reference frame theory notes

��

Reference Frame Theory ��. �. �. ��

•

2

•

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I�� . � �� I� �� , �� . �� ; �� . B�� . I� �� , �� . � ��

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2

•

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I�� . � �� I� �� , �� . �� ; �� . B�� . I� �� , �� . � ��

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3 •

•

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P�� P�� . ��. �� . ��

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P�� F�� :

 E��

�� P�� D�� I�� C��

 ��

N�� C�� C��

��

�

L��

 F�� F�� D��

E�� C��  C�� B� �� M�� AC Q�� B�� DC

Q��  F�� M�� C��

�� •

�

1929: P�� �� M��; �� F��

� 1938: �� I�� M��; �� F��

� 1951: K�� I�� M��; �� F��

� 1957: B�� I�� M��; �� F��

� 1965: K�� A�� F��

�� 2�ф �� /� •

�� –

� –

–

–

•

��

�� B�� I�� . �. �. ��

��

��

��

��

�� 900 �� α �� θ� �� .

��

•

•

��

N�� 1 & �2. �� & �� .

• The equations are :

�

vqs

=

Rqiqs + ρ ( Lqqiqs ) + ρ ( Lqd ids ) + ρ ( Lqα iα ) + ρ ( Lqβ iβ )

vds

=

ρ ( Ldqiqs ) + Rd ids + ρ ( Ldd ids ) + ρ ( Ldα iα ) + ρ ( Ld β i β )

vα

=

ρ ( Lα q iqs ) + ρ ( Lα d ids ) + Rα iα

v β

=

ρ ( Lβ q iqs ) + ρ ( Lβ d ids ) + ρ ( Lβα iα ) + Rβ iβ

•

ρ ( Lαα iα ) + ρ ( Lαβ iβ ) +

ρ ( Lββ iβ )

�� , �� & �� .

Lαα ��

+

=

Lββ

=

Lrr ;

Ldd

=

Lqq

=

Ls

•

� •

��

M�� 900. ��, Lαβ = Lβα = 0 �� L�� = L�� = 0 M�� θ�. ��, �� L�� , �� , Lα d

=

Ldα

=

Lsr cos θ r ;

Lβ d

=

Ld β

=

Lsr sin θ r

Lα q

=

Lqα

=

Lsr sin θ r ;

Lβ q

=

Lq β

= − sr

L cos θ r

• The resulting equations are as follows :

10

vqs

=

( Rs + Ls ρ ) iqs + Lsr ρ ( iα sin θr ) − Lsr ρ ( iβ cos θ r )

vds

=

( Rs + Ls ρ ) ids + Lsr ρ ( iα cos θr ) + Lsr ρ ( iβ sin θ r )

vα

=

Lsr ρ ( iqs sin θ r ) + L sr ρ ( ids cos θ r ) + ( Rrr

v

= −

Lsr ρ ( iqs cos θ r ) + L sr ρ ( ids sin θ r ) + ( Rrr where Rs

•

��

+

=

Rq

=

Rd and Rrr

=

Rα

=

Lrr ρ ) iα +

Lrr ρ ) i β

Rβ

�� .

�� 11

��

��

•

idrr  cos θ r i  =   qrr   sin θ r

12

•

•

•

��

F�� α �� β �� , �� : sin θ r  iα 

i  i.e., idqrr = Tαβ  iαβ  − cos θ r  β  

A�� , �� . F��, �� αβ = �αβ�1 �� . �αβ �� .

•

A�� α �� β �� , �� , 0  Rs + Ls ρ  vqs   Rs + Ls ρ v   0  ds  =  . vqrr   Lsr ρ − Lsr θ r    . v  drr   L θ Lsr ρ  sr r

13

Lsr ρ

0 Rrr

+

Lrr ρ .

Lrr θ r

  iqs   Lsr ρ    ids .   − Lrr θ r   iqrr   idrr  Rrr + Lrr ρ  0

.

where θ r is the time derivative of θ r •

��

�� & � �� .

• Steps involved ar e : Rr 1�

iqr

=

=

2

a Rrr ; Lr iqrr a

a=

•

=

2

a Lrr ; vqr

idrr

=

avqrr ; vdr

=

avdrr

where

a stator effective turns / phase

=

rotor effective turns / phase

k w1T 1 k w 2T 2

lso, Magnetizing and Mutual inductances

are : ��

; idr

=

Lm

∝

T12 ; Lsr

∝

T1T2 ;

∴

Lm

=

aLsr

•

1�

��, �� :

0 Rs + Ls ρ   vqs   Rs + Ls ρ v   0  ds  =  . vqr   Lm ρ − Lm θ r    . v  dr   L θ Lm ρ  m r →

•

��

 i  qs   0 Lm ρ  ids  .   − Lr θ r Rr + Lr ρ  iqr  .  idr  Lr θ r Rr + Lr ρ  Lm ρ

0

system equation of induction mot or

N�� .

�� •

1�

•

•

��

A �� 3�� . E�� . A�� 3�� 1 ��/�� , �� 2�� 3�1/2 ��/�� .

1�

��

��

• Relationship between dq 0 and abc currents are :

1�

iqs  i  =  ds   i0 

 cos θ c  2  sin θ c 3  1   2

2π  2π     cos  θ c −  cos θ c +  3 3     ias  2π  2π       sin  θ c −  sin  θ c +   ibs  3  3    ics   1 1  2 2 

Thus, iqd 0 •

��

=

[T abc ] iabc

C�� 0 �� , � �� .

•

��

iabc 1�

=

[T abc ]

iqd 0

wher e

  cos θ c    2π 1 [T abc ] =  cos  θ c − 3     2π cos  θ c + 3   −

��

1

−

sin θ c 2π     sin  θ c −  3    2π     sin  θ c +  3   

 1   1   1 

•

�� (��) �� (��0) ��.

20 •

•

��

�� 0. B�� (�� ) �� .

•

•

21

F�� , ��, �� . F�� , �� , �� & �� . ��

v0 s •

��

=

( Rs + Lls ρ ) i0 s and v0r

=

( Rr + Llr ρ ) i0r

N�� .

�� 22

•

A�� .

•

H��, θ� = 0 �� , ��

S abc =

T

��

 1  2 0  3  1 2 

−

1 2 3

−

2 1 2

−

1 2

 3 for abc to qd 0  2  1  2 

�� 23

��

A� �� : 5��, 200�, 3��, 60H�, 4��, �� ; �� = 0.277Ω; �� = 0.183Ω; L� = 0.0538H L� = 0.0553H; L� = 0.056H E�� = 3. �� . F�� , �� I��, I��, Iα & Iβ �� . �� .

•

�� :

v as 2�

=

200 3

×

2 sin ωs t

=

163.3sin ω s t

2π  2π    v bs = 163.3sin  ωs t −  & vcs = 163.3sin  ω s t +  3  3   

vqs  vas     S  vbs • The d and q axes voltage s are : v s = Tabc      v0   v cs  Hence, vqs ��

=

2

 vas − ( vbs + vcs )   3 2  1

•

F�� 3�� , �� + �� + �� = 0

•

�� vqs

2�

  vas  = vas 32  1

( vcs − v bs ) and v0 = 0

• Similarl y, v ds

=

Thus, vqs

163.3sin ω s t

and v ds ��

=

23

=

=

vas 1 3

=

=

3

=

163.3 00

( vcs − vbs ) = 163.3cos ω s t

163.3 900

=

j163.3 V

=

163.3 V

.

•

•

2�

•

�� θ r

��

0

F�� , ρ = �ω� = �2π� � = �2π �60 = �377 ��/�� 0  vqs   Rs + Ls ρ v   0 Rs + Ls ρ  ds  =   0   Lm ρ 0    Lm ρ 0  0

•

=

 iqs   i  0 Lm ρ   ds   iqr  0 Rr + Lr ρ   0 Rr + Lr ρ  idr  Lm ρ

0

N�� .

•

N�� . The currents are : i qs =35.37-j108.18=113.81 -71.90

2�

•

•

��

ids =108.18+j35.37=113.8118.10

i qr =-34.88+j103.63=109.34 108.60

i dr =-103.63-j34.88=109.34 -161.40 N�� 900 �� 2�� . �� .

• The phase currents are :

 1  i  =  −1/ 2  bs    i cs   −1 / 2 

2�

1  i  qs

113.8 −71.90     0  3 / 2 1 ids =  113.8 168.1    0      i 113.8 48.1 3 / 2 1  0    0

 i as 

−

• Rotor currents are : i qrr •

i drr

=

aiqr aidr

= =

328.02 108.6

328.02 −161.4

0

�� α �� β ��, �� θ� = 0 ��

 iα  cos θ r i  =   β   sin θ r ��

=

0

sin θ r   i drr  328.02 −161.40  i  =   0 − cos θ r   qrr   −328.02108.6 

2) D�� . •

2�

v as

�� . =

Vm sin ωs t ; vbs

=

 

Vm sin  ωs t −

2π   & vcs 3 

=

 

Vm sin  ω s t +

2π 



3 

 Vm 00        0 S  Thus, v ds = Tabc vbs = Vm cos ω s t = Vm 90         v0   v cs     0  0 vqs 

vas 

Vm sin ω s t 

In steady State, ρ =jω s and vqr ��

=

v dr

=

0

•

0  V m   Rs + jωs Ls  jV   Rs + jωs Ls 0  m =   0   jωs Lm −ωr Lm    jωs Lm  0   ωr Lm

30

•

•

��

�� , jω s Lm

0 Rr

+

jωs Lr

Lr ωr

 iqs   i  jω s Lm   ds   i qr  − Lr ωr   Rr + jωs Lr  i dr  0

�� = �� = �� ,

V s

=

( Rs + jωs Ls ) I s + jω s Lm I r

0 = L m (ωs 31

•

+

( Rr + j (ωs − ωr ) L r ) I r

 Rr L mωs I s +   S

+

 jω s Lr  I r 

�� , �� .

Al so, L s ��

ωr ) I s

�� ω�� = ω� � ω� = �ω�. ��, ��

0= •

−

=

Lm

+

Lls and Lr

=

Lm

+

Llr

�� • 3 − phase instantaneous power input is 32

Pi

=

t vabc iabc

vas ias

=

+

vbs ibs

+

vcs ics

• On transforming to q and d − axes, we get , Pi

=

=

= ��

([T

]

abc

t qd 0

v

3

1

−

t

vqd 0

([T

abc

v ( ( 2

i

qs qs

1

−

]

+

) [T ) [T

abc

t

abc

1

−

]

1

−

]

iqd 0 iqd 0

vds ids ) + 2v0i0

)

→

(1)

•

Thus, Pi

33 •

•

•

��

�� . =

3

v ( 2

i

qs qs +

vds ids )

�� . �� . �� . �� (1) �� .

reference frame theory notes

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