Unified Theory of Torque Production in Switched and Synchronous Reluctance Motors

8/19/2019 Unified Theory of Torque Production in Switched and Synchronous Reluctance Motors

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UNIFIED THEORY OF TORQ UE PRODUCTION IN SWITCHED AND SYNCHRONOUS

RELUCTANCE M OTORS

D

A

Staton W L Soong C Cossar T J E Miller

University

of

Glasgow,

UK

ABSTRACT

One of the difficulties in comparing the capabil ities of

the switched reluctancemotor with those of AC motors

is the lack of a unified theory of toque production, in

which the same equation can be used for both motors

so

that the comparison can be reduced to the comparison of

coefficients.

The main impediment to the development of such a

theory has been the fact that the switched reluctance

motor does not satisfy the necessary conditions

for

the

dq-axis transformation, because of its double saliency.

Essentially it produces toque in impulses and it is

analysed by calculating

the

energy converted from

electrical to mechanical during

one

stroke. This is equal

to the area enclosed by the trajectory of t k operating

point in the phase flux-linkage versus phase-current

diagram [1-31. In this paper the toque production in the

synchronous reluctance motor is

analysed by the same

energy-conversion loop method [4]. The new concept of

the

ellijxe

diagram not only allows a direct comparison

with the switched reluctance motor, but also enables the

calculationof torque ripple, this not being possible using

conventional dq-axis theory.

THE

SWlTCHED RELUCTANCEMOTOR

The switched reluctance motor

SR

motor) while having

several advantages compared with other motor types (i.e.

simple construction, low rotor losses,

a

high torque to

inertia ratio, etc.), does have a reputation for high levels

of toque ripple and acoustic noise. Both deficienciesare

due

to

the doubly-salient construction and the fact that

toque is produced in impulses.

The toque production mechanism of the

SR

motor is

well documented [l-31.

T h e

average toque (TAv)

is

given by:

where

q

= number of phases,

N,

= number of rotor

poles, and

W

s the energy converted from electrical to

mechanical on

one

'working stroke'. The number of

working strokes per revolution is qN,.

The SR

motor is

analogous o a multicylinder internal combustion

engine

and

W

s analogous to the enclosed area of the 'indicator

diagram'. In the

SR

machine the indicator diagram is a

closed locus

on

coordinates of phase flux-linkage vs.

phase current, traced out by the 'operating point' [ i g ]

a s the rotor rotates through one 'stroke', Fig. 1.

1

I* I _

I . #

U ..-

I

*.¶a

1.10

L'M

OW

0.60

0.U

LL

2.00

9.m 4-00 U 8 '.Do 7.w

A LbI

Fig. 1:

Locus of

operating point

in the

i-Q

plane,

showing energy

W

-converted in each step.

The instantaneous torque

of the SR motor is given by:

aw (i,e)

ae

= -

where

W

is the

co-energy

and 8 defines the rotor

position.

In

general,

W' is a

function of all the currents

but this equation

is

still valid for calculating the toq ue

if

'i '

is replaced by the vector of all phase currents.

In

most SR machines the scalar phase

current 'i '

is used,

because mutual coupling :ffects are usually assumed to

be negligible. This assumption isnot necessarily justified

[SI,

ut it is very common.

Graphically, for a constant current of 10 Amps and

moving from

25

to 30

Oo =

un-alignment;

45'

=

alignment), the electromechanical energy converted

is

equal to the shaded area (X ) shown in Fig. 3. The

instantaneous torque

is equal to the ratio AWIAO, where

A0 = 5 .

It is clear from this diagram that the

instantaneous toque of the SR motor varies with the

spacing between the magnetization curves,

so

that the

production of constant toque is difficult, even with

constant current. Over a range of average toques, the

toque per ampere is also liable

to

vary widely.

When

two phases a n conducting simultaneously or a fraction

of each stroke, the toque contributions add in a

nonlinear way.



68

The

torque

capabilityof the SR motor clearly depends on

the available area of the i- v diagram.To achieve

a

high

specific output, it is important to have a large inductance

ratio and

a

high aligned saturation flux-linkage. P e

inductance ratio

is

the ratio of the unsaturated aligned

and unaligned inductances).

In

order to calculate the instantaneous toque of a SR

motor, not only must the aligned and unaligned

magnetisation curves be known, but also magnetisation

curves at intermediate otor positions. The magnetisation

curves can be calculated using analytical functions [3] or

finite-element (FE) analysis. Fig.

2 shows

a 2-

dimensional FE flux-plot for a typical

6:4 SR

motor with

its rotor 30 from the un-aligned position. This motor

has

been

modeled with

0

vaned between

0

and

45

in

5

increments. The

FE

magnetisation curves are

compared with measured characterist ics in Fig. 3. 'he

FE magnetisation curves have been input to

PCSRD

(a

simulationpackage developed at Glasgow University for

the design of SR motors). This package has the facility

to interpolate between magnetisation curves

131,

and

has

been used to calculate the motor's static torque

characteristic. The predicted static toque is compared

with measured characteristics in Fig. 4.

I

Fig. 2: SR flux-plot e = 30 ).

2 4 6

LO

12

CURRE

[APMSl

Fig. 3: Measured and FE predictions of phase flux-

linkage

chamcter ist ics.

I

Fig.

4

Mesured and FE ta t ic torque (SR motor).

THE

SYNCHRONOUS RELUCTANCE MOTOR

The synchronous reluctance motor(SYNCHRELmotor) is

a

sinewave a.c. motor. It

has

cylindrical surfaces on both

sides of the airgap. The stator is a conventional

polyphase a.c. stator, while the rotor has internal flux

barriers shaped to maximise the ratio of d-axis (high

inductance axis) to q-axis (low inductance axis)

reactance. Compared with the

SR

motor, it can be

more

easily designed to give reduced levels of torque ripple

and acoustic noise due to its cylindrical construction and

sinewave a.c. operation. Due to the operational

advantages, there has been a great deal of recent interest

in the

SYNCHREL

motor

[6-14].

While the design parameters that are important in

optimising the design of the SYNCHREL motor are well

known 161, there has been no attempt to compare the

toque production mechanism of the motor directly with

that of the SR motor. D- and q-axis flux plots for a

7.5kW

D132 single-banier

SYNCHREL

motor are shown

in Fig. 5. It has been shown that the single-barrier

construction has a poor performance compared with an

axially-laminated construction [6]. However, due to the

large component of torque ripple associated with the

single-barrier design, it has been modelled in this paper

in order to demonstrate the abilities of the non-linear

CO

energy method in calculating average and instantaneous

toque.

Torque Calculated using DQ-Axis Theory

The electromagnetic torque of the synchronous

reluctance motor is traditionally calculated using dq axis

theory, i.e.:

(3)

where Id and I, are components of the r.m.s. phase

current ,,, resolved along the d- and q-axes of the

phasor diagram and correspond to the space-vector



69

ig. 5: SYNCHREL d- and q- axis

flux

plots.

components of stator m.m.f along the d- and q-axes of

the rotor. Here m is the number of phases, p is the

number of pole-pairs, and

Ld

and Lq re the direct- and

quadrature-axis synchronous inductances, respectively.

Fig.

6

shows the conelation between measured and

FE

predictions of the variation in Ldand

Lq

with I, and

I,

respectively. Ld and Lq re equal to phase A inductance

(La)

when phase A cunent

is

at maximum (all three

phases energised) and the rotor d- and q-axes are aligned

with the MMF-axis of phase A respectively. These are

only truly equal to the synchronous inductances

if

the

winding is sinusoidally distributed and airgap permeance

function is sinusoidal [

131.

The SYNCHREL motor under investigation is supplied

with the following sinusoidal currents:

i,

=

I,,cos €3 y)

(4)

i,

= IPkcos O+ y - 120 )

5)

i =

Ipkcos €3

y

120')

6)

where the current angle y is the angle between stator

MMF-axis and the rotor d-axis.

Fig. 7 compares the measured torque per ampere with

that calculated using equation (3). y

= 45

and the FE

values of Ld nd

Lq

shown in Fig.

6

being used.

The self and mutual inductances associated with the

three stator phases are assumed to be sinusoidal

functions of rotor position [141.

1

P W E C u R R e K T

[AMRI

Fig. 6:

Measured and FE predictions of L,

vs

I, ani

- -

Lq-vs

Iq for the D132 side -barrier motor.

mecuRReKT IAMBI

T i . 7: Single-barrier motor torque-per-ampere.

Stator Self Inductance:

In

a magnetically linear

SYNCHREL

motor, the self inductance of a phase winding

is always positive and has a second-harmonic variation

with rotor position

€3)

due

to

the different permeance

values associated with the q- and d-axes. The self

inductances

La,,

L nd

L

are given by

[141:

where Lpo

is

the average component of self inductance

due to self flux-linkage crossing the airgap, LsL

is

the

phase leakage inductance and is due to self flux-linkage

which does not cmss the airgap (slot leakage and end-

winding), and Lp2 is the magnitude of the second

harmonic component.

Stator Mutual Inductance: The mutual inductance

between stator phases also exhibits a second-harmonic

variation with €3 due

to

the rotor geometry. The mutual

inductance between phases A and B is evaluated by



70

considering the airgap

flux

linking phase A when only

phase B is excited. The resulting equations are

[14]:

&Axis

and

Mi

nductance: From dq-axis theory

(141,

he d- and q-axis inductance values are equal to:

L,

=

L

+;CL )

+

Lp2)

Lq =

LaL - L A

(14)

(13)

Rearranging in

order

to obtain L* and4:

In the case of the single-bamer motor, the

FE

predictions of

Ld,

L and 4. at h = 15 Amps and

y =

45

a n

89.6mH, U).lmH

and

23mH

respectively. The

resultant predicted s inusoidalvariation n self and mutual

inductance is shown in Fig. 8.

I

I

ANOLB [F452 DffiReeFI

Fig.8: Self and mutual inductance variation with 8.

Torque from Co-Energyusing Linear Analysis

It has been shown that in a magnetically linear 3-phase

SYNCHREL

motor, the variation

in

self and mutual

inductance with

8 is

sinusoidal in nature. The self flux-

linkage

v b

and

V%)

variation with

8

and

y

can be

calculated by multiplying&a by i,,

L

y i, and

L

by

i,.

Similarly the variation in mutual flux-linkage

Vab,

calculated by multiplying

kb

y i,, y i,,

La,

by i

Vh,

vac,

V V k and

V d

with

e

and Y can be

L by i,,

For ,, = 15

Amps

and y

=

45 , the variation in phase A

flux-linkage components, i.e. total

q,),

self

(Vm)

and

mutual

( Q , ~ (U,&,

with

8 is

shown in

Fig 9.

The self

and mutual wmwnents

are

nonainusoidal, but add

to

by

i,

and

L*

by i,.

form

a

sinusoidai total nux-linkage waveform.

-BA w u x ~ e v u t w m w ~

O

%

100

110 m0 250 300

310

AUGLB

[E tE .

DffiBeESl

-1Sd

Fig.

9 Phase flnx-linkage variation with

8

If sinusoidal 3-phase currents, of the form given in

equations

(4)

o

6),

are fed to the phases of the motor,

the i-Cv locus for each phase has the form shown by the

solid line in Fig. lo., i.e. an ellipse. The flux-linkage

of

phase A has

two

components, one due

to

its own current

and the other due to the currents in phases

B

and C.The

self-component produces a flux-linkage which, when

plotted against in, gives the locus shown by the dashed

line in Fig.

10.This

locus

is

bound by the aligned and

unaligned magnetisation curves and is, in fact, tangent to

them. The area enclosed is

less

than that which would be

enclosed with rectangular current waveform having the

same peak current. The mutual component of flux-

linkage, plotted against i,, gives the locus shown by the

dot-dash line in Fig. 10. This

is

made up of two

components, i.e. one due

to

the current in phase

B

and

the

other due

to

current in phase C. These

two

terms are

plotted in Fig.

11.

Note that areas enclosed in a counter-

clockwise direction give r ise to positive (motoring)

toque, while areas enclosed in a clockwise direction

give rise

to

negative toque. The sum of the

two

flux-

linkage components for each phase give the ellipsoidal

total flux-linkage for that phase. The energy conversion

associated with any phase is the area enclosed within the

locus for that phase, and is the a m within the

ellipse.

Clearly

it is not bound d

by themagnet*otbn curves

m

the

same sense

as

is

for

the

SR

motor.

Note

that one traverse of the el lipse corresponds

to

the

passage

of two

rotor poles past the axis of the phase

winding. Consequently

the

basic unit of energy

conversion is half the area of the ellipse

0

nd the

number

of

'strokes' per revolution is equal to the

product 2mp, where m

i s

the phase number and p the



71

m .IS -10 .I

o J

IO IS

zo

CuBRexp AMRI

Fig. 11: Phase A mutual i v oci.

-1mal

. .

.

I

number of pole-pairs. The average toque is:

T = X x W (17)

n

This toqu e is numerically equal to that calculated using

dq axis theoty, i.e equation

(3).

Fig. 12 plots a full

family of el lipses, for y = 45 and values of ,, equal to

1, 2.5,

5

7.5, 10, 12.5 and 15 Amps. The orientation of

the main axis

of

the ellipse changes with current, and is

due to saturation, both Ldand

Lq

eing functions of IPh.

Toque rom Co-Energy using Non-Linear Analysis

In order to verify the linear analysis given in the

previous section,

FE

analysis has been used to model the

variation in phase flux-linkage with

8

and

ph for

the

D132 single-barrier motor. This was achieved by

performing

FE

calculations at

7

alues of

,,

between

1

and 15 Amps, while moving e in 2 increments over a

30

range, the value

of

y

remaining constant at 45'

throughout. The values of

va,

vb

nd

vc

t any given

value of

0

and

ph

is simply determined from the flux-

linkage with the respective phase winding. Fig. 13 shows

a

typical

FE

flux plot obtained during the analysis (the

particular plot is for 0 = 20' and

ph

= 15 Amps).

.m .IS -IO .J

o

s

IO

IS 20

CURREM'

[AMPS]

- I d '

'

Fig.

12:

Linear analysis iyr loci at 1,

2.5,

5, 7.5 11

1 2 5 and 15 Amps (Non-linearanalysb i-y,

locus

at

15

Amps also shoh).

I

I

Fig. 13: FE flux-plot e

= 20

and I,, = 15 Amps).

Fig.

9

compares the non-linear prediction of II against e

with that calculated using linear analysis ( ,, = 15

Amps), the non-linear waveform being a close

approximation to the theoretical sinusoidal waveform.

Fig. 12 compares the linear and non-linear i-y, loci at ,

= 15 Amps. A full set of non-linear ellipses is shown in

Fig. 14.

Also

plotted are the set of non-linear

magnetisation curves at 2 (mechanical degrees) ntervals

of 0.The average torque can be calculated from the total

co-energy enclosed by the non-linear ellipses and

equation (17). This is compared with the measured

toque-per-ampere in Fig. 7. Compared to dq-axis

theory, non-linear analysis gives closer agreement to the

measured average toque, however, the

errom

associated

with linear analysis are not appreciable. This is due

to

the total co-energy of the non-linear ellipse being

virtually equal to that of the linearcltipse.

Even though the

total

co-energy and average to que

of

the non-linear ellipse is virtually equal to that of the

.



72

linearellipse, the spacing of

the

non-linear magnetisation

curves is such that the instantaneous torque

is

far from

constant. At any rotor position, the imtantaneous toque

of phase A

is

proportional to

the

cocnergy between

magnetisation curves at that position. The instantaneous

toque

of

phases B and

C can

be

calculated in a similar

fashion, but with a 5120' offset.

The

otal instantaneous

toque

is the

sum of all three phases. This

is

compared

with the measured toque ripple in Fig.

15.

I

.m .IS -IO 5 o

J

to IS

20

C r [AMPSI

.IYIOl

Fig. 14:

Non-linear analysis ig

oci

at 1,

2.5,

5,

7.5,

10, 12.5 and 15 Amps Non-linear magnetisation

curves also shown).

I . . I

ANGlP U?ClRlW DECREE31

Oo m

10

M 80

LW

120

140 IM

IBO

pig. 15: Measured and FE oque ripple.

CONCLUSIONS

A method has

been

proposed for calculating the average

and instantaneous torque of the SYNCHREL

motor

from

a knowledge of the trajectory of the phase flux-linkage

versus phase current waveform, i.e. the samemethod as

used with the SR motor. This allows a direct comparison

between toque production in the two motors to be

made. It has been shown that while the locus of the

operating point

on

the flux-linkage/current plane

is

limited to the first-quadrant in the SR motor,

it

encompassesall four quadrants in the SYNCHREL motor.

Further it has been

shown

that, neglecting saturation and

slotting, the locus

is

ellipsoidal in shape. Even with an

ellipsoidal [i,y] locus, however, constant (ripple-free)

toque is not guaranteed. A complete comparison of the

toque

producing capabilities of a SR and axially-

laminatedSYNCHREL motor, having he same volume and

airgap, will

be

presented in a future paper. The method

can

be

extended to include

PM

synchronous and

induction motors.

REFERENCES

Miller,

TIE

: Switched reluctana

motors

and their

control.

Published by Magna Physics Corporation, Hillsboro. OH, 1992

Miller TIE :BnrphleJs pnmancnt-magnet and reluctaaa motor

drives, Oxford University

Ress,

1989.

Miller TJE, McGilp M : Nonl inar

ihuuy

of the switebcd

r e l u a a n a motor for rapid computer-aidcd design, IEE

Proceed ings, Vo1.137,

R B ,

No.6. Nov.

1990.

Woodsoa E and Melcben E

:

Electromechanical Dynamics,

Wiley (New York), 1968.

Preston

MA

and Lyons JP

;

A switchedrelu daoa motor

model

with

mutual

coupling and multi-phase exatation. IEEE

Transactions

on

Magoctira.

Vo1.27,

N0.6, Nov. 1991.

Staton

DA, Miller TJE and Wood SE :Optimisation

of

the

synchronous reluctana motor

geometry,

IEE Conferena on

Electrical Machines and Drives, London, Sept.

1991.

Xu L, Xu

X.

tip0 TA and Novotny DW : Control of a

synchronous reluctana motor ncluding sahwation and iron

loss.

IEEE

Traosaaioos. Vol

IA-27.

No.5, SepVOct

1991,

pp. 977-

985.

Ida I, Fu W nd Nasar SA : To que vector

control

( n C )

of

axially-laminated anisotropic (ALA) rotor reluctaoa

synchronous motors. Electric machines and Power Systems.

Vol.19, 1991, pp.381-398.

Platt D :Reluctance motor with strong anisotropy,

C o o t

Rec.

of IEEE-lAS Annual Meeting, Seanle (USA), Oct. 1990,

pp.225-229.

Fram

A and Vagati A :Axially laminated reluctancemotor:An

analytical approach to the magnetic behaviour, Interoational

Conference

on

Electrical Machines, Pisa (Italy), Sept. 1988,

Vol.111, pp.l-6.

t ip0 TA

Vagnti

A,

Malcsani

L,

Fukao

T :

Synchmnous

Reluctana Motors and D rives

-

A New Alternative, Tutorid at

the

IEEE-IAS

Annual meeting,

Houston,

Oct. 1992

Beh. RE and Miller TIE :

Aspecs

of the control of

synchronous reluctance machines, European PO WNE l c d m i c s

Conferena. EPE91. Flonna, Sept

1991.

Chiha A and Fukao T

:

Inductances of cageless relud.na-

synchronous machines having aoosinusoidal s p a distributions.

IEEE

Transactions

on Industrial Applications, Vo1.27. No.1.

JardFeb

1991.

pp.44-51.

Fitzgerald

AE

and Kingsley JR

:

Elecwical Machinery

~

n e

dynamics and statics of electromechanical energy conversion,

2nd Ed., McGraw-Hill. 1961.

Unified Theory of Torque Production in Switched and Synchronous Reluctance Motors

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Transcript of Unified Theory of Torque Production in Switched and Synchronous Reluctance Motors