A New Breed of Electric Machines - Basic Analysis and Applications of DualMechanical Port Electric Machines

Longya Xu, Fellow IEEEDepartment of Electrical and Computer Engineering, The Ohio State University,

2015 Neil Avenue, Columbus, OH 43210, USAEmail: xu. 12(osu.edu

Abstract - Conceptually, mechanical port of an electricmachine can be doubled and the concept of dual-mechanical-port (DMP) will then give birth to a new breedof electric machines. In this paper, the various possiblestructures of DMP electric machine are discussed. Basicmodeling and analysis issues related to the DMP electricmachine are presented. An exemplary design of DMPmachine is given and verified by FEM results. Potentialapplications and future research work are given to concludethe paper.

I. Introduction

Any electric machine, DC or AC, is considered as aphysical device to accomplish electromechanical energyconversion. In this regard, an electric machine isdescribed as a coupled electromagnetic field with oneelectrical and one mechanical port. As shown in Fig. 1,the electrical port of a conventional electric machine ischaracterized by its terminal voltages and currents whilethe mechanical port its shaft speed and torque. This is thetraditional single-in and single-out port configuration.

of electric machine withillustrated in Fig. 2 (b).

dual mechanical ports is

DualElectical

Ports Mechanical

CoupledEM Field

(a) Doubly Excited Electric Machine

K

P.2

CoupledMecanca

Pnl2 A ~~EM Fied

Fig. 2 (b) Conceptual Electric Machine with DualMechanical Shafts

lI Mechanical

( , 4 , 1)Torque Electical

0 0 0 ~~~~~~~~~EMField

Speed

Fig. 1 Conventional Electric Machine

A single-in and -out port electric machine sometimes can

be designed in a doubly excited configuration, resulting ina doubly-fed electric machines. A typical example ofdoubly excited electrical machine is the so-called slipenergy recovery induction machine where the stator androtor windings are both electrically excited, participatingin electromechanical energy conversion. The doubly fedelectric machine is quite attractive for high-powervariable-speed applications where special operation modeand high energy efficiency is needed, such as those usedin wind power generation. A doubly fed electric machineis illustrated in Fig. 2 (a). Conceptually, the mechanicalport of an electric machine can also be doubled and theconcept of dual-mechanical-port (DMP) will then givebirth to a family of new electric machines. The concept

In this paper, various possible structures of a DMPelectric machine will be explored and operation featuresdiscussed. Further, the modeling and analysis issuesrelated to the DMP electric machine are to be investigated.To fully illustrate operational features, explore design andanalysis principles, and discuss application potentials, an

exemplary design of DMP machine is given and verifiedby FEM results.

II. Basic Structure and Operational Principles

A basic structure of a DMP electric machine can beillustrated by its cross-section as shown in Fig. 3. Asshown, a DMP electric machine typically has three basicparts, separated by two air-gaps and magnetically coupled.In general but not necessarily, the three parts can allequipped with windings mutually coupled throughmagnetic flux lines. For example, Parts #1 and #2 sharethe same magnetic flux lines with the same pole numbersand, when in operation, the 2 magnetic fields from Parts#1 and #2 are synchronized. Similar magnetic couplingbetween Parts #2 and #3 is also required. In addition, thefollowing important features can be identified in a DMPmachine:

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Vabc

Case 1 Case 2 Case 3Part #1 (01=0 stator (01>0 rotor (01>0 rotorPart #2 02>0 rotor 02=0 stator (02>0 rotorPart #3 303>0 rotor 303>0 rotor 303=0 stator

Fig. 3 Basic structure ofDMP electric machine

a) Among the three parts, at least two will be allowed tomove mechanically. In effect, the two movable partsconstitute the two mechanical ports, characterized bynone-zero mechanical torque and speeds at the port.

b) The two mechanical movable parts or mechanicalports can be assigned arbitrary among the threethough certain assignment of movable parts may bemore favorable than the others in terms ofconstruction, controlling and application.

c) The stationary part of the three will have to haveelectrical terminals as the electrical port. This isbecause the stationary part is at zero speed and, thus,not qualified as a mechanical port. To participate inelectromagnetic torque interactions, the energy flowsolely goes through the electrical terminals.

d) The middle part is sandwiched by the inner and outerair-gaps, both contributing electromagnetic torque tothe piece; hence, the overall electromagnetic torqueof the center part is the sum from both inner andouter air-gap.

e) As the dual mechanical ports the two moving partsare linked to and have external independentmechanical speeds and torque that are determined bythe external loads or engine connected.

f) Depending on the speed of a moving part relative tothe rotating field, electrical terminals may be alsoneeded on the moving part so that the energy flow isbalanced not only for the overall machine but alsoforeach individual part.

In Table 1, various possible moving part assignments arelisted and their speeds are specified. In all cases, themachine is a DMP, or a so-called dual-rotor electricmachine.

Table 1 Possible moving part assignment ofDMP

Let us start to examine Case 1 where Part #1 is stationary(stator), and the other two rotating (rotors). It is especiallyinteresting to investigate the operation modes withrelative speeds between the 2 rotors. Three operatingconditions discussed below:

i. If 01)2=03, the two rotors are of the same speed and norelative speed between the 2 rotors and, thus, norelative speed EMF induced in the windings on the 2rotors. In this condition, no electromechanicalenergy flow involved between the 2 rotors andequivalently we can treat the 2 rotors stickingtogether as one and the machine essentially is simplya conventional single-rotor machine with 2 shafts.

ii. If 01)2>01)3 or 01)2<01)3, in either condition, there is a slipbetween the two rotors. To have meaningful torqueinteractions between the 2 rotors, one of the rotorshas to be excited with the slip frequency so that the 2magnetic fields of the rotors are synchronized. If wedecide to compensate Part #3 (inner rotor), then theinner rotor will need excitation windings with a slipelectric frequency (relative to outer rotor) plus thefrequency of the outer rotor. The energy flowthrough the winding is proportional to the slipbetween the two rotors.

iii. The energy flows through the excited rotor due to slipmay be negative or positive depending on whetherthe slip is positive or negative.

iv. The excitation of slip frequency can also be donethrough excitation control over the outer rotor withthe inner rotor as the reference base. In such a case,the slip frequency excitation will be applied to thewinding terminals on the outer rotor.

v. In either slip compensation conditions, additionalcontrol can be given to the stator through the statorterminals to affect the overall electromechanicalenergy flow of the machine.

vi. In addition to the electric port, the externalmechanical energy flow are always allowed to beinput/output independently to the 2 mechanical ports(shafts) and the only law to observe for the DMPmachine is energy conservation.

We can observe that in terms of electromechanical energyflow, Condition i) is identical to that of a conventionalelectric machine with a single-in and -out ports; that is,the energy flow-in or -out from the single electrical portalways equals to that of the single mechanical port(neglecting losses). However, for other conditionsdiscussed in ii), the analysis becomes much complicatedsince we have three ports involving in energy flow in theDMP machine. Overall, the energy flow is balanced.However, depending on the energy flow directions among

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the dual mechanical ports and electric port, the operationmodes are very versatile.

For only Case 1 in Table 1, we have presented manypossibilities of operation modes. For Cases 2 and 3,another array of operation modes can be obtained, usingsimilar methods in analysis. Though all possiblestructures and operation modes are equally interesting intheory, it is the specific application objectives that dictatethe choice of one over the rest.

III. An Exemplary Design and Analysis

a) DMP machine structure

Fig. 4 shows an exemplary design of a DMP machine of 4poles. The outmost cylinder is chosen as the stator with 3phases of windings installed, acting as the pure electricalport, and the dual rotors the other 2 cylinders inside.

unified the two magnetic fields with the same number ofpoles.

F ODID

# of turns

stator17012168

outer rotor12099PM

inner rotor983568

Table 2 Main dimensions (in mm) and winding ofexemplary DMP

b) Dynamic modeling and equivalent circuit

Because of several similarities between the DMP and awound rotor doubly fed induction machine, we willpresent the DMP equivalent circuit based on that of aconventional doubly fed wound rotor induction machine.First, a brief review is in order of the steady stateequivalent circuit of a wound rotor induction machine asshown in Fig. 5.

rs Li L' r',/s

+ +Is I t M InrVs V rlS

L

Fig. 5 Equivalent Circuit for Wound Rotor Induction Machine

Fig. 4 Exemplary design ofDMP machine

We anticipate the 2 rotors with different speeds inoperation in general and choose the inner rotor as the slippower compensation port. So, the inner rotor will notonly be one of the two mechanical ports but also part ofthe electrical port. The two electrical ports, one on thestator and the other the inner rotor, can share a commonDC bus to form the single electrical port for the overallmachine (as shown in Fig. 3). Since the outer rotor is thereference base for controlling slip power compensation, itdoes not need to have active excitation control, becominga pure mechanical port. For that reason, we have usedpermanent magnets to build the outer rotor. The maindimensions and the winding specifications of the DMPare listed in Table 2.

In this equivalent circuit, the joint currents of the statorand rotor winding produce the air gap flux mutuallycoupling to both stator and rotor windings. Note that theconventional doubly fed induction machine has dualelectric ports that are represented by the stator and rotorvoltage terminals. The single mechanical port of themachine is represented by te element, r2/s, meaning thatthe power going through this virtual resistor is theconverted power from electrical to mechanical, or viceversa. To be mentioned in the equivalent circuit is thefactor of "1/s" on the rotor voltage. Here, "v," is theactually applied voltage (with stator-rotor turn ratioconsidered) to the rotor windings and the factor "1/s"accounts for the rotor speed difference referred to thesynchronous rotating field. If the rotor is synchronizedwith the rotating field, "s" equals to zero, or else if therotor standstill, "s" unity. For the new breed of DMPmachine, we propose the equivalent circuit by adding aparallel connected current source Ipm to the magnetizingbranch as shown in Fig. 6.

As discussed in earlier section, the two magnetic fieldsaround the two air-gaps can be designed separately withdifferent pole numbers but can also be unified to reducesize of the machine. In the exemplary design, we have

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L, L'i r'r/s

Vs

Fig. 6 Equivalent Circuit for DMP with PM Outer I

To explain the modification of the equivalent circneed to compare the electromagnetic structureDMP to that of the wound rotor doubly fed incmachine. If the DMP outer rotor were removreplaced by a piece of slid iron ring, the DMP essewould have degenerated to a wound rotor incmachine. In other word, if the outer rotor were retthe DMP machine would have become identicwound rotor induction machine with a huge air galsurprising, the equivalent circuit of a woundinduction machine applies to the DMP in this casethat the magnetizing inductance becomes verybecause of the vastly enlarged air gap. Altematibthe polarized permanent magnet outer rotor of thewere replaced by a piece of none-polarized, none-,solid iron ring, the DMP also would have becwound rotor induction machine since the solid ironneither polarized nor salient - its mechanical rotanot electromechanical energy related. Comparedremoval of the PM outer rotor to its replacemensolid iron ring, the later makes the DMP air gapsmaller and therefore, much larger magninductance. As discussed in both cases where theouter rotor were either removed or replaceequivalent circuit would have become that of arotor induction machine.

As we discussed in previous section, the outer PM rotor+

not only contributes to the main magnetic flux generation,but also acts as one of the dual mechanical ports of the

v"/S DMP machine. To explicitly exhibit the role of the outerrotor in electromechanical energy conversion, wetransform its current source representation into thevoltage source counterpart by Thevine equivalent. Finally,the equivalent circuit of the DMP in Fig. 7 arrives with

Rotor two electrical ports and two mechanical ports. Note thatthe voltage source embedded in the magnetic branch

uit, we drives magnetic flux in the magnetizing branch. The jointof the currents from both the stator and rotor interact with theluction voltage source, resulting in the electromagnetic torqueted, or and power. The power represents the electromechanicalmntially power conversion through the outer rotor, the secondluction mechanical port of the DMP machinemoved,.ally ap. Not

rotor r',/sexcept r k

small +vely, if V

V

1

- fDMPsalient,,ome aring istion isto the

Lt by amuch

ietizing- DMPd, thewound

However, if the PM outer rotor is placed in the air gapand rotating synchronously with the main field, the DMPmagnetic structure changes very differently. First of all,the total magnetic field of the air gap is now contributedand driven by three MMFs, one from the stator excitation,another the inner rotor excitation, and the third the outerrotor permanent magnets. We can regard the abovecircuit an intermediate equivalent circuit to account forMMF contribution by the permanent magnets. In thisintermediate circuit we have defined a current source Ipmto represent the function of the permanent magnets. Herethe permanent magnets are described by a magnetizingcurrent source, in parallel connection to the magnetizingbranch of the wound rotor induction machine equivalentcircuit. To qualify the permanent magnets in such adescription we need to ensure that the permanent magnetsare rotating synchronously with the main field and createmagnetic flux linking to both the stator and rotorwindings. Examining the magnetic structure of the DMP,it is clear the permanent magnet outer rotor satisfies theconditions. The magnitude of Ipm is directly proportionalthe magnetizing strength of the magnets.

Fig. 7 Equivalent Circuit for DMP for Steady State

Once the equivalent circuit of the DMP machine isestablished, we can find the physical meaning and therelative size of each parameter as listed in Table 3.

F RIR2

L1 =L2Lm

0.27 ohms0.49 ohms15.4x 10-4H Em I 243 A

0.374 wbCOX.m

The transient model of the DMP is derived briefly andpresented in Appendix. The following are the majorequations:

Vqs = qsrs + dt + Ct) dsd2e*e ds2 Avj, = idsrs+ dt w,qs

d2ey e =1 re + qr +(c o.)lqr qr r dt dr

Vdr = ierr + d -(co c

where

(1)

(2)

(3)

(4)

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-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

ze =Lje +L je

Zqs = qs miqre~ =Z +L Je +Lmi;ids im + tSds + tdr

qr r qr m qs

Zdr =in rtrdr intds

For the DMP, there are three torque equations

Te,stator 2 2 (Xmiqs + dr qs Xqrids)for the stator,

Te,pm-rotor = 2 Am (iqs + iqr)'e,m 22for the PM rotor, and

Te,wd-rotor 2 2 (Amiqr + Adsiqr pAqsdr )forthew t2 2

for the wound rotor.

Note that the dynamic Eqs. (1) through (11) are allreferred to the PM rotor reference frame, that is, the d-axis of the reference frame is aligned with that of thepermanent magnet rotor.

c) Modes of operation

Utilizing the derived equivalent circuit and dynamicmodel, we can investigate the power flow and operationmodes achievable by the DMP. Let us start to investigatethe power flow through the stator windings. The powerinput or output through the stator terminals is purelyelectrical since the stator is stationary. Once the electricalpower reaches to the stator terminal and continues acrossthe air gap between the stator and PM rotor, it becomeselectromechanical power to be handled by the outer rotor.Similarly, at the inner wound rotor, the electrical andmechanical power is combined and net amount goesacross the air gap between the inner wound and outer PMrotors. The net power arrives on the outer PM rotor fromthe stator and inner rotor finally determines the net powerin or out of the mechanical port of the outer PM rotor.The equation to illustrate the overall power balance of theDMP machine is (losses neglected):

± Pe, stator ± Pm, pm-rotor ± Pe, wd-rotor ± Pm, wd-rotor 0 ( 1 2)

In the equation, the " " indicate the power flow in orout. The first subscript "m" or "e" represents mechanicalor electrical power related. The second subscriptspecifies if the power is related to the stator, outer PMrotor or inner wound rotor. As can be observed, thissingle equation is multi-variable based and theoretically,has infinite sets of mathematic solutions and eachcorresponds to one operation mode. In practice, of course,many physical constraints apply. Nevertheless, theappropriate solutions, each corresponding one operationmode, are much more than those of a single-in single-outconventional electrical machine. We will show several of

the solutions to the equation and attempt to interpretapplication situations corresponding to them.a. + Pe, stator - Pm, out-rotor - Pm, in-rotor + Pe, in-rotor 0In this case, we input electrical power to the stator, andoutput mechanical power to both the inner and outerrotors. The inner wound rotor may a have lower or highermechanical speed than that of the main field, and hence+ Pe, in-rotor will take care of the balance due to the slip ofthe inner wound rotor. In this case, we call the DMPmachine as a power splitter because the input electricalpower is splitted into and output from the 2 mechanicalpower ports. This is basically a motoring operation mode.

b. - Pe, stator + Pm, out-rotor + Pm, in-rotor + Pe, in-rotor= 0Opposite to (a) we can change the signs of power relatedto the stator and rotors. That is, we input mechanicalpower from both rotors, and output electric power to thestator. The inner rotor is allowed to have a lower orhigher speed than that of the main field, the+ Pe in-rotorwill take care of the balance due to the slip of the innerrotor. In this case, we can call the DMP machine as apower combiner because mechanical power is combinedand converted into electrical power, mainly in thegenerating operation mode.

c. - Pm, out-rotor + Pm, in-rotor =-i Pe, stator ± Pe, in-rotor 0In this mode, additional constrain is imposed, that is, notonly is (11) satisfied but also are the electrical andmechanical subset of the equations balanced and equal tozero. This is a so-call variable ratio gearbox operationmode in which we want the mechanical power balancedfrom both rotors but allow them to have different speeds.Since the two rotor speeds are different with the powerequals to each other, the torques of the two rotors will beinversely proportional to the speeds. According to theequation, we need to involve in the stator and inner rotorelectrical power in the process in such a way that theyare also balanced. We may call the DMP machinefunctioning as a variable E-gearbox because we useelectromechanical device to achieve variable gearboxresults.

d. - Pm, out-rotor + Pm, in-rotor = JL Pe, stator ± Pe, in-rotor = CCase (c) can be extended slightly from "0" to a constant"C", implying that the mechanical power of the tworotors is not completely balanced and the remaining canbe absorbed by the electrical terminals and sent to thecommon DC bus, for example a battery bank, for energystorage or draining. This operation mode is very similarto a hybrid traction system for electrical vehicles.

We can derive many other modes of operation. Inaddition, if the rotating parts are assigned differently, wecan derive another family of operation modes andversatility and flexibility of DMP machine is trulyextraordinary.

d) Finite Element Analysis

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In evaluating the electromagnetic design, Finite ElementAnalysis (FEM) is used to compute magnetic fluxdistribution, torque production, and controllability of thedesigned DMP machine. Shown in Fig. 8 (a) are thecurrent waveforms in the stator and rotor windings. Inorder to focus on the DMP machine torque production,the windings of the stator are excited with 3 phasecurrents with a peak value of 1OA and the windings of theinner rotor 5A. As expected and shown in Fig. 5 (b),when the 2 rotors are stationary and the 2 windings of theDMP are excited with sine currents, the torque productionis also in sine function in time. The FEM result indicatesthat if the MMFs of both rotors are controlled insynchronization, a maximum and constant torque of 49Nm will be achieved. Compared to AC machines ofsimilar size, a DMP machine seems to be very impressive

emEind rt s Tie

10

and controlled so that all MMFs are synchronized toproduce a unified magnetic field.

Fig. 9 Magnetic flux distribution

IV. Conclusions and future work

0

-5

-10in -que production in addition to its highly flexibleoperational modes.

(a) Stator and inner rotor current

Tor ue vs Time

Fig. 8iet1X

(b) Torque production

Fig. 9 shows the magnetic flux distribution at the momentof maximum torque production. As indicated in thefigure, the magnetic fields of the machine separated bythe two air-gaps are unified; that is, the continuous fluxlines go through both air-gaps and link all three parts, thecores of one stator and two rotors. The flux lines aredriven by three sources, the stator MMF, the outer rotorpermanent magnets, and the inner rotor MMF. In variousconditions if the outer and inner rotors are of differentspeeds, the electrical frequencies and phase angle of thestator and inner of rotor windings have to be coordinated

Though AC electric machines have evolved for more thana century and arrived at very sophisticated level today, asingle electric machine with dual mechanical ports is arelatively new concept. When the mechanical ports of anelectric machine are multiplied, a whole family of newelectric machines can be achieved. Although this breed ofnew electric machines essentially follow the basicelectromagnetic and electromechanical principlesgoverning the conventional ones, the new breed electricmachine still give electric machine design, control andapplication engineers enough new territory to explore,given the newly developed power electronics and digitalcontrol environment. The application potentials of theDMP machine are vast, from multi-source hybrid traction,integrated starter and generator, to variable gearboxes,just to list a few. The research work on DMP machine isin progress and more results are to be reported in futurepapers.

ACKNOWLEDGMENT

The finite element analysis presented in this paper isconducted by Jingchuan Li using Maxwell softwareprovided by Ansoft Inc. Their excellent work and help isacknowledged and greatly appreciated

REFERENCES

[1] P. Hammond, "Energy Methods in Electromagnetism," (Book),Clarendon Press, Oxford, 1981.

[2] S. A. Nasar, L. E. Unnerhr, "Electromechanics and EletricMachines, " (Book), John Wiley and Sons, 1983.

[3] Y. Tang and L. Xu, "A Flexible Active and Reactive Power ControlStrategy for a Variable Speed Constant Frequency Wind-PowerGenerating System", IEEE Transactions on Power Electronics,Vol.10, No. 4, July/Aug., 1995, pp.472-478.

[4] L. Xu and W. Cheng, "Torque and Reactive Power Control of aDoubly-Fed Induction Machine by Position Sensorless Scheme"

29

IEEE Transactions on Industry Applications, Vol. 31, No. 3,May/June 1995, pp. 636-642

[5] Martin J. Hoeijmakers, Jan A. Ferreira, "The ElectricalVariableTransmission", IEEE-IAS Annual Meeting Proceedings, Oct. 2004,Seattle

[6] Yuusuke Minagawa, Masaki Nakano, Minoru Arimitsu and KanAkatsu, "New Concept Motor with Dual Rotors Driven byHarmonic", SAE TECHNICAL, PAPER SERIES 2002-01-2857,NISSAN Motor Co., LTD

APPENDIX

Modeling and the derivation of differential equations forDMP dynamics are presented briefly in the appendix.First we derive voltage equations and conduct referenceframe transformations for the stator windings:

IVabcI s I = Iiabc,s ]rs + dt

Hyabc,s ] = [jm + [Ls ] [Iabc,s ] + [Lm ] IIabc,r ]

(A-1)

(A-2)

where [L,] is the inductance among stator windings and[Lm] the stator and rotor winding:

Laa Lab[Ls]=L:::: La4

LLca Lcb

Lac- 1

Lbc =L, -0.5

Lcc- L-- 05

-0.5 -0.51 -0.5 (A-3)

-0.5 1 J

[Lm]

in which dT(O)r% ] 0) dT(O)r s ] iscalledt]dt Rabc,s dOIVacspeed voltage component and can be write out in thescalar form explicitly:

dT(O) s -] ds (A.dO Rabc,s] Xqs

L0]j

[Vqdois =[iqdo,sirs + dt

- 1ds

-CO ,zeqCt) jq

,he

-9)

(A-10)

For simplicity , we have neglected leakage inductance sofar. If we write the stator voltage equations in scalarforms, it follows that

Vqs = qsrs +t + () ds

d2<eVdS =dlA+e d

dt qs

(A- 1)

(A-12)

At this point, the stator voltage equations appear in thesame forms as those of a general AC machine. However,if we also apply abc-qdo transformation to the fluxlinkage equations, we will reveal the essential differenceof the DMF machine from a conventional AC machine.

ze =Lje +L je

qs = qs miqre= Xe + Lti + Lm e

(A-13)

(A-14)LaA LaB L,C

LbA LbB LbC =

LC LC LjCJcA LcB ,cC-

cos Or

Lm cOs(Or- 22/3)

LcOS(Or + 2ff/3)

COS(Or+ 22T/3) COS(Or -227/3)COsOr COS(Or + 227/3)

COS(Or -2ff/3) COsOr

Applying abc- qdo transtormaton and alignmg the d-axisof the reference frame to that of the permanent magnetrotor, that is,

d[2bs]T(O)[VabC¶s ] = T(O)[iasc,s ]rs + T(O) csI

we then have

[Vqdoj, ] = Iqdo,s ]r5 + T(O) dt

In the above stator voltage equation, we can furth(out that

T(O) d[abcs] dT(O)[Iabc,s] d]dt dt t

The stator voltage equations become

r (0)

dIt [abc,s ] (A )

[Vqdo,s] [I r+dqdo,s]Ts+([b dTO Iavbc,s] (A-8)

[qder + dLZed, ] dT(O)rSLqdos4's dt dt- [L'abc,s]

(A-4) In Eqs. (A-13) and (A-14), it is clearly shown that themain field of the DMP is contributed by three components,one from the stator, the second the inner rotor and thethird the PM rotor.

Voltage equations for the inner rotor can be derived in asimilar way, that is

[Vabc,r] [abc,r ]rr +d[)abc,r]

dt(A-15)

[aibc,r] [2 +Lrr Iabc,r] + Lm[Iabcj (A-16)After reference frame transformation and write theequations in scalar forms, we have

(A-6) Ve e dXq,rer find die

re =Lje +L_ )

lqr riqr m qse =Ze +L je +LmeZdr m r dr m ds

(A-17)

(A-18)

(A-19)

(A-20)

Both the DMP stator and rotor voltage equations can bederived further to obtain steady state equations to form itsequivalent circuit as that shown in Fig. 6. Detailedderivation is omitted here.

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-L- -7- -A--' J A-1-

Cll Cll

The DMP machine has very different torque equations,compared to a conventional AC machine. This is due tothe fact that the DMP machine has two mechanical portsand two sets of windings, resulting in different speed-torque characteristics for each mechanical shaft. Theelectromagnetic torque equations are derived based onmagnetic flux interactions with respect to the excitationcurrent vectors for the stator, PM rotor and inner rotorrespectively. For the stator, the torque equation is

Te,stator 2 2 (miqs + Adrlqs - Aqrlds) (A-21)

For the PM rotor

Te,pm-rotor 2 2 m (iqs + iqr) (A-22)

chosen as that of the permanent magnet rotor. Therefore,the d-axis currents in the stator and inner rotor will notgenerate electromagnetic torque on the permanent magnetrotor. This fact is reflected into the torque equation forthe PM rotor. For the inner wound rotor:

Te,in-rotor 2 2 (Amiqr + Ad,iqr cAqsdr) (A-23)

With Eqs. (A-1) through (A-23), combined with 2 motionequations (not show) for the two rotors, we can simulatethe electric and electromechanical dynamics of the DMPmachine. The overall system equations is summarized bya block diagram shown in Fig. A-1. In addition, thepower electronic converters, energy storage elements,current and torque controllers, and speed regulators canalso be added to form a large scale DMP based system.

Note that we have assumed in the abc-qdo referenceframe transformation that, the d-axis of reference frame is

Fig. A- I Overall System Block Diagram for DMP Dynamics and Control

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