AN706-00032-1v0-E

32-BIT MICROCONTROLLER

MB9B100A/MB9B300A/MB9B400A/MB9B500A Series

Coordinate transform

TM

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AN706-00032-1v0-E

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Revision History

Rev Date Remark

1.0 Jul. 04, 2011 First Edition

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Table of Contents

Revision History....................................................................................................................2

Table of Contents..................................................................................................................3

1 Introduction....................................................................................................................4

2 Technical Background ...................................................................................................4

2.1 Overview ................................................................................................................4

3 Clarke transform ............................................................................................................5

3.1 Theory ....................................................................................................................5

3.1.1 Clarke transform theory...................................................................................5

3.1.2 Inverted Clarke transform theory .....................................................................8

3.2 Application..............................................................................................................9

4 Park transform .............................................................................................................10

4.1 Theory ..................................................................................................................10

4.1.1 Park Transform theory...................................................................................10

4.1.2 Inverted Park Transform theory .....................................................................13

4.2 Application............................................................................................................14

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1 Introduction

This application note describes the coordinate transform which with the Clarke, Park,

Inverse Clarke and Inverse Park transformation.

This application note describes the coordinate transform’s Theory, Block, Function, Flow,

Sample and Parameter in the ARM Inverter Platform.

.

2 Technical Background

Why need use coordinate transform?

2.1 Overview

In motor control, the motor’s start, stop, and speed and so on, all of them need to change

the motor magnetic torque, the motor torque with armature current is described in the

formulae:

DC motor: (1) ΦI

AC motor: (2) ΦI cosφ

Where Te is magnetic torque, CT    is torque modulus, I  is armature current, Φ is flux, φ  is

rotor power factor.

As DC motor, indirectly determine and control Φ and I to control motor torque. The flux and

the current have direct proportion with motor torque. Changes the current will change the

motor torque. It’s very simply.

But as AC motor, the motor torque control not only need I and Φ, the φ is also important.

The φ will change with the rotor current frequency. The Φ come from stator current and

rotor current, it will change along with the motor load change, so AC motor in dynamic

running, it’s controlled more difficult than DC motor.

AC motor should have the parameter relation like the DC motor.

Transvector Control could simulate AC motor into DC motor, and simply the control.

The base theory: Use the 3-phase AC motor rotating magnetic field transform into like DC

motor 2-phase rotating magnetic field, then control the 2-phase current to control the torque.

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3 Clarke transform

Clarke transform ‘s theory

3.1 Theory

3.1.1 Clarke transform theory

In motor theory, balanced 3-phase AC motor has 3 fixed windings. Through 3-phase

balanced AC current ia,ib,ic will bring a rotating magnetic field Φ with the speed ω.

Figure 3-1

.

ω

α

β

F

iα

iβ

Figure 3-2

ω

A

B

C

F

ia ib

ic

Figure 3-1

Not only balanced 3-phase fixed windings could bring rotating magnetic field, 2-phase

symmetry windings ( phase=90°) through 2-phase balance AC current iα,iβ, also could

bring rotating magnetic field. Figure 3-2

When balanced 3-phase fixed windings and 2-phase symmetry windings bring rotating

magnetic field Φ value and speed are equality, the 3-phase windings equivalent with

2-phase windings.

Clarke transform is converts balanced 3-phase (ia,ib,ic) into balanced 2-phase

(iα,iβ).Figure3-3

βClarke

α

c b

a

Figure 3-3

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Clarke transform must keep the power fixedness, in Figure3-4,

iα k1 ia ibcos 60° iccos 60° k1 ia12ib

12ic         

iβ k1 ibsin 60° icsin60° k1√32ib

√32ic

(3)

Where is the 3-phase current is 2-phase current, is the balanced

coefficient.

ia , ib , ic iα , iβ k1

Add a zero-axis value: (4) i0   k1k2 ia ib ic

Change the math formulae into motor ap , h e the matrix formulae plication av

iαiβi0

k1

112

12

0√32

√32

k2 k2 k2

iaibic

CTiaibic

(5)

Where

CT k1

112

12

0√32

√32

k2 k2 k2

(6)

If the power fixedness, the formulae transf ro m:

CT1 k1

1 0 k212

√32

k2

12

√32

k2

(7)

CT1CT k1

2

1 0 k212

√32

k2

12

√32

k2

112

12

0√32

√32

k2 k2 k2

1 0 00 1 00 0 1

(8)

β

Iα

IsIβb

αa

c

Figure 3-4

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Then k2

1

√2，k1

23

Take into the k2, k1 CT1

,

CT23

112

12

0√32

√32

1

√2

1

√2

1

√2 (9)

This formula is the Clarke transform matrix.

If take into the k2, k1 CT1

CT1 2

3

1 012

12

√32

1

√212

√32

1

√2

(10)

This formula is the Inverted Clarke transform matrix.

In motor theory, the balanced 3-phas r current have: e AC moto

sinsin 2 /3sin 2 /3

(11)

Take k2

1

√2，k1

23

and formula (11) into formula (5)

iαiβi0

23

112

12

0√32

√32

1

√2

1

√2

1

√2

sinsin 2 /3sin 2 /3

Here, if defined ; iα ia

The formula can transformed to

iα ia sin

iβ√3ia3

2√3ib3

sin /2

(12)

This transformation course use wave shown in Figure 3-5 below:

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iβ iα ic ib ia

Clarke β

α

bc

a

Figure 3-5

3.1.2 Inverted Clarke transform theory

In motor theory, when have two current component vectors in the stationary α-β axis,

through complementary inverse transforms to get back to the 3-phase stationary A,B,C

axis. This transformation uses the Inverse Clarke transformation, Figure 3-6.

a

Through the Inverted Park matrix and the Figure 3-4, the Inverted Clarke transform

formula:

CT1

β

β √3 α /2

β √3 α /2

(13)

α

βbInverted

cClarke

Figure 3-6

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3.2 Application

Function Name: ClarkAmplitude

C file name: ClarkAmplitude.C, ClarkAmplitude.H

<Function interface> void ClarkAmplitude ( volatile _stDataInThreeAxis *stDataInThreeAxis, //Inputs: 3-axis system

volatile _stDataInFixAxis *stDataInFixAxis //Outputs: fixed 2-axis system ) typedef struct { Q15_VAL32 a_Q15;// phase-a variable Q15_VAL32 b_Q15;// phase-b variable Q15_VAL32 c_Q15;// phase-c variable }_stDataInThreeAxis; _stDataInThreeAxis *stDataInThreeAxis typedef struct { Q15_VAL32 alpha_Q15;// stationary d-axis variable Q15_VAL32 beta_Q15;// stationary d-axis variable }_stDataInFixAxis;

_stDataInFixAxis *stDataInFixAxis

Table 1

Item Name Description Format

a phase-a of balanced 3- phase Q15_VAL32

b phase-b of balanced 3- phase Q15_VAL32 Inputs

c phase-c of balanced 3- phase Q15_VAL32

alpha alpha - alpha of fixed 2- phase Q15_VAL32 Outputs

beta alpha - beta of fixed 2- phase Q15_VAL32

Intput handle: stDataInThreeAxis, Output handle: stDataInFixAxis.

The following code is example for this module. Main() { } void example_Clarke() { stDataInThreeAxis.a_Q15=INa;//input phase-a stDataInThreeAxis.b_Q15=INb;// input phase-b ClarkAmplitude(&stDataInThreeAxis, pCurrentInFixAxis);// calculate clarke OUTa=CurrentInFixAxis. alpha_Q15;//Output alpha OUTb=CurrentInFixAxis. beta_Q15;//Output beta }

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4 Park transform

Park transform‘s theory

4.1 Theory

4.1.1 Park Transform theory

2-phase symmetry windings ( phase=90°) through 2-phase balance AC current iα,iβ, and

keep the windings stop, will bring rotating magnetic EMF F1 with speed ω. Figure 4-1.

2-phase symmetry windings ( phase=90°) through 2-phase balance DC current id,iq, and

rotate windings at speed ω, will bring rotating magnetic EMF F2, Figure 4-2.

If EMF F1=F2, stationary system 2-phase through AC current equivalent with rotating

system 2-phase through DC current.

The Park transformation convert the stationary 2-phase (iα,iβ) system into rotating 2-phase

(id,iq) system. Figure 4-3

Where the iα,iβ come from Clarke transform; and the θ come from the rotor is displaced

from the direct axis of the stator reference frame by the rotor angle θ. Because it can be

ω

α

F

iα

iβ

β

Figure 4-1

ω F

id

iq

q

d

Figure 4-2

α dβ

Park qθ

Figure 4-3

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seen that the angle between the real axis (x) of the general reference frame and the real

axis of the reference frame rotating with the rotor is θ,

Figure 4-4

id iαcos θ iβsin θiq iαsin θ iβcos θ (14)

As shown in Figure 4-4,

Where is the rotating 2-phase current, is stationary 2-phase current, is

the angle between iα and iβ.

id , iq iα , iβ θ

Change the math f r t mo r l n, ve the matrix formulae: o mulae in o to app icatio ha

idiq

cos θ sin θsinθ cos θ

iαiβ

CPiαiβ

(15)

CPcos θ sin θsin θ cos θ (16)

Where:

This matrix convert the stationary 2-phase(α, β) system into rotating 2-phase(d, q) system,

it’s called Park transform.

If the power fixedness, the formulae transform:

CP1 cos θ sin θ

sinθ cos θ

(17)

This matrix convert rotating 2-phase(d, q) system into stationary 2-phase(α, β) system, it’s

called Inverted Park transform. This transform course use wave shown in Figure 4-1 below:

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iα iβ id

iq

α

β

θ

d

qPark

Figure 4-5

4.1.2 Inverted Park Transform theory

In motor theory, these are two current component vectors in the rotating d-q axis, through

the complementary inverse transformation to get back to the 2-phase stationary α-β axis.

This transformation uses the Inverse Park Transform, Figure 4-5.

Through the Inverted Park matrix 4-2, the Inverted Park formula: CP1

and the Figure

(20)

iα idcoiβ idsin

s θ iqsinθθ iqcos θ

Figure 4-6

dα

q

θ

Inverted

Park β

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4.2 Application

Function Name: Park`

C file name: Park.C, Park.H Function interface: void Park(_stDataInFixAxis *stDataInFixAxis, //fixed 2-axis system _stDataInRotAxis *stDataInRotAxis //rotational 2-axis system) typedef struct { Q15_VAL32 alpha_Q15; Q15_VAL32 beta_Q15; }_stDataInFixAxis; _stDataInFixAxis *stDataInFixAxis typedef struct { Q15_VAL32 d_Q15; Q15_VAL32 q_Q15; Q15_VAL32 theta_Q15; }_stDataInRotAxis; _stDataInRotAxis *stDataInRotAxis

Table 2 Item Name Description Format

alpha phase- alpha of fixed 2- phase Q15_VAL32 beta phase- beta of fixed 2- phase Q15_VAL32 Inputs

Theta Phase angle between stationary and rotating frame Q15_VAL32

d alpha - alpha of rotational 2-axis system Q15_VAL32 Outputs q alpha - beta of rotational 2-axis system Q15_VAL32

The following code is example for this module. Main() { } void example_Park() { CurrentInFixAxis. alpha_Q15= INa;//Input alpha CurrentInFixAxis. beta_Q15= INb;//Input beta stDataInRotAxis. theta_Q15=Intheta;//input Phase angle Park(&CurrentInFixAxis, stDataInRotAxis); OUTa= stDataInRotAxis. d_Q15; OUTb= stDataInRotAxis. q_Q15; }

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Function Name: ParkInv

C file name: ParkInv.C, Park.H

Function interface: void InvPark(_stDataInRotAxis *stDataInRotAxis, // rotating 2- phase system

_stDataInFixAxis *stDataInFixAxis // fixed 2- phase system) typedef struct { Q15_VAL32 d_Q15; Q15_VAL32 q_Q15; Q15_VAL32 theta_Q15; }_stDataInRotAxis; _ stDataInRotAxis * stDataInRotAxis typedef struct { Q15_VAL32 alpha_Q15; Q15_VAL32 beta_Q15; }_stDataInFixAxis; _stDataInFixAxis *stDataInFixAxis

Table 3 Item Name Description Format

d alpha - d of rotating 2- phase Q15_VAL32 q alpha - q of rotating 2- phase Q15_VAL32

Inputs θ

Phase angle between stationary and rotating frame

Q15_VAL32

alpha alpha - alpha of fixed 2- phase Q15_VAL32 Outputs beta alpha - beta of fixed 2- phase Q15_VAL32 Intput handle: stDataInRotAxis, Output handle: stDataInFixAxis.

The following code is example for this module. Main() { } void example_ParkInv() { stDataInRotAxis.d_Q15=INd;//input phase-d stDataInRotAxis.q_Q15=INq;// input phase-q stDataInRotAxis.theta_Q15=Intheta;//input Phase angle InvPark (&stDataInRotAxis, pCurrentInFixAxis);// calculate Inverte Park OUTa=CurrentInFixAxis. alpha_Q15;//Output alpha OUTb=CurrentInFixAxis. beta_Q15;//Output beta }

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