Abstract—In this paper is presenting a reactive homopolar

brushless synchronous machine (RHBSM) and a reactive homo-heteropolar brushless synchronous machine (RHHBSM) with stator excitation destined to operate as low power generator or servomotor with variable speed. Hereby are presented two mathematical models, the orthogonal one and the one of the spatial phasors. Based on these, is designed the control system and are presented the results obtained by simulation. The work is presented as an integrated design of machines, drive and controller.

Keywords—orthogonal model, simulations, sensorless control, space-phasor model.

I. INTRODUCTION N THIS paper is described the design, construction and control of a reactive synchronous homo-heteropolar and

homopolar brushless synchronous machines (RHHBSM, RHBSM) generator/motor for variable speed.

For modeling and simulation were used two methods: the field tubes method – which brings acceptable simplifications, taking into account the magnetic saturation of the ferromagnetic core, and the types of electric windings and 3D FEM analysis with specialized software [1] – [3].

The classic synchronous machines have the following main disadvantage: the armature rotor excitation which determines a great rotor inertia and weight, and involves the brushes and slip rings.

In [4] is presented a form of heteropolar linear synchronous machine that is able to provide both thrust and lifting force at relatively high efficiencies and power factor.

Starting form this idea, there were developed two rotating reactive models with stator excitation, one homopolar and the

Manuscript received September 27, 2010. Sorin Ioan Deaconu is with Electrical Engineering Department,

“Politechnica” University of Timisoara, Revolutiei str., no. 5, Hunedoara, 331128, Romania (phone: 0040254207529; fax: 0040254207501; e-mail: sorin.deaconu@fih.upt.ro).

Lucian Nicolae Tutelea is with Electrical Engineering Department, “Politechnica” University of Timisoara, V. Parvan str., no. 1-2, corp D, etaj 1, Timişoara, Romania (e-mail: luci@lselinux.upt.ro).

Gabriel Nicolae Popa is with Electrical Engineering Department, “Politechnica” University of Timisoara, Revolutiei str., no. 5, Hunedoara, 331128, Romania (e-mail: popa.gabriel@fih.upt.ro).

Tihomir Latinovic is with Robotics Department, University of Banja Luka, Bosnia and Hercegovina (e-mail:tiho@inecco.net).

other homo-heteropolar [1] – [3], which removes the disadvantages of the classic synchronous machines.

Although not widely used in practice, synchronous homopolar machine has been researched for a variety of applications. They are sometimes referred to as homopolar inductor generator/motors’ [5], [6], or simply as homopolar motors’ [7], [8]. The defining feature of this machine is the homopolar or homo-heteropolar d-axis magnetic field created by a field winding [5], [6], [8] - [10] and permanent magnets and windings [7].

However, in case of the synchronous homopolar machine, the field winding is fixed to the stator and generally encircles the rotor rather than being placed on the rotor. There are several advantages to having the field winding in the stator. Among these is the elimination of slip rings and greatly simplified rotor construction, making it practical to construct the rotor from a single piece of high-strength steel. The other rotor designs feature laminations [8], permanent magnets [7], or other non-magnetic structural elements to increase strength and reduce winding age losses [6]. Other advantages of having the field winding in the stator include ease in cooling and increased available volume [1], [11], [12].

The first part of the paper presents a description of the synchronous homopolar and homo-heteropolar machines. The second parts is focused on orthogonal model and space-phasor model.

The third part presents the machines’ dynamics and control algorithms’ development, and simulations results for the control systems.

II. THE CONSTRUCTIVE ELEMENTS The RHBSM and RHHBSM which we’ll analyze further

are rotary machines. In fig. 1 is presenting a cross-section and longitudinal section of RHBSM, and in fig 2 a longitudinal section of RHHBSM.

The excitation coil has a ring shape and is placed in the windows of the U-shaped laminations stack (fig. 1), respectively E-shaped laminations stack (fig. 2), and, at passing of the rotor poles, the field is closing, having by this a rectangular variation form. When the rotor pole is not under the laminations stack, the field is practically null [2], [3].

Mathematical models and the control of homopolar and homo-heteropolar reactive

synchronous machines with stator excitation Sorin Ioan Deaconu, Lucian Nicolae Tutelea, Gabriel Nicolae Popa and Tihomir Latinovici

I

Advances in Communications, Computers, Systems, Circuits and Devices

ISBN: 978-960-474-250-9 78

Field coil

Γ

Rotor pole

x v Induced coil

Stator core

Fig. 1 Magnetic circuit of RHBSM

Stator core (laminations

stack)

Excitation coil

Airgap

δ

Rotor pole

Fig. 2 Longitudinal magnetic circuit’s section of RHHBSM

By this representation, it results a homopolar inductor

magnetic field in the area of the left leg, respectively right leg of the stator laminations stack which is positive under left and negative under right leg (fig. 1 and fig. 2) and a heteropolar inductor magnetic field in the area of the central leg of the stator laminations stack [2].

For the RHBSM the armature’s winding is in two layers, with 8-shaped coils (fig. 3), and is placed in the open slots. This winding type allows the elimination of non-uniformities that might appear if it would be achieved separately on each leg of the laminations stack [2].

2 181716

15

14

13

12

11

10

9

8

7

6

5

4

3

1

2 181716

15

14

13

12

11

10

9

8

7

6

5

4

3

1

A

X

Fig. 3 The armature winding of RHBSM Fig. 4 and 5 present a 3D representation of the magnetic

circuit and windings of the stator and the rotor of RHBSM in fig. 6 and 7 for RHHBSM [2], [3].

Excitation coil

Laminations stack

Fig. 4 3D representation of the stator magnetic circuit with excitation coil of RHBSM

Isolated cylinder

Rotorpole

Fig. 5 3D representation of the RHBSM rotor

Laminationsstack

Excitation coil

Armature’s winding

Fig. 6 3D representation of stator magnetic circuit with field and armature winding coils of RHHBSM

Similar with RHBSM, the RHHBSM’s armature winding is

placed in the open slots, formed between the laminations stack. In each slot there are two sides of the coil (fig. 6). The winding is distributed in three layers. Lateral coils of one phase are connected in series and the resulting group is connected in parallel with the respective phase coils of the central leg. The coils of one phase are distributed in different

Advances in Communications, Computers, Systems, Circuits and Devices

ISBN: 978-960-474-250-9 79

layers for each leg of magnetic circuit coils on all phases occupy a layer (of three) leading in this way to a uniformity of dispersion reactances [3].

R otor pole

Isolated cylinder

Fig. 7 3D representation of the rotor of RHHBSM

Similar with RHBSM, the RHHBSM’s armature winding is

placed in the open slots, formed between the laminations stack. In each slot there are two sides of the coil (fig. 6). The winding is distributed in three layers. Lateral coils of one phase are connected in series and the resulting group is connected in parallel with the respective phase coils of the central leg. The coils of one phase are distributed in different layers for each leg of magnetic circuit coils on all phases occupy a layer (of three) leading in this way to a uniformity of dispersion reactances [3].

III. ORTHOGONAL MODEL OF RHBSM AND RHHBSM The orthogonal model is identically for the two machine

types (homopolar and homo-heteropolar) with other inductivity values that intervene in the mathematical relations [13].

The excitation winding and the armature’s winding are found in the stator, and the rotor poles are massive, provided with a damping cage.

In fig. 8 is presented the real machine, and in fig. 9 the d-q model obtained by transformation [8]. The complete equivalence between the real machine and the d-q model assumes the existence of the homopolar components U0 and i0.

q

q c i C U C

U a

ia

a

θ

Q ib

iQ

i D

D

U b

U E

b d d

ω r

3 2 π

iE

Fig. 8 The real machine’s equivalent diagram

U q

d

i q

q q

Q

i Q

D

i D

U E i E

U d

i d

E d

ω b = ω 1

d

q

Fig. 9 The machine’s d-q model

At the star-connection this component does not intervene. It should be taken into account the equality of the momentary powers, losses, the couple and the stored magnetic energy.

Using the Park’s transformation for fluxes and currents, are obtained the expressions of the fluxes by axes d and q, ψd and ψq where LSσ is the own dispersion inductance, L0 the inductance’s constant component, L2 the inductance dependent on the rotor’s position, M0 the coupling constant inductance, ME, MD, MQ the coupling maximum inductances between a stator phase and the excitation respectively damping winding D and Q [9], [10]:

DDEEddSd iMiMiLMLiL ⋅+⋅+⋅⎟⎠⎞

⎜⎝⎛ +−+⋅=ψ σ 2

323

200 , (1)

QQqqSq iMiLMLiL ⋅+⋅⎟⎠⎞

⎜⎝⎛ −−+⋅=ψ σ 2

323

200 . (2)

If the dispersion coupling inductance between the stator and

cage D (LdDσ = 0) is neglected, are obtained the synchronous inductances, longitudinal Ld and transversal Lq [11]:

200 23 LMLLd −−= , (3)

200 23 LMLLq +−= , (4)

20

0LM −= . (5)

Angle θ has the expression, where ωr is the rotor’s angular

speed and θ0 the initial value:

∫ θ+⋅ω=θ 0dtr . (6)

Advances in Communications, Computers, Systems, Circuits and Devices

ISBN: 978-960-474-250-9 80

The binding relation between the orthogonal model’s currents id, iq and i0 and the real machine’s currents ia, ib and ic is [9]:

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

c

b

a

abcdqq

d

iii

P

i

ii

0

0

, (7)

where:

2 2cos( ) cos( ) cos( )3 3

2 2 2sin( ) sin( ) sin( )3 3 3

1 1 12 2 2

abcdqP

π πθ θ θ

π πθ θ θ

⎡ ⎤+ −⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤ = + −⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

, (8)

the same relation being valid also for fluxes,

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ψψψ

⋅=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ψ

ψψ

c

b

a

abcdqq

d

P 0

0

. (9)

From the previous relations and taking into account the

relation:

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

0

0

i

ii

Piii

q

dT

abcdq

c

b

a

, (10)

is obtained:

( ) DDEEdSd iMiMiLLL ⋅+⋅+⋅⎥⎦⎤

⎢⎣⎡ ++=ψ σ 2

323

20 , (11)

( ) QQqSq iMiLLL ⋅+⋅⎥⎦⎤

⎢⎣⎡ −+=ψ σ 2

323

20 , (12)

[ ] 00000 2 iLiMLL SS ⋅≅⋅++=ψ σσ . (13)

Considering a single rotor cage by each axis, the voltages

UD = UQ = 0 and the brushes’ speed ωb = ωr, there are obtained the general equations of the orthogonal model of the homo-heteropolar synchronous machine, where RS is the stator resistance, RE the excitation’s resistance, RD and RQ the rotor cages’ resistances, Me the electromagnetic torque and p the number of pole pairs:

qrd

dSd tURi ψ⋅ω+

∂ψ∂

−=− , (14)

drq

qSq tURi ψ⋅ω+

∂

ψ∂−=− , (15)

qrE

EEE tURi ψ⋅ω+

∂ψ∂

−=− , (16)

tRi D

DD ∂ψ∂

−= , (17)

tRi Q

QQ ∂

ψ∂−= , (18)

( )dqqde iipM ⋅ψ−⋅ψ= . (19)

IV. THE SPACE-PHASOR MODEL

We define the stator current space-phasor, si in stator coordinates [14]:

( )cbas

s iaiaii ⋅+⋅+⋅= 2

32 . (20)

For distributed windings (q ≥ 2) all the stator- self (Laa),

mutual (Lad, Lac, Laf), and stator-rotor inductances (Ladr, Laqn) are dependent of the rotor position θer, and rotor inductances are independent of this.

The phase a flux linkage λa is:

rqraqrdradfafcacbabaaaa iLiLiLiLiLiL +++++=λ . (21)

Making use of the inductance definition we find:

( ) ( ) ( )( ) ( ) ( )erj

rqr

rsqerjr

rdrsderj*

faf

erj*ssssla

eijLReeiReLeiReL

eiReLiReLiReL

θθθ

θ

−++

+++=λ

2

220

23

23

23

. (22)

The stator flux space-phasor λ s is:

( )cbas

s aa λ+λ+λ=λ 2

32 , (23)

where λb, λc are similar as in (21).

The stator and rotor equation in d-q coordinates becomes [14]:

,dt

dirV qrd

dsd λω−λ

+⋅= (24)

drq

qsq dtd

irV λω+λ

+⋅= , (25)

,iL ;dt

dirV fflfqr

ffff ⋅=λλ⋅ω−

λ+⋅= (26)

Advances in Communications, Computers, Systems, Circuits and Devices

ISBN: 978-960-474-250-9 81

dmrdlrdrdrd

rdrd iL ;dt

dir λ+⋅=λ

λ+⋅=0 , (27)

qmrqlrqrqrq

rqrq iL ;dt

dir λ+⋅=λ

λ+⋅=0 . (28)

The torque Te is [14]:

( ) ( )dqqd*

sse iipijRepT ⋅λ−⋅λ=⋅λ=23

23 . (29)

Finally, the d-q variables are related to the abc variables by

the Park transformation:

( ) 322

32 /jerj

cbaqds ea ;eVaaVVjVVV π⋅θ− =⋅++=+= . (30)

Note also that all rotor variables are reduced to the stator:

22d

rdrl

drld

rdr

dr KLL ;

Krr == , (31)

22q

rqrl

qrlq

rqr

qr KL

L ;Kr

r == . (32)

The equations obtained based on the orthogonal model and

the one of the spatial phasors lead us to the idea that the second model is more complete, therefore we’ll use it for designing the control system.

V. CONTROL ALGORITHMS’ DEVELOPMENT The operation of the RHBSM and RHHBSM synchronous

machines as generator or servomotor with variable speed presents a special importance at conceiving the adjustment system’s structure. Based on the spatial phasors’ theory, this system should be treated unitary. The used model takes in consideration also the machines’ saturation [15].

A. Basic Control Scalar control (V / f) is related to sinusoidal current control

without motion sensors (sensorless) (fig. 10) [14]:

RHBSM (RHHBSM)

DC – DC Converter

Int. Control

PI Control

if

Vref -

+

if

ifref

- +

+

Fig. 10 Scalar control for RHBSM (RHHBSM) with torque angle

increment compensation

For faster dynamics applications, vector control is used (fig. 11) [11], [14].

Speedcontroller ej er PWM

generator PWM

inverter

Vd

Positionand speedestimation

i a ibVa Vb

i a i b

θr

p

1/s

a b

b

a

er

referencespeed

θ

r*

-i

id

*

*

q i*a

i*b

i*c r

measuredor estimated

measured or estimated

encoder

DC – DC Converter

Int. Control

PI Control

if

ω

ω θ

i d

id*

RHBSM (RHHBSM)

if

ifref

+ -

Fig. 11 Basic vector control of RHBSM (RHHBSM)

a - with encoder; b – without encoder

B. Torque Vector Control To simplify the motor control, the direct torque and flux

control (DTFC for induction machines) has been extended to RHBSM (and to RHHBSM) as torque vector control (TVC). Again, fast flux and torque control may be obtained even in sensorless drive (fig. 12) [11], [14], [15]:

Speedcontroller Commutation

table V (T)

PWM inverter

Vd

Flux, torqueobserver

and speedobserver

ia ibVa Vb

r

p

1/s

b

a

referencespeed

r*

-

T

s

*

*

e

r

encoder

i

r^

Te^ s

^-

-

α

DC – DC Converter

Int. Control

PI Control

if i d

id*

RHBSM (RHHBSM)

Reftransf

Int. Control

ω ω

ω

θ

θλs

λ

λ

if

ifref

- +

Fig. 12 Torque vector control (TVC) of RHBSM (RHHBSM) a - with encoder; b – without encoder

C. Simulation Results The voltage source inverter used in simulations was a

Danfoss VLT 3005, 5 KVA one, working at 7 KHz switching frequency. The RHBSM has the following parameters: PN = 2.5 kW, UN = 400 V, Y, IN = 5.5 A, fN = 50 Hz, p = 3, Rs = 2 Ω, Ld = 0.023 H, Lq = 0.017 H, IEN = 6 A.

Further, the simulation results are presented with sensorless control of RHBSM at different speeds and during transients (speed step response and speed reversing) using DTC and SVM control strategies [14].

To eliminate the nonlinear effects produced by the inverter, the dead time compensation is 2.5 μs.

Steady state performance of RHBSM drive with DTFC control strategy at 25 rpm is presented in figure 13 (in “a” is represented estimated torque and in “b” the estimated speed).

Advances in Communications, Computers, Systems, Circuits and Devices

ISBN: 978-960-474-250-9 82

For DTFC-SVM control strategy the performance are presented in figure 14 (a - estimated torque, b - estimated speed).

The reduction in torque and current ripple by SVM is obvious.

15

12

9

6

3

0

-3

-6

-9

-12

-15

Estim

ated

torq

ue [N

m]

0 150 300 450 600 750 900 1050 1200

Time [ms] 0 150 300 450 600 750 900 1050 1200

Time [ms]

75

50

25

0

Estim

ated

spee

d [r

pm]

a) b) Fig. 13 Direct torque and flux control of RHBSM at 25 rpm steady

state a) – estimated torque; b) – estimated speed

15

12

9

6

3

0

-3

-6

-9

-12

-15

Estim

ated

torq

ue [N

m]

0 150 300 450 600 750 900 1050 1200

Time [ms]

0 150 300 450 600 750 900 1050 1200Time [ms]

75

50

25

0

Estim

ated

spe

ed [r

pm]

a) b) Fig. 14 DTFC-SVM control of RHBSM at 25 rpm steady state

a) – estimated torque; b) – estimated speed

16

12

8

4

0

-4

-8

-12

-16 0 150 300 450 600 750 900 1050 1200

Time [ms]

Estim

ated

torq

ue [N

m]

1000

900

800

700

600

500

400

300

200

100

0 0 150 300 450 600 750 900 1050 1200

Time [ms]

Estim

ated

spee

d [r

pm]

a) b) Fig. 15 Torque and speed transients during no load deceleration

from 900 to 60 rpm with DTFC a) – estimated torque; b) – estimated speed

Comparative results have been presented in fig. 15 for DTFC

and DTFC - SVM control during transients (reference speed step from 900 rpm to 60 rpm). A slightly faster torque response has been obtained with a DTFC control strategy.

VI. CONCLUSION In this work was obtained an orthogonal and a space-

phasor model for RHBSM and RHHBSM. Based on this models was develop a control strategy. A

sensorless control scheme that does not require an estimator for rotor position or flux was presented. With the combined DTFC-SVM strategy, low torque ripple operation has been obtained with RHBSM. Further improvement of the rotor

position estimation is necessary in order to obtain very fast response with this machine.

Further experimental results will be developed.

REFERENCES [1] S.I. Deaconu, L.N. Tutelea, G.N. Popa, I. Popa and C. Abrudean,

“Optimizing the Designing of a Reactive Homopolar Synchronous Machine with Stator Excitation”, IECON 2008, 34th Annual Conference of the IEEE Industrial Electronics Society, Orlando, Florida, USA, 10-12 November, 2008, pp. 1311-1318.

[2] S.I. Deaconu, M. Topor, L.N. Tutelea, G.N. Popa and C. Abrudean, “Mathematical Model of a Reactive Homopolar Synchronous Machine with Stator Excitation”, EPE-PEMC 2009, Barcelona, Spain, 8-10 September, 2009, pp. 2269-2277.

[3] S.I. Deaconu, M. Topor, L.N. Tutelea, G.N. Popa and C. Abrudean, “Modeling and Experimental Investigations of a Reactive Homo-Heteropolar Brushless Synchronous Machine”, IECON 2009, 35th Annual Conference of the IEEE Industrial Electronics Society, Porto, Portugal, USA, 3-5 November, 2009, pp. 1209-1216.

[4] M.J. Balchim, and J.F. Eastham, “Characteristics of heteropolar linear synchronous machine with passive secondary”, Electric Power Application, vol. 2, no. 8, pp. 213-218, December 1979.

[5] J. He and F. Lin, “A high frequency high power IGBT inverter drive for 45 HP/16.000 RPM brushless homopolar inductor motor”, Conference Record of the IEEE IAS Annual Meeting, 1995, pp. 9-15.

[6] M. Siegl and V. Kotrba, “Losses and cooling of a high-output power homopolar alternator, IEEE Fifth International Conference on electrical Machine and Drives (Conf. Publ. No. 341), London, U.K., 11-13 Sept. 1991, pp. 295-299.

[7] G. P. Rao, J. L. Kirtby, D. C. Meeker Jr. and K. J. Donegan, Hybrid permanent magnet/homopolar generator and motor, U. S. Patent 6097124, Aug. 1, 2000.

[8] O. Ichikawa, A. Chiba and T. Fukao, “Development of Homo-Polar Type of Bearingless Motors”, Conference Record of the 1999 IEEE IAS Annual Meeting, pp. 1223-1228.

[9] M. Hippner and R. G. Harley, “High speed synchronous homopolar and permanent magnet machines comparative study”, Conference Record of the IEEE IAS Annual Meeting, 1992, pp. 74-78.

[10] H. Hofman and S. R. Sanders, “High speed synchronous machine with minimized rotor losses”, IEEE Transactions on Industry Applications, vol. 36, no. 2, Mar. 2000, pp. 531-539.

[11] P. Tsao, M. Senesky and S. Sanders, “A Synchronous Homopolar Machine for High-Speed Applications”, Conference Record of IEEE IAS Annual Meeting, 2002, pp. 406-416.

[12] L. Vido, M. Gabsi, M. Lecrivain, Y. Amara and F. Chabot, “Homopolar and Bipolar Hybrid Excitation Synchronous Machine”, IEEE Transactions on Energy Conversion, vol. 21, no. 3, September, 2006, pp 1212-1218.

[13] S.I. Deaconu, The study of control brushless electrical generator, PHD Thesis (in Romanian), Politechnica University of Timisoara, Romania, 1998, unpublished.

[14] I. Boldea and S.A. Nasar, Electric drives, 2nd edition, CRC Press, Taylor and Francis Group, 2005, pp. 256-375, New York.

[15] A. Kelemen and M. Imecs, Field-Oriented Control of AC Machines (in Romanian)., Academic Publishing House, pp. 136-240, ISBN 973-27-0032-7, Bucuresti, 1989, Romania.

Sorin I. Deaconu (M’07) was born in Orastie, Romania, in 1965. He received the B. S. degree in electrical engineering in 1989 and Ph.D. degree in electrical machines in 1998 from “Politechnica” University of Timisoara, Romania.

He is currently Associate Professor at the Department of Electrical Engineering and Industrial Informatics, Engineering Faculty of Hunedoara, “Politechnica” University of Timisoara.

He has authored almost 160 international papers in the field of electrical machines, electrostatics, electric arc furnaces and renewable energy.

Since 1994, he has collaborated with Bee Speed Automation Ltd, Timisoara, where he is involved in several industry projects regarding industrial automation, machines and drives.

Advances in Communications, Computers, Systems, Circuits and Devices

ISBN: 978-960-474-250-9 83