AD2S100

REV. A

–5–

To relate these stator current to the reference frame the rotor

currents assume the same rectangular coordinates, but are now

rotated by the operator e

j , where ej = Cos

+ jSin

.

Here the term vector rotator comes into play where the stator

current vector can be represented in rotor-based coordinates or

vice versa.

The AD2S100 uses e

j

as the core operator. Here

represents

the digital position angle which rotates as the rotor moves. In

terms of the mathematical function, it rotates the orthogonal ids

and iqs components as follows:

ids' + jiqs' = (Ids + jIqs) e

j

where ids', iqs' = stator currents in the rotor reference frame. And

e

j

= Cos

+ jSin

= (Ids + jIqs)(Cos

+ jSin

)

The output from the AD2S100 takes the form of:

ids' = Ids Cos

– Iqs Sin

iqs' = Ids Sin

+ Iqs Cos

The matrix equation is:

[ids' ] = [Cos – Sin ][Ids]

iqs'

Sin

Cos

Iqs

and it is shown in Figure 2.

ids

iqs

ids'

iqs'

φ

ejφ

Figure 2. AD2S100 Vector Rotation Operation

DIGITAL

φ

LATCH

3

φ + 2φ

TRANSFORMATION

SINE AND

COSINE

MULTIPLIER

(DAC)

SINE AND

COSINE

MULTIPLIER

(DAC)

Cos(

θ + φ)

Cos(

θ +(120

° + φ))

Cos(

θ +(240

° + φ))

PARK

OUTPUT CLARK

COS

θ

COS

θ + 120°

COS

θ + 240° SINθ

2

φ–3φ

INPUT CLARK

LATCH

LATCH

Figure 3. Converter Operation Diagram

THEORY OF OPERATION

A fundamental requirement for high quality induction motor

drives is that the magnitude and position of the rotating air-gap

rotor flux be known. This is normally carried out by measuring

the rotor position via a position sensor and establishing a rotor

reference frame that can be related to stator current coordinates.

To generate a flux component in the rotor, stator current is ap-

plied. A build-up of rotor flux is concluded which must be

maintained by controlling the stator current, ids, parallel to the

rotor flux. The rotor flux current component is the magnetizing

current, imr.

Torque is generated by applying a current component which is

perpendicular to the magnetizing current. This current is nor-

mally called the torque generating current, iqs.

To orient and control both the torque and flux stator current

vectors, a coordinate transformation is carried out to establish a

new reference frame related to the rotor. This complex calcula-

tion is carried out by the AD2S100 vector processor.

To expand upon the vector operator a description of a single

vector rotation is of assistance. If it is considered that the mod-

uli of a vector is OP and that through the movement of rotor

position by

, we require the new position of this vector it can

be deduced as follows:

Let original vector OP = A (Cos

+ jSIN ) where A is a

constant;

so if

OQ = OP e

j

(1)

and: e

j

= Cos

+ jSin

OQ = A (Cos ( + ) + jSin ( +

))

= A [Cos

Cos

φ – Sin Sin φ + jSin Cos φ + jCos Sin φ]

= A [(Cos

+ jSin ) (Cos

+ jSin

)]

(2)

θ

φ

θ + φ

Q

P

O

a

d

Figure 1. Vector Rotation in Polar Coordinate

The complex stator current vector can be represented as is = ias

+ aibs + a

2i

cs where a = e

j 2

π

3

and a

2 = e

j 4

π

3

. This can be re-

placed by rectangular coordinates as

is = ids + jiqs

(3)

In this equation ids and iqs represent the equivalent of a two-

phase stator winding which establishes the same magnitude of

MMF in a three-phase system. These inputs can be seen after

the three-phase to two-phase transformation in the AD2S100

block diagram. Equation (3) therefore represents a three-phase

to two-phase conversion.