field oriented control of induction...

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Chapter 3

FIELD ORIENTED CONTROL

OF INDUCTION MOTOR

3.1. Introduction

The control of AC machine is basically classified into scalar and vector

control. The scalar controls are easy to implement though the dynamics are

sluggish. The objective of FOC is to achieve a similar type of controller with

an inner torque control loop which makes the motor respond very fast to the

torque demands from the outer speed control loop. In FOC, the principle of

decoupled torque and flux control are applied and it relies on the

instantaneous control of stator current space vectors. Control of induction

motor is complicated due to the control of decoupled torque and flux

producing components of the stator phase currents. There is no direct

access to the rotor quantities such as rotor fluxes and currents. To

overcome these difficulties, high performance vector control algorithms are

developed which can decouple the stator phase currents by using only the

measured stator current, flux and rotor speed.

In this chapter, the mathematical model of induction motor based on

space vector theory and the principle of indirect FOC are presented. The

simulation model of the induction motor drive is developed using the

principle of indirect FOC.

3.2. High Performance Drive

A system employed for motion control using electric motor as a prime

mover is called electric drive. The function of an electric drive system is the

controlled conversion of electrical energy to a mechanical form and vice

versa through a magnetic field.

study requiring proper integration of knowledge of electrical machines,

actuators, power electronic converters, sensors and instrumentation, control

hardware and software and communicatio

drive system showing the same performance can be designed in various

ways, like other engineering designs.

drive system’s ability to offer precise control in addition to a rapid dynamic

response and a good steady state response. High performance drives are

considered for critical applications due to their precision of control.

Several control str

speed drive industry, which includes i) open loop inverter with fixed V/f

control, ii) open loop inverter with flux vector control, iii) closed loop inverter

with flux vector control and iv)

linear control and Predictive Control

loop feedback control to obtain high precision, good dynamics and steady

state response. FOC predominantly relies on the mathematical modeling of

17


gh a magnetic field. Electric drive is a multi-disciplinary field of



hardware and software and communication links as shown in Fig. 3.1.


ways, like other engineering designs. High performance drive refers to the




Fig. 3.1 Electric drive system

Several control strategies as shown in Fig.3.2 are found in the variab


inverter with flux vector control, iii) closed loop inverter

with flux vector control and iv) DTC. The controls, namely, FOC,

edictive Control (PC) are to be implemented with closed


FOC predominantly relies on the mathematical modeling of


disciplinary field of



n links as shown in Fig. 3.1. The


High performance drive refers to the




are found in the variable


inverter with flux vector control, iii) closed loop inverter

DTC. The controls, namely, FOC, DTC, non-

are to be implemented with closed-


FOC predominantly relies on the mathematical modeling of

AC machine, while DTC makes direct use of physical int

place within the integrated system of the machine and its supply.

c)

Fig. 3.2 Electrical drive control technique

18

AC machine, while DTC makes direct use of physical interaction that takes


a) DC drive

b) AC drive – scalar control

c) AC drive – Field Oriented Control

d) AC drive – Direct Torque Control

Fig. 3.2 Electrical drive control technique

eraction that takes


Fig. 3.2 Electrical drive control techniques

19

3.3. Induction Motor Drive

3.3.1. Physical Layout of Induction Motor

In an induction motor induction refers the field in the rotor is induced

by the stator currents and asynchronous refers that the rotor speed is not

equal to the stator speed. The rotor of the squirrel cage three phase

induction motor is cylindrical in shape and have slots on its periphery. The

slots are not made parallel to each other but are a bit skewed to prevent

magnetic locking of stator and rotor teeth and make the working of motor

more smooth and quiet. The magnetic path comprises a set of slotted steel

laminations pressed into the cylindrical space inside the outer frame. The

magnetic path is laminated to reduce eddy currents, lower losses and lower

heating. The squirrel cage rotor consists of aluminum, brass or copper bars,

this aluminum, brass or copper bars are called rotor conductors and are

placed in the slots on the periphery of the rotor.

The rotor conductors are permanently shorted by copper or aluminum

rings called the end rings. In order to provide mechanical strength, these

rotor conductors are braced to the end ring and hence form a complete

closed circuit resembling a cage and also the squirrel cage rotor winding is

made symmetrical. As the bars are permanently shorted by end rings, the

rotor resistance is very small and it is not possible to add external

resistance. Even though the aluminium rotor bars are in direct contact with

the steel laminations, practically all the rotor current flows through the

aluminum bars and not through the laminations. It is necessary to keep the

bars tightly in the slots because loose bars can be damaged quickly by

mechanical vibrations and thermal cycling.

The only parts of the squirrel cage motor that can wear are the

bearings. The absence of slip ring and brushes make the construction of

squirrel cage three phase induction motor very simple, robust, requires less

maintenance and eliminates sparking. These motors are widely used in

industrial drives because they are rugged, reliable, economical and have the

advantage of adapting any number of pole pairs. Fig. 3.3 shows the cut

sectional view of a typical induction motor.

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Fig. 3.3 Cut sectional view of a typical induction motor

(Source: www.ctiautomation.net)

3.3.2. Dynamic Model in Space Vector Form

Mathematical description of induction motor is based on space vector

notation. When describing a three phase induction motor by a system of non

linear equations, following assumptions are made:

i. The three phase motor is symmetrical,

ii. Only the fundamental harmonics is considered, while the higher

harmonics of the special field distribution and of the Magneto Motive

Force (MMF) in the airgap are disregarded,

iii. The spatially distributed stator and rotor windings are replaced by a

concentrated coil,

iv. Effects of anisotropy, magnetic saturation, iron losses and eddy

currents are neglected,

v. Coil resistance and reactance are taken to be constant,

vi. In many cases, especially when considering steady state, the current

and voltages are taken to be sinusoidal.

Considering the above assumptions, the stator and rotor voltage

equations can be written as:

( ) aa s a

dV t R i

dt

ψ= +

urv v

(3.1)

( ) bb s b

dV t R i

dt

ψ= +

urv v

(3.2)

21

( ) cc s c

dV t R i

dt

ψ= +

urv v

(3.3)

To describe the model of the induction motor, generally the space

vector method is adopted. This approach has the advantages like: i) analysis

is possible at any supply voltage and ii) number of dynamic equations can

be reduced.

3.3.3. Space Vector Definition

The three phase symmetric system represented in a natural

coordinate system by phase quantities such as currents, voltages and flux

linkages of AC motors can be analyzed in terms of complex space vectors

[77]. Any three time varying quantities, which always sum to zero and are

spatially separated by 120° can be expressed as space vector. The space

vector can be defined by considering the instantaneous values ua, ub, uc. A

three phase system defined by ua(t), ub(t), uc(t)) can be represented uniquely

by a rotating vector. The space vector u may represent the motor variables

(voltage, current and flux). The vector control principle on AC motor take the

advantages of transforming the variables from the physical three phase a-b-c

system to a stationary coordinate α-β or rotating reference frame d-q [78],

which is equivalent to the armature and field currents of a DC motor. Space

vector and its component are shown in Fig. 3.4.

Fig. 3.4 Space vector representation for three phase variables

22

The complex stator current vector ‘u’, which represents the three

phase sinusoidal system, is represented as:

( )2 /3 2 /32( ) ( ) ( )

3

j ja b cu u t u t e u t e

π π−= + +

(3.4)

where,

�

� is the non power invariant transformation constant (normalization

factor) and ua(t), ub(t) and uc(t) are arbitrary phase quantities in a

system of natural coordinates satisfying the condition,

( ) ( ) ( ) 0a b cu t u t u t+ + =

(3.5)

3.3.4. Circuit Model on a Stationary Reference Frame

Equations for a two pole induction machine with a short circuited

rotor in stator reference frame using space phasor rotation are [79] shown

below. By applying space vector Vs, the stator voltage equation written in

stator axis can be written as in (3.6). The squirrel cage induction motor rotor

is shorted and so there is no rotor excitation, the rotor voltage equals zero.

Figs. 3.5 and 3.6 show the stator current space vector and equivalent circuit

of an induction motor. From this equivalent circuit it is clear that each

motor winding has two current paths.

Magnetizing path: Each stator winding has an iron core, thus will have a

high inductance. The inductances of each winding are important to the

operation of the motor, because when drawing current they generate the

rotating magnetic field essential to the operation of the motor. The

magnetizing current is reactive, i.e., it lags behind the applied voltage by

90⁰.

Load path: This current path transforms from the stator to the rotor by

transformer action, and flows through the rotor bars. The more load on the

motor, the higher the slip, and the higher the load current. Load current is

real, i.e., it is in phase with the applied voltage.

Total current: The total current in each winding of a motor is the vector sum

of the load current and the magnetizing current. Generally the magnetizing

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component is constant and does not change with load. It ensures that the

motor always runs at a lagging power factor.

Using the space vector method the induction motor model can be

written as:

( ) ss s s

dV t R i

dt

ψ= +

urv v

(3.6)

( ) rr r r

dV t R i

dt

ψ= +

urv v

(3.7)

Fig. 3.5 Stator current space vector

Fig. 3.6 Equivalent circuit of Induction motor

From Fig. 3.5, the flux current equations are represented as:

24

js s s m rL i L i e

εψ = +uur ur ur

(3.8)

r r r m sL i L iψ = +uur ur ur

(3.9)

Complete set of motor equation is obtained by transforming the above

equations into a common rotating reference frame and bringing the rotor

values into the stator side,

( )( )

js r

s s s s m

di d i eV t R i L L

dt dt

ε

= + +

v vv v

(3.10)

( )( ) sr

r r r r m

d idiV t R i L L

dt dt= + +

vvv v

(3.11)

( )0

j j srr r r m

d idiR i e L e L

dt dt

ε ε= + +

vvv

(3.12)

Equation of the dynamic rotor rotating with an angular speed ωr , can

be represented as:

( )( ) ( )d L mrT t T t Bd

dt J

ωω − −=

(3.13)

r

d

dt

εω =

(3.14)

where, Ls = Lm(1+σs)

Lr = Lm(1+σr)

In further consideration viscous coefficient will be negated as B=0. The

electromagnetic torque is expressed by:

2( ) ( ) Im ( )*

3 2

jrd L m s r

d PT t J T t L i i e

dt

εω = + = v v

(3.15)

The applied space vector method is used as a mathematical tool for

the analysis of the electric machines. The complete set of equations can be

expressed in the stationary coordinate α-β system. The motor model

equations defined with respect to α-β reference frame is written as:

( ) ss s s

dV t R i

dt

αα α

ψ= +

urv v

(3.16)

( )s

s s s

dV t R i

dt

ββ β

ψ= +

urv v

(3.17)

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0 rr r r r

dR i

dt

αα β

ψω ψ= + +

uruuuvv

(3.18)

0r

r r r r

dR i

dt

ββ α

ψω ψ= + −

uruuuvv

(3.19)

( )( ) ( ) 1 2

3 2

d L mrs s s s L

r

T t T t Ld Pi i T

dt J J Lα β β α

ωψ ψ

−= = − −

(3.20)

where, s s s m rL i L iα α αψ = +

s s s m rL i L iβ β βψ = +

r r r m sL i L iα α αψ = +

r r r m sL i L iβ β βψ = +

The complex space vector is resolved into components of α and β.

s s si i ji

α β= +

r

(3.21)

r r ri i ji

α β= +

r

(3.22)

s s sV V jV

α β= +

uur

(3.23)

s s sj

α βψ ψ ψ= +uur

(3.24)

r r rj

α βψ ψ ψ= +uur

(3.25)

In the stationary α-β coordinate system, the input to the motor is the

stator voltage. The above equation is transformed into:

( )ss s s

dV t R i

dt

αα α

ψ= −

urv v

(3.26)

( )s

s s s

dV t R i

dt

ββ β

ψ= −

urv v

(3.27)

rr r r r

dR i p

dt

αα β

ψω ψ= − −

urv

(3.28)

r

r r r r

dR i p

dt

ββ α

ψω ψ= − +

ur

v

(3.29)

( )1 2

3 2

mrs s s s L

r

Ld Pi i T

dt J Lα β β α

ωψ ψ

= − −

(3.30)

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The output signals such as flux, speed and torque depend on both the

inputs, by the orientation of the coordinate system to the stator or rotor flux

vectors decoupling of flux and torque can be achieved.

3.3.5. Equivalent Circuit on a d-q Reference Frame

The principle of vector control of AC machine can be controlled to give

dynamic performance comparable to the separately excited DC motor. There

are at least three fluxes, rotor, airgap and stator and three currents, stator,

rotor and magnetizing in an induction motor. For high dynamic response,

interactions among current, fluxes, and speed must be taken into account

in determining appropriate control strategies. Independent control of motor

flux and torque can be obtained by this method and it is possible by

connecting coordinate system with rotor flux vector. Fig. 3.7 shows the

vector diagram of induction motor in stationary α-β and rotating d-q

coordinates. The rotor synchronous speed is equal to the angular speed of

the rotor flux vector. The reference frame d-q is rotating with the angular

speed equal to rotor flux vector angular speed ωe, which is defined as

follows:

e

d

dt

θω =

(3.31)

Fig. 3.7 Vector diagram in stationary and rotating reference frame

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The voltage and current complex space vector is resolved into

components of d and q as:

( ) ( )1( )jj

s s sd sqV t e V e V jVγ θθ − −− = = +

v v

(3.32)

( ) ( )jjs s s s s sd sqR i e R i e R i ji

γ θθ − −− = = +v v

(3.33)

( ) ( )j

r r r rd rqR i e R i jiε θ− −

= +v

(3.34)

Induction motor model equation in d-q reference frame is written as

follows:

( )( )

jj j j js r

s s s m

di d i eV t e R ie L e L e

dt dt

εθ θ θ θ− − − −= + +

v vv v

(3.35)

( ) ( ) ( )( )0

jj j jsr

r r r m

d i ediR i e L e L e

dt dt

εε θ ε θ ε θ

−− − −

= + +

vvv

(3.36)

( )( ) ( ) 1 2

3 2

d L mrrd sq rq sd L

r

T t T t Ld Pi i T

dt J J L

ωψ ψ

−= = − −

(3.37)

The stator flux linkages are given by:

sq s sq m rqL i L iψ = +

(3.38)

sd s sd m rdL i L iψ = +

(3.39)

( )2 2ˆ

s sd sqψ ψ ψ= +

(3.40)

The rotor flux linkages are:

0rq r rq m sq

L i L iψ = = +

(3.41)

rd r r rd m sdL i L iψ ψ= = +

(3.42)

( )2 2ˆ

r rd rqψ ψ ψ= +

(3.43)

The airgap flux linkages are:

mq m sq m rqL i L iψ = +

(3.44)

md s sd m rdL i L iψ = + (3.45)

( )2 2ˆ

m md mqψ ψ ψ= +

(3.46)

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The motor torque can be expressed by rotor flux magnitude and stator

current component, if the rotor can be kept constant as in the case of DC

machine, then the torque control can be accomplished by controlling the

current component.

( )2

3 2

md rd sq rq sd

r

LPT i i

Lψ ψ= −

(3.47)

( )1 2

3 2

mrrd sq rq sd L

r

Ld Pi i T

dt J L

ωψ ψ

= − −

(3.48)

Dynamic equivalent circuit for d and q axis is shown in Fig. 3.8 (a)

and (b) and block diagram of induction machine in d-q coordinate system is

shown in Fig. 3.9. The complete motor dynamic equation can be obtained by

separating the real and imaginary components of the voltage and current

complex space vector as:

sdsd s sd s m rd s e sq m e rq

di dV R i L L i L i L i

dt dtω ω= + + − −

v

(3.49)

sq

sq s sq s m rq s e sd m e rd

di dV R i L L i L i L i

dt dtω ω= + + + +

v

(3.50)

( ) ( )0 rdr rd r m sd r e r rq m e r sq

di dR i L L i L i L i

dt dtω ω ω ω= + + − − + −

(3.51)

( ) ( )0rq

r rq r m sq r e r rd m e r sd

di dR i L L i L i L i

dt dtω ω ω ω= + + + − + − (3.52)

(a) (b)

Fig. 3.8 Dynamic equivalent circuit (a) d-axis circuit (b) q-axis circuit

29

Fig. 3.9 Block diagram of Induction machine in d-q coordinate system

The rotational voltage term across stator and rotor is expressed as:

( )sd e s sd m rd eL i L iψ ω ω= +

(3.53)

( )sq e s sq m rq eL i L iψ ω ω= +

(3.54)

( ) ( )( )rd e r r rd m sd e rL i L iψ ω ω ω ω− = + −

(3.55)

( ) ( )( )rq e r r rq m sq e rL i L iψ ω ω ω ω− = + −

(3.56)

The machine dynamic voltage in matrix form is represented as:

( ) ( )( ) ( )

0

0

sds s e m m e

sqe s s m e m

m e r m r r e r r rd

e r m m e r r r r rq

iR pL Ls pL LVsd

iLs R pL L pLVsq

pL L R pL L i

L pL L R pL i

ω ω

ω ω

ω ω ω ω

ω ω ω ω

+ − − + = − − + − −

− − +

(3.57)

3.3.6 Field Oriented Control Concept of Separately Excited DC Machine

In separately excited DC machine, the axis of the armature and field

current are orthogonal to one another. Ideally a vector controlled induction

motor drive operates like a separately excited DC motor drive as shown in

Fig. 3.10. This means that the magneto motive forces established by the

currents in these windings are also orthogonal. If iron saturation and

armature reaction effect are ignored, developed torque can be expressed as:

30

d f aT kI I=

(3.58)

where, If - field current

Ia - armature current

Fig. 3.10 Block diagram of separately excited DC motor

The field flux ψf produced by the current in the field coils is

perpendicular to the armature flux ψa. The ampere turns resulting from the

armature current has no effect on the field flux because the spatial direction

in which the armature mmf oriented has an angular displacement of π/2

radians with respect to the spatial direction of the field flux. Therefore

changes in armature current irrespective of whether they are caused by the

controller or by changes in load do not affect field flux. It is for this reason

that DC machines are said to have decoupled or independent control over

torque and flux. These stationary space vectors are orthogonal and

decoupled in nature. This decoupling that naturally exists in the DC motor

between the field flux and the armature current, irrespective of the angular

position of the shaft, that gives the machine its high dynamic performance

capability. Induction motors are coupled non linear multivariable systems

whose stator and rotor fields are not held orthogonal to one another. In

order to achieve decoupled control over the torque and flux producing

components of the stator currents, a technique known as Field Oriented

Control (FOC) is used. The main drawback of this technique is the reduced

reliability of the DC motor – the fact that brushes and commutators wear

down and need regular servicing. DC motors can be costly to purchase since

they require encoders for positional feedback.

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3.3.7 Description of Field Oriented Control

The principle of FOC system of an induction motor is that the d-q

coordinate reference frame is locked to the rotor flux vector, this results in

decoupling of the variables so that flux and torque can be separately

controlled by stator direct axis current isd and quadrature axis current isq

respectively like in the separately excited DC machine. Performance of DC

machine can also be extended to an induction motor if the machine is

considered in a synchronously rotating reference frame where the sinusoidal

variables appears as DC quantity in steady state. The induction motor with

the inverter and vector control in the front end is shown in Fig. 3.11.

Fig. 3.11 Field oriented control of induction motor

With FOC, direct axis component of the stator current is analogous to

field current and quadrature axis component of stator current is analogous

to armature current of a DC machine, therefore torque can be expressed as:

'

d sq sdT k I I=

(3.59)

Basic equations describing the dynamic behaviour of an Induction

machine in a rotating reference frame aligned to the rotor flux axis is

described above. For obtaining linear relationship between control variables

and torque, coordinate transformation to new field coordinates are of prime

importance in vector control. The induction motor model is often used in

vector control algorithms, for this, the reference frame may be aligned with

the stator flux linkage, rotor flux linkage or the magnetizing space vector.

The most accepted reference frame is the frame attached to the rotor flux

linkage, this can be achieved by deciding ωr to be the speed of rotor and

locking the phase of the reference system so that the rotor flux is aligned

32

with the d axis. In this, the torque can be instantaneously controlled by

controlling the current isq after decoupling the rotor flux and torque

producing component of the current components [80]. To perform the

alignment on a reference frame revolving with rotor flux requires information

on the position of the rotor flux. The reference frame d-q aligned with the

rotor flux is shown in Fig. 3.12. A condition for elimination of transients in

rotor flux and the coupling between the two axes is to have the flux along

the q axis must be zero, thus the field orientation concept in rotating

reference frame is,

0rqψ =

(3.60)

rd rψ ψ=

(3.61)

Fig. 3.12 Field orientation in d-q reference frame

0rqd

dt

ψ=

(3.62)

0rq r rq m sqL i L iψ = + =

(3.63)

Maximum peak of torque per ampere is attained when the magnetizing

current imr, which is responsible for the magnetizing flux generation, is equal

to the torque producing component of the stator current isq at steady state

condition. The rotor inductance can be expressed in terms of mutual

inductance and rotor leakage coefficient σr as:

( )1r r m r m m m rL l L L L Lσ σ= + = + = +

(3.64)

( )1rd r rd m sd m r rd m sd m mrL i L i L i L i L iψ σ= + = + + =

(3.65)

33

( )1

mr sdrd

r

i ii

σ

−=

+

(3.66)

Up to the rated speed rotor magnetizing current is kept constant to get

the fast control over electromagnetic torque of the machine because the

dynamics of the magnetizing current involves a big time constant. The

magnetizing current is responsible for the magnetizing flux generation. From

the voltage loop equation the magnetizing current dependency on the d

component of stator current is obtained as:

0rdr rd

dR i

dt

ψ+ =

(3.67)

mrr mr sd

dii i

dtτ + =

(3.68)

where,

rr

r

L

Rτ =

(1 )

sdmr

r

ii

sτ=

+

3.3.8 Determination of the Rotor Flux Angle

Knowledge of the rotor flux angle is essential for accurately applying

the Clarke and Park transformations. If this angle is incorrect, the flux and

torque producing components of the stator current are not decoupled and

true field oriented control is not achieved. Induction motors are

asynchronous machines so the flux speed is not equal to the mechanical

speed of the rotor due to the effect of slip. In this reference frame, the d axis

is moving at the same relative speed as the rotor phase ‘a’ winding and

coincides with its axis. When the rotor is at standstill, rotor phase current

and rotor d axis current are at fundamental frequency, but at normal

running speed it gradually change to slip frequency, which is useful in

studying transient phenomena in the rotor. In induction motor, mechanical

speed is defined as the difference between rotor flux speed and slip angular

speed and rotor flux linkage can be with the physical quantity as:

r e slipω ω ω= −

(3.69)

34

( ) 0rq e r rd r rq

dR i

dtψ ω ω ψ+ − + =

(3.70)

( ) 0rd e r rq r rd

dR i

dtψ ω ω ψ+ − + =

(3.71)

Rotor currents ird and irq can be written in terms of isq and isd as:

rq m

rq sq

r r

Li i

L L

ψ= −

(3.72)

rd mrd sd

r r

Li i

L L

ψ= −

(3.73)

Rotor current ird and irq can be eliminated from (3.70) and (3.71) as

( ) 0mrrq e r rd rq r sq

r r

LRdR i

dt L Lψ ω ω ψ ψ+ − + − =

(3.74)

( ) 0mrrd e r rq rd r sd

r r

LRdR i

dt L Lψ ω ω ψ ψ+ − + − =

(3.75)

Substituting the expressions for irq and ird into (3.62) and (3.64) gives

2m m

sq s sq rq

r r

L LL i

L Lψ ψ

= − +

(3.76)

2

m msd s sd rd

r r

L LL i

L Lψ ψ

= − +

(3.77)

Slip speed is calculated based on the following two equations:

( ) 0rd e r r rqR iψ ω ω− + =

(3.78)

0rqψ =

(3.79)

To determine the rotor flux angle, first we need to calculate the slip

using the following equation:

1sq sqmslip

r rd r mr

i iL

iω

τ ψ τ= =

(3.80)

1 sq

e r slip r

r mr

i

iω ω ω ω

τ= + = +

(3.81)

Then rotor flux angle can be calculated as:

35

edtθ ω= ∫ (3.82)

2

3 2 (1 )

mrmr sq L

r

Ld PJ i i T

dt

ω

σ= −

+

(3.83)

The fundamental equations for vector control, which allows the

induction motor to act like a separately excited DC machine with decoupled

control of torque and flux making the induction motor to operate as a high

performance four quadrant servo drive. The expression for the

electromagnetic torque of the machine becomes:

2

3 2 (1 )

md mr sq

r

LPT i i

σ=

+

(3.84)

Rated torque of the motor is obtained by selecting the magnetizing

current to achieve the maximum torque per ampere ratio. If the magnetic

saturation is not taken into consideration, the maximum peak of torque per

ampere is achieved when the magnetizing current is equal to the torque

producing component of the stator current at steady state condition for all

permitted ranges of stator currents.

3.4. Basic Scheme of Field Oriented Control of Induction

Motor

Field Oriented Control or Vector Control (VC) techniques are being

used extensively for the control of induction motor. This technique allows a

squirrel cage induction motor to be driven with high dynamic performance

comparable to that of a DC motor. In FOC, the squirrel cage induction motor

is the plant which is an element within a feedback loop and hence its

transient behavior has to be taken into consideration. This cannot be

analyzed from the per phase equivalent circuit of the machine, which is valid

only in the steady state condition. The induction motor can be considered as

a transformer with short circuited and moving secondary where the coupling

coefficients between the stator and rotor phases change continuously in the

course of rotation of rotor. The machine model can be described by

differential equations with time varying mutual inductances but such model

is highly complex. For simplicity of analysis, a three phase machine which is

supplied with three phase balanced supply can be represented by an

36

equivalent two phase machine. The problem due to the time varying

inductances is eliminated by modeling the induction motor on a suitable

reference frame. The basic idea behind the FOC is to manage the

interrelationship of the fluxes to avoid the issues mentioned above, and to

squeeze the most performance from the motor. The basic principle of FOC is

to maintain a desired alignment between the stator flux and rotor flux.

In FOC, the stator currents are transformed into a rotating reference

frame aligned with the rotor, stator or air-gap flux vectors to produce d-axis

component of current and q-axis component of current. Torque can be

controlled by the q-axis component of stator current space vector and flux

controlled by the d-axis component of current vector. The basis of FOC is to

use rotor flux angle to decouple torque and flux producing components.

Basically, there are two different types of FOC methods depending on the

calculation of this rotor flux angle. Field orientation achieved by direct

measurement of flux is termed Direct FOC (DFOC). The flux orientation

achieved by imposing a slip frequency derived from the rotor dynamic

equations is referred to as Indirect FOC (IFOC). IFOC is preferred to DFOC

since the fragility of Hall sensors detracts the inherent robustness of an

induction machine.

3.4.1. Direct Field Oriented Control

In DFOC, an estimator or observer calculates the rotor flux angle.

Inputs to the estimator or observer are stator voltages and currents. In

DFOC, rotor flux vector orientation can be measured by the use of a flux

sensor mounted in the air gap like Hall-effect sensor, search coil and other

measurement techniques introduces limitations due to machine structural

and thermal requirements or it can be measured using the voltage

equations. Saliency of fundamental or high frequency signal injection is the

other flux and speed estimation technique, but this method fails at low and

zero speed level. The method may cause torque ripples and mechanical

problems when applied with high frequency signal injection. The advantage

of this method is that the saliency is not sensitive to actual motor

parameters. Flux sensor is expensive and needs special installation and

37

maintenance. Rotor flux cannot be directly sensed by this method but from

the directly sensed signal it is possible to calculate the rotor flux, which may

result in inaccuracies at low speed due to the dominance of stator resistance

voltage drop and due to variation of flux level and temperature and makes it

expensive. Fig. 3.13 shows DFOC drive system.

Fig. 3.13 DFOC drive system

3.4.2. Indirect Field Oriented Control

The field orientation concept implies the current components supplied

to the machine should be oriented in such a manner as to isolate the stator

current magnetizing flux component of the machine from the torque

producing component. This can be obtained by the instantaneous speed of

the rotor flux linkage vector and the d axis of the d-q coordinates are exactly

locked in rotor flux vector orientation. In IFOC, the rotor flux angle is

obtained from the reference currents, rotor flux vector is estimated by using

the field oriented control current model equations and requires a rotor speed

measurement. In this, flux position can be calculated by considering

terminal quantities in motor model such as voltage and currents, but it is

very sensitive to rotor time constant. When rotor time constant is not

accurately set, detuning will takes place in the machine and the loss of

decoupled control of torque and flux causes a sluggish performance. IFOC of

the rotor currents can be implemented using instantaneous stator currents

and rotor mechanical position. It does not have inherent low speed problems

and is preferred in most applications.

The flux control through the magnetizing current is by aligning all the

flux with d axis and aligning the torque producing component of the current

with the q axis. The torque can be instantaneously controlled by controlling

the current isq after decoupling the rotor flux and torque producing

38

component of the current components. The flux along the q axis must be

zero and the mathematical constraint is,

0rqψ =

(3.85)

The rotational position information is measured from slip frequency.

The flux and torque can be controlled independently by providing the slip

frequency. The block diagram of IFOC drive system is shown in Fig. 3.14.

Fig. 3.14 IFOC drive system

Properties of the FOC methods are,

i. It is based on the analogy to the control of separately excited DC

motor,

ii. Coordinate transformations are required,

iii. PWM algorithm is needed,

iv. Current controllers are necessary,

v. Sensitive to rotor time constant,

vi. Rotor flux estimator is essential in DFOC and

vii. Mechanical speed is required in IFOC.

The goal of FOC is to perform real time control of torque variations

demand, to control the mechanical speed and to regulate phase currents. To

perform these controls, the equations are projected from a three-phase non-

rotating frame into a two coordinate rotating frame. FOC uses a pair of

conversions to get from the stationary reference frame to the rotating

reference frame which is known as Clarke transformation and Park

transformation. This mathematical projection (Clarke and Park

39

transformation) greatly simplifies the expression of the electrical equations

and removes their time and position dependencies.

The control task can be greatly simplified by first using the Clarke and

Park transforms to perform a two-step transformation on the stator

currents. The first is from a three phase to a two phase system with the

Clarke transform, and then translating them into the rotor reference frame

with the Park Transform. This enables the controllers to generate voltages to

be applied to the stator to maintain the desired current vectors in the so-

called rotor reference frame. The voltage command is then transformed back

by the inverse Park and Clarke transform to voltage commands in the a-b-c

stator reference frame, so that each phase can be excited via the power

converter.

3.4.3. Clarke Transformation

This transformation block is responsible for translating three axes to

two axes system reference to the stator. Two of the three phase currents are

measured because the sum of the three phase currents equal to zero.

Basically the transformation shift from a three axis, two- dimensional

coordinate system attached to the stator of the motor to a two axis system

referred to the stator. The measured current represents the vector

component of the current in a three axis coordinate system which are

spatially separated by 120°. Clarke transformation transforms the rotating

current vector in a two axis orthogonal coordinate system, so that the

current vector is represented with two vector components which vary with

time. The space vector can be transformed to another reference frame with

only two orthogonal axis called α-β, where the axis ‘α’ and axis ‘a’ coincide

each other. Fig. 3.15 shows the stator current space vector and its

component in stationary reference frame and the Clarke transformation

module.

The projection that modifies the three phase system into two

dimensional orthogonal system is expressed as:

( )2 /3 2 /32Re ( ) ( ) ( )

3

j js a b ci i t i t e i t e

π πα

− = + +

(3.86)

40

( )2 /3 2 /32Im ( ) ( ) ( )

3

j js a b ci i t i t e i t e

π πβ

− = + +

(3.87)

Fig. 3.15 Stator current space vector in stationary reference frame

1 112 22

3 3 302 2

as

bs

c

ii

ii

i

α

β

− − = −

(3.88)

3.4.4. Park Transformation

It is used to rotate the two axis coordinate system so that it is aligned

with the rotating motor and this projection modifies a two phase orthogonal

α-β system in the d-q rotating reference frame. The stator reference frame is

not suitable for the control process. The space vector is is rotating at a rate

equal to the angular frequency of the phase currents, the components

change with time and speed. In order to gain a complete decoupling of

torque and flux, the current phasor is transformed into two components of a

rotating reference frame rotating at the same speed as the angular frequency

of the phase currents, these components do not depend on time and speed.

In FOC this is the most important transformation, the component of stator

current which is responsible for the rotor flux can be fix to the d axis. These

components depend on the α-β current vector components and the rotor flux

position. The separate flux and the torque components of stator current

vector in two coordinate time invariant system can be expressed by (3.89).

Direct torque control is possible and becomes easy with the flux component

41

isd aligned with the d axis representing the direction of the rotor flux and

torque component isq aligned with the q axis perpendicular to the rotor flux.

Fig. 3.16 shows the stator current space vector and its component in

rotating reference frame and the Park transformation module.

s sd sqi i ji= +r

(3.89)

cos sin

sin cos

sd s

sq s

i i

i i

α

β

θ θ

θ θ

= −

(3.90)

Fig. 3.16 Stator current space vector in rotating reference frame

3.4.5. Inverse Park Transformation

In this transformation, the stator voltages represented in the d-q

rotating reference frame are transformed to a two phase orthogonal α-β

system, from which we can obtain the reference vector components to be

applied to the motor phases through space vector modulation technique.

The projection that modifies the d-q rotating reference frame to two phase

orthogonal system is expressed by (3.91). Fig. 3.17 shows the Inverse Park

transformation module and the stator voltage space vector and its

component in stationary orthogonal system.

cos sin

sin cos

sds

sqs

VV

VV

α

β

θ θ

θ θ

− =

(3.91)

42

Fig. 3.17 Stator voltage space vector from d-q to α-β

3.5. Implementation of FOC in Induction Motor Drive System

The main aspect of field oriented control method is the coordinate

transformation. Basic Field Oriented induction motor drive system is shown

in Fig. 3.18. The current vector is measured in stationary reference

frame α-β, where the components of currents isα and isβ must be transformed

to the rotating co-ordinate system d-q known as Park transformation.

Similarly the reference stator voltage vector components Vsα and Vsβ must be

transformed from the d-q system to α-β known as inverse transformation.

These transformations require a rotor flux angle ‘θ’.

Fig. 3.18 Field oriented Induction motor drive system

43

The transition from the stationary reference frame to the rotor rotating

reference frame requires the determination of position of rotor. The position

estimation can be done through sensorless control. In sensorless control, an

estimator block is needed. Two of the three phase currents are measured

because the sum of the three phase currents is equal to zero. This current is

fed to the Clarke transformation module, the output obtained from this

block is designated as isα and isβ. These two components of current act as the

input to the Park transformation block, which gives current in the rotating

reference frame. Calculation of the two components in the rotating reference

frame isd and isq is possible by finding the exact rotor flux angular position.

These components of currents are compared with the flux reference current

isd,ref and torque reference current isq,ref. The portability from asynchronous

to synchronous drive can be obtained by simply changing the flux reference

and determining the rotor flux position. The torque command isq,ref is

obtained from the speed regulator output. The output of the current

regulators are Vsd,ref and Vsq,ref, they are acting as the input to the inverse

Park transformation, where the conversion from d-q to α-β takes place. The

output of this projection gives the component of the stator vector voltage in

the α-β stationary reference frame as Vsα,ref and Vsβ,ref. The rotor flux position

is necessary for Park and Inverse Park transformations. Here space vector

modulation techniques are used, which is a sophisticated PWM method that

provides advantages such as higher DC bus voltage utilization and lower

total harmonic distortion.

3.6. Simulation Model of FOC Induction Motor Drive System

In this field oriented control simulation model, a 1.5 kW induction

motor is used, where three phase voltages are converted into two phase

reference frame voltages using Clarke transformation module. Park

transformation is used to obtain the voltages. From these voltages,

associated flux and current are calculated and then applied to

electromechanical torque equations to obtain torque speed responses. Fig.

3.19 shows the system configuration of IFOC induction machine with sensor

and the simulation is implemented using MATLAB/Simulink.

44

Fig. 3.19 System configuration of IFOC induction machine with sensor

3.7. Summary

In this chapter, review of the dynamic model of the induction motor in

space vector form and characteristic features of the FOC scheme were

presented. Mathematical transformations are carried out using Clarke and

Park transformations to decouple variables and to facilitate the solutions of

complicated equations with time varying coefficients. The simulation of the

FOC scheme is described and the simulation results are presented in the

next chapter along with the description of inverter.

field oriented control of induction...

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