Nano and Microelectromechanical Systems P5

INDUCTION MOTORSIn this section, the following variables and symbols are used: u uas, bs and ucs are the phase voltages in the stator windings as, bs and cs; u uqs, ds and uos are the qu

3.4 INDUCTION MOTORS

In this section, the following variables and symbols are used:

u uas, bs and ucs are the phase voltages in the stator windings as, bs and cs;

u uqs, ds and uos are the quadrature-, direct-, and zero-axis components of

stator voltages;

i ias, bs and ics are the phase currents in the stator windings as, bs and cs;

i iqs, ds and ios are the quadrature-, direct-, and zero-axis components of

stator currents;

ψ ψas, bs and ψcs are the stator flux linkages;

ψ ψqs, ds and ψos are the quadrature-, direct-, and zero-axis components of

stator flux linkages;

uar, ubr and ucr are the voltages in the rotor windings ar, br and cr;

u uqr, dr and uor are the quadrature-, direct-, and zero-axis components of

rotor voltages;

iar, ibr and icr are the currents in the rotor windings ar, br and cr;

i iqr, dr and ior are the quadrature-, direct-, and zero-axis components of

rotor currents;

ψ ψar, br and ψcr are the rotor flux linkages;

ψ ψqr, dr and ψor are the quadrature-, direct-, and zero-axis components of

rotor flux linkages;

r

r

TL is the load torque applied;

s

r and rr are the resistances of the stator and rotor windings;

Lss and Lrr are the self-inductances of the stator and rotor windings;

ms

ls

P is the number of poles;

m

B is the viscous friction coefficient;

3.4.1 Two-Phase Induction Motors

Figure 3.4.1 Two-phase symmetrical induction motor

To develop a mathematical model of two-phase induction motors, wemodel the stator and rotor circuitry dynamics As the control and state

and br) windings, as well as the stator and rotor currents and flux linkages.

Using Kirchhoff’s voltage law, four differential equations are

uabs r is abs d abs

where uabs as

bs

u u

stator and rotor resistances

Studying the magnetically coupled motor circuits, the following matrixequation for the flux linkages is found

abs abr

2

; Lsr is the matrix of the

N N

r abr

N N

'

' '

0 0

Therefore, the circuitry differential equations (3.4.1) are rewritten as

uabs r is abs d abs

N

N

N N

r r

'

' '

Assuming that the self- and mutual inductances L Lss, rr' , Lms are invariant and using the expressions for the flux linkages, one obtains a set ofnonlinear differential equations to model the circuitry dynamics

L L

r L

'

'

' '

'

'sin cos

L L

r L

' ' '

'

'

' '

'

'cos sin

L L

r L L

L L

r L L

Newton’s second law, we have

electrical angular velocity ωr and the number of poles P In particular,

The self-inductances Lss and L'rr

, as well as the leakage inductances

Lls and Llr'

torsional-mechanical equations of motion, one obtains

Augmenting differential equations (3.4.3) and (3.4.5), the following set

of highly nonlinear differential equations results

sincos

cossin

' '

' '

' '

'

'

' '

' 2

'

br r ms ar r ms as rr r rr

r r r br

rr

ms

r rr

r r r ar rr ms r bs ms as

s

rr

as

u L

L u L

L u L

L L

r i

L

L L i L

L i

θ θ

ω

θ θ

ω ω

Σ Σ

Σ Σ

Σ Σ

Σ

+

−+

+

−

=

,cossin

cossin

sincos

' '

' '

' '

'

'

' '

' 2

'

br r ms ar r ms bs rr r rr

r r r br

rr

ms

r rr

r r r ar rr ms r as ms bs

s

rr

bs

u L

L u L

L u L

L L

r i

L

L L i L

L i

θ θ

ω

θ θ

ω ω

Σ Σ

Σ Σ

Σ Σ

Σ

−

−+

cos

sincos

cossin

' '

br

ms

r ss

s r r bs ss ms r ss

s r r as ss ms ar

r

ss

ar

u L

L u L

L u L

L i

L

L

L

r i

L L L L

r i

L L L i

Σ Σ

Σ Σ

−

=

θ θ

ω

θ θ

ω θ

θ ω

,cos

sin

cossin

sincos

' '

ar

ms

r ss

s r r bs ss ms r ss

s r r as ss ms br

r

ss

br

u L

L u L

L u L

L i

L

L

L

r i

L L L L

r i

L L L i

Σ Σ

Σ Σ

Σ

+

−+

−

=

θ θ

ω

θ θ

ω θ

θ ω

L r L

L r L

B J

as bs ar br r r

' '

ω θ

L L

L i

r L L

L i

L L

L i

r L

L L

L i

r L

'

' '

' '

' '

L L

L i

r L

L L

L i

r L

L

L i P

u u u u

0000

20

(3.4.7)

Modeling Two-Phase Induction Motors Using the Lagrange Equations

The mathematical model can be derived using Lagrange’s equations.The generalized independent coordinates and the generalized forces are

sin

sin cos

2 5 2 1 2 4 ' 2 1 2 3 ' 2

1 5 4 2 5

3

2

2 2 2

1 5 4 1 5

3 1 2

1

2

1

q J q L q L q q q L q

q

q

L

q L q q q L q q q L

q

L

rr rr

ms ms

ss ms

ms ss

+ +

+

+

− +

& & '& '& &

.Thus,

( ) ( )

sin cos

cos sin

5 3 2 4 1 5 4 2 3 1

5 4 2 5

3

2

5 4 1 5

3 1 5

q q q q q q q q q

q

L

q q q L q q

q

L

q q q L q q q L

q

ms

ms ms

ms ms

−

=

− +

= , ∂

∂

Π

q30

= , ∂

∂

Π

q40

= , ∂

∂

Π

q50

∂

D

q &5 = B qm&5.Taking note of q &1= ias, q &2 = ibs, q &3= iar' , q &4 = ibr' and q &5= ωr, oneobtains

differential equations, as found in (3.4.6), result

Control of Induction Motors

The angular velocity of induction motors must be controlled, and thetorque-speed characteristic curves should be thoroughly examined Theelectromagnetic torque developed by two-phase induction motors is given byequation (3.4.4) To guarantee the balanced operating condition for two-phase induction motors, one supplies the following phase voltages to thestator windings

( )

u tas( ) = 2 uMcos ωft , u tbs( ) = 2 uMsin ( ) ωft ,

and the sinusoidal steady-state phase currents are

i tas( ) = 2 iMcos ωft − ϕi and i tbs( ) = 2 iMsin ( ωft − ϕi)

as and bs stator currents; ωf is the angular frequency of the applied phasevoltages, ωf = 2 π f ; f is the frequency of the supplied voltage; ϕi is thephase difference

The applied voltage to the motor windings cannot exceed the admissiblevoltage uM max That is,

P

varies the magnitude of the applied voltages as well as the frequency Thetorque-speed characteristic curves of induction motors must be thoroughlystudied Performing the transient analysis by solving the derived differential

plotting the angular velocity versus the electromagnetic torque developed.The following principles are used to control the angular velocity ofinduction motors

Voltage control By changing the magnitude uM of the applied phasevoltages to the stator windings, the angular velocity is regulated in the stable

frequency are shown in Figure 3.4.2.b

Voltage-frequency control The angular frequency ωf is proportional to

applied voltages applied to the stator windings should be regulated if thefrequency is changed In particular, the magnitude of phase voltages can bedecreased linearly with decreasing the frequency That is, to guarantee the

constant volts per hertz control one maintains the following relationship

voltage-frequency patterns, one shapes the torque-speed curves For example, the

f

u

i

and frequency f of the supplied voltages To attain the acceleration and settling

time specified, overshoot and rise time needed, the general purpose (standard),soft- and high-starting torque patterns are implemented based upon therequirements and criteria imposed (see the standard, soft- and high-torquepatterns as illustrated in Figure 3.4.2.d) That is, assigning ωf = ϕ ( uM) withdomain uMmin < uM < uMmax and range ωfmin < ωf < ωfmax, onemaintains u

characteristics, as documented in Figure 3.4.2.e, can be guaranteed

High Torque Pattern

Soft Torque Pattern

Stable Operating Region

Variable Voltage Frequency Control u

f u

Mi i Mi fi

Figure 3.4.2 Torque-speed characteristic curves ω r= ΩT( )T e :

a) voltage control; b) frequency control;

c) voltage-frequency control: constant volts per hertz control;

d) voltage-frequency patterns;

e) variable voltage-frequency control

S-Domain Block Diagram of Two-Phase Induction Motors

To perform the analysis of dynamics, to control induction machines, as

diagrams For squirrel-cage induction motors, the rotor windings are

1 s

r' r

L rr L' rr

X

r' r

L rr L' rr

X

X X

L ms

L ms

X X

X X

X X X X

1

sJ + B m P

2

X

X

-PL ms 2

+

-+ +

3.4.2 Three-Phase Induction Motors

Dynamics of Induction Motors in the Machine Variables

Our goal is to develop the mathematical model of three-phase induction

bs

as bs'

br

br' ar

Figure 3.4.4 Three-phase symmetrical induction motor

Studying the magnetically coupled stator and rotor circuitry, Kirchhoff’s

linkages through the set of differential equations

For magnetically coupled stator and rotor windings, we have

linkages are used as the variables, and in matrix form equations (3.4.8) arerewritten as

uabcs r is abcs d abcs

u u u

i i i

i i i

r r r

1 2 1

2

1 2 1

2

1 2

1 2 1

2

1 2 1

2

1 2

2 3

2 3 2

3

2 3 2

3

2 3

N N

r abcr

N N

The inductances are expressed as

2 3 2

3

2 3 2

3

2 3

1 2 1

2

1 2 1

2

1 2

r lr

cos

cos cos

cos

cos cos

cos

cos cos

cos

cos cos

cos

cos cos

cos

' ' '

' 2

2 3

3

2 '

2 3

3

2 2

' 3 3

3 3

2 2

3 3

2 2

3 3

2 2

−

− +

−

− +

−

− +

+

−

− +

−

− +

−

− +

ms lr ms

ms r

ms r

ms

r

ms

ms ms

lr ms

r ms r ms r

ms

ms ms

ms lr r

ms r ms

r

ms

r ms r

ms r ms ms ls ms

ms

r ms r ms r

ms ms ms

ls ms

r ms r ms r ms ms

ms ms

L L L

L L

L

L

L L

L L

L L

L

L L

L L L

L

L

L L

L L L L

L

L L

L L L

L

L

L L

L L

L L

L

θ π

θ π

θ

π θ θ

π

θ

π θ π θ θ

θ π

θ π θ

π θ θ

π θ

π θ π θ θ

Using (3.4.9) and (3.4.11), one obtains

d dt

d dt

d dt

d dt

d dt

d dt

' ' '

' ' ' '

( )

, cos

cos

'

2 1 2

1

dt

i d L dt

i d L dt

di L dt

di L L

i

r

u

r cr ms r

br ms r

ar

ms

cs ms bs ms as ms ls

as

s

as

π θ π

θ

+

+ +

+

−

− +

cos

3 2 '

2 1 2

1

dt

i d L dt

i d L dt

di L L dt

di L

i

r

u

r cr ms r br ms r

ar

ms

cs ms

bs ms ls

as ms

bs

s

bs

π θ θ

π

+ +

− +

− +

3 2 '

3 2 '

2 1 2

1

dt

i d L dt

i d L dt

di L dt

di L

i

r

u

r cr ms r

br ms r

ar

ms

cs ms ls bs ms as ms

cs

s

cs

θ π

θ π

θ

+

− +

+ +

+ +

cos

' 2 1 ' 2 1 '

'

3 2 3

2 '

'

dt

di L dt

di L dt

i d L dt

i d

br ms

ar

ms

lr

r cs ms r

bs ms r as ms

+

+ +

− +

cos

' 2 1 ' '

'

2

1

3 2 3

2 '

'

dt

di L dt

di L L

i d L dt

i d

ar

ms

r cs ms r bs ms r

as ms

br

r

br

− +

+

−

− +

+

+ +

cos

' '

' 2

2 '

'

dt

di L L dt

di L

i d L dt

i d

br ms

ar

ms

r cs ms r

bs ms r

as ms

cr

r

cr

+ +

−

−

+

+ +

− +

Cauchy’s form differential equations, given in matrix form, are found tobe

1 2 1

2

1 2 1

2

1 2

1 2

1 2

1 2 1

2

1 2

as bs cs ar br cr

Σ

Σ

' ' '

cos cos cos

r L

i i i i i i

cos

' ' '

i i i i i i

1 2

2 3

2 3 1

2

1 2

2 3

2 3

cos cos cos

cos cos cos

as bs cs ar br cr

'

'

'

' ' '

equations, and the expression for the electromagnetic torque must beobtained

For P-pole three-phase induction machines, as one finds the expression

for coenergy Wc( iabcs, iabcr' , θr), the electromagnetic torque can be

∂θ

For three-phase induction motors we have

Matrices Ls and L'r

, as well as leakage inductances Lls and L'lr

, arenot functions of the electrical displacement θ r Therefore, we have

2 3 2

3

2 3 2

3

2 3

(3.4.14)

equations are found to be

Augmenting differential equations (3.4.13) and (3.4.15), the resulting

Mathematical Model of Three-Phase Induction Motors in the

Arbitrary Reference Frame

The abc stator and rotor variables must be transformed to the quadrature, direct, and zero quantities To transform the machine (abc)

zero-axis components of stator voltages, currents and flux linkages, the direct

Park transformation is used In particular,

uqdos = K us abcs, iqdos = K is abcs, ψqdos= Ksψabcs, (3.4.16)

2 3 2

3

2 3 1

2

1 2

1 2

and zero-axis components of rotor voltages, currents, and flux linkages are

2 3 2

3

2 3 1

2

1 2

1 2

From differential equations (3.4.12)

uabcs r is abcs d abcs

3

2 3

1 1 1

3

2 3

1 1 1

Multiplying left and right sides of equations (3.4.20) by Ks and Kr,one has

d dt

d dt

d dt

K r Ks s s−1= rs

and K r Kr r' r− 1= rr'

.Performing differentiation, one finds

2 3 2

3

2 3

0 0 0

2 3 2

3

2 3

0 0 0

One obtains the voltage equations for stator and rotor circuits in the

arbitrary reference frame when the angular velocity of the reference frame

From (3.4.22), six differential equations in expanded form are found tomodel the stator and rotor circuitry dynamics In particular,

ψabcs = L is abcs + L i'sr abcr'

and ψ' abcr = L i'sr T abcs+ L i'r abcr'

zero quantities Employing the Park transformation matrices one has

Ks−1ψqdos= L K is −s1 qdos+ L K i'sr r−1qdor'

s qdos r r abcr

.Thus

ψqdos = K L K is s s− 1 qdos+ K L K is 'sr r− 1qdor'

' 2

2

qr qs RM r dr RM ds

qr r ds RM

SM qs s RM RM

SM

qs

Mu u L i

L Mi M

i Mr i M L L i r L M L

L

dt

di

− +

' 2

2

dr ds RM r qr RM qs

dr r ds s RM qs RM

SM RM

SM

ds

Mu u L i

L Mi M

i Mr i r L i M L L M L

L

dt

di

− +

+ +

' 2 '

2 '

qr SM qs r dr RM ds SM

dr RM

SM qr r SM qs s RM

SM

qr

u L Mu i

L Mi L

i M L L i r L i Mr M L

' '

2 2

'

dr SM ds r qr RM qs SM

dr r SM qr RM

SM ds s RM

SM

dr

u L Mu i

L Mi L

i r L i M L L i Mr M L

−

−

− +

−

=

ω ω

By performing multiplication of matrices, the following formula results

by differential equations (3.4.26) and (3.4.29), the model for three-phase

We have a set of eight highly coupled nonlinear differential equations

' 2

2

qr qs RM r dr RM ds

qr r ds RM

SM qs s RM RM

SM

qs

Mu u L i

L Mi M

i Mr i M L L i r L M L

L

dt

di

− +

' 2

2

dr ds RM r qr RM qs

dr r ds s RM qs RM

SM RM

SM

ds

Mu u L i

L Mi M

i Mr i r L i M L L M L

L

dt

di

− +

+ +

' 2 '

2 '

qr SM qs r dr RM ds SM

dr RM

SM qr r SM qs s RM

SM

qr

u L Mu i

L Mi L

i M L L i r L i Mr M L

' '

2 2

'

dr SM ds r qr RM qs SM

dr r SM qr RM

SM ds s RM

SM

dr

u L Mu i

L Mi L

i r L i M L L i Mr M L

−

−

− +

−

=

ω ω

The last differential equation in (3.4.30) can be omitted in the analysisand simulations if induction motors are used in electric drive applications.That is, for electric drives one finds

' 2

2

qr qs RM r dr RM ds

qr r ds RM

SM qs s RM RM

SM

qs

Mu u L i

L Mi M

i Mr i M L L i r L M L

L

dt

di

− +

' 2

2

dr ds RM r qr RM qs

dr r ds s RM qs RM

SM RM

SM

ds

Mu u L i

L Mi M

i Mr i r L i M L L M L

L

dt

di

− +

+ +

' 2 '

2 '

qr SM qs r dr RM ds SM

dr RM

SM qr r SM qs s RM

SM

qr

u L Mu i

L Mi L

i M L L i r L i Mr M L

' '

2 2

'

dr SM ds r qr RM qs SM

dr r SM qr RM

SM ds s RM

SM

dr

u L Mu i

L Mi L

i r L i M L L i Mr M L

−

−

− +

−

=

ω ω

L r L

ls s

SM RM

SM r SM

i i i i i i

M Mi L i

RM s

SM RM

SM r

SM RM

r lr m

qs ds os qr dr or r

ω

ω

' ' '

' ' '

2 2

ω

ω ω

M L

M

L M

P J

T L.

The block diagram for three-phase induction motors, modeled in the

arbitrary reference frame is developed using (3.4.31) Applying the Laplace

1 s(L SM L RM - M 2 )+ L RM r s

i ' dr

+ +

X X

M 3P 4

P 2

M

L SM

X X X X

Mr s

Mr '

1 s(L SM L RM - M 2 )+ L RM r s

1 s(L SM L RM - M 2 )+ L SM r '

Figure 3.4.5 Block diagram of three-phase squirrel-cage induction motors in the arbitrary reference frame

Micro- and miniscale induction motors are squirrel-cage motors, and therotor windings are short-circuited To guarantee the balanced operatingconditions, one supplies the following balanced three-phase voltages

The quadrature-, direct-, and zero-axis components of stator voltages

are obtained by using the stator Park transformation matrix as

2 3

2 3 2

3

2 3 1

2

1 2

1 2

The stationary, rotor, and synchronous reference frames are commonlyused For stationary, rotor, and synchronous reference frames, the reference

conditions for stationary, rotor, and synchronous reference frames one finds

θ = 0, θ θ = r and θ θ = e Hence, the quadrature-, direct-, and zero-axis

components of voltages can be obtained to guarantee the balance operation

of induction motors

Mathematical Model of Three-Phase Induction Motors in the

Synchronous Reference Frame

The most commonly used is the synchronous reference frame Themathematical model of three-phase induction motors in the synchronousreference frame is found by substituting the frame angular velocity in the

' 2

2

e qr

e qs RM r

e dr RM

e ds

e qr r

e ds e RM

SM

e qs s RM RM

SM

e

qs

Mu u

L i

L Mi M

i Mr i M L

L i r L M L

L

dt

di

− +

' 2

2

e dr

e ds RM r

e qr RM

e qs

e dr r

e ds s RM

e qs e RM

SM RM

SM

e

ds

Mu u L i

L Mi M

i Mr i r L i M L L M L

L

dt

di

− +

+ +

' 2 '

2 '

e qr SM

e qs r

e dr RM

e ds SM

e dr e RM

SM

e qr r SM

e qs s RM

SM

e

qr

u L Mu i

L Mi L

i M L L i r L i Mr M L

' '

2 2

'

e dr SM

e ds r

e qr RM

e qs SM

e dr r SM

e qr e RM

SM

e ds s RM

SM

e

dr

u L Mu i

L Mi L

i r L i M L L i Mr M L

−

−

− +

−

=

ω ω

form, using (3.4.32), we have the following differential equation for electricdrives

ls s

i i i i i i

SM r

SM RM

e s

SM RM

SM RM

r lr m

qs e

ds e

os e qr e dr e or e r

'

' ' '

' ' '

SM RM

qs e RM qr

e r

SM RM

SM ds e RM dr

e r

SM RM

SM qs e RM qr

e r

SM RM

qs e dr

e

ds e qr e

J T

qs e

ds e

os e qr e dr e or e

L

' ' '

.

0 0 0 0 0 0 2

The quadrature, direct and zero voltages uqs e

, uds e

and uos e

to guaranteethe balanced operation of induction motors are found from

Taking note that θ θ = e in

2 3

2 3 2

3

2 3 1

2

1 2

1 2

2 3 2

3

2 3 1

2

1 2

1 2

2 3 2

3

2 3 1

2

1 2

1 2

sin sin

)

(

, cos

cos cos

2 3

2

3 2 3

2 3

2

cs bs as

e

os

e cs e

bs e as

e

ds

e cs e

bs e as

e

qs

u u u

t

u

u u

u

t

u

u u

u

t

u

+ +

=

+ +

− +

=

+ +

− +

θ θ

π θ π

θ θ

Taking note of a balanced three-phase voltage set

zero stator voltages must be supplied to guarantee the balance operation

0 ) ( , 0 ) ( , 2

e

components of stator and rotor voltages, currents, and flux linkages have dc

voltage u tqs e( ) is regulated because u tds e( ) = 0 and u tos e( ) = 0

1 s(L SM L RM - M 2 )+ L RM r s

i e qs

- +

i 'e dr

+ +

X X

M 3P 4

P 2

M

L SM

X X X X

Mr s

Mr '

1 s(L SM L RM - M 2 )+ L RM r s

1 s(L SM L RM - M 2 )+ L SM r '

Figure 3.4.6 Block diagram for three-phase squirrel-cage induction

motors modeled in the synchronous reference frame

3.5 MICROSCALE SYNCHRONOUS MACHINES

In this section, the following variables and symbols are used:

u uas, bs and ucs are the phase voltages in the stator windings as, bs and cs;

u uqs, ds and uos are the quadrature-, direct-, and zero-axis stator voltage

components;

i ias, bs and ics are the phase currents in the stator windings as, bs and cs;

i iqs, ds and ios are the quadrature-, direct-, and zero-axis stator current

components;

ψ ψas, bs and ψcs are the stator flux linkages;

ψ ψqs, ds and ψ0s are the quadrature-, direct-, and zero-axis stator flux

linkages components;

permanent-magnets;

ωr and ωrm are the electrical and rotor angular velocities;

θr and θrm are the electrical and rotor angular displacements;

TL is the load torque applied;

Bm is the viscous friction coefficient;

rs is the resistances of the stator windings;

Lss is the self-inductances of the stator windings;

axes;

md

axes;

Ns is the number of turns of the stator windings;

P is the number of poles;

Micro- and miniscale synchronous machines can be used as motors andgenerators Generators convert mechanical energy into electrical energy, whilemotors convert electrical energy into mechanical energy A broad spectrum of

systems applications We will develop nonlinear mathematical models, andperform nonlinear modeling and analysis of synchronous machines

3.5.1 Single-Phase Reluctance Motors

We consider single-phase reluctance motors to study the operation ofsynchronous machines, analyze important features, as well as to visualizemathematical model developments It should be emphasized that micro- andminiscale synchronous reluctance motors can be easily manufactured A single-

Figure 3.5.1 Microscale single-phase reluctance motor

The quadrature and direct magnetic axes are fixed with the rotor, which

the angular velocity of synchronous machines is equal to the synchronous

the initial conditions are zero Hence, the angular displacements of the rotor

)()

ω

θ

r m

s r

m

N L

θ

θ

ℜ

per one revolution of the rotor and has minimum and maximum values, and

max

2 min

π π θ

ℜ

=

r m

s m

N

, , , min

2 max

s m

N

e

r,T ω

Axis Magnetic Quadrature

as

i

as ψ

ω

θ

θ r=

Assume that this variation is a sinusoidal function of the rotor angulardisplacement Then,

half of amplitude of the sinusoidal variation of the magnetizing inductance.The plot for L m( ) θr is documented in Figure 3.5.2

Figure 3.5.2 Magnetizing inductance Lm( ) θr

The electromagnetic torque, developed by single-phase reluctance

2

1

2 cos

r m m ls as r

r as

θ

2 sin 2

1

M m r

r as m av

The mathematical model of the single-phase reluctance motor is found

by using Kirchhoff’s and Newton’s second laws

dt

d i

dt

d J T B

L r

, 2 cos 1

2 sin 2

cos

2 2

cos

as r m m

ls

r r as r m m ls

m as

r m m ls

s as

u L

L

L

i L

L L

L i

L L L

r dt

di

θ

θ ω θ θ

−

− +

3.5.2 Permanent-Magnet Synchronous Machines

Permanent-magnet synchronous machines are brushless machinesbecause the excitation flux is produced by permanent magnets

Permanent-Magnet Synchronous Machines in the Machine Variables

Three-phase two-pole permanent-magnet synchronous motors andgenerators are illustrated in Figures 3.5.3 and 3.5.4

t r d

asm cs ascs bs asbs as asas

bsm cs bscs bs bsbs as bsas

csm cs scs c bs csbs as csas

From (3.5.1), one finds

uabcs r is abcs d abcs

dt

u u

r r r

i i i

d dt d dt d dt

as bs cs

s s s

as bs cs

established by the permanent magnet, one has

ψasm = ψmsin θr,ψbsm= ψmsin ( θr −23π ),ψcsm = ψmsin ( θr +23π ).Self- and mutual inductances for three-phase permanent-magnetsynchronous machines can be derived Equations for the magnetizing

quadrature and direct inductances are

N L

ℜ

maximum value of Lasas occurs at θr = 21π , 23π , 52π ,

md ls asas

m ls

the half of amplitude of the sinusoidal variation of the magnetizinginductance

and for three-phase synchronous motors, one obtains

L =23 − ∆ and Lmd =23( Lm+ L∆m)

md

s mq

s m

N N

L

2 2 3

=

∆

mq

s md

s m

N N L

2 2 3

cos

3 1 2

1

3 1 2

1

r m cs r

m

m

bs r

m m as

r m m

ls

as

i L

L

i L

L i

L L

L

θ ψ π

θ

π θ θ

ψ

+ +

− +

cos

3 2 2

1

3 2 3

1 2

1

π θ ψ θ

π θ π

θ ψ

−+

m

bs r

m m ls as r

m m

bs

i L

L

i L

L L i L

cos

3 2 3

2

2 1 3

1 2

1

π θ ψ π θ

θ π

θ ψ

++

m m

ls

bs r m m as

r m

m

cs

i L

L

L

i L

L i

L L

(3.5.2)From (3.5.2), one has

as bs cs m r r r

sin sin sin

.

2 2

1 2

1 3 1 2

1 3 1

2

1 3

2 3

1 2 1

2

1 3

1 2

2 3

It was shown that Lm and L∆m are expressed as

=

md

s mq

s m

N N

L

2 2 3

=

∆

mq

s md

s m

N N L

2 2 3 1

.Permanent-magnet synchronous machines are round-rotor electrical

md

s mq

s

m

N N

L

ℜ

= ℜ

=

3

2 3

and L∆m = 0

Therefore, the inductance matrix is