AC Motor Control and EV Applications

Rotating Field Theory

Chapter 2

Wound Stator of an Induction Machine

Cross Sectional View of AC Machine

b`

a`

a

c

c`

b

a-axis

b-axis

c-axis

(a)

a

b

c

(b)

Phase windings are separated by 120o.

Apply right hand rule to determine current axis.

a1a2a3a4

a1a2a3a4 a1 ’a2 ’a3 ’a4 ’

a1a2a3a4 a1 ’a2 ’a3 ’a4 ’ a1a2a3a4

Cut and Stretch

a1 ̀

a1

a2 ̀

a2

a3 ̀

a4 ̀

a3a4

Superposition of MMFs of Distributed Winding

Fourier Series Expansion of MMF

MMFa

• x• x

+a -ax

x

-a -a +a -a

Stator core

(a)

(b)

Fourier Series Expansion of MMF

MMF Made by Sinusoidally Distributed Winding

a

a’

High Order Space Harmonics

High Order Space Harmonics

Negative sequence

Positive sequence

Rotation of Current Vector to Three Phase Current

(1, - ½, - ½ ) (-½,1, -½) (-½, -½,1) (½,-1, ½)(½, ½, -1) (-1, ½, ½ )

1

-1/2

(1, - ½, - ½)

Current

Current vectorcomponents

Vector sum

Space Vector Representation

Mapping into the complex plane

Example

Transformation into Rotating Frame

Mapping into the rotating frame

DQ Transformation

Complex Variable

Break-down into Two Steps (Matrix Formulation)

Two Steps

In Stationary d-q axes

In Synchronous d-q axes

• abc dq (Stationary)

Coordinate Transformation into Stationary DQStep 1

Coordinate Transformation into Rotating DQ

• dq stationary dq synchronous

(Exciting Frame)

Step 2

An Example

In the synchronous reference frame, the vector looks as a constant vector.

Inverse Transformation

Coordinate Change Formulae

Stator Inductance Matrix

Thus,

Stator Inductance Matrix

Thus,

Power Relations