minimum sensitivity cogging
`rejection feeds, desired control currents Ia, lb and Ic, an
`exemplary Beml waveform and an exemplary torque output
`multiplied by 10 for a system including the machine having
`the characteristics of FIGS. 2 and 3.
`
`FIG. 11 illustrates FFT"s for exemplary idealized control
`currents that may be used to control the exemplary motor
`characterized by the data of FIGS. 2 and 3.
`FIG. 12 provides a plot of exemplary idealized phase A
`control currenLs that may be used to control the exemplary
`motor characterized by the plots of FIGS. 2 and 3.
`FIG. 13 provides a plot of the output torques produced for
`the control currents of FIG. 12.
`
`JEJ
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`FIG. '14 provides a plot of the percentage ripple vs. sensor
`error in electrical degrees for two minimum sensitivity
`solutions and the minimum RMS current solution that may
`be used to control the exemplary motor represented by the
`plots of FIGS. 2 and 3.
`FIGS. 15A, 15B and 16 illustrate exemplary controllers ’
`that utilize active cogging rejection to reduce or eliminate
`undesirable torque irregularities.
`FIG. 17 provides an illustration of representative torque
`rejection (or anti-cogging) currents that may be produced by
`the controllers of FIGS. 15 and 16.
`
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`
`FIG. 18 provides exemplary illustrations of representative
`torque producing currents that may be produced by the
`controllers of FIGS. 15 and 16.
`FIG. 19 illustrates an exemplary controller constructed in
`accordance with certain teachings of this disclosure in which
`the torque rejection feed added to the voltage outputs from
`a control law element.
`
`FIG. 20 provides exemplary current solutions that may be
`used with the machine having the cogging and torque
`signatures illustrated in FIGS. 2 and 3 to produce a desired
`-4 N*m output.
`ignoring the 0
`li, plane plot
`FIG. 21 illustrates a (1,
`sequence currents for various current solutions in a rotating
`reference frame.
`[3, 0 based control
`FIG. 22 illustrates an exemplary or,
`system for providing active torque irregularity reduction that
`is constructed in accordance with certain teachings con-
`tained in this disclosure.
`
`FIG. 23 represents a plotting of the Q (real part) and D
`(imaginary part) waveforms (the 0 sequence component is
`zero) for two exemplary output torques (U N*m and 2 N*m)
`for a motor operated by a controller operating in one
`particular Q, D, D rotating reference frame.
`FIG. 24 illustrates various torque outputs that may be
`obtained through use of the control system of FIG. 22.
`FIGS. 25 and 26 provides a general
`illustration of a
`control system utilize a rotating QDO reference frame that is
`constructed in accordance With certain teachings found in
`this disclosure.
`
`FIG. 27 illustrates an exemplary controller constructed in
`accordance with certain teachings of this disclosure for
`controllers motors operating on non-linear regions.
`DETAILED DESCRIPTION OF THE
`INVENTION
`
`to FIG. I, a
`in particular,
`Turing to the drawings and,
`system 10 constructed according to certain teachings of this
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`disclosure is illustrated. The illustrated system 10 actively
`controls the electric power supplied to an electromagnetic
`machine such that
`the negative consequences of torque
`irregularities that would otherwise be produced by the
`machine are reduced or eliminated.
`System 10 includes an eiectromagnetic machine 12 and a
`drive 14 that provides electric power to the electromagnetic
`machine. Electromagnetic machine 12 may be any rotating
`machine that exhibits torque irregularities. For example, in
`the FIG. 1 machine 12 is a permanent magnet motor of
`conventional construction that includes a rotating compo-
`nent (a “rotor") and a stationary component (a “stator”).
`Wound about the stator are a number of energizablc stator
`windings which may he energized through the application of
`electric power to terminals I5, 16 and 17.
`While machine 12 is described as a permanent magnet
`motor, those of ordinary skill in the art having the benefit of
`this disclosure will recognize that other forms of electro-
`magnetic machines may be used without departing from the
`teachings contained herein.
`Drive 14 is coupled to provide electric power to terminals
`15, 16 and 17 of motor 12. Drive 14 is also coupled to
`receive feedback from motor 12 in terms of rotor position
`information 18 and energization feedback 19. Other feed-
`back information may be provided to the drive 14. While
`drive 14 is illustrated in exemplary form as providing any
`three power terminals to motor 12, it should be understood
`that more or fewer power terminals may be provided to
`accommodate motors or machines with greater than three
`phases, less than three phases or if various types of inverters
`(e.g., with neutral connections) are used.
`Energization feedback 19 should provide an indication of
`the operational characteristics of the motor 12 and may, for
`example, include feedback concerning the currents flowing
`in the statorwinrlirtgs andfor the voltages at the terminals 15,
`16 and I7. The position and energization parameters may be
`detected through conventional detectors such as standard
`rotor position detectors andfor standard currentfvoltage sen-
`sors. Alternate embodiments are envisioned in which the
`
`rotor position and feedback parameters are not detected
`directly but are calculated or estimated through known
`techniques. For example, embodiments are envisioned
`where only the terminal voltages are known or sensed along
`with the currents flowing through the stator windings of
`motor 12 and the sensed current and voltage values are used
`to derive rotor position infonriation.
`Drive [4 also receives an input command signals 13 that
`corresponds to a desirable output parameter of motor 12
`such as rotor speed, output torque etc.
`As describe in more detail below, drive 14 controls the
`application of electric power to motor 12 in such a manner
`that the dillerence between the input command signal and
`the corresponding output of motor 12 is minimized. Drive 14
`also actively controls application of power to motor 12 as a
`function of rotor position in such a manner that
`torque
`irregularities (including Cogging torque) are reduced or
`eliminated.
`
`In general, drive 14 receives rotor position information
`and, based on that information develops a control signal that
`actively adjusts the energization of motor 12 such that any
`torque irregularities that would exist at that rotor position are
`reduced. This control may be accomplished by first
`charaeterizingfmodeling motor 12 to obtain information
`about the torque irregularities that would exist in the absence
`of such active control of the energization and then using that
`characterization information to develop a controller as
`described above.
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`5
`The use ofdrive 14 to actively reduce torque irregularities
`as opposed to attempting to reduce such irregularities
`through complex rotor or stator constructions, results in a
`better performing system in that, for example, conventional,
`low cost motors and motor construction techniques may be
`used without the undesirable torque irregularities commonly
`associated with such motors.
`As briefly described above, before developing a control
`system in accordance with certain ele ments of the teachings
`described herein, it is beneficial to characterize the machine
`to be controlled to obtain information about
`its potential
`torque irregularities. This may be accomplished by charac-
`terizing the machine to obtain information concerning: (1)
`the torque producing characteristics of the machine, which
`is a function of the back electromagnetic force (“Bemf”) of
`the machine; and (ii) the torque irregularities produced by
`the motor absent active control as discussed herein.
`An understanding of the torque characteristics for a given
`machine may be obtained by:
`(1) calculation andfor estimation; (2) empirical testing of
`the given machine; (3) testing of an appropriate sample
`of one or more machines of a given construction and a
`processing (e.g., averaging) of the obtained results; or
`(4) a combination of any of the foregoing. In general,
`it
`is believed that actual
`testing of a machine or a
`number of machines of a given type is believed to
`produce the most desirable results. As such.
`that
`methorl of determining the Bemf and torque irregulari-
`ties will be discussed in greater detail. Those of ordi-
`nary skill in the art will understand that other methods
`of obtaining torque data for a given machine may be
`used without departing from the teachings contained in
`this disclosure.
`this disclosure refers in many
`For ease of discussion,
`instances to “motors.” Such references are exemplary only
`and those of ordinary skill in the art will understand that
`such references are intended to include all electromagnetic
`machines that exhibit
`torque irregularities as described
`herein. Moreover, in many instances this disclosure refers to
`“cogging torque" or “coggiiig.” Those of ordinary Skill in
`the art should understand that such references are intended
`
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`repeat at a rate corresponding to the electrical cycle of the
`machine. As such, the Bemf waveform will typically be
`expressed in terms ofelectrical degrees. The torque- irregu-
`larity waveform—or cogging torque waveform-
`however—will
`typically be a
`repeating waveform that
`repeats at a rate corresponding to the rotational speed of the
`rotor. This is because the physical phenornenaahat pro-
`duces such irregularities is amociatcd primarily with the
`physical—not the electrical—characteristics of a machine.
`Once the Bemf and cogging torque information for a
`given motor
`is calculated or determined through
`measurement, it is desirable to represent both the Bemf and
`the cogging torque in terms of a common parameter or
`parameters to allow for each of control. One exemplary
`approach is to develop representations of the Bemf and the
`cogging torque as functions of the angular position of the
`rotor with respect to the stator.
`Representing the Bemf of a motor as a function of one or
`more fit parameters (e.g., angular rotor position) may be
`accomplished by: (l) defining a harmonic series that can be
`fitted to the Bemf information‘, (2) defining the derivative of
`the harmonic series as a function of the parameters to be
`fitted;
`(3) defining a fit vector that
`includes the Bemf
`harmonic series and its partial derivatives with respect to
`each parameter to be fitted; and (4) utilizing the lit vector to
`select an appropriate representation of the Bemf as a func-
`tion of the parameters to be fitted.
`Utilizing the data reflected in FIG. 2, and assuming that
`the only fit parameter is the angular rotor position, the
`following provides an example of an approach that may be
`used to develop a generation of the Bemf data of FIG. 2 as
`a function of the angular position of the rotor:
`First, a series (hereinafter the “Bemf function”) is defined
`that will be tilted to the measure line to neutral Bemf data.
`Because of the nature of the Bemf data, a harmonic series
`may be used:
`
`Bemf 1(r. kl !=k0 +t'c1-cos(J'c;3rl+ kg-sintkurj +
`k3 -eo.=.(3-!c1,r)+k4 -sin-13-Jq3r)+
`kg, -cos(5 -K131) + kg, -sin(5-klgrl +
`-cos(7 ‘Kurt + kg -sint'.v'-Kurt
`
`Bcmf_2tr. kt :=kg -eos(‘JAiq3r) 4-km -sin(9~Iq3r) 4-
`It.” -cos(ll -kUr}+k12 -sintll -kl; -1)
`Bemf(t. kt] := Bemfgltt, it) +Berrtt',flt, tltj
`
`the derivative of the Bemf function is defined
`Second,
`with respect to the fitting parameters. The fitting parameters
`being each harmonic coeflicient and the electrical angular
`frequency.
`In the described example the Bemf data was
`measured at a rotationalspeed of about I000 RPM, although
`the precise rotor shaft speed at which the Bemf data was
`collected is not known precisely. Accordingly, to accommo-
`date this fact, a generalized fit with the electrical angular
`frequency as a parameter is used:
`
`dl3emf_dlL] 3_1(t,l:_):--k, -sin(k, 3-I]-t+k;-ccs(k. 3 -t)~t-3.1:;
`
`sin(3'k,3't)'H-3'k_,'cos{3'k13't)-t—5-k5-
`
`si.tt(5'lc13't)'t+5'lC¢'cDS[5'li1-3'I)'t
`
`dBeml'_r:lk,'l3_2(t.k):--'7-lr ,-sin[7-l:,5-t)-t+7'kB-cos(7-km-L)-l—
`
`sluts-ic..,-r)-c+9~k..,-costa-is,-r)-:-i1-k..-
`
`sin1’_ll-kn-2)-t+ll-km-r:os(ll-kl;-l_l-L
`
`32
`
`to include all torque irregularities associated with rotating
`machines.
`One acceptable approach for determining the Bemf of a
`given motor is to simply measure the Bemf using conven-
`tional techniques and analytical devices. Such techniques
`and devices are known in the art and will not be discussed
`in detail herein. FIG. 2 provides a representation of mea-
`sured Bemf data (line to neutral) for one phase of a three-
`phase twelve slot, eight-pole permanent magnet motor
`(12s8p PM machine). In the illustrated example, the data
`represents data taken at approximately 1000 RPM. The
`selection of a three phase 125813 PM motor is for illustrative
`purposes as the methods and apparatus discussed herein are
`applicable to machines of varied construction including
`machines ranging from motors having only two phases to _
`machines having significantly more than three phases.
`The Bemf waveform of FIG. 2 represents one complete
`electrical cycle of the motor under analysis.
`Known approaches similar to those described above for
`detecting the Bemf of a motor may be used to obtained
`information about the oogging torque characteristics of a
`motor. FIG. 3 provides a representation of measured cogging
`torque over 4 electrical cycles (one mechanical cycle) for the
`1258p PM machine used to generate the data reflected in
`FIG. 2.
`It should he noted that the detected Bemf waveform of a
`rotating machine will be a repeating waveform that will
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`US 6,326,750 B1
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`7
`d.Bemi_dk_13(t.k)'.=dBemi' _dk_13_l(t,k}+dBemi__r1lc,,,13_2(t,
`k)
`
`Third, having defined the Bemt function and its relevant
`partial derivatives, a fit vector of functions (Bemf__function
`(t, k)_) is established that includes the Bemf function and its
`partial derivatives with respect to each parameter to be fitted
`(in this case each harmonic coefliciertt and the electrical
`angular frequency). To allow for ease of calculation, initial
`guemes (vg) for each of the parameters to be determined are
`provided:
`
`5
`
`JD
`
`8
`the primary cogging components
`typically ensures that
`contain many cycles for one mechanical revolution, an FFI‘
`series can be used lo determine what components are needed
`for a good fit. Calculating the l"lv‘l‘ of the cogging data may
`be accomplished by considering the following:
`
`CoggingKFFT:=CFFI'[Coggirtg_datn“"')
`
`fll._index::U .
`
`.
`
`. rows Clogging FFT)
`
`Berrtfjuncrion (r. k} :=
`
`Bemfir, kl
`l
`
`cosfku -I)
`sintkig - rt
`Ct.)§l'3'fi‘-13'-fl
`sil1[3 -kn -r_I
`C0-3(5 ‘M3 '1)
`si11€5 ring -I)
`t:tJS(7‘k1_3-fl
`siI1(7'~'i-H -I]
`cost‘) -km -r)
`sirtl9 -ICJJ ‘ll
`costll -kl; v{J
`sit\(!.1-Ru-n
`d]3ernt‘_dlcm t Bu, it]
`
`'1
`
`._._.._.._._._.._.,_._.,_,_,_.o
`
`vg :=
`
`Next, the input data vectors for the fitting function are
`defined:
`
`vx :=Bem[_r.lata‘°",vy:=BemLdala‘”’
`
`and an appropriate approach is used to solve for the fit
`parameters.
`in the present example a generalized least
`squares flt of the Bemf function to the data is selected
`although other solutions may be used.
`
`Bcrrtf
`
`f'Lt__paramr.ters1-genfi.t(v3t,vy,vg,Bemt'
`
`function)
`
`To more precisely determine the various fit parameters, it
`is often helpful to consider the Flemf data and its fit divided
`by 100 along with the lit error. A plot of such information is
`provided as an example in FIG. 4.
`From the Bcmf fit, a representation of the Bemf as a
`function of angular position of the rotor and the fit param-
`eters can he defined:
`
`Bemt‘_I (B_E_.k)I-l{1'CO5[0_6)+l(:'Sll'l(fi_e)+l§3‘
`
`cos(3-8, , c)+k,,-sin{3-9_e)+ir5-cos(5-El
`
`.C)+k6'
`
`sit:t(5'6_e)+lr-_.-cos('n'-0_ el+lr.B'sitt('o"El_e}
`
`Bemf __2(B_e,k):=k,,-cos(9-e,e)+lt,a-sin(9-t)__e)+ku-cos(11-B _e)+
`kn-sin(5-6 ejl
`
`Bemf(6_e,k):-Bernt'_l (t3_e,k)+Bcn1f_2((:‘,e,k)
`
`Having developed a representation of the Benztf as a
`function of the angular position of the rotor,
`it
`is next
`desirable to develop a corresponding representation of the
`Cogging torque data. in general, the same approach may be
`used.
`
`First, at series representation of the cogging torque should
`be developed. Since the cogging data will typically corre-
`spond to one complete mechanical revolution of the rotor
`shaft, and since the torque irregularity producing mechanism
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`Au exemplary plot of the magnitude of the l~‘l"l‘ of the
`cogging torque data represented by FIG. 3 is provided in
`FIG. 5. From the plot at FIG. 5, the important frequencies of
`the cogging torque can be determined. Any component with
`a magnitude of more than 0.001 will be used in the fit. The
`components of interest are therefore, Sxtumcch, l.6><(1)IIlt?.Ch,
`24>-twmech, 25><wmeeh, 4-7><tnmech, 48:-oiomech. DC may be
`considered in the fit, but can be ignored when defining the
`cogging function without significant negative impact on the
`system. The same is true of the Bemf fit. Thus, an appro-
`priate series for representing the cogging torque may he
`defined as follows where:
`
`Clogging _l{|.,k):=l-r,J+.l{1-cos[8-lr13't)+l(3-si.rt(3'ku't)+
`k-3‘C05[] 5'l(,_1't]+k4'SiJ1(1i3'kl3‘ t')+lr5-cos(24-kn-t_]+ko-sin(24-km-1.)
`
`Coggir1g__2(t_.k:l:-k7-cos(25'k,3‘ l.)'+kg'Sill(25'kLa'i)+kg‘CU3(47'k1313+
`km-sin(47-k,,-tytiu-cosras-k,,-t)+i,, 'siu(48'lt.1:;t)
`
`Cogging(l,k):='CIZIggirtg__l (t,k)+Cugging_2(t,k)
`
`Second, the partial derivative of the coggidg function with
`respect to the fitting parameters is defined. In the illustrated
`example,
`the harmoni