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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 16

controlled conversion of electrical energy to a mechanical form and vice versa through gh a magnetic field. Electric drive is a multi-disciplinary disciplinary field of study requiring proper integration of knowledge of electrical machines, actuators, power electronic converters, sensors and instrumentation, control hardware and software and communication communication links as shown in Fig. 3.1. The drive system showing the same performance can be designed in various ways, like other engineering designs. High performance drive refers to the 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.

Fig. 3.1 Electric drive system

Several control strategies str as shown in Fig.3.2 are found in the variable variab 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) DTC. The controls, namely, FOC, DTC, nonlinear control and Predictive edictive Control (PC) are to be implemented with closedclosed loop feedback control to obtain high precision, good dynamics and steady state response. FOC predominantly relies on the mathematical modeling of 17

AC machine, while DTC makes direct use of physical interaction interaction that takes place within the integrated system of the machine and its supply.

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 techniques technique 18

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.

19

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: ur v v dψ a Va (t ) = Rs ia + dt

(3.1)

ur v v dψ b Vb (t ) = Rs ib + dt

(3.2)

20

ur v v dψ c Vc (t ) = Rs ic + dt

(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 21

The complex stator current vector ‘u’, which represents the three phase sinusoidal system, is represented as:

u=

2 ua (t ) + ub (t )e j 2π /3 + uc (t )e− j 2π /3 3

(

)

(3.4)

where, ଶ ଷ

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,

ua (t ) + ub (t ) + uc (t ) = 0

(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

22

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: ur v v dψ s Vs (t ) = Rs is + dt ur v v dψ r Vr (t ) = Rr ir + dt

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: 23

(3.6) (3.7)

uur

ur

ur

ψ s = Ls is + Lm ir e jε uur

ur

(3.8)

ur

ψ r = Lr ir + Lm is

(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, v v v v dis d (ir e jε ) Vs (t ) = Rs is + Ls + Lm dt dt v v v v d (is ) dir Vr (t ) = Rr ir + Lr + Lm dt dt v v v jε d (is ) dir jε e + Lm 0 = Rr ir e + Lr dt dt

(3.10) (3.11) (3.12)

Equation of the dynamic rotor rotating with an angular speed ωr , can be represented as: d ωr (Td (t ) − TL (t ) − Bωm ) = dt J

ωr =

dε dt

(3.13) (3.14)

Ls = Lm(1+σs)

where,

Lr = Lm(1+σr) In further consideration viscous coefficient will be negated as B=0. The electromagnetic torque is expressed by:

Td (t ) = J

v v d ωr 2P + TL (t ) = Lm Im  is (ir e jε ) * dt 32

(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: ur v v dψ sα Vsα (t ) = Rs isα + dt ur v dψ sβ v Vsβ (t ) = Rs isβ + dt 24

(3.16)

(3.17)

ur uuuv v dψ rα 0 = Rr irα + + ωr ψ r β dt ur uuuv dψ r β v 0 = Rr irβ + − ωrψ rα dt

(3.18)

(3.19)

 d ωr Td (t ) − TL (t ) 1  2 P Lm ψ sα isβ −ψ sβ isα − TL  = =  dt J J  3 2 Lr 

(

)

(3.20)

ψ sα = Ls isα + Lmirα

where,

ψ sβ = Lsisβ + Lmir β

ψ rα = Lr irα + Lmisα ψ r β = Lr ir β + Lmisβ The complex space vector is resolved into components of α and β.

r is = isα + jisβ

(3.21)

r ir = irα + jir β

(3.22)

uur Vs = Vsα + jVsβ

(3.23)

uur (3.24)

ψ s = ψ sα + jψ sβ uur

(3.25)

ψ r = ψ rα + jψ r β

In the stationary α-β coordinate system, the input to the motor is the stator voltage. The above equation is transformed into: ur v dψ sα v = Vsα (t ) − Rs isα dt

ur dψ sβ dt

(3.26)

v v = Vsβ (t ) − Rs isβ

(3.27)

ur v dψ rα = − Rr irα − pωrψ r β dt

ur dψ r β dt

(3.28)

v = − Rr ir β + pωrψ rα

(3.29)

 d ω r 1  2 P Lm ψ sα isβ −ψ sβ isα − TL  =  dt J  3 2 Lr 

(

25

)

(3.30)

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:

ωe =

dθ dt

(3.31)

Fig. 3.7 Vector diagram in stationary and rotating reference frame 26

The voltage and current complex space vector is resolved into components of d and q as:

v v − j γ −θ V s ( t ) e − jθ = V s e ( 1 ) = V sd + jV sq

(

v v − j γ −θ Rs is e − jθ = Rs is e ( ) = Rs isd + jisq

(

v − j ε −θ R r ir e ( ) = Rr ird + jirq

(

)

(3.32)

)

(3.33)

)

(3.34)

Induction motor model equation in d-q reference frame is written as follows: v v v v − jθ dis − jθ d (ir e jε ) − jθ − jθ Vs (t )e = Rs ie + Ls e + Lm e dt dt v v v j (ε −θ ) d (is e − jε ) j (ε −θ ) dir j (ε −θ ) + Lr + Lm 0 = Rr ir e e e dt dt  d ω r Td (t ) − TL (t ) 1  2 P Lm ψ rd isq − ψ rq isd − TL  = =  dt J J  3 2 Lr 

(

)

(3.35)

(3.36) (3.37)

The stator flux linkages are given by:

ψ sq = Lsisq + Lmirq

(3.38)

ψ sd = Ls isd + Lm ird

(3.39)

(ψ

(3.40)

ψˆ s =

2 sd

+ ψ sq2 )

The rotor flux linkages are:

ψ rq = 0 = Lr irq + Lmisq

(3.41)

ψ rd = ψ r = Lr ird + Lm isd

(3.42)

+ ψ rq2 )

(3.43)

ψ mq = Lm isq + Lm irq

(3.44)

ψ md = Ls isd + Lmird

(3.45)

(ψ

(3.46)

ψˆ r =

(ψ

2 rd

The airgap flux linkages are:

ψˆ m =

2 md

2 + ψ mq )

27

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. Td =

2 P Lm ψ rd isq −ψ rq isd 3 2 Lr

(

)

(3.47)

 d ωr 1  2 P Lm ψ rd isq −ψ rq isd − TL  =  dt J  3 2 Lr 

(

)

(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:

v di d Vsd = Rs isd + Ls sd + Lm ird − Lsωeisq − Lmωeirq dt dt

(3.49)

v disq d Vsq = Rs isq + Ls + Lm irq + Lsωeisd + Lmωeird dt dt

(3.50)

0 = Rr ird + Lr

0 = Rr irq + Lr

dird d + Lm isd − Lr (ωe − ωr ) irq + Lm (ωe − ωr ) isq dt dt dirq dt

+ Lm

d isq + Lr (ωe − ω r ) ird + Lm (ωe − ωr ) isd dt

(a)

(b)

Fig. 3.8 Dynamic equivalent circuit (a) d-axis circuit (b) q-axis circuit 28

(3.51)

(3.52)

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

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

ψ sd ωe = ( Lsisd + Lmird ) ωe

(3.53)

ψ sqωe = ( Ls isq + Lmirq ) ωe

(3.54)

ψ rd (ωe − ωr ) = ( Lr ird + Lmisd )(ωe − ωr )

(3.55)

ψ rq (ωe − ωr ) = ( Lr irq + Lmisq ) (ωe − ωr )

(3.56)

The machine dynamic voltage in matrix form is represented as: Vsd   Rs + pLs Vsq   ωe Ls  =  0   pLm     0  (ωe − ω r ) Lm

−ωe Ls

pLm

Rs + pLs

Lmωe

− (ωe − ωr ) Lm pLm

Rr + pLr (ωe − ωr ) Lr

 isd   i  pLm   sq  − (ωe − ωr ) Lr  ird    Rr + pLr   irq  − Lmωe

(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: 29

Td = kI f I a

where,

(3.58)

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. 30

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: Td = k ' I sq I sd

(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 31

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,

ψ rq = 0

(3.60)

ψ rd = ψ r

(3.61)

Fig. 3.12 Field orientation in d-q reference frame

dψ rq dt

=0

ψ rq = Lr irq + Lm isq = 0

(3.62) (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:

Lr = lr + Lm = σ r Lm + Lm = Lm (1 + σ r )

ψ rd = Lr ird + Lmisd = Lm (1 + σ r ) ird + Lmisd = Lmimr 32

(3.64) (3.65)

ird =

imr − isd (1 + σ r )

(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:

dψ rd + Rr ird = 0 dt

τr

where,

(3.67)

dimr + imr = isd dt

τr = imr =

(3.68)

Lr Rr

isd (1 + sτ r )

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:

ωr = ωe − ωslip 33

(3.69)

d ψ rq + (ωe − ωr )ψ rd + Rr irq = 0 dt

(3.70)

d ψ rd + (ωe − ωr )ψ rq + Rr ird = 0 dt

(3.71)

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

ψ rq

irq =

Lr

ird =

ψ rd Lr

−

Lm isq Lr

(3.72)

−

Lm isd Lr

(3.73)

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

L R d ψ rq + (ωe − ωr )ψ rd + r ψ rq − m Rr isq = 0 dt Lr Lr

(3.74)

L R d ψ rd + (ωe − ωr )ψ rq + r ψ rd − m Rr isd = 0 dt Lr Lr

(3.75)

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



ψ sq =  Ls − 



ψ sd =  Ls − 

Lm2  Lm  isq + ψ rq Lr  Lr

(3.76)

Lm2  Lm  isd + ψ rd Lr  Lr

(3.77)

Slip speed is calculated based on the following two equations:

ψ rd (ωe − ωr ) + Rr irq = 0

(3.78) (3.79)

ψ rq = 0

To determine the rotor flux angle, first we need to calculate the slip using the following equation:

ωslip =

Lm isq

τ r ψ rd

=

1 isq τ r imr

ωe = ωr + ωslip = ωr + Then rotor flux angle can be calculated as: 34

1 isq τ r imr

(3.80)

(3.81)

θ = ∫ ωe dt

J

(3.82)

d ωr 2 P Lm = imr isq − TL dt 3 2 (1 + σ r )

(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: Td =

2 P Lm imr isq 3 2 (1 + σ r )

(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 35

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 36

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 37

component of the current components. The flux along the q axis must be zero and the mathematical constraint is,

ψ rq = 0

(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 nonrotating 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

38

projection

(Clarke

and

Park

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 socalled 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  isα = Re  ia (t ) + ib (t )e j 2π /3 + ic (t )e− j 2π /3  3 

(

)

39

(3.86)

2  isβ = Im  ia (t ) + ib (t )e j 2π /3 + ic (t )e− j 2π /3  3 

(

)

(3.87)

Fig. 3.15 Stator current space vector in stationary reference frame

  isα  2 1 i  =   sβ  3 0 

−1

2

3 2

−1

2 − 3 2

 ia   i   b   ic 

(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 40

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. r i s = isd + jisq

 isd  i  =  sq 

 cos θ  − sin θ 

sin θ   isα    cos θ   is β 

(3.89)

(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. Vsα   cos θ V  =   s β   sin θ

− sin θ  Vsd    cos θ  Vsq  41

(3.91)

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 42

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.

43

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.

44