ν

d q d qk k− ⇒ −ψ ψ δ δ θ θk r sdk sd sqk sq k k ki i i i= = = = =; ; ; ;

d q d qk k− ⇒ −λ λ

ψ ψ δ δ θ θλ λ λ λk s sdk sd sqk sq k ki i i i= = = = =; ; ; ;

53

d q d qk k m m− ⇒ −ψ ψ δ δ θ θk m sdk sdm sqk sqm k m k mi i i i= = = = =; ; ; ;

is* θ s

*

α−β .

54

σ

55

σσ

/2/3

i

i

i

i

i

i

i

i

i

sa

sb

sc

s

s

so

s s

s s

so

= −− −

⋅

= −− −

⋅⋅⋅

1 0 1

1 2 3 2 1

1 2 3 2 1

1 0 1

1 2 3 2 1

1 2 3 2 1

/ /

/ /

/ /

/ /

cos

sinα

β

θθ

isdk

isqk θk α−β

56

57

/P)

i i i

arctgi

i

s sdk sqk

ksqk

sdk

* * *= +

=

δ

d qk k− → −α β

i

i

i

is

s

k k

k k

sdk

sqk

α

β

θ θθ θ

=

−

⋅

cos sin

sin cos

α β− → −d qk k

i

i

i

isdk

sqk

k k

k k

s

s

=

−

⋅

⋅

cos sin

sin cos

θ θθ θ

α

β

58

α−β

u u u u us sa s sb scα β= = −; ( )1

3

u u ju u es s s

dj

ν αν βν

ν π ν

ν= + = =

=

−2

31 6

0 0 7

1 3( ) / , , . .. , ,

, , ,

u U Ud d sn=∧

/

ω πs t = / 3 π / 3

ν

uν

59

u1 1 0 0( , , )

u 0 u 7

60

α−β

u u r i ldi

dt

l

l

d

dtss

s ss

ns

ss

n

m

r

rs

= = + +ν ω ω

ψ1 1'

ψ ψ ψω ωrs

rj t

rj te es s= ⋅ = ⋅

d

dt

d

dte j e

rs

r j ts r

j ts sψ ψ

ω ψω ω= ⋅ + ⋅

u u r i ldi

dtjf

l

lss

s ss

ns

ss

s rs m

r

= = + +ν ωψ1 '

uν u udν =2

3e jf

l

lrs

s rs m

r

= ψ

4.2.2.2. .

uν

u u

u u

u u

u u

1 0

2 3

7 4

6 5

1 0 0 0 0 0

11 0 0 1 0

111 0 11

1 0 1 0 0 1

( , , ) ( , , )

( , , ) ( , , )

( , , ) ( , , )

( , , ) ( , , )

→

→

→

→

61

u 0 u 7

4.2.2.3.

uν er

∆ ∆is

∆∆

tl i

u en

s s

r

≅ ⋅−

1

ω ν

'

∆ uν er

/∆

62

±H

.

α−β:

63

is* is ∆is ∆ ∆ ∆

∆is

∆±H

is*

is*

64

isa* isb

* isc* isa

isb isc usa* usb

* usc*

i I Id d sn=∧

/

65

α−β

i i i i is sa s sb scα β= = −; ( )1

3

i i jii e

s s sd

j

ν αν βν

νπ π ν

ν= + =

=

=

+2

31 6

0 7 8 9

3 6( / / ) , , . .. , ,

, , ,

66

ω πs t = / 3

π/ i1

67

is

si

m ie sq~ sasbsa iii ,,

si /6

π/

si ψ r is1

ω11 ~ sqe im si

π/

is1

si

68

ψ r

α−β.

..

α−β

ψ

69

α β ψ ψα βm m,

/ ),( ss iu ν

θ

Modelul tensiunii - curentului statoric ),( ss iu

α−β

u r id

dt

u r id

dt

s s sn

s

s s sn

s

α αα

β ββ

ωψ

ωψ

= +

= +

1

1

( , )u is s

70

Modelul curentului statoric - turatiei ( , )is ν in coordonate statorice

α−β

1

1ω

ψνψ

ψ

ωψ

νψψ

αβ

α α

βα

β βn

rr r

r m s

r

n

rr r

r m s

r

d

dtr

l i

ld

dtr

l i

l

= − −−

= −−

( , )is ν

νδ ψ=< ( , )r si

[28],[96]. [41], [51], [95]

71

Modelul curentului statoric - turatiei ( , )is ν in coordonate de camp

( , )is ν

α−β

α, β

δ

Estimarea cuplului electromagnetic in coordonate statorice

α−β

m i i ie s s s s s s= ℑ ⋅ = −[ ]*ψ ψ ψα β β α

ψ s ψ r

ml

li

l

li ie

m

rs s

m

rr s r s= ℑ ⋅ = −[ ] ( )*ψ ψ ψα β β α

72

Estimarea unghiului de cuplu in coordonate statorice

/P)δ.

δ δ

73

sin

cos[ ]*

δψ

ψ ψψ

δψ

ψψ ψ

ψ

α β β α

α α β β

= = =−

= =ℜ ⋅

=+

i

i

l

l

m

i

i i

i

i

i

i

i

i i

i

sq

s

r

m

e

r s

r s r s

r s

sd

s

r s

r s

r s r s

r s

δ δ.

- Estimarea cuplului electromagnetic in coordonate de camp

ml

lie

m

rr sd= ψ

* *

74

* * (

ψr* ψ νr f* ( )=

ψ*

< 2000 Hz)

75

isd* *

isα*

β*

d q− → −α β isα*

β* sunt

isa* *, isc

* /3),

d q− → −α βsin , cosθ θk k

δ

76

ν ν* −*

**.

*, * *, δ*

* δ*,

θ θ *

δ.

δ .

77

δ

δ .

- Bucla de reglare a amplitudinii curentului statoric

78

~ νδ

- Bucla de reglare a unghiului de cuplu

ω .

Τµ

79

α−β

π/3

rψ

is*

*, δ* θs*

δ12=δ δ12 δ*,

* θsi*.

θsi* *δ23 δ23 δ*

* θsi* θsi*

δ δ/ δθH

Τµ

80

81

:

f

TT

T

H

IIn

H n

i

max ( )θ

ωθπµ

δ

δ

=+

1

4 121

θH ,

δ δ/ δ

82

ν ν* −*

*

.ctrnr == ψψ*. is ,ν

α−β

/3

rψ ψ s

ψψ

ω

ψψ

ω

sds

sq

nsqssq

sqssd

nsdssd

f+dt

d1+ir=u

f-dt

d1+ir=u

il+ll= ssr

r

ms σψψ

sψ

ixf+x

xf+dt

d

x

x1+

dt

idx

1+ir=u

ixf-x

xf-dt

d

x

x1+

dtid

x1

+ir=u

sdssrdr

ms

rq

r

m

n

sqs

nsqssq

sqssrqr

ms

rd

r

m

n

sds

nsdssd

σψψ

ωσ

ω

σψψ

ωσ

ω

ψ r rd r rq

rq= ; = 0 ;

d

dt= 0 ψ ψ ψ

ψ

83

ixf+x

xf+dt

idx

1+ir=u

ixf-dt

d

x

x1+

dtid

x1

+ir=u

sdssrr

ms

sqs

nsqssq

sqssr

r

m

n

sds

nsdssd

σψσω

σψ

ωσ

ω

ct=rψ r m sd= x iψ

sd s sdn

ssd

s s sq

sq s sqn

ssq

s s sd

u = r i +1

xd i

dt- f x i

u = r i +1

xd i

dt+ f x i

ωσ σ

ωσ

+=−=

eq,sqsdss

,sqsq

ed,sdsqss

,sdsd

uuixf+u=u

uuixf-u=u σ

↑

α−β

is ,ν

θ α−β,

fm r r

l

l ir

e r

r

r

r

m sq

r

= =ψ ψ2

[44], [53], [55], [75], [87].

84

85

θ * α-β , *,

ψ * ν

θ ω ν ωk n r

t

n s

t

f dt f dt* * *( )= + =∫ ∫0 0

θ δ θs k* * *= +

θ

86

isd*, isq* me* ψr* (

α, isβd q− → −α β

(2/3).

ψ r

lr/rr* lr/rr lr/rr

* lr/rr

87

lr/rr* lr/rr lr/rr

* lr/rr

lr/rr)* lr/rr lr/rr

* lr/rr.

*

r r*

r r r*r = r + r = k r∆

ψψψ

er

*rr

*rr

er

*rrr

rr*sm

er

fj+irk=fj+ir=0

ix+ix=

ri

reψ

re

m s* r

*m r

r r*

r*

rs*

mr*

r

r r*

r*

rsd*

sq*= x i -

j f x x

k r + j f xi = x (1-

j f x

k r + j f x)( i + j i )ψ

m ; = ; r ; f ; i=ij+i=i *e

*rd

*r

*r

*rs

*sq

*sd

*s ψψ

ri

88

sd*i i*sq

re

mr*

r

r r*

r*

r

r*

m

r*

r*

r

m r*

r r*

r*

r

r r*

r*

rrd*

rde

rqe= x (1-

j f x

k r + j f x)(

x+ j

f x

x r)=

k ( r + j f x )

k r + j f x= + jψ

ψ ψψ ψ ψ

*re

rd*=ψ ψ

r e,mψ r*

e*,mψ

ψψ

ψψ

*rd2

r*r

2*rr

r*r

*rrre

rq

*rd2

r*r

2*rr

2r

*rr

2*rre

rd

)xf(+)rk(

xrf1)-k(k=

)xf(+)rk(

)xf(k+)rk(=

rrr r r

*r*

r

r r* 2

r r*

r2

= arctgk ( k -1) f r x

( k r ) + k ( f x )ε

rde

r<ψ ψ

rde

r>ψ ψ

89

*

{ }i=m conjr

ere ⋅ℑψ ri

iconjr *

rr*

m

r r*

r*

rs* r

* 2r r

*r r

*r r

* 2r*

r2

r*

r r* 2

r*

r2 rd

*i = -

j f x

k r + j f xi = -

( f ) x r [ 1- k ] + j f [ k ( r ) +( f x ) ]

r [ ( k r ) +( f x ) ]ψ

ψ

ψ

*rd2

r*r

2*rr

*r

2r

*r

2*rr

*r

rq

*rd2

r*r

2*rr

*r

r*rr

2*r

rd

])xf(+)rk[(r

])xf(+)r(k[f-=i

])xf(+)rk[(r

]k-[1rx)f(-=i

{ }

*e

2r

*r

2*rr

2r

*rr

2*rr

22r

*r

2*rr

2r

*r

*r

2rr

*r

*r

2*rd

rqerdrd

erq

conjr

ere

m

])xf(+)rk(

)xf(k+)rk(+

])xf(+)rk[(

)xrf()1-k(k[r

f)(=

=i-i=i=m

↑

⋅ℑ

ψ

ψψψ

90

[37] [53];

[35];

[ 55]; [61].

91

ers

α β

e u r il di

dtrs

ss

s ss s

n

ss

= − −'

ω

( ) *ers ( ) *ψ r

s

( )*( ) **

el

l

d

dtrs

n

m

r

rs

=

1

ω

ψ

ψ ψ θψ ψ θ

α

β

r r k

r r k

* * *

* * *

cos

sin

==

e e

e er r k

r r k

α

β

θθ

* * *

* * *

sin

cos

= −=

e fl

lr sm

rr

* **

*=

ψ

∆r

lr

r

92

∆r

le e e er

rr r r r= −α β β α

* *

∆r

le er

rr k r k= −α βθ θcos sin* *

ψ m

isdm*, isqm* ale

isqm*, isdm*, aim* .

* *).

93

01

= + +r id

dtjfr r

n

rr rω

ψψ

ψ ψm m m m m ml i l i= = = i i im s r= +

ψ ψ σrr

mm r s

l

ll i= −

01 1

= − + − + −r i i ldi

dtl

di

dtjf l i jf l ir m s

nr

m

nr

sr r m r r s( )

ω ω σ σ

il

r

di

dti

l

r

di

dtf

l

ri

f

il

r

di

dt

l

ri

l

ri

sdmn

r

r

sdmm

n

r

r

mr

r

rsqm

r

sqmr

r n

sqm

r

rm

r

rsdm

+ = + +

=+

−

1 1

1ω ω

ω

σ σ

σ

σ

* *, *

ψ νm f= ( )

94

[9],

θ

* *), ψ

95

96

ψ s

ψ s

λ λ

λ

λ λ

λ λ

ψ ψs m ms s m msl i l i= = = il

li i

l

li ims

s

ms r

s

ms m= + + = +( )σ σ1

ψ ψ σrr

ms s s

l

ll i= −[ ]

01

= − + − + −r il

li l

di

dt

l

l

di

dtjf l i

l

lir ms

s

ms

nr

ms s

m

sr r ms

s

ms( ) ( ) ( )

ωσ σ

il

r

di

dt

l

li

l

r

di

dtf

l

ri

f

il

r

di

dtl

l

l

ri

l

ri

sdn

r

r

sd m

sms

n

r

r

msr

r

rsq

r

sqr

r n

sq

m

s

r

rms

r

rsd

λλ

λ

λλ

λ

ωσ

ωσ

σω

σ

+ = + +

=+

−

1 1

1

( )

λ* λ*, *

ψ νs f= ( )

97

98

<

λ* λ*), ψ

99