1. EEEB443 Control & Drives Dynamic Model of Induction Machine By
Dr. Ungku Anisa Ungku Amirulddin
Department of Electrical Power Engineering
College of Engineering
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives
Dr. Ungku Anisa, July 2008

2. Outline Introduction Three-phase Dynamic Model
Space Phasors of Motor Variables
Three-phase to Two-phase Transformation (Stationary)
Two-phase (Stationary) Dynamic Model (in dsqs frame)
Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Voltage equations
Torque equation
Commonly-used Induction Motor Models
Stationary (Stator) Reference Frame Model
Rotor Reference Frame Model
Synchronously Rotating Reference Frame Model
Equations in Flux Linkages
References
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

3. Introduction
Per phase equivalent circuit model only useful for analysing IM performance in steady-state
all transients neglected during load and frequency variations
used in scalar control drives which do not require good transient response
example: drive systems for fans, blowers, compressors
Dynamic model used to observe dynamic (steady-state and transient) behaviour of IM since:
Considers instantaneous effects of varying:
Voltages and currents
Stator frequency
Torque disturbances
machine is part of the feedback loop elements to control the dynamics of the drive system
High performance drive control schemes are based on dynamic model of IM
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

4. Introduction
Dynamic model – complex due to magnetic coupling between stator phases and rotor phases
Coupling coefficients vary with rotor position and rotor position vary with time
Dynamic behavior of IM can be described by differential equations with time varying coefficients
Complexity of dynamic model can be reduced by employing space vector equations
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

5. Three-phase Dynamic Model
Magnetic axis of phase B
Magnetic axis of phase C
IM consists of three-phase windings spaced at 120 apart
Model windings using simplified equivalent stator winding located on the magnetic axis of each phase.
ics
ibs a b b’ c’ c a’
Magnetic axis of phase A
Simplified equivalent stator winding
ias
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

6. Three-phase Dynamic Model
Similar model is applied to represent the rotor ‘windings’
Rotor rotates at speed r
Rotor phase ‘a’ winding displaced from stator phase ‘a’ winding by angle r
stator, b
stator, c
stator, a
rotor, b
rotor, a
rotor, c r
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

7. Three-phase Dynamic Model
Voltage equation for each stator phase:
Similarly, voltage equation for each rotor phase:
These equations can be written in a compact form.
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

8. Three-phase Dynamic Model
Stator voltage equation (compact form):
Rotor voltage equation (compact form):
where:
 abcs = stator flux linkage (flux linking stator windings )
abcr = rotor flux linkage (flux that links rotor windings)
(1)
(2)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

9. Three-phase Dynamic Model
The displacements between 3 phase stator (rotor) windings are non-quadrature (i.e. not 90)
Magnetic coupling exists between the 3 stator (or rotor) phases, i.e. the flux linkage each stator (or rotor) phase is sum of:
fluxes produced by the winding itself
fluxes produced from the other two stator (or rotor) windings
fluxes produced by all three rotor (or stator) windings
Example: Flux linkage for stator phase ‘a’ is sum of:
Fluxes produced by stator phase ‘a’ winding itself
Fluxes produced by stator phase ‘b’ and stator phase ‘c’
Fluxes produced by rotor phase ‘a’, ‘b’ and ‘c’
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

10. Three-phase Dynamic Model
In general, the stator flux linkage vector:
(3)
Flux linking stator winding due to stator currents
Flux linking stator winding due to rotor currents
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

11. Three-phase Dynamic Model
In general, the rotor flux linkage vector:
(4)
Flux linking rotor winding due to rotor currents
Flux linking rotor winding due to stator currents
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

12. Three-phase Dynamic Model
Self inductances in (3) and (4) consists of magnetising inductance and leakage inductance:
For stator:
For rotor:
Due to symmetry in windings, mutual inductances between stator phases in (3)(and rotor phases in (4)) can be written in terms of magnetising inductances:
Stator leakage inductances
Rotor leakage inductances
Note:
Subsrcipt ‘s’ is replaced with ‘r’ for rotor phase leakage inductances, currents and flux linkage
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

13. Three-phase Dynamic Model
Mutual inductances between the stator and rotor windings depends on rotor position r:
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

14. Three-phase Dynamic Model
Equations (1) – (4):
completely describe dynamic characteristics of 3-phase IM
consists of 6 equations (3 for stator and 3 for rotor), i.e. large number of equations
all equations are coupled to one another
Magnetic coupling complicates dynamic model in 3-phase!
Better to develop model based on space phasors:
reduces number of equations
eliminates magnetic coupling between phases
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

15. Space Phasors of Motor Variables
If xa, xb, and xc are the 3-phase IM quantities, whereby:
The space phasor in the 3-phase system is obtained from the vectorial sum of the 3-phase quantities, i.e.:
(5)
x is called the space phasor or complex space vector
, where a = ej2/3
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

16. Three-phase to Two-phase Transformation (Stationary)
Any three-phase machine can be represented by an equivalent two-phase machine using Park’s transformation
Two-phase equivalent
Three-phase
easier way to obtain dynamics of IM
There is magnetic coupling between phases
There is NO magnetic coupling between phases (due to 90 angle between phases)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

17. Three-phase to Two-phase Transformation (Stationary)
Dynamic model of IM usually obtained using the two-phase equivalent machine r
stator, qs
rotor, 
stator, ds
rotating r
Two-phase equivalent
rotor, 
stator, b
rotating r
rotor, b
rotor, a r
stator, a
Three-phase
rotor, c
stator, c
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

18. Three-phase to Two-phase Transformation (Stationary)
All 3-phase system quantities have to be transformed to 2-phase system quantities
Equivalence between the two systems is based on the equality of mmf produced and current magnitudes, i.e.:
MMF produced by 2-phase system = MMF of 3-phase system
current magnitude of 2-phase system = current of 3-phase system
The use of space phasors enables the transformations from 3-phase to 2-phase system.
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

19. Three-phase to Two-phase Transformation (Stationary)
In the stationary 2-phase system, the space phasor is defined as :
(6)
The space phasor in the 2-phase system must equal that in the 3-phase system.
Hence, by comparing (5) and (6):
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

20. Three-phase to Two-phase Transformation (Stationary)
The abc  dsqs transformation is given by:
(7)
Under balanced conditions, the zero-sequence component adds to zero, i.e.:
Zero-sequence components, which may or may not be present.
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

21. Three-phase to Two-phase Transformation (Stationary)
Assuming balanced conditions, the abc  dsqs transformation:
(8)
The inverse transform (dsqs  abc transformation) is given by:
(9)
where:
(10)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

22. Three-phase to Two-phase Transformation (Stationary)
Transformation equations (8) and (9) apply to all 3-phase quantities of the IM (i.e. voltages, current and flux linkages)
Transformation matrices (Tabc and Tabc-1) given by (10) causes the space phasor magnitude to be equal to peak value of the phase quantities, i.e.:
This is one of many abc  dsqs and dsqs  abc transformation matrices in literature, eg.:
space phasor magnitude to be equal to 1.5 times peak value of the phase quantities ( )
space phasor magnitude to be equal to rms value of the phase quantities ( )
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

23. SPACE VECTORS
Space vector representation of the mmf distribution in an AC machine created by balanced
positive-sequence three-phase sinusoidal currents.
Each of the ABC (RGB) space vectors pulsates along its respective axis.
The resultant vector (in black), of 1.5 magnitude, rotates at the excitation frequency.
Source:
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

24. SPACE VECTORS
This animation shows the motion of space vectors for the case of a balanced three-
phase sinusoidal signal: fA = cos(ωt), fB = cos(ωt-α), fC = cos(ωt+α) where α = 2π/3.
The corresponding space vector is obtained from fR = (fA + γ fB + γ2 fC) = ejωt where γ = ejα.
Source:
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

25. Example 1 - Space Phasor & 3-2 phase transformation
An induction motor has the following parameters:
Parameter
Symbol
Value
Rated power
Prat
30 hp (22.4 kW)
Stator connection
Delta ()
No. of poles P 6
Rated stator phase voltage (rms)
Vs,rat
230 V
Rated stator phase current (rms)
Is,rat
39.5 A
Rated frequency
frat
60 Hz
Rated speed
nrat
1168 rpm
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

26. Example 1 - Space Phasor & 3-2 phase transformation (contd.)
Assuming that the motor is operating under rated conditions, with stator and rotor current phasors of:
Calculate the values the following stator current values at time t = 0:
ias, ibs and ics (3-phase stator phase currents)
isds and isqs (2-phase stator phase currents)
iar, ibr and icr (3-phase rotor phase currents)
irds and irqs (2-phase rotor phase currents)
Show that the magnitude of isds and isqs is equal to the peak stator phase current
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

27. Two-phase (Stationary) Dynamic Model (in dsqs frame)
From the three-phase dynamic model (eq. 1 and 2):
Applying the transformation given by (8):
Note:
dsqs – stator equivalent two-phase winding
 - rotor equivalent two-phase winding r
stator, qs
rotor, 
stator, ds
rotating r
rotor, 
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

28. Two-phase (Stationary) Dynamic Model (in dsqs frame)
The  rotor winding rotates at a speed r
Hence, need to transform the rotor quantities from the  to the stationary dsqs frame. r
stator, qs
rotor, 
stator, ds
rotating r
rotor,  r r   xr  qs ds
Bring all rotor and stator quantities to be on the same axis!
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

29. Two-phase (Stationary) Dynamic Model -   dsqs frame transformxr  qs
xr
xr
On the  frame:
On the dsqs frame:
The angle between the two frames is r
xrds r r   xr  qs ds
xrqs
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

30. Two-phase (Stationary) Dynamic Model -   dsqs frame transformxr  qs
xr
xr ds
Therefore:
More elegantly :
(full derivation   dsqs frame transform)
(11) r r   xr  qs ds
xrqs
(12)
xrds
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

31. Two-phase (Stationary) Dynamic Model (in dsqs frame)
Two-phase dynamic model:
To transform rotor quantities from the   dsqs frame, from equation (12):
Substituting these into the rotor voltage equation above..
Expressed in rotating frame
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

32. Two-phase (Stationary) Dynamic Model (in dsqs frame)
Hence:
Therefore, the two-phase dynamic model in the stationary dsqs frame:
Expressed in rotating frame
Expressed in stationary frame
(13)
(14)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

33. Two-phase (Stationary) Dynamic Model (in dsqs frame)
The flux linkages in (13) and (14) are given by:
where and
Note that equations (13)-(16) each consists of two equations.
One from equating real quantities
One from equating imaginary quantities
Final dynamic equations in the stationary dsqs frame is given by substituting (15) and (16) into (13) and (14) and separating the real and imaginary equations.
(15)
(16)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

34. Two-phase (Stationary) Dynamic Model (in dsqs frame)
Final dynamic equations in the stationary dsqs frame:
Lm = mutual inductance
Lr’ = rotor self inductances referred to stator
Rr’ = rotor resistance referred to stator
Ls = stator self inductance
vrd, vrq, ird, irq are the rotor voltages and currents referred to stator
S = derivative operator
(17)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

35. Example 2 – Dynamic Model of Induction Motor in dsqs frame
The induction motor from the Example 1 has the following additional parameters:
Parameter
Symbol
Value
Rated torque
Te,rat
183 Nm
Stator resistance Rs
0.294 
Stator self inductance Ls H
Referred rotor resistance
Rr’
0.156 
Referred rotor self inductance
Lr’ H
Mutual inductance Lm
0.041 H
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

36. Example 2 - Dynamic Model of Induction Motor in dsqs frame (contd.)
Using the values of stator and rotor currents obtained in the Example 1 , calculate the stator flux s and rotor flux r vectors at time t = 0.
Given that
Calculate the torque produced by the motor using:
Stator flux s and stator current Is vector
Rotor flux r and stator current Is vector
Stator current  Is and rotor current Ir vector
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

37. SPACE VECTOR DECOMPOSITION
Space vectors under balanced sinusoidal conditions, appears as constant amplitude vectors rotating at the excitation frequency (2f).
In the stationary dsqs (αβ in the diagram) frame:
dsqs components are
time varying sinusoidal
signals at stator frequency
(2f)
Source:
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

38. SPACE VECTOR DECOMPOSITION
If we want the Induction Motor to behave like a DC motor, the two-phase components must be constant values.
This can be achieved by having a two-phase frame that rotates together with the space vector.
A rotating dq frame!
Source:
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

39. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
The dq referance frame is rotated at an arbitrary speed g
On the dsqs frame:
On the dq frame:
The angle between the two frames is g where: g g d q x qs ds
xqs
xds
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives
Dr. Ungku Anisa, July 2008

40. Two-phase Dynamic Model in Arbitrary Rotating Frame (dsqs  dq frame transform)
Therefore:
More elegantly :
(full derivation dsqs  dq frame transform) g g d q x qs ds
xqs g
(18) xd xq
xds
(19)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

41. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Hence, space vectors in the stationary dsqs frame will have to be transformed into the rotating dq frame using:
Equation (20) will have to be employed onto equations (13) –(16) to obtain the IM dynamic model in the rotating dq frame. g g d q x qs ds
xqs
(20)
xds
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

42. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Subst. (20) into (13), stator voltage equation:
Subst. (20) into (14), rotor voltage equation:
Expressed in stationary frame
Expressed in arbitrary rotating frame
(21)
Expressed in stationary frame
Expressed in arbitrary rotating frame
(22)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

43. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Subst. (20) into the flux linkages equations of (15) and (16):
Stator flux linkage:
Rotor flux linkage:
Airgap flux linkage:
where and
(23)
(24)
(25)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

44. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Note that equations (21)-(25) each consists of two equations.
One from equating real quantities
One from equating imaginary quantities
Final dynamic equations in the arbitrary rotating dq frame is given by substituting (23)-(24) into the (21)-(22) and separating the real and imaginary equations.
Final dynamic equations in the arbitrary dq frame:
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives
(26)

45. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Torque equation in the arbitrary dq frame:
Product of voltage and current conjugate space vectors:
It can be shown that for ias + ibs + ics = 0,
Input power to the IM:
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

46. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Torque equation in the arbitrary dq frame:
Input power to the IM:
If and then:
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

47. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Torque equation in the arbitrary dq frame:
The IM equation given by (26) can be written as:
Note that the matrices:
[R] = consists of resistive elements
[L] = consists of coefficients of the derivative operator S
[G] = consists of coefficients of the electrical rotor speed r
[F] = consists of coefficients of the reference frame speed g
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

48. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Torque equation in the arbitrary dq frame:
Hence, the input power is given by:
Power associated with g – upon expansion gives zero
Power
Losses in winding resistance
Rate of change of stored magnetic energy
Mechanical power
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

49. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Torque equation in the arbitrary dq frame:
The mechanical power is most important:
By observing equation (26), [G] consists of terms associated with r :
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

50. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Torque equation in the arbitrary dq frame:
Therefore, mechanical power:
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

51. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame)
Torque equation in the arbitrary dq frame:
Since m = r / (P/2), hence the electromagnetic torque:
(27)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

52. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) - Summary
Final dynamic equations in the arbitrary dq frame:
Torque equation in the arbitrary dq frame:
(26)
(27)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

53. Commonly-used Induction Motor Models - Stationary (Stator) Reference Frame
Speed of reference frame:
Dynamic model in the stationary reference frame:
Torque equation in the stationary reference frame:
(28)
(29)
(30)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

54. Three-phase to Two-phase Transformation (Stationary)
Assuming balanced conditions, vsds, vsqs, isds, isqs can be obtained from:
(31)
The inverse transform is given by:
(32)
where:
and (33)
vrds, vrqs, irds, irqs obtained from transforming vabcr  vr  vrdqs using equations (31), (32), (33) and (11)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

55. Commonly-used Induction Motor Models - Stationary (Stator) Reference Frame
This model is used when:
stator variables are required to be actual (i.e. same as in the actual machine stator)
rotor variables can be fictitious
Allows elegant simulation of stator-controlled induction motor drives
phase-controlled and inverter-controlled IM drives (i.e. this IM model is used for variable voltage control at constant frequency)
Input variables are well defined and can be used to find vsd and vsq easily
Reduce computations leading to real-time control applications
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

56. Commonly-used Induction Motor Models – Rotor Reference Frame
Speed of reference frame:
Dynamic model in the rotor reference frame:
Torque equation in the rotor reference frame:
(34)
(35)
(36)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

57. Two-phase Transformation (from stationary stator dsqs frame to rotor drqr frame)
Assuming balanced conditions, vsdr, vsqr, isdr, isqr can be obtained from vsds, vsqs, isds, isqs using:
(37)
The inverse transform is given by:
(38)
Note: vsds, vsqs, isds, isqs can be obtained from vabcs and iabcs using (31), (32) and (33)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

58. Commonly-used Induction Motor Models - Rotor Reference Frame
vrdr, vrqr, irdr, irqr obtained from transforming vabcr  vrdqr using equations (31), (32) and (33)
This model is used when:
Switching elements and power are controlled on the rotor side
Example: for simulations of slip-energy recovery scheme
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

59. Commonly-used Induction Motor Models – Synchronously Rotating Reference Frame
Speed of reference frame:
Dynamic model in the synchronously rotating frame:
Torque equation in the synchronously rotating frame:
(39)
(40)
(41)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

60. Two-phase Transformation (from stationary stator dsqs frame to synchronously rotating dq frame)
Assuming balanced conditions, vsd, vsq, isd, isq can be obtained from vsds, vsqs, isds, isqs using:
(42)
The inverse transform is given by:
(43)
Note: vsds, vsqs, isds, isqs can be obtained from vabcs and iabcs using (31), (32) and (33)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

61. Commonly-used Induction Motor Models - Synchronously Rotating Reference Frame
vrd, vrq, ird, irq obtained from transforming vabcr  vr  vrdqs  vrdq using equations (31), (32), (33), (11) and (42)
Synchronous reference frame:
transforms sinusoidal inputs into dc signals
provides decoupled torque and flux channels
Hence, IM control similar to separately excited DC motor achieved by employing vector control schemes
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

62. SPACE VECTOR DECOMPOSITION
Space vectors under balanced sinusoidal conditions, appears as constant amplitude vectors rotating at the excitation frequency (2f).
In the stationary αβ (dsqs) frame:
αβ (dsqs) components are time
varying sinusoidal signals at
stator frequency (2f)
In the rotating synchronous dq frame:
dq components are constant
values depend on the
orientation of the space vectors
with respect to the dq axes.
Source:
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

63. Equations in Flux Linkages
All dynamic equations presented are consists of 4 variables, i.e. vsdq, vrdq, isdq and irdq
Note that in squirrel-cage IM: vrdq = 0 at all times
If the equations are required to contain flux linkages (i.e. either sdq, rdq or odq), the dynamic model can be obtained by substituting irdq using the following equations respectively:
(23)
(24)
(25)
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

64. References
Krishnan, R., Electric Motor Drives: Modeling, Analysis and Control, Prentice-Hall, New Jersey, 2001.
Bose, B. K., Modern Power Electronics and AC drives, Prentice-Hall, New Jersey, 2002.
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

65.   dsqs frame transformxr  qs
xr
xr ds
xrqs
xrds
Back
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives

66. dsqs  dq frame transformg g d q x qs ds
xqs g xd xq
xds
Back
Dr. Ungku Anisa, July 2008
EEEB443 - Control & Drives