Hetdocumentac Motor Drives

1. Dynamic Model of AC Machines

1.1 Introduction

Although traditional per-phase equivalent circuits have been widely used in steady-state analysis and design of ac machines, it is not appropriate to predict the dynamic performance of the motor. In order to understand and analyze vector control of ac motor drives, a dynamic model is necessary. As the application of ac machines has continued to increase over this century, new techniques have been developed to aid in their analysis. The significant breakthrough in the analysis of three-phase ac machines was the development of the reference frame theory. Using these techniques, it is possible to transform the machine model to another reference frame. By judicious choice of the reference frame, it proves possible to simplify vastly the complexity of the mathematical machine model. While these techniques were initially developed for the analysis and simulation of ac machines, they are now invaluable tools in the digital control of such machines. As digital control techniques are extended to control current, torque and flux of ac machines, the need for compact and accurate machine models is obvious. Much of the analysis has been carried out for the treatment of the well-known induction machine. The induction motor, which is the most widely used motor type in industry, has been favored because of its good self-starting capability, simple and rugged structure, low cost and reliability, etc. Along with variable frequency ac inverters, induction motors are used in many adjustable speed applications, which do not require fast dynamic response. The concept of vector control has opened up a new possibility that induction motors can be controlled to achieve dynamic performance at least as good as that of dc motors. In order to understand and analyze vector control, the dynamic model of the induction motor is necessary. It has been found that the dynamic model equations developed on a rotating reference frame is easier to describe the characteristics of induction motors. Fortunately, the developed theory of reference frames is equally applicable to synchronous machines, e.g. the Permanent Magnet Synchronous Machine (PMSM). The PMSM, sometimes known as sinusoidal brushless machines or brushless ac machine, is very popular as a high-performance servo drive due to its superior torque-to weight ratio and its high dynamic capability.

2 Chapter 1

Over the years, many different reference frames have been proposed for the analysis of ac machines. The most commonly used are the so-called stationary reference frame, the rotor reference frame for synchronous machines and the synchronously rotating reference frame for induction machines. Considering synchronous machines, the rotor reference frame is equal to the synchronous reference frame, since the supply frequency equals the electrical rotor speed. The aim of this chapter is to introduce the essential concepts of reference frame theory and to derive and explain ac machine models in relatively simple terms. It will be shown that when we choose a synchronous reference frame in which rotor flux lies on the d-axis, the dynamic equations of both induction and synchronous machines are simplified and analogous to a dc motor.

1.2 Transformation between Reference Frames

A clear and comprehensive description of the dynamic behavior of ac machines is a key point for their application in speed or torque controlled drive systems. Steady-state analysis of induction machines or PMSM are typically presented by means of the so called per phase equivalent circuit. However, these models are inadequate when applied to dynamic conditions, as needed in variable frequency drives. The pertaining methods of dynamic analysis concerning field-oriented control have been developed decades ago. Today, they form part of the fundamentals in electrical engineering being documented in numerous publications and books, e.g. [Bose 97], [Hen 92], [Leo 85], [Vas 92]. In the original three phase reference frame the motor windings (a,b,c) are displaced at 120 deg (electrical) in space and both current and voltage equations are time-dependent variables. The idea of simplifying the motor equations by transforming the stator machine variables to the rotor reference frame was first introduced in 1899 by Blondel [Vas 92]. In the late 1920’s, R.H. Park further developed the idea to transform the stator machine variables as currents, voltages and flux linkages to another rotating reference frame [Park 29] described by 2 rotating phase variables, which are perpendicular. This technique was first applied to a synchronous machine transforming the stator variables to a fictitious frame rotating with the rotor speed. The method was extended by [Kron 42] to be applicable to any type of ac machine. Considering squirrel-cage induction motor or PMSM, the transformations are usually based on following assumptions:

• Space harmonics of the flux linkage distribution are neglected. • Slot harmonics and deep bar effects are not considered. • Iron losses are not taken into account. • Saturation is introduced on a macroscopic basis. • Permanent magnets behave linearly. • Neutral point is isolated.

Dynamic Model of AC Machines 3

The assumption, that ac machines are linear (no saturation) and MMF-harmonic free is an oversimplification, which cannot describe the behavior of the machine in all operation modes. However, in a majority of applications, the machine behavior can be adequately predicted with this simplified representation. Considering the design of practical machines, the rotor conductors of squirrel-cage induction machines are often skewed. The conductors are not placed in the plane parallel to the axis of rotation, but skewed slightly with the axis of rotation. This arrangement helps to reduce the magnitude of torque harmonics (ripples) due to the harmonic content of the MMF-waves. Furthermore, the applied transformations are quite general, since the flux and MMF-waves may be considered as a sum of a fundamental and higher harmonics. The restriction to a magnetic linear system without saturation may be dropped, if the stator and rotor voltage equations are written in terms of the flux linkages. The transformation of flux linkages is valid even for asymmetrical or magnetic nonlinear systems and an extensive amount of work can be avoided by transforming the flux linkages directly. This is especially true in the analysis of ac machines where the inductances are function of the rotor position. Only the current-flux relation, represented by affiliated inductances, must be adapted in the case of asymmetry or non-linearity. Also iron losses are more or less considered. Due to the braking effect of iron losses, they are subsequently treated as an equivalent speed-dependent load torque.

1.3 Transformation to the α/β-reference frame

The applied approach eliminates the redundancy of poly-phase windings substituting these by their two-axes equivalent. Such a redundancy is given for both line currents and line voltages of three-phase ac machines. By definition, a three-phase machine with isolated neutral point (figure 1.1) restricts its phase currents to a plane constructed by two perpendicular static phasors, since the sum of all phase currents has to be zero at all times according to Kirchoff’s law.

0=++ cba iii (1.1)

0=++ cabcab uuu (1.2)

inverter ac motor

uab

ubc uca

ia

ib

ic

Udc

ua0

Not connected!reference-potential ϕo

Figure 1.1: Three-phase machine with isolated neutral point.

4 Chapter 1

According to (1.1)-(1.2), the input of three-phase ac motors is completely described by two line voltages and two currents since the third value equals the negative sum of the two others. Thus, describing the dynamic behavior of ac machines, the poly-phase motor windings can be reduced to a set of two-phase windings. The most suitable choice, describing the motor behavior by a two-axes equivalent, is a set of field coils, having their magnetic axes arranged in quadrature as shown in figure 1.2. This approach eliminates the mutual magnetic coupling of the three-phase windings, since the flux of one winding of the two-axes equivalent does not interact with the perpendicular winding and vice versa. The simplified diagram of a three-phase ac motor (figure 1.2) shows only the stator windings for each phase displaced 2π/3 radians in space. Of course, the same transformation is applied to the rotor. The transformation of rotor and stator variables to a 2-phase reference frame (indicated by subscripts α, β), where the coils are perpendicular, guarantees that there is no interaction between perpendicular windings as long as there is no saturation, referred to as cross saturation.

jβ

α ia

ic

ib

jβ

α iα

iβ

a

a' c

c'

b

b'

α

β

a

c

b

Figure 1.2: Three-phase windings and two-axes equivalent (α/β transformation). Due to the redundancy in (1.1)-(1.2), it is obviously that the voltage and current equations of any ac machine are reducible to a set of each two appropriate variables in the α/β-reference frame. One straightforward way, but unusual option of transferring variables to the α/β-reference frame, is a simple vector addition (subscript VA) of the three-phase variables. According to figure 1.2, a geometrical calculation yields:

[ VAcba iiii ,)240cos()120cos( α] =°+°+ (1.3)


Solving the trigonometric functions and replacing ic according to (1.1) results in:

VAbab

a iiiii ,2)(

2 α=−−

−−⇒ (1.4)

aVA ii23

, =⇒ α (not used!) (1.5)

Similarly for the β-axis:

[ VAcb iii ,)150cos()30cos( β] =°+° (1.6)

VAbab iiii ,)(23

23

β=−−−⇒ (1.7)

baVA iii 323

, +=⇒ β (not used!) (1.8)

However, above transformation, equally applied on both phase current and voltages, is not power invariant:

VAVAVAVAccbbaa iuiuiuiuiutP ,,,,)( ββαα +≠++= (1.9)

[ ]

[ ] )(23

23

)()(23

23

2333

3233

23

23

23

,,,,

tPiuiuiu

iiuuiuiu

iuiuiuiu

iiuuiuiuiu

ccbbaa

bababbaa

abbabbaa

babaaaVAVAVAVA

=++=

−−−−++=

+++=

+

++=+ ββαα

(1.10)

Note, the variables in the α/β-reference frame are no real values, they are fictitious values, preferably describing the real electrical and dynamic behavior of the machine. Obviously, the original current phasor can be equally described by two other perpendicular vector components multiplied by a constant scaling factor. According to (1.10), the power in the α/β-reference frame reduced by the scaling factor 2/3 guarantees power invariance. At first sight, this scaling factor can be distributed on the α/β-current and voltages in different ways, e.g. the α/β-current remains and the voltages uα, uβ are reduced by the factor 2/3 (⇒ uα = ua ) or vice versa (⇒ iα = ia ). However, the air-gap flux is proportional to the voltage whereas the magneto-motive force (MMF) is proportional to the current. Thus, only a symmetrical distribution, both α/β-current and voltages are reduced by the same

6 Chapter 1

factor, yields, compared to the original system, identical magnetic characteristics in the α/β-reference frame. Postulating symmetrical distribution of the scaling factor 2/3, following equations are used when quantities in the three-phase reference frame are transformed to the α/β-reference frame [Hen 92]. For reasons of simplicity, the coil axis ia coincides with the coil axis iα, perpendicular to iβ (figure 1.2):

[ αiiii cba =°+°+ )240cos()120cos(32 ] (1.11)

αiiiiii abab

a ==

−−

−−⇒23

32

2)(

232 (1.12)

aii23

=⇒ α (1.13)

Similarly for the β-axis:

[ βiii cb =°+° )150cos()30cos(32 ] (1.14)

βiiii bab =

−−−⇒ )(

23

23

32 (1.15)

ba iii 221

+=⇒ β (1.16)

In matrix form, the transformation to the α/β-reference frame is:

=

=

b

a

b

a

ii

ii

ii

22

1

023

32Tβ

α (1.17)

The inverse transformations, calculating the phase currents by α/β-values, are as:

−=

=

−

β

α

β

α

ii

ii

ii

b

a

21

61

032

132T (1.18)


Equally, the voltage and flux distribution of a three-phase ac machine behaves as phasors, identically described by two perpendicular vector components. Thus, the transformation of voltage and flux linkages to the α/β-reference frame is equivalent to the current transformation.

=

=

b

a

b

a

uu

uu

uu

22

1

023

32Tβ

α Condition: (uc = - ua - ub) (1.19)

−=

=

−

β

α

β

α

uu

uu

uu

b

a

21

61

032

132T Condition: (uc = - ua - ub) (1.20)

Obviously, the relation of the transformation matrices is:

=

−

=−

1001

21

61

032

22

1

023

13232 TT (1.21)

Unfortunately, while the sum of the line voltages is zero, the phase voltages or phase-to-neutral voltages may consist of a homopolar component dependent on the PWM method used. The origin of the homopolar component is exploited later in the inverter-analysis presented. However, the effect of the homopolar component on a machine with isolated neutral point is equal to a change of the reference potential (e.g.: changing the potential ϕ0 in figure 1.1). In other words, the potential of all three phases is varied uniformly and simultaneously (third harmonics). Fortunately, the sum of the line voltages is always zero. Thus, this homopolar component is unseen by the motor terminals and therefore not reflected in motor behavior. However, since motor voltages are often calculated by means of the reference voltages (e.g. ua0 in figure 1.1) containing such a homopolar component, the transformation according to (1.25) must be used for the α/β-voltage calculation. The transformation matrix is obtained similarly to (1.11)-(1.16) by geometrical calculations without replacing the phase voltage uc. For completeness, the zero-component is also given. According to figure 1.3, the zero-component indicates a displacement of the reference potential ϕ0 with respect to the center m of the line voltages. Obviously, the phase voltages (ua, ub, uc) are equal to the affiliated line-to-neutral voltages (uam, ubm, ucm) plus the zero-component um0. Since the sum of the line-to-neutral voltages is zero, following relation is valid:

8 Chapter 1

0)()()(!

000 =−+−+−=++ mcmbmacmbmam uuuuuuuuu (1.22)

⇒ )(31

0 cbam uuu ++=u (1.23)

um0

uc

ua

ub

m

b c

a

uam

ucmubm

ϕ0

m

b c

a

ua

uc

uab

ub ubc

uca

Figure 1.3: Zero-component of the phase voltages. Left: Phase and line voltages without zero component. Right: Phase voltages and displacement of the neutral point.

Scaling by the power-invariance factor yields the zero-component in the α/β-reference frame:

)(21

32)(

31

23

0 cbacba uuuuuuu ++=++= (1.24)

⇒

−

−−

=

=

c

b

a

c

b

a

uuu

uuu

uuu

21

21

21

23

230

21

211

32

0

320Tβ

α

(1.25)

The inverse transformation is:

−−

−=

=

−

001

23

21

123

21

101

32

uuu

uuu

uuu

c

b

a

β

α

β

α1

320T (1.26)

However, the zero-component is required neither for motor control nor for describing the dynamic motor behavior. Usually in ac motor control, only equation (1.17) is applied for current transformation. Most inverter control strategies


and PWM techniques circumvent an additional voltage back transformation, e.g. space vector modulation directly specifies the switching signals using the α/β-voltages. Sometimes, the back-transformation (1.20) is applied for the calculation of the reference voltages, e.g. being the input of a sinusoidal PWM. Special applications, e.g. phase voltage estimation by means of the reference voltages, require the transformation (1.25) due to the non-linearity of the inverter and arising homopolar component. If line voltages are given (e.g. measurement), the α/β-transformation is:

=

bc

ab

uu

uu

210

61

32

β

α (1.27)

According to (1.2), a homopolar component in phase voltages vanishes in the line voltages of any ac machine with isolated neutral point.

1.3.1 Properties of the power invariant α/β-transformation

Note that α/β transformations with different scaling factors are used in literature as well: e.g. the non power-invariant form [Vas 97], also termed Clark transformation. Here, all transformations are based on (1.17)-(1.27) guarantying power invariance:

abbabbaa

bababbaaccbbaael

iuiuiuiuiiuuiuiuiuiuiuP

+++=−−−−++=++=

22)()( (1.28)

Replacing the phase quantities by the affiliated α/β-quantities yields:

αβααβααα

αββαββαααα

αβαβαα

βαβααα

iuiuiuiu

iuiuiuiuiu

iuuiiu

iiuuiuPel

31

31

31

31

62

62

62

34

32

21

61

21

61

32

21

61

21

612

32

322

+−+−

−−++=

+

−+

+

−+

+

−

+

−+=

(1.29)

⇒ ββαα iuiuPel += (1.30)

10 Chapter 1

The α/β-transformation eliminates the redundancy of poly-phase windings substituting these by their two-axes equivalent. The current and voltage quantities are each described by two components in direction of the α-axis and β-axis, respectively. According to (1.13), the relation of the mean values is:

phasephase II 32 23

= (1.31)

phasephase UU 32 23

= (1.32)

However, describing the electrical behavior of the ac machine by two perpendicular windings requires an adaptation of the fictitious two-phase winding number (figure 1.4), since the resulting flux quantities and magneto-motive force (MMF) must be identical to the original values. According to [Bel 92], [Hen 91], the MMF (as well as the magnetic induction B) is proportional to the current, the phase number m and the coil windings w scaled down by the winding factor ξ due to the spreading of the windings and the shift of a two layer winding. Identical MMF, generated by three respectively two coils, is given, if following relation is valid:

phase

phase

I

I

22phase2phase

2phase

!

33phase3phase

3phase

2p

w)(ξπ2

m4

2p

w)(ξπ2

m4

=

Θ==Θ (1.33)

⇒ 23

32

23

mm

w)(ξw)(ξ

22phase

33phase

3phase

2phase ===phase

phase

II

(1.34)

According to [Hen 91], the air-gap flux is proportional to the induced voltage and power supply frequency ωs. Obviously, applying identical (symmetrical) transformations on current and voltage quantities yields the same relation between the coil windings as in (1.34):

s

phase

s

phase UUωω 2phase

22phase

!

3phase

33phase w)(ξ

ˆ2w)(ξ2

=Φ==Φ (1.35)

⇒ 23

w)(ξw)(ξ

3

2

3phase

2phase ==phase

phase

UU

(1.36)

Thus, a transformation with the coil ratio according (1.34) yields identical flux quantities and identical MMF. Also the current density in the α/β-system is identical to the original value:


CuqIJ = (qCu = conductor cross-section) (1.37)

The conductor cross-section qCu expressed in terms of the total electrically effective copper cross-section ACu yields:

mw2Aq Cu

Cu = (1.38)

⇒ CuA

m2wIJ = (1.39)

⇒ 132

23

23

mm

ww

3phase

2phase

3phase

2phase

3

2

3

2 ===phase

phase

phase

phase

II

JJ

(1.40)

Accordingly, the magnetic quantities of a three-phase ac machine are identically and the electrical quantities are equally described by their two-axes equivalent as shown in figure 1.4.

ia

a

c

b iα

-iα

iβ -iβ

α

β

coil ratio:

-ia

ic ib

-ic -ib 23

ξ)(wξ)(w

3phase1

2phase1 =

Figure 1.4: Three-phase windings and two-axes equivalent. Additionally, the resistive losses and the stored magnetic energy in the α/β-reference frame are equal to the values of the original system:

22

2

223, 2

3233 phasesphasesphaseslossesR IRIRIRP =

== (1.41)

To fulfill (1.41), the resistance and inductance values in the two-phase motor model must remain identical. The stator resistance is proportional to the averaged conductor length lm, number of windings w and the conductor cross-section qCu:

12 Chapter 1

Cu

m

qlwρ=sR (1.42)

Applying (1.38) yields:

Cu

m2

Alwm2ρ=sR (1.43)

The inductances are proportional to the phase number m, the number of windings w scaled down by the winding factor ξ [Hen 91]:

pπδlτ4µ

w)(ξ2m

2p02=L (1.44)

Postulating equal resistance and inductance values in the original (m3phase = 3) and in the α/β-reference frame (m2phase = 2) yields the same coil ratio as in (1.36):

23phase3phase

22phase2phase w)(ξmw)(ξm = (1.45)

⇒ 23

mm

w)(ξw)(ξ

2phase

3phase

3phase

2phase == (1.46)

More straightforward it can be argued, that all impedances, proportional to the affiliated voltage drop divided by the current, remain unchanged, since the transformation of current and voltage quantities is equivalent. Therefore, the current-voltage relation stays constant:

phasephasex

phasex

phasex

phasexphase x

IU

IU

x 33,

3,!

2,

2,2 === (1.47)

Thus, a transformation with the coil ratio 23

ξ)(wξ)(w

3phase

2phase = is:

Symmetrical Power invariant Inductance and resistance invariant Flux, magnetic field and induction, MMF and current density are identical

Note: The flux linkages are coupled to the magnetic induction/field distribution via the winding number of the three-phase machine, respectively two-phase equivalent.


Thus, the transformation of the flux linkages contains the coil ration (1.46). Accordingly, the transformation of flux linkages is equivalent to the transformation of current and voltage quantities.

1.3.2 Voltages and flux linkages in the α/β-reference frame

A general three-phase ac machine is completely described by the stator and rotor voltage equations with Rs and Rr representing the resistance of stator and (stator related) rotor windings, respectively:

+

=

sc

sb

sa

sc

sb

sa

s

sc

sb

sa

dtd

iii

Ruuu

ψψψ

(1.48)

+

=

rc

rb

ra

rc

rb

ra

r

rc

rb

ra

dtd

iii

Ruuu

ψψψ

(1.49)

In the case of synchronous machines, the rotor equation (1.49) vanishes. In the case of induction machines, the phase voltages of the rotor are equal to zero and the quantities in (1.49) are related to the real rotor quantities via the winding ratio of stator and rotor [Hen 91]:

rotor

stator, ξ)(w

ξ)(wrealrr ii = (1.50)

rotor

stator, ξ)(w

ξ)(wrealrr uu = (1.51)

2

rotor

stator, ξ)(w

ξ)(w

= realrr RR (1.52)

A majority of induction machines is not equipped with coil-wound rotor windings. Instead, the current flows in copper or aluminum bars, uniformly distributed and embedded in a ferromagnetic material, with all bars terminated in a common ring at each end of the rotor. This type of rotor is referred to as a squirrel-cage rotor. However, the uniformly distributed rotor winding is adequately described by its fundamental sinusoidal component and is represented by an equivalent 3-phase winding [Hen 91].

14 Chapter 1

According to the introduced α/β-transformation, the electrical behavior of a three-phase ac machine is equally described by their two-phase equivalent (figure 1.5). Applying the transformation matrix T32 on (1.48)-(1.49) yields:

+

=

β

α

β

α

β

α

ψψ

s

s

s

ss

s

s

dtd

ii

Ruu

(1.53)

+

=

β

α

β

α

β

α

ψψ

r

r

r

rr

r

r

dtd

ii

Ruu

(1.54)

a

usb, isb usc, isc

usa, isa

b c

θ = ∫ωr dt

ura, ira urc, irc

urb, irb

jβs

αs

usα, isα

usβ, isβ

urα, irα

urβ, irβ

θ = ∫ωr dt

αr

jβr

coil ratio:

23

ξ)(wξ)(w

3phase1

2phase1 =

Stator

Rotor

Figure 1.5: Three-phase windings and two-axes equivalent. The α/β representation eliminates the mutual magnetic coupling and the redundancy of the poly-phase windings, but the time-dependence of both current and voltage equations still remains. Furthermore, the equation set (1.53)-(1.54) is based on two different reference frames, one stationary for the stator and another for the rotor rotating at electrical motor speed ωr. In steady state, the stator current/voltage frequency equals the supply frequency, the frequency of the rotor windings equals the supply frequency minus electrical rotor speed ωr. According to figure 1.6, the mutual magnetic coupling of rotor and stator depends on the current position of the rotor reference frame. The rotating flux phasor, generated by the rotor currents, can be divided in a flux vector parallel and perpendicular to the α-axis of the stator. Obviously, only the parallel flux is magnetically linked to the affiliated stator winding and vice versa. For reasons of simplicity, the main inductance L1h is assumed to be constant, i.e. there is no rotor or stator asymmetry. This restriction will be dropped in a subsequent paragraph. Applying the mentioned simplification, the stator and rotor flux linkages in the α/β-reference frame are obtained by simple vector addition according to figure 1.6:


)sincos(1 θθψ βααα rrhsss iiLiL −+= (1.55)

)cossin(1 θθψ βαββ rrhsss iiLiL ++= (1.56)

)sincos(1 θθψ βααα sshrrr iiLiL ++= (1.57)

)cossin(1 θθψ βαββ sshrrr iiLiL +−+= (1.58)

jβrαr

irα

irβ

jβs

αsisα

isβ θ = ∫ωr dt L1h(θ)

irβ i rβ co

s θ

θ

irα

irα cosθ

i rα si

n θ

i sβ

isα isβ cosθ

isβ sinθ isα cosθ

-isα sinθ θ

L1h(θ)

-irβ sinθ

Figure 1.6: General two-axes ac machine model and mutual magnetic coupling between stator and rotor. Top: Stator and rotor windings. Bottom: Geometrical calculations.

Defining the matrix

−=− θθ

θθcossinsincos

θT (1.59)

with the inverse matrix

−

=−− θθ

θθcossinsincos1

θT (1.60)

yields the flux linkages in matrix form:

+

=

−

β

α

β

α

β

α

ψψ

r

rh

s

ss

s

s

ii

Lii

L θT1 (1.61)

16 Chapter 1

+

=

−−

β

α

β

α

β

α

ψψ

s

sh

r

rr

r

r

ii

Lii

L 1θT1 (1.62)

Annotation: The negative sign in T-θ is introduced to guarantee symbol analogy with following transfer matrixes. Furthermore, the rotor reference frame rotates at first sight away from the stator reference frame, whereas the matrix T-θ transfers the rotor currents contrary to the direction of rotation, i.e. back to the stationary stator reference frame.

1.4 Rotating reference frames

Traditionally in analysis and design of ac motor drives, a "per-phase equivalent circuit" has been widely used. Note that in this equivalent circuit, all motor parameters and variables are not actual quantities but are quantities referred to the stator. Although the per-phase equivalent circuit is useful in analyzing and predicting steady-state performance, it is not appropriate to predict the dynamic performance of ac motors. In order to understand and analyze vector control, the dynamic model of the motor is necessary. As already mentioned, the equation set (1.53)-(1.54) is based on two different reference frames, one stationary for the stator and another for the rotor rotating at electrical motor speed ωr. At the transformation to the α/β-reference frame, the α-axis of the stator reference frame was aligned to the first motor phase. Just as well, the α-axis aligned to the second or third phase would have been a suitable choice. Generally, the electrical behavior of the ac machine can be equally described by two other perpendicular windings, being, with respect to the original α/β-system, spatially rotated by the arbitrary angle γ. Naturally, the arbitrary angle γ may be also time-dependent, e.g. itself rotating at the electrical speed ωr. Similarly to the "per-phase equivalent circuit", the theory of the rotating reference frames is applied to transfer the stator and rotor variables and equations to one common reference frame. It has been found that the dynamic model equations developed on one common reference frame is easier to describe the dynamic behavior of ac motors. The choice of the common reference frame is the key element that distinguishes the various vector control approaches from each other. It is instructive to take a preliminary look at the reference frames commonly used in analysis of electrical machines and power system components. Information regarding each of these reference frames, namely the stationary, rotor, synchronous and arbitrary reference frame, is summarized in table 1.1.


Table 1.1: Commonly used reference frames.

Reference frame speed Interpretation Notation Application

0 Variables referred to the stationary reference frame

d/q subscripts and 's' superscript

DTC and steady-state analysis

ωr Variables referred to a reference

frame fixed to the rotor d/q subscripts

and 'r' superscript Vector control of

synchronous machines

ωγ Variables referred to an arbitrary

rotating reference frame d/q subscripts

and , superscript General reference frame

ωµ Synchronously rotating reference

frame d/q subscripts no superscripts

Vector control of induction machines

Although the transformation to the arbitrary reference frame involves all other transformations as a special case, it is didactically preferable to consider at a first step only the transformation to the stationary reference frame and then modify this analysis to other reference frames.

1.4.1 Stator fixed reference frame

In principal, the transformation to the stationary reference frame has already been introduced. In (1.61), the rotor currents are transformed via matrix T-θ to current components in α/β-direction of the stationary stator reference frame. Expressing the rotor current phasor ir in terms of the stationary α/β-components is based on the transformation shown in figure 1.7. The axes of the stationary α/β-components are subsequently denoted as d-and q-axis to indicate a common reference frame of stator and rotor. The superscript 's' indicates the stator-fixed reference frame. The rotor currents expressed in components of the stator-fixed reference frame are:

θθ βα sincos rrsrd iii −= (1.63)

θθ βα cossin rrsrq iii += (1.64)

With the transformation matrix T-θ and its inverse T-

-1θ

−=− θθ

θθcossinsincos

θT (1.65)

−

=−− θθ

θθcossinsincos1

θT ( (1.66) ):hint θ1θ TT =−

−

equations (1.63)-(1.64) can be rewritten in matrix form. Back-transformation with the inverse matrix yields the rotor current in α/β-components:

18 Chapter 1

=

−

β

α

r

rsrq

srd

ii

ii

θT (1.67)

=

−− s

rq

srd

r

r

ii

ii 1

θTβ

α (1.68)

irβ i rβ co

s θ

θ

irα

irα cosθ

i rα si

n θ -irβ sinθ

θ

irα

isrd= irα cosθ - irβ sinθ

is rq=

i rα si

n θ +

i rβ

cos θ

ir irβ

d

q

Figure 1.7: Transformation of the rotor currents to the stationary reference frame.

Equally, the rotor voltage phasor and the rotor flux phasor are expressed as a vector sum of components in the stationary stator reference frame:

=

−

β

α

r

rsrq

srd

uu

uu

θT (1.69)

=

−

β

α

ψψ

ψψ

r

rsrq

srd

θT (1.70)

The rotor voltage equation is obtained by multiplying the voltage equation (1.54) with T-θ:

+

=

+

=

−−−−

β

α

β

α

β

α

β

α

β

α

ψψ

ψψ

r

r

r

rr

r

r

r

rr

r

r

dtd

ii

Rdtd

ii

Ruu

θθθθ TTTT (1.71)

Due to the relation,

=

++

=

−

−

=−−−

1001

sincos00sincos

cossinsincos

cossinsincos

22

22

θθθθ

θθθθ

θθθθ

θ1θ TT

(1.72)


the rotor flux vector multiplied with T-θ

-1 T-θ remains unchanged. Using (1.72) in (1.71) and applying the partial derivative yields:

+

+

=

+

=

+

=

−−−

−−−

−−−

−−−−

srq

srd

srq

srd

srq

srd

r

srq

srd

srq

srd

r

r

rsrq

srd

rsrq

srd

dtd

dtd

ii

R

dtd

ii

R

dtd

ii

Ruu

ψψ

ψψ

ψψ

ψψ

β

α

1θθ

1θθ

1θθ

θ1θθ

TTTT

TT

TTT

(1.73)

With applying the partial derivative of the transfer matrix T-θ,

rdtd

dtd

dtd

dtd

dd

dtd

ωθθθ

θθ

θθθ

θθθθθθ

θθθ

θθθθ

−

=

−−+

=

−−

−

−=

−−

−=

=

−−−−

−−−

0110

0sincossincos0

sincoscossin

cossinsincos

sincoscossin

22

22

θ1θθ

1θθ TTTTT

(1.74)

the transformed rotor voltage equation consists of three parts, namely the resistive voltage drop, the time-dependent flux variation and the induced voltage due to the rotation with the electrical motor speed ωr.

−

+

+

=

srq

srd

rsrq

srd

srq

srd

rsrq

srd

dtd

ii

Ruu

ψψ

ωψψ

0110

1001

(1.75)

Since the used reference frame is equal to the original α/β-reference frame of the stator, all stator variables and quantities remain unchanged. Only the notation is adapted to guarantee symbol analogy. Summarizing, following voltage equations are valid in the common stationary reference frame:

+

=

ssq

ssd

ssq

ssd

sssq

ssd

dtd

ii

Ruu

ψψ

(1.76)

−+

+

=

srd

srq

rsrq

srd

srq

srd

rsrq

srd

dtd

ii

Ruu

ψψ

ωψψ

(1.77)

20 Chapter 1

According to (1.61)-(1.62), the flux linkage can be also expressed as:

+

=

+

=

=

− s

rq

srd

hssq

ssd

sr

rh

s

ss

s

sssq

ssd

ii

Lii

Lii

Lii

L 11β

α

β

α

β

α

ψψ

ψψ

θT (1.78)

+

=

+

=

+

=

=

−−−−

−−−−

ssq

ssd

hsrq

srd

r

s

sh

r

rr

s

sh

r

rr

r

rsrq

srd

ii

Lii

L

ii

Lii

L

ii

Lii

L

1

1

1

β

α

β

α

β

α

β

α

β

α

ψψ

ψψ

1θθθ

1θθθ

TTT

TTT

(1.79)

With respect to (1.78)-(1.79), the stator and rotor currents in the d-axis are only active in the affiliated windings of the d-axis. The same applies for the q-axis. Thus, the mutual magnetic coupling between d- and q-axes is eliminated. The voltage equations in the stationary reference frame are easily adapted to the well known "per-phase equivalent circuit", especially useful in analyzing and predicting steady-state performance.

1.4.2 Rotor fixed reference frame

In the late 1920s, R.H. Park [Park 29] introduced a new approach to ac motor analysis. The variables (voltages, currents and flux linkages) associated with the stator windings are replaced with variables associated with fictitious windings rotating with the rotor. In other words, the stator variables are transformed to a reference frame fixed to the rotor. As will be shown, Park's transformation, which revolutionized electrical motor control, has the unique property of eliminating all time-varying inductances from the voltage equations of the synchronous machine occurring due to both electric circuits in relative motion and electric circuits with varying magnetic reluctance. In this subsection, the common reference frame is attached to the rotor rotating at the electrical speed ωr. For some reasons, it may be more instructive to imagine the whole motor rotating at the electrical speed ωr contrary to the direction of motion. Then, the α/β-rotor reference frame stands still and a similar condition as in the previous paragraph is given. Only the transfer direction is opposite, i.e. all stator variables are transformed in direction of original rotor rotation and all rotor variables remain unchanged.


Expressing the stator current phasor is in terms of rotating α/β-components, corresponding to the axes used for the rotor quantities (figure 1.6), is based on the transformation shown in figure 1.8. The axes of the rotating α/β-components are subsequently denoted as d-and q-axis to indicate a common reference frame of stator and rotor. The superscript 'r' indicates the rotor-fixed reference frame. In this paragraph, the electrical rotor angle θ is replaced by the more general angle γ to guarantee symbol analogy with the subsequent transformation to the arbitrary reference frame. According to figure 1.8, the stator currents expressed in components of a rotor-fixed reference frame are:

γγ βα sincos ssrsd iii += (1.80)

γγ βα cossin ssrsq iii +−= (1.81)

With the transformation matrix Tγ and its inverse Tγ

-1

−

=γγγγ

cossinsincos

γT (1.82)

−=−

γγγγ

cossinsincos1

γT (1.83)

equations (1.80)-(1.81) can be rewritten in matrix form. Back-transformation with the inverse matrix yields the stator current in α/β-components:

=

β

α

s

srsq

rsd

ii

ii

γT (1.84)

=

−rsq

rsd

s

s

ii

ii 1

γTβ

α (1.85)

i sβ

isα

irsq = -isα sinγ + isβ cosγ

γ

i sβ

isα isβ cosγ

isβ sinγ isα cosγ

-isα sinγ γ

is

irsd = isα cosγ + isβ sinγ

d q

Figure 1.8: Transformation of the stator currents to the rotor-fixed reference frame.

22 Chapter 1

Equally, the stator voltage phasor and the stator flux phasor are expressed as a vector sum of components in the rotor reference frame:

=

β

α

s

srsq

rsd

uu

uu

γT (1.86)

=

β

α

ψψ

ψψ

s

srsq

rsd

γT (1.87)

Multiplying the stator voltage equation (1.53) with Tγ yields:

+

=

+

=

β

α

β

α

β

α

β

α

β

α

ψψ

ψψ

s

s

s

ss

s

s

s

ss

s

s

dtd

ii

Rdtd

ii

Ruu

γγγγ TTTT (1.88)

The rotor flux vector multiplied with Tγ

-1 Tγ remains unchanged:

=

++

=

−

−=−

1001

sincos00sincos

cossinsincos

cossinsincos

22

22

γγγγ

γγγγ

γγγγ

γ1

γ TT

(1.89)

Using (1.89) in (1.88) and applying the partial derivative yields:

+

+

=

+

=

+

=

−−

−

−

rsq

rsd

rsq

rsd

rsq

rsd

s

rsq

rsd

rsq

rsd

s

s

srsq

rsd

srsq

rsd

dtd

dtd

ii

R

dtd

ii

R

dtd

ii

Ruu

ψψ

ψψ

ψψ

ψψ

β

α

1γγ

1γγ

1γγ

γ1

γγ

TTTT

TT

TTT

(1.90)

With applying the partial derivative of the transfer matrix Tγ,


γωγγγ

γγ

γγγγγ

γγγγ

γγγγγγ

γ

−=

+−−

=

−−−

−

=

−−−

=

=

−−

0110

0sincossincos0

sincoscossin

cossinsincos

sincoscossin

22

22

dtd

dtd

dtd

dtd

dd

dtd

γ1

γγ1

γγ TTTTT

(1.91)

the transformed stator voltage equation consists of three parts, namely the resistive voltage drop, the time-dependent flux variation and the induced voltage due to the rotation with the electrical speed ωγ:

−+

+

=

rsq

rsd

rsq

rsd

rsq

rsd

srsq

rsd

dtd

ii

Ruu

ψψ

ωψψ

γ 0110

1001

(1.92)

Since the used reference frame is equal to the original α/β-reference frame of the rotor, all rotor variables and quantities remain unchanged. Only the notation is adapted to guarantee symbol analogy. Summarizing, following voltage equations are valid in the common rotor-fixed reference frame:

−+

+

=

rsd

rsq

rsq

rsd

rsq

rsd

srsq

rsd

dtd

ii

Ruu

ψψ

ωψψ

γ (1.93)

+

=

rrq

rrd

rrq

rrd

rrrq

rrd

dtd

ii

Ruu

ψψ

(1.94)

According to (1.61)-(1.62) and considering T-θ = Tγ

-1 and T-θ-1 = Tγ (valid only for

the rotor-fixed reference frame with θ = γ), the flux linkage can be also expressed as:

+

=

+

=

+

=

=

−

−

rrq

rrd

hrsq

rsd

s

r

rh

s

ss

r

rh

s

ss

s

srsq

rsd

ii

Lii

L

ii

Lii

L

ii

Lii

L

1

1

1

β

α

β

α

β

α

β

α

β

α

ψψ

ψψ

1γγγ

1γγγ

TTT

TTT

(1.95)

24 Chapter 1

+

=

+

=

=

rsq

rsd

hrrq

rrd

r

s

sh

r

rr

r

rrrq

rrd

ii

Lii

L

ii

Lii

L

1

1β

α

β

α

β

α

ψψ

ψψ

γT

(1.96)

With respect to (1.95)-(1.96), the flux generation is caused only by current components parallel to the flux direction. Thus, d- and q-axis are completely decoupled.

1.4.3 Arbitrary rotating reference frame

In the last two paragraphs, each stator and rotor quantities and equations are transformed to another reference frame. Just as well, a transformation to any arbitrary reference frame is possible. However regarding drive control, there is no practical application applying the transformation to two different reference frames for stator and rotor, respectively. Only a common reference frame contains all advantages as e.g.: constant main inductance, no mutual magnetic coupling between d- and q-axes and eliminating time-dependence of current/voltage equations. In this subsection, the common reference frame may rotate at any constant or varying angular velocity or it may remain stationary. The angular velocity ωγ associated with the change of variables is unspecified. Only the angular displacement γ of the rotating reference frame must be continuous. Usually, this constriction is automatically fulfilled, since the angle is the time-integral of the speed.

)0(0

γωγ γ += ∫ dtt

(1.97)

Again, it may be easier to imagine the whole motor rotating at a speed ωγ contrary to the direction of rotation (figure 1.9). Then, the imaginary reference frame stands still and the stator windings are rotating with -ωγ and the rotor windings with -(ωγ - ωr). A comparison with figure 1.8 shows, the stator quantities and equations are transformed using the same transformation matrix as in equation (1.82). Only the transformation angle γ is no longer fixed to the rotor position. Obviously, applying the same transformation matrix and geometrical calculations on the stator variables and equations yields the same stator equations in any rotating reference frame. Thus, the results of the previous paragraph are directly used. As seen from the new reference frame, the rotor rotates at the electrical speed -(ωγ - ωr) (figure 1.9). Contrary to the stator-fixed reference frame, the applied


transformation is not contrary to the direction of rotation but in direction of the rotating reference frame. Thus, the transformation of the rotor variables and affiliated rotor equations is obtained by simply replacing the angle -θ of the transformation matrix T-θ in (1.65) by the angle γ - θ. Strictly replacing the angle in all rotor equations representing the stationary reference frame yields the rotor equations in the arbitrary rotating reference frame.

a

a' c

c'

b

b'

αS

βS

γ = ∫ωγ dt

αr

βr

d

q

θ = ∫ωr dt

γ - θ= ∫(ωγ - ωr) dt

αr

βr

d

q

γ - θ= ∫(ωγ - ωr) dt

αs

βs

d

q

γ = ∫ωγ dt

stator:

rotor:

Figure 1.9: Transformation to an arbitrary reference frame. Summarizing, following transformations and equations are valid in any arbitrary rotating reference frame:

−

=γγγγ

cossinsincos

γT (1.98)

=

β

α

s

s

sq

sd

ii

ii

γT,

,

(1.99)

−−−−−

=− )cos()sin()sin()cos(

θγθγθγθγ

θγT (1.100)

=

−

β

α

r

r

rq

rd

ii

ii

θγT,

,

(1.101)

The transformation of voltages and flux linkages is equivalent. The inverse transformations, calculating the α/β-currents by the d/q-values, are as:

26 Chapter 1

−=−

γγγγ

cossinsincos1

γT (1.102)

=

−,

,

sq

sd

s

s

ii

ii 1

γTβ

α (1.103)

−−−−−

=−− )cos()sin(

)sin()cos(θγθγθγθγ1

θγT (1.104)

=

−− ,

,

rq

rd

r

r

ii

ii 1

θγTβ

α (1.105)

Considering the speed of the rotating reference frame in (1.77) and (1.93), the voltage equations in the arbitrary reference frame are:

−+

+

=

,

,

,

,

,

,

,

,

sd

sq

sq

sd

sq

sds

sq

sd

dtd

ii

Ruu

ψψ

ωψψ

γ (1.106)

−−+

+

=

,

,

,

,

,

,

,

,

)(rd

rqr

rq

rd

rq

rdr

rq

rd

dtd

ii

Ruu

ψψ

ωωψψ

γ (1.107)

As mentioned previously, the voltage equations for all reference frames may be obtained from those in the arbitrary reference frame. A comparison of the different reference frames shows, the stator and rotor reference frames are just special cases of the arbitrary reference frame: Setting the speed ωγ in (1.106)-(1.107) to zero yields the stationary equations (1.76)-(1.77). If the speed of the reference frame equals the rotor speed ωr, the equations of the rotor reference frame (1.97)-(1.30) are obtained. Generally, the transformation for a specific reference frame is obtained by substituting the appropriate reference-frame speed for ωγ into (1.97) to obtain the angular displacement. In most cases, the initial or time-zero displacement γ(0) is selected equal to zero. Equally to the stator and rotor reference frames, the flux linkages of d- and q-axis are completely decoupled: The flux generation is caused only by current components parallel to the flux direction. According to (1.61)-(1.62) and considering


1γ

1θγθ

T

TT

−

−−−

=

−=

−+−−−+−−−−−−−−

=

−−−−−

−=

γγγγ

θγθθγθθγγθγθθγθθγθθγθθγθ

θγθγθγθγ

θθθθ

cossinsincos

)cos(cos)sin(sin)sin(cos)cos(sin)cos(sin)sin(cos)sin(sin)cos(cos

)cos()sin()sin()cos(

cossinsincos

⇒ (1.108) ( ) 1TTTTTTTT 1γγ

1θγθγ

1θγθγ === −−

−−−−−

⇒ ( ) ( ) 1TTTTTTTTTTT 1θγθ

1θθγ

1θγθ

1θθγ

1γ

1θθγ === −

−−−−−

−−−

−−−

−−−− (1.109)

the flux linkage can be also expressed as:

+

=

+

=

+

=

=

−−−

−−−−

−

,

,

1,

,

1

1,

,

rq

rdh

sq

sds

r

rh

s

ss

r

rh

s

ss

s

s

sq

sd

ii

Lii

L

ii

Lii

L

ii

Lii

L

1θγθγ

θγ1θγθγγ

θγγ

TTT

TTTTT

TTT

β

α

β

α

β

α

β

α

β

α

ψψ

ψψ

(1.110)

⇒

+

=

,

,

1,

,

,

,

rq

rdh

sq

sds

sq

sd

ii

Lii

Lψψ

(1.111)

+

=

+

=

+

=

=

−−−−

−−−−−

−−−−

,

,

1,

,

1

1,

,

sq

sdh

rq

rdr

s

sh

r

rr

s

sh

r

rr

r

r

rq

rd

ii

Lii

L

ii

Lii

L

ii

Lii

L

1γ

1θθγ

γ1

γ1θθγθγ

1θθγθγ

TTT

TTTTT

TTT

β

α

β

α

β

α

β

α

β

α

ψψ

ψψ

(1.112)

⇒

+

=

,

,

1,

,

,

,

sq

sdh

rq

rdr

rq

rd

ii

Lii

Lψψ

(1.113)

The most practical applied arbitrary rotating reference frames are rotating with synchronous electrical speed. In steady state, the real stator voltage, current and flux quantities are equally represented by phasors, rotating with the supply frequency.

28 Chapter 1

The frequency of the real existing rotor quantities equals the supply frequency minus electrical rotor speed. As seen from a reference frame rotating with synchronous speed, all stator and rotor phasors stand still (figure 1.10). Thus, dc values, very practical regarding drive control strategies, are obtained in steady state.

jβ

α

id

iq

d

q is

iβ (t)

iα (t)

γ = ωµ t

ωµ

iβ (t)

t

iα (t)

t

jβ

α

id

iq

d

q

is

iq

t

t

id

proj

ectio

n on

β-a

xis

projection on α-axis

projection on q-axis

proj

ectio

n on

d-a

xis

Figure 1.10: Current phasor, α/β- and d/q-representation and affiliated time charts.

Considering reference frames synchronously rotating with the supply frequency, several ways of aligning the zero angular position of the d-axis and perpendicular q-axis are practically used: rotor-flux-oriented, stator-flux-oriented and magnetizing-flux-oriented [Vas 97]. In steady state, the speed of the affiliated reference frames is equal, only a load-dependent angle characterizes the different systems. As seen in figure 1.11, the angular displacement of stator, rotor and main flux is marginal. However, aligning the zero angular position of the d-axis to the rotor flux yields considerable advantages regarding control strategies. The final objective of the vector control philosophy is to be able to control the electromagnetic torque in a way equivalent to that of a separately excited dc machine. As shown in the next chapter, field-oriented control enables control over both the excitation flux-linkage and the torque-producing current in a decoupled way. Furthermore, the practical method determining the transformation angle γ will be shown. However, only the rotor-flux-oriented control yields complete decoupling of torque and flux. Choosing a different flux orientation may outweigh the lack of complete decoupling for some special applications [Xu 92].


0 0.02 0.04 0.06 0.08 0.1-1.5

-1

-0.5

0

0.5

1

1.5

t [s]

Ψα [V

s]0 0.02 0.04 0.06 0.08 0.1

-1.5

-1

-0.5

0

0.5

1

1.5

t [s]

Ψβ [V

s]

0 0.02 0.04 0.06 0.08 0.10.8

1

1.2

1.4

t [s]

Ψd [V

s]

0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

t [s]

Ψq [V

s]

Figure 1.11: Rotor-, stator- and main flux of an induction machine (1,5 kW; n = 1500 rpm; p = 2) using FOC. Left: Flux quantities in a rotor-flux-oriented reference frame. Right: Flux quantities in the

stationary stator reference frame.

1.4.4 Power and electromagnetic torque in the arbitrary reference frame

Since any ac machine with isolated neutral point contains no homopolar current component, the total power expressed in the αβ variables equals, according to (1.30), the total power expressed in abc variables. Applying the transformation to a rotating reference frame on the αβ variables and solving the trigonometric functions yields:

,,,,

2,,2,,2,,2,,

,,,,

,,,,

cossinsincos

)cossin()cossin(

)sincos()sincos(

)(

sqsqsdsd

sqsqsdsdsqsqsdsd

sqsdsqsd

sqsdsqsd

ssssccbbaael

iuiu

iuiuiuiu

iiuu

iiuu

iuiuiuiuiutP

+=

+++=

+++

−−=

+=++=

γγγγ

γγγγ

γγγγββαα

(1.114)

Although the waveforms of the d/q-voltages, currents and flux linkages are dependent upon the angular velocity of the reference frame, the waveform of the total power is the same and independent of the reference frame in which it is evaluated. In literature (e.g. [Hen 92]), the torque equation is usually obtained by the derivation of the magnetic system energy as a function of the mechanical angular displacement. Here, a different approach is introduced, showing the energy flow and motor losses

ψrsα

ψssα

ψhsα

no load 80% of rated load no load 80% of rated load

ψs,d

ψr,d

ψh,d

ψssβ ψr

sβ ψs

,q

ψh,q

ψhsβ

ψr,q

30 Chapter 1

considered in the dynamic equations. Considering a doubly fed ac motor, i.e. power is supplied at both stator and rotor terminals, the total electrical power input is:

,,,,,,,,, )( rqrqrdrdsqsqsdsdtotalel iuiuiuiutP +++= (1.115)

Usually, the rotor voltages are zero (e.g. squirrel-cage induction motor and PMSM). Then, the universal-valid equation (1.115) equals (1.114). Replacing the voltages by applying (1.106)-(1.107), the total input power can be rewritten:

)(

)()(

)()()(

,,,,

,,,,,

,,

,2,2,

,,,,,

,,

,2,2,,

rdrqrqrdr

rqrdrdrqrq

rqrd

rdrqrdr

sqsdsdsqsq

sqsd

sdsqsdstotalel

ii

iidt

di

dtd

iiiR

iidt

di

dtd

iiiRtP

ψψω

ψψωψψ

ψψωψψ

γ

γ

−+

−+++++

−++++=

(1.116)

Using the flux representation (1.111)-(1.113), following correlation is valid:

,,,,

,,,,,,,,

,,,,,,,,,,,,

sdrqrqrd

sdhrqrqrrdsqhrdrdrrq

rqhsdsqssdrdhsqsdssqsqsdsdsq

ii

iLiiLiiLiiLi

iLiiLiiLiiLiii

ψψ

ψψ

−=

−−+=

−−+=−

(1.117)

Applying (1.117), the power equation (1.116) can be simplified:

)(

)()()(

,,,,,

,,

,

,,

,,2,2,2,2,

,

rdrqrqrdrrq

rqrd

rd

sqsq

sdsdrqrdrsqsdstotalel

iidt

di

dtdi

dtd

idt

diiiRiiRtP

ψψωψψ

ψψ

−+++

+++++= (1.118)

Obviously, the first two terms in (1.118) are resistive losses. According to figure 1.12, the current multiplied by the time-derivation of the flux linkages is the power used for the magnetic field generation:

∑ ∫∫ ∑ ∫==

=

==

n

vvv

t n

v

tv

vfieldfield

v

didtdt

didtPtE

1 00 1 0

)(ψ

ψψ

(1.119)


i

ψ

E*

E

Figure 1.12: Magnetic field energy E and co-energy E*. As illustrated in figure 1.13, the interaction of flux linkages and perpendicular current generates electromagnetic torque. Thus, the total electrical input power (1.118) consists of resistive losses PR,loss, power Pfield stored as magnetic field energy in the windings and mechanical output power Pmech. Truly, (1.118) contains not all losses of the motor drive. Among the resistive losses, also friction and iron losses are active. However, considering the braking effect of iron and friction losses, they are acting like an equivalent load: e.g., a PMSM with open terminals driven in generator mode behaves as a load with no electrical output power but with friction and iron losses. Thus, the iron and friction losses are handled as a part of the mechanical output power Pmech.

mechfieldlossRtotalel PPPtP ++= ,, )( (1.120)

with:

)()( 2,2,2,2,, rqrdrsqsdslossR iiRiiRP +++= (1.121)

dtd

idt

di

dtd

idt

diP rq

rqrd

rdsq

sqsd

sdfield

,,

,,

,,

,, ψψψψ

+++= (1.122)

)( ,,,,rdrqrqrdrmech iiP ψψω −= (1.123)

The mechanical output power of any ac machine with p pole-pairs can be also written as a function of the electromagnetic torque Tel and mechanical rotor speed ω:

pTTP r

elelmechω

ω == (1.124)

Comparison of (1.123) and (1.124) yields the electromagnetic torque as a function of electrical quantities. Thus, the electromagnetic torque is produced by the interaction of rotor flux linkages and rotor currents:

)( ,,,,rdrqrqrdel iipT ψψ −= (1.125)

32 Chapter 1

According to the natural counter-action of forces, the electromagnetic torque is produced as well by the interaction of stator flux linkages and stator current. As illustrated in figure 1.13, this interaction can be pictorially represented by electromagnetic forces Fel on current-carrying conductors in a magnetic field. Applying (1.117) on (1.125), the electromagnetic torque expressed in stator quantities is:

)( ,,,,sqsdsdsqel iipT ψψ −= (1.126)

Position of the reference frame:

isq

-isq

isd

-isd

ψsq

ψsd

Fel,2

Fel,2

Fel,1

Fel,1

Tel ~ (isq ψsd - isd ψsq)

Tel ~ (Fel,1 - Fel,2)

isd

isq

q

d

Figure 1.13: Interaction of stator flux linkages and stator current in an arbitrary reference frame. The electromagnetic torque and the rotor speed are related by the dynamic equation

dtd

pJ

dtdJTT r

loadelωω

==− (1.127)

where J is the inertia of both rotor and load.

1.4.5 Transformation of stationary machine variables

The electrical circuit of ac machines is completely described by stator and rotor voltage equations with Rs and Rr representing the resistance of stator and (stator related) rotor windings, respectively. A linear set of rotor and stator inductances represents the relation of current and flux linkages. For reasons of simplicity, all motor parameters have been assumed to be constant, i.e. the rotor and stator geometry as well as all windings are completely symmetrical. From now on, this restriction will be dropped. It is convenient to treat resistive and inductive circuit elements separately.


If the phase resistances are unequal, the representation by a scalar is incorrect and the voltage equation (1.106) in the arbitrary reference frame is adapted by the correct transformation of the resistive voltage drop

=

=

=

−

−

,

,

00

00

00

sq

sd

s

s

s

s

s

s

ii

ii

RR

ii

RR

ii

RR

dqγ1

γγ

γ1

γγγ

RTTT

TTTT

β

α

β

α

β

α

β

α

β

α

β

α

(1.128)

= 1

with the resistance matrix in dq variables containing sinusoidal functions of γ:

1γsγdq TRTR −= (1.129)

However, all stator phase windings of either synchronous or induction machines are designed to have the same resistance. Similarly, transformers, transmission lines and usually all power-system components are designed so that all phases have equal resistances. Even power-system loads are distributed between phases so that all phases are loaded nearly equal. If the nonzero elements of the diagonal resistance matrix are equal

=

s

s

RR0

0sR (1.130)

following relation is valid:

s1

γγ

1γ

1γ

1γ

1γsγdq

RTT

TT

TTRTR

==

−

=

−

=

−

==

−

−−

−−

s

sss

ss

s

s

R

RRRRR

RR

γγγγ

γγγγ

γγγγ

cossinsincos

cossinsincos

00

cossinsincos

(1.131)

Thus, the resistance matrix associated with the arbitrary reference variables is equal to the resistance matrix associated with the actual variables if each phase of the actual circuit has the same resistance. Obviously, the same applies for the rotor resistance. Equally, the rotor and stator inductances of symmetrical machines, e.g. induction machines or cylindrical synchronous machines, are not dependent on the rotor position and thus independent on the reference frame used. In contrast, the

34 Chapter 1

inductances of most PMSM and synchronous machines with salient poles (figure 1.14) depend on the actual rotor position, since the magnetic path in direction of the rotor poles is different to the magnetic path in perpendicular direction, i.e. the air-gap between rotor and stator is not constant.

a' c

c'

b

b'

α

β

dq

ωr

Permanent Magnets

a

a'c

c'

b

b'

α

β

d q

θ = ∫ωr dt θ = ∫ωr dt

Salient-pole SM: Ld > Lq PMSM: Ld < Lq

a

Figure 1.14: PMSM with inset magnets and salient-pole synchronous machine.

In the case of a magnetic linear system, it is customary to express the flux linkages as a product of inductances and current matrices before performing the transformation. However, the transformation of flux linkages is valid even for asymmetrical or magnetic nonlinear systems and an extensive amount of work can be avoided by transforming the flux linkages directly. This is especially true in the analysis of ac machines where the inductances are function of the rotor position. Thus, the stator and rotor voltage equations introduced are universal. Only the current-flux relation must be adapted in the case of asymmetry or non-linearity. The electrical behavior of PMSM and salient-poles synchronous motors is comparable, only the method of rotor field generation is different. The set of voltage and flux equations introduced simplifies, since the rotor of synchronous machines contains no three-phase windings and the rotor flux linkage, generated by permanent magnets or dc current in a separately exited winding, is not affected by the stator flux linkage. Due to the high permeability of air, the rotor flux linkage ψr is completely active only in direction of the poles. Considering the power-invariant transformation factor, the distribution of the rotor flux linkage in α/β-components of the stator reference frame yields:

=

θψθψ

ψψ

β

α

sincos

23

r

rsr

sr (1.132)

Since the voltage equations expressed in terms of flux linkages are still valid, only the current-flux relation is adapted. If the zero angular displacement θ(t=0) is chosen


in direction of the poles (according to figure 1.14), the stator flux linkage of the original three-phase synchronous motor is:

+−+

=

)32cos()32cos(

cos

πθψπθψ

θψ

ψψψ

r

r

r

sc

sb

sa

ccbcac

cbbbab

cabaaa

sc

sb

sa

iii

LLLLLLLLL

(1.133)

Due to the magnetic anisotropy of the rotor, the self-inductances of the original three-phase windings are not constant but rotor position dependent (figure 1.15). Assuming uniformly and sinusoidal distributed stator windings, the self-inductances are:

( )θ2cos20 LLLaa −= (1.134)

( 3/22cos20 )πθ +−= LLLbb (1.135)

( 3/22cos20 )πθ −−= LLLcc (1.136)

0 π 2π

L0

L2

θ

Laa

0 π 2π

L0

L2

θ

Laa

Figure 1.15: Self-inductance Laa as a function of the electrical rotor angle θ. Left: PMSM with inset magnets. Right: Salient-pole synchronous motor.

Figure 1.16 illustrates the ideal field distribution as a function of the electrical rotor angle of a non-excited salient-pole synchronous motor in the α/β-reference frame. To guarantee a clear illustration, the rotor field as well as the leakage field is not drawn. As seen by the field distribution, the α- and β-axis are not decoupled. Dependent on the rotor position, the field generated by the current iα is also active in the β-axis and the current iβ induces in the windings of the α-axis.

36 Chapter 1

θ = 0

iβ

-iα

iα

iβ

θ = π/2

iβ

-iα

iα

-iβ

θ = π/2

ψsα

ψsαψsα

ψsβ ψsβ

ψsβ

Figure 1.16: Schematic main field distribution (no leakage) as a function of the electrical rotor angle of a

non-excited (ψr = 0) salient-pole synchronous motor in the α/β reference frame. Thus, the stator flux linkage in the α/β-reference frame contains inductances describing the magnetic coupling:

+

=

β

α

β

α

βαβ

βαα

β

α

ψψ

ψψ

r

r

s

s

s

s

ii

LLLL

(1.137)

with:

=

θψθψ

ψψ

β

α

sincos

23

r

r

r

r (1.138)

θα 2cos1 LLL ∆+= (1.139)

θβ 2cos1 LLL ∆−= (1.140)

θβααβ 2sinLLL ∆== (1.141)

Transformation of (1.137) to the arbitrary reference frame yields:

+

=

+

=

+

=

=

−

−

,

,

,

,

,

,

rq

rd

sq

sd

r

r

s

s

r

r

s

s

s

s

sq

sd

ii

ii

ii

ψψ

ψψ

ψψ

ψψ

ψψ

β

α

β

α

β

α

β

α

β

α

1γαβγ

γγ1

γαβγ

αβγγ

TLT

TTTLT

LTT

(1.142)

Obviously, the time-varying inductances, i.e. rotor-position dependent inductances of a synchronous machine are eliminated only if the reference frame is fixed to the


rotor. Consequently, the arbitrary reference frame does not offer the advantages in the analysis of synchronous machines that it does in the case of induction machines. Naturally, the rotor-fixed reference frame is equal to the synchronously rotating reference frame for synchronous motor drives. According to (1.142), the transformation of the stator quantities to the rotor-fixed reference frame with the zero angular displacement θ(t=0) chosen in direction of the poles yields:

+

=

=

−

rrq

rrd

rsq

rsd

s

srsq

rsd

ii

ψψ

ψψ

ψψ

β

α 1θαβθθ TLTT (1.143)

The rotor flux linkage in terms of the original rotor field ψr is active only in the d-axis:

+−+

=

−

=

=

θθθθθθ

ψ

θψθψ

θθθθ

ψψ

ψψ

β

α

cossincossinsincos

23

sincos

23

cossinsincos

22

r

r

r

r

rrrq

rrd

θT

(1.144)

⇒

=

023 r

rrq

rrd ψ

ψψ

(1.145)

Reformulation of the inductance matrix in (1.143) exhibits the complete magnetic decoupling and the elimination of time-varying inductances in the rotor fixed reference frame:

∆−

∆+=

∆−∆

∆∆+= −−

LLLL

LLLLLL

1

1

1

1

00

2cos2sin2sin2cos 1

θθ1

θαβθ TTTLTθθ

θθ

(1.146)

⇒ (1.147)

== −

q

d

LL0

01θαβθdq TLTL

Summarizing, the stator flux linkage of synchronous machines in the rotor-fixed reference frame is:

rrsdd

rsd iL ψψ

23

+= (1.148)

38 Chapter 1

rsqq

rsq iL=ψ (1.149)

with constant inductances:

LLLd ∆+= 1 (1.150)

LLLq ∆−= 1 (1.151)

Again, the sign in (1.150)-(1.151) depends on the reluctance in pole direction. For common PMSM with inset magnets, the d-axis inductance is smaller than the q-axis inductance (Ld < Lq.). Unfortunately, the original inductances (1.134)-(1.136) are not suitable to calculate directly the α/β-inductances (1.139)-(1.141), respectively d/q-inductances (1.150)-(1.151), since a practical measurement does not enable a splitting of the leakage inductances in parts regarding different causes, e.g. leakage due to the end winding and slot-leakage, respectively. However, the d/q-inductances (1.150)-(1.151), required for motor control, are directly measured using the motor model in the rotating reference frame. Without restriction respecting the electrical behavior of torque controlled synchronous machines (i.e. the slip frequency is zero) a damper cage has not yet been considered. Only in open-loop control mode, e.g., the synchronous motor is directly connected to the electrical grid, the damper cage must be considered to calculate the transient behavior [Hen 92].

1.5 Conclusions

This chapter introduces and explains a general dynamic model of ac machines as required for high-performance motion control. The developed theory of reference frames is equally applicable to various forms of synchronous machines and induction machines, respectively. From all proposed transformations, it can be deduced that the mutual magnetic coupling between d- and q-axes is eliminated. The stator and rotor currents in the d-axis are only active in the affiliated windings of the d-axis. The same applies for the q-axis. The only difference in the equations is the variable indicating the speed of the rotating reference frame. The transformed stator and rotor voltage equations consist of three parts, namely the resistive voltage drop, the time-dependent flux variation and the induced voltage due to the relative rotation with respect to the rotating reference frame. The stator and rotor voltage equations introduced are universal. Only the current-flux relation must be adapted in the case of asymmetry or non-linearity. The most commonly used reference frames are rotating with synchronous electrical speed. As seen from a reference frame rotating with synchronous speed, all stator


and rotor phasors stand still and all time varying variables become constant (in steady state). Thus, dc values, very practical regarding drive control strategies, are obtained. Another nice effect of applying the transformation to a synchronous reference frame is, that the rotor-position dependence of inductances due to magnetic asymmetries, e.g. in salient-pole synchronous machines or in PMSM, is eliminated from the voltage equations. Considering synchronous machines, the synchronous reference frame is equal to the rotor reference frame, since the supply frequency equals the electrical rotor speed. As shown in the next chapter, choosing a synchronous reference frame in which rotor flux lies entirely in the d-axis, the dynamic equations of both induction and synchronous machines are simplified and analogous to a dc motor. The Field-oriented control, using the motor model in a synchronous reference frame, enables control over both the excitation flux-linkage and the torque-producing current in a decoupled way.

2. Field Oriented Control

2.1 Introduction

Adjustable speed drives with torque control are required in many applications, e.g. manufacturing and transportation. Using ac motors and high performance ac drives that are capable of controlling torque and flux independently offer several advantages over dc motors and drives such as lower maintenance, smaller size and higher speeds. Compared to dc drives, the higher cost of ac drives is in part compensated by a lower ac machine cost. Compared to uncontrolled ac motors, supplied by a regular grid, the efficiency of inverter-controlled drives can be vastly increased by, e.g. flux optimization. The resulting energy saving, which depends on the respective application, may be crucial. It might be expected that in the medium term future, as ac drives become a mature product state, cost of the additional power electronics will no longer be an issue. This chapter provides the fundamentals of high-performance motion control employing the FOC technique. Suitable motor models are derived and different control approaches discussed. Motor models are rewritten in a slightly different form providing an increased performance of the entire control system. The basic speed control scheme is refined systematically in subsequent paragraphs by, e.g. advanced speed control and flux optimization. As shown, the electrical and mechanical behavior of squirrel-cage induction motor and synchronous machines, e.g. the PMSM, are very similar. Thus, apart from modeling, they are handled simultaneously within this text. Considering FOC, a comprehensive and clear description of the controller design is given in this chapter. Special care has been taken for the viability of the real-time implementation. The design concepts differ slightly from literature, e.g. [Leo 85], but they are proven to result in a very robust and high-performance drive control. This chapter is rather suitable for engineers implementing FOC in real-time than just simulating the drive with simplifying assumptions. Differences to common presentations described in literature are special flux control schemes, current anti-windup with automatic flux adaptation and the introduction in advanced speed control.

42 Chapter 2

2.2 Entire Control System using FOC

In this chapter, only the rotor-flux-oriented type of vector control, also termed “Field-Oriented Control” (FOC), is considered. Diagram 2.1 shows the basic speed control scheme for ac motor drives with FOC [Bla 72]. In this figure, interactions between the different modules are neglected. However, they are considered by decoupling within subsequent paragraphs. The control of induction motors and synchronous machines is handled simultaneously. As will be shown, their electrical and mechanical behavior is very similar. The goal of FOC is to maintain the amplitude of the rotor flux linkage Ψr at a fixed value, except for field-weakening operation or flux optimization, and only modify a torque-producing current component in order to control the torque of the ac machine. Considering a complete decoupling of torque and flux, a linear relation between torque Tel and torque producing current iq is achieved and the torque in the ac machine can be expressed as [Vas 97]:

qrel icT Ψ= (2.1)

Thus, the electromagnetic torque generated by the motor can be controlled by controlling the q-axis current. In speed control mode, the torque reference Tel

* is calculated by a speed controller. As shown later, the rotor flux can be controlled directly by controlling the d-axis current.

uq*

load

SVM

PWM generation

PWM Inverter

ub*

uc*

ac motor

ua*

Power supply

d,q

α,β uβ

*

uα*

ud*

d,q

α,β iβ

iα

γ

id

iq

id*

iq* Park-1 T.

Park T.

α,β

a,b,cib

ia

3 ⇒ 2

Udc

ω

ω* Tel*

Flux reference

iq* = f(Tel

*)

id* = f(Ψr

*) Ψr

*

Speed control

Current mapping

Current mapping

id - control

iq - control

Digital control system

Figure 2.1: Basic speed control scheme for ac motor drives. Motor currents are measured in two phases. These measurements feed the ‘3 ⇒ 2-transformation’ module. The outputs of this projection are designated iα and iβ. These two current components are the inputs of the Park transformation giving the current in the d/q rotating reference frame. The current components id, iq are compared to the reference currents id

* and iq* controlling flux and torque generation,

respectively.

Field Oriented Control 43

At this point, this control structure shows an interesting advantage: it can be used to control either synchronous or induction machines by simply changing the flux reference and the method of obtaining rotor flux position. As the rotor flux, fixed to the absolute rotor position due to characterizing poles, of synchronous permanent magnet motors is generated by the magnets or by separately exited windings in the case of synchronous machines, there is no need to create one via the stator current. Hence, when controlling synchronous machines, id

* can be set to zero. Advanced control schemes may exploit additionally the reluctance torque, if available. As induction motors need a rotor flux creation in order to operate, the flux reference Ψr

* must not be zero. The FOC conveniently solves one of the major drawbacks of the “classic” control structures: the portability from induction to synchronous motor drives. In speed control mode, the torque command Te

*l is the output of the speed

controller mapped into a current reference iq*. The current controllers calculates the

voltages ud* and uq

* in the d/q reference frame; they are applied to the inverse Park transformation. The outputs of this projection are uα

* and uβ*, being the components

of the stator voltage in the α/β stationary orthogonal reference frame and used as inputs of the space vector PWM. The outputs of this block are the signals driving the inverter. Note that both Park and inverse Park transformations need the rotor flux position γ. Obtaining the rotor flux position depends on the ac machine type used (synchronous or induction machine). The different control loops as shown in figure 2.1 are described in detail in subsequent paragraphs. The basic speed control scheme is further refined in following subsections adding step by step additional features as, e.g. advanced speed and flux control/estimation approaches.

2.3 Park Transformation (α/β ⇒ d/q)

The α/β representation eliminates the mutual magnetic coupling of the phase-windings, but the time-dependence of both current and voltage equations still remains. Observing the current in a reference frame rotating with the same speed as the current state phasor, pictorially expressed “sitting on the current space state phasor”, dc values are obtained in steady state. The so-called Park transformation is the most important transformation in the FOC. Figure 2.2 presents the relation between two reference frames for the stator current state phasor. In this chapter, the d-axis of the rotating frame is aligned with the rotor flux. The real axis is further denoted as the direct axis (d-axis), and the imaginary axis as the quadrature axis (q-axis). According to the introduced transformation to the arbitrary rotating reference frame, the stator current is:

−

=

β

α

γγγγ

ii

ii

q

d

cossinsincos

(2.2)

44 Chapter 2

Of course, voltages and flux linkages are transformed in the same way. The inverse Park transformation (2.3) is usually used for the calculation of the reference voltages (figure 2.1) being the input of the PWM module.

−=

*

*

*

*

cossinsincos

q

d

uu

uu

γγγγ

β

α (2.3)

jβ

α

id

iq

a

a' c

c'

b

b'

α

β

dq

γ

d

q ωµ

is

iβ

iα

γ = ∫ ωµ dt

Ψr ⇔

Ψr

Permanent Magnets for PMSM Rotor flux linkage for induction and synchronous machines

Figure 2.2: Park transformation of the α/β current (d/q transformation).

Here, only the transformations to a rotor flux reference frame are considered [Bla 72], [Hen 92]. The angle of the transformation from stator to rotor reference frame coincides with the rotor flux angle γ rotating at synchronous speed ωµ. Thus, the rotor flux lies entirely on the d-axis (figure 2.2). In steady state, the frequency of the rotating reference frame equals the frequency of both flux and stator voltage/current. In the case of an induction machine, the speed ωµ may differ from the rotor speed ωr depending on the slip due to load.

slipr ωωωµ += (2.4)

In the case of synchronous machines, the transformation angle is fixed to the position of the permanent magnets or to the pole direction of separately exited windings. Thus, the speed ωµ equals the electrical rotor speed ωr. Then, under linear magnetic conditions, all time-varying (rotor-position dependent) inductances are eliminated from the voltage equations of the salient-pole machine. This aspect is especially important when control strategies are to be employed from the corresponding differential equations. When this reference frame is used, the physical picture is such that the transformed quantities rotate with the rotor and thus see invariable magnet paths. Consequently, the inductances Ld and Lq, corresponding to d-axis and q-axis respectively, are constant.


The introduced transformations are first applied to the squirrel cage induction motor to obtain a clear and comprehensive description of the dynamic behavior. The results are then easily transformed to the PMSM by neglecting the rotor equations and considering a possible asymmetry of the rotor geometry. Regarding FOC, the dynamic and electrical behavior of cylindrical and salient-pole synchronous machines is similar to the PMSM. Differences due to the excitation and damper cage, if existing, are treated separately.

2.4 Induction Motor Model

A general three-phase ac machine is completely described by the stator and rotor voltage equations with Rs and Rr representing the resistance of stator and (stator related) rotor windings, respectively:

+

=

sc

sb

sa

sc

sb

sa

s

sc

sb

sa

dtd

iii

Ruuu

ψψψ

(2.5)

+

=

rc

rb

ra

rc

rb

ra

r

rc

rb

ra

dtd

iii

Ruuu

ψψψ

(2.6)

In the case of squirrel-cage induction motor, the phase voltages of the rotor are equal to zero. According to (1.106)-(1.107), the transformation to a reference frame rotating at the arbitrary angular velocity ωµ results in voltage equations consisting of resistive voltage drops and induced voltages due to both time variation and rotation of the flux [Kron 42], [Hen 92]. The angular velocity of the rotor is ωr. As seen from the rotor, the reference frame rotates at (ωµ - ωr) and hence the voltage equations are:

−+

+

=

sd

sq

sq

sd

sq

sds

sq

sd

dtd

ii

Ruu

ψψ

ωψψ

µ (2.7)

( )

−−+

+

=

rd

rqr

rq

rd

rq

rdr dt

dii

Rψψ

ωωψψ

µ00

(2.8)

The flux linkages are proportional to the currents linked via stator, rotor and main inductances Ls, Lr and L1h:

46 Chapter 2

+

=

rq

rdh

sq

sds

sq

sd

ii

Lii

L 1ψψ

(2.9)

+

=

rq

rdr

sq

sdh

rq

rd

ii

Lii

L1ψψ

(2.10)

From now on, the applied transformation is fixed to the rotor flux linkage aligned to the d-axis rotating at synchronous speed ωµ. Thus, the rotor flux lies entirely in the d-axis and the transformation angle γ coincides with the rotor flux angle. Subsequently, this is indicated by disregarding the indices ‘s’ of stator variables. As seen from the rotating d-axis, representing the rotor flux direction, the perpendicular flux linkage ψrq equals zero by definition.

0!=rqψ (Rotor flux lies entirely in the d-axis) (2.11)

Defining a (fictitious) magnetizing current iµ, representing the rotor flux linkage,

µψ iL hrd 1= (2.12)

and using (2.10), the rotor current can be written as:

−−

=

q

d

r

h

rq

rd

iii

LL

ii µ1 (2.13)

Substituting (2.10) and applying (2.11)-(2.13) to (2.8), the rotor equations in the field-oriented reference frame are obtained:

( )

−+

+

−−

=

µµ

µµ ωωiL

iLdtd

iii

LL

Rh

rh

q

d

r

hr

1

11 000

0 (2.14)

⇒ )()(0 11

µµ iLdtdii

LL

R hdr

hr +−= (2.15)

⇒ ( ) µµ ωω iLiLL

R hrqr

hr 1

10 −+−= (2.16)

According to (2.15), the relation between the magnetizing current iµ and flux producing current id is a first-order linear transfer function with the rotor time constant τ2 as motor parameter:


diidtdi

=+ µµτ 2 (2.17)

r

r

RL

=2τ (2.18)

Comparison of (2.4) and (2.16) yields the relation of the slip frequency ωslip and the torque producing current iq:

µµ τ

ωωi

iqr

2

+= (2.19)

Substituting (2.13) and (2.9) to (2.7), the stator voltage equations in the field-oriented reference frame are:

( ) ( )

( )

qr

hs

r

hd

r

hsds

qr

hhqsd

r

hhdsds

rqhqsrdhdsds

sqsddsd

iLL

Ldt

diLL

dtdi

LL

LiR

iLL

LiLiiLL

LiLdtdiR

iLiLiLiLdtdiR

dtdiRu

µµ

µµ

µ

µ

ω

ω

ω

ψωψ

−−+

−+=

−−

−++=

+−++=

−+=

21

21

21

11

11

11

(2.20)

( ) ( )

( )

µµµ

µµ

µ

µ

ωω

ω

ω

ψωψ

iLL

iLL

Ldtdi

LL

LiR

iiLL

LiLiLL

LiLdtdiR

iLiLiLiLdtdiR

dtdiRu

r

hd

r

hs

q

r

hsqs

dr

hhdsq

r

hhqsqs

rdhdsrqhqsqs

sdsqqsq

21

21

21

11

11

11

+

−+

−+=

−++

−+=

++++=

++=

(2.21)

With the leakage coefficient σ (Blondel-coefficient) and the stator time constant τ1 as motor parameter, the stator voltage equations (2.20)-(2.21) rewritten in state form are:

dtdi

iRui

dtdi

qs

dd

d µµ τσωστστ 111 )1( −−+=+ (2.22)

48 Chapter 2

µµµ ωτσωστστ iiRu

idtdi

ds

qq

q111 )1( −−−=+ (2.23)

s

s

RL

=1τ (2.24)

rs

h

LLL2

1−=σ (2.25)

The electrical behavior of the induction motor in continues time is completely described by two voltage equations of the stator (2.22)-(2.23), two rotor equations (2.17)-(2.19) and a torque equation. The electromagnetic torque is produced by the interaction of rotor flux linkages and rotor currents (or stator flux and stator current) [Bose 97]. Considering ψrq = 0 and applying (2.12)-(2.13) to (1.125) yields:

( ) qr

hqrd

r

hrqrdrdrqel ii

LLpi

LLpiipT µψψψ

211 ==−= (2.26)

The torque equation (2.26) clearly shows the desired torque control property of providing a torque proportional to the torque command current iq. A block diagram of the induction motor model used is shown in figure 2.3. The mechanical behavior of the drive is:

dtd

pJ

dtdJTT r

loadelωω

==− (2.27)

γ = ∫ωµ

id1/τ2

1 στ1

1/Rs

στ1

στ1

p

1 στ1

(1-σ)τ1

1/Rs pL1hLr

1/J

Tload

L1h

a,b

d,q uq

iq

iµ

ωµ

ωslip

1/τ2 ωr

ω

uq

ub

ua

(1-σ)τ1

Tel

ψrd

Figure 2.3: Block diagram of the induction motor.


Figure 2.4 shows typical transients of an induction motor supplied by a stiff grid calculated by a motor model according to figure 2.3. Obviously, the high torque and current transients can overheat the motor and are not suitable considering high-performance motion control. Subsequently, a system is required controlling the current and electromagnetic torque. Since the current is directly associated with the applied voltage, variable ac voltages, e.g. generated by an inverter, are required.

0 0.2 0.4 0.6 0.8 1

-500

0

500

t [s]

u ab [V

]

0 0.2 0.4 0.6 0.8 1-50

0

50

100

t [s]

i a [A]

0 0.2 0.4 0.6 0.8 1-50

0

50

t [s]

T el [N

m]

0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

t [s]

n [rp

m]

0 500 1000 1500 2000-40

-20

0

20

40

60

n [rpm]

T el [N

m]

Load step

Figure 2.4: Transients of a 1,5 kW induction motor (IN = 3,7A) supplied by a stiff grid (U = 400V).

Left: Line voltage uab, current ia and electromagnetic torque Tel. Right: Motor speed n and torque-speed characteristic.

2.4.1 Flux Calculation

The transformation of the voltage equations to the field-oriented reference frame requires the information of the rotor flux position γ. Furthermore, the magnitude of the rotor flux, represented by the magnetizing current iµ, is needed for control purposes. As mentioned earlier, the flux identification can be implemented as indirect or direct (measurement) method. In this chapter, as also in both industry and all recent research, only the indirect method is used. In the case of sensorless control schemes, the flux identification is based on estimation approaches employing terminal quantities such as voltages and currents in a motor model to calculate the rotor flux. Employing speed measurement, the indirect method uses the slip relation to estimate flux position relative to the rotor. Figure 2.5 illustrates this concept and shows how the rotor flux position can be obtained by adding the slip frequency ωslip to the sensed rotor speed ωr. Slip frequency and magnetizing current are calculated according to the rotor equations in the field-oriented reference frame:

50 Chapter 2

rq

slip ii

ωωτ

ω µµ

−==2

(2.28)

diidtdi

=+ µµτ 2 (2.29)

Special care has to be taken at the division of the q-axis current iq by the magnetizing current iµ: the output of the division block is set to zero if iµ is smaller than a predetermined value (e.g. 1% of the rated value). Of course, this is only critical at the initial start-up of the drive.

ib

ia

γ

a,b,c

α,β

1/τ2 α,β

d,q

1/τ2

iq

id

ωµ

ωr

iµ iµ

id

iq

ωr

ωslip

iβ

iα

Figure 2.5: Park transformation and rotor flux calculation.

2.4.2 Flux Weakening and Flux Optimization

Except in the case of the minimum loss control, the basic control strategy is to operate the induction machine with constant (rated) flux, represented by the magnetizing current iµ,R, up to the base speed ωb and at a nearly fixed terminal voltage above the base speed. The variation of the magnetizing current reference required to implement this strategy is shown in figure 2.6.

iµ*

ω

Pel,max = const Tel,max ~ 1/ω

Tel,max = const

ωmax -ωmax

iµ,R

ωb -ωb

ωr

iµ,R

iµ*

≥

≤

brr

b

br

ωωωω

ωω

:for

:for1

Figure 2.6: Flux reference as a function of the motor speed. At constant flux, the maximum electromagnetic torque is limited by the q-axis current and thus constant. The operation at constant terminal voltage beyond the base speed is carried out by reducing the flux reference inversely proportional to the


motor speed. In literature, the flux reference value is sometimes fixed to the reference speed. This is an unsuitable choice since the reference speed may change directly while the real speed changes much slower: Braking down the motor from a high speed range to zero may result in a return of the rated flux while the motor speed is still in the flux weakening range. The inverse applies at acceleration into the flux weakening range. Thus, the flux reference should be linked to the real motor speed. A major advantage of induction motors compared to PMSM is the ability to adjust the flux to meet different operating requirements. Flux weakening in the high-speed range, providing constant power mode as in figure 2.6, is widely known [Bose 97]. Less appreciated is the ability to operate above the rated flux at low speed to enhance the torque per amp and thus better use the available power supply current [Lor 90]. This is feasible since the iron losses are negligible at low motor speed. Thus, operating at a higher saturated level is acceptable and desirable if the torque per amp relation, accounting for the major portion of losses in low speed operation, is improved. In steady state, the d-axis current id equals the magnetizing current iµ and the rms stator current irms is:

3

22qd

rms

iii

+= (2.30)

According to (2.26) and postulating a maximum allowed current Imax, maximum electromagnetic torque is achieved when the current limit is distributed equally among magnetizing current iµ and maximum allowed torque producing current iq. The approach can be further refined by flux optimization. At partial load, the flux can be reduced in order to increase efficiency of the drive. Of course, the dynamic performance is reduced simultaneously since the field must return to its rated value, which is governed by the d-axis current via the relative large rotor time constant, before maximum electromagnetic torque again can be generated by the motor. This special control approach is implemented in real-time by using the output of the speed controller, determining the torque reference, as reference for both q- and d-axis current. Note that the flux reference must be positive, which can be achieved by neglecting the sign of the reference. The experimental results of such an approach are presented in figure 2.7 showing the response of an induction motor drive with flux optimization (FL-O) and constant flux control at steps of speed and load torque, respectively. Figure 2.7 clearly exhibits the reduced dynamic performance during speed or load steps. However, the enhanced torque per amp relation leads to a higher possible acceleration of the induction motor. In any case, the efficiency is vastly increased during partial load. Contrary to the assertions in literature, e.g. [Bose 97], [Van 98], [Vas 92], this feature makes the induction motor superior compared to the PMSM in

52 Chapter 2

a wide operation range when efficiency is considered. This is especially true in the range where iron losses are dominant. However, the choice of a suitable flux control strategy depends on the respective application. In recent drive systems, it can be switched over easily and in real-time to different flux control strategies.

0 0.5 10

50

100

150

t [s]|U

| [V]

0 0.5 1

0 200 400 600 800

1000

t [s]

n [r

pm]

0 0.5 1

-100

0

100

t [s]

u a, u

b, u c

[V]

0 0.5 1

-100

0

100

u a, u

b, u c

[V]

t [s]

FL-O

FL-O

FL-O iµ = const

Figure 2.7: Response of FOC with flux optimization and constant flux control at steps of speed and load

torque, respectively. Top: Speed and applied voltage. Bottom: Applied (fundamental) phase voltages.

2.4.3 Flux Control Loop

According to (2.29), the magnetizing current follows the d-axis current id via the rotor time constant τ2. Thus, the rotor flux can be adjusted by simply setting the d-axis current equal to the reference of the flux-proportional magnetizing current. However, a control loop is required to achieve high-dynamic flux control as required for advanced flux optimization schemes. The realization of the discrete flux controller for real-time applications, considering the delay of the d-axis current control loop, is shown in figure 2.8.

Ψr*

id*

Flux reference

id

eqsτ+11

Induction motor

Digital flux control

Ts /τµ Kµ

z-1

iµ*

|id*| < Imax

Integrator with anti-windup

|id*| < Imax

1 1+ τ2 s

1/L1h

iµ

iµ

Figure 2.8: Digital flux control with equivalent current control loop and current anti-windup.


Simplifying the current control loop by a first order system with an equivalent time constant τeq,

eqd

d

sii

τ+=

11

* (2.31)

and employing a PI controller, the open-loop transfer function is defined by:

11

111

20 ++

+=

ssss

KGeqτττ

τ

µ

µµ (2.32)

Within such systems, the time constant τµ of the PI controller is usually chosen equal to the largest time constant in the loop [Mayr 91]:

τµ = τ2 (2.33) This yields following closed-loop transfer function of the magnetizing current:

1

12

*

++=

sK

sK

ii

eq

µ

µ

µ

µµ

µ

τττ (2.34)

The damping factor ζ of the closed-loop transfer function can be adjusted by the controller gain Kµ. Considering second order systems as given by (2.34), the optimum value of the damping factor [Mayr 91] and resulting closed-loop gain are:

21

=ς (2.35)

⇒eq

Kττ µ

µ 2= (2.36)

⇒122

122* ++

=ssi

i

eqeq ττµ

µ (2.37)

Figure 2.9 presents an exemplary start-up and the response of the flux control loop of a 1,5 kW induction motor drive using FOC. At start-up, the d-axis current id generates a magnetic field in the rotor windings, represented by the magnetizing current iµ. According to figure 2.8, the d-axis current is limited to the maximum allowed motor current and the magnetizing current follows the d-axis current with the time constant τ2. After reaching the flux reference, the applied stator currents are dc values. Obviously, they induce no rotor current, but the generated field linked to

54 Chapter 2

the rotor windings is not zero. At time t = 0,4s, a load step is applied. In spite of a reference speed n* = 0, the frequency of the applied stator current is not zero, but equal to the slip frequency. Since the stator current frequency is not zero, they induce current in the still-standing rotor windings compensating the load torque.

0 0.25 0.5 0.75 10

2

4

6

t [s]

i µ [A

]

0 0.25 0.5 0.75 1-6

-3

0

3

6

t [s]

i s [A]

0 0.25 0.5 0.75 1-2

0

2

4

6

t [s]

i q [A]

0 0.25 0.5 0.75 1-60

-40

-20

0

20

t [s]

n [rp

m]

id

Load step iµ

ia ib ic

Figure 2.9: Flux control loop, start-up of a 1,5 kW induction motor drive using FOC and response to a load step (80% rated toque) with reference speed n* = 0. Top: Magnetizing current iµ, flux producing

current id, and torque producing current iq. Bottom: Applied stator current and motor speed.


2.5 Model of the PMSM

The stator windings of permanent magnet synchronous motors are almost equally arranged as the stator windings of induction machines. However, similar to the skewed rotor bars of squirrel cage induction machines, the stator slots are often skewed to avoid (very high) torque ripples due to slot-reluctance torque. Figure 2.10 shows the stator and rotor layout of a 3 kW prototype-PMSM with nearly sinusoidal back-EMF [Hen 98]. This motor type, referred to as a surface-inset PMSM [Seb 87], combines some advantages of both surface mounted and interior permanent magnet motors. From the rotor geometry, it can be seen that the direct axis inductance Ld is smaller than the quadrature axis inductance Lq. As will be shown, this results in an additional reluctance torque and allows for maximum torque-per-amp control and an extended flux weakening range [Jah 86].

1.95 1.83 1.71 1.59 1.46 1.34 1.22 1.10 0.98 0.86 0.73 0.61 0.49 0.37 0.24 0.12 0.00

B [T]d-axis

q-axis

magnets

-a

c

c

c

c

b

b

b b

a a

a

Figure 2.10: Stator and rotor geometry (flux plot) of a 6-pole prototype inset PMSM with the d-axis aligned with the field direction of the permanent magnets.

Permanent magnet synchronous motors are usually modeled in the rotor reference frame, i.e. Park’s d-q model. The angle of the transformation from the stator to rotor reference frame coincides with the center position of the surface magnet as the permanent magnet flux linkage lies entirely in the d-axis (figure 2.10). As seen from the rotating rotor, the magnetic path is constant and not longer rotor-geometry dependent. Thus, modeling the machine in a synchronously rotating reference frame, the inductances are no longer functions of rotor position. Mathematical models describing the PMSM motor dynamics in a rotor flux reference frame are well known [Jah 86], [Hen 92]. The model of the PMSM can also be easily derived from the equations of the squirrel-cage induction motor (2.22)-(2.23) by neglecting the rotor equations and considering the flux rotating at the synchronous motor speed ωr. A possible asymmetry of the rotor geometry is described by Ld and Ld representing constant inductances of respective directions. Applying the expression of the stator

56 Chapter 2

flux linkages (1.148)-(1.149) to stator voltage equation in the rotor fixed reference frame (1.93), the voltage equations of the PMSM are:

rsqqrr

rsdd

rsds

rsd iLiL

dtdiRu ωψ −

++=

23 (2.38)

+++= r

rsddr

rsqq

rsqs

rsq iLiL

dtdiRu ψω

23)( (2.39)

The rotor flux linkage in terms of the original rotor field ψr is active only in the d-axis. Due to the power invariant transformation to the α/β system, the factor

2/3 has to be introduced in the calculation of the flux. The flux generated by the permanent magnets is assumed to be constant. Thus, the time-derivation is zero. In the case of synchronous machines, the rotor reference frame is equal to the synchronous reference frame and the rotor voltage equation vanishes. Subsequently, the superscripts and the subscripts representing stator values are dropped. Defining the magnet flux linkage ΨMd in the reference frame used proportional to the original flux linkage of the permanent magnets ψr = ΨM,

MMd Ψ=Ψ23 , (2.40)

the PMSM without rotor damper cage is generally described by following equations:

qqrd

ddsd iLdtdiLiRu ω−+= (2.41)

Mdrddrq

qqsq iLdtdi

LiRu Ψ+++= ωω (2.42)

With the time constants τd and τq, equations (2.41)-(2.42) can be rewritten in a state form as required by the current control:

s

dd R

L=τ (2.43)

s

qq R

L=τ (2.44)


qqrs

dd

dd i

Ru

idtdi

τωτ +=+ (2.45)

s

Mdrddr

s

qq

qq R

iRu

idtdi Ψ

−−=+ ωτωτ (2.46)

The electrical behavior of PMSM is completely described by two voltage equations of the stator (2.41)-(2.42) and a torque equation. The electromagnetic torque is produced by the interaction of stator flux linkages and stator currents (or rotor flux and rotor current). Applying the expression of the stator flux linkages (1.148)-(1.149) to the torque equation (1.126) yields:

−−=

−

+=−=

ddqrq

qqdrddqsqsdsdsqel

iLLip

iLiiLipiipT

)(23

23)( ,,,,

ψ

ψψψ

(2.47)

⇒ ( )[ ddqMdqel iLLip −−Ψ= ]T (2.48)

The model of the PMSM used (figure 2.11) neglects rotor damping and iron losses. However, it can be adapted taking stator iron losses into account [Hon80], [Mel 91b].

γ = ∫ωr

idLq-Ld

1 τd 1/Rs

τq

τd

p

1 τq 1/Rs 1/J

Tload

a,b

d,q uq

iq

ωr

ω

uq

ub

ua

Tel p

1/Rs

ΨMd

Figure 2.11: Block diagram of the permanent magnet synchronous motor. In a PMSM with surface-mounted magnets (Ld = Lq), torque control can be achieved very simply, since the instantaneous electromagnetic torque can be expressed similarly to that of the dc machine as product of the q-axis current iq and magnet

58 Chapter 2

flux ΨMd. In the case of interior permanent magnets (Ld ≠ Lq), an additional reluctance torque can be exploited. Of course, the mechanical behavior of the PMSM is identical to the mechanical behavior of the induction machine:

dtd

pJ

dtdJTT r

loadelωω

==− (2.49)

In contrast to the induction motor control, the rotor flux reference frame (with transformation angle γ) is fixed to the mechanical rotor position. Employing an incremental encoder for speed/position measurement, the magnet position is indicated by an index pulse resetting all registers and guarantying the correct transformation to the d/q plane. However, the rotor must first rotate (up to one revolution) to find this index pulse. Therefore, a special start-up strategy is required [Ter 02]. As implemented by recent sensorless control approaches, the magnet direction can be detected also by an observer exploiting a possible rotor asymmetry. As the rotor flux is fixed in synchronous permanent magnet motors (determined by the magnets), there is no need to create one. Hence, when controlling a PMSM, id

* can be set zero. However, advanced control schemes may additionally exploit the reluctance torque, if available (Ld ≠ Lq). Furthermore, high-speed operation of the PMSM is constrained by the back-EMF due to a maximum inverter output voltage. However, the possible speed range can be expanded by flux weakening adding a negative d-axis current to counteract the positive magnet flux. The calculation of suitable d-axis current command values id

* is described in the next two paragraphs.

2.5.1 Optimum Torque Control

The electromagnetic torque of the PMSM is controlled by the amplitudes and phase angles of the stator currents with respect to the rotor magnet orientation. Instantaneous torque control is conveniently achieved by controlling the q-axis current iq and setting the d-axis current id to zero [Jah 86]. Both current components are dc quantities under steady-state conditions. However, the optimal control of a PMSM with Ld ≠ Lq takes advantage of the reluctance torque by introducing a suitable direct axis current component id. This results in a maximum torque-per-amp trajectory. According to the torque equation,

( )[ ]ddqMdqel iLLipT −−Ψ= (2.50)

the d-axis current id must be positive if Ld > Lq. However, a positive current increases the field, responsible for the iron losses, and it is therefore not obvious, that the efficiency increases simultaneously. Here, only Ld ≤ Lq, typical for PMSM with magnet placing of the inset-type (Ld < Lq) or surface-mounted type (Ld = Lq), is considered. With Ld < Lq, a positive contribution of the reluctance torque is achieved


by introducing a negative d-axis current component id. Considering both torque components, (2.54) yields a maximum torque-per-amp trajectory in the id - iq plane (figure 2.12), almost equivalent to the maximum drive efficiency [Jah 87], [Mor 90], [Van 98]. The optimum operation points, corresponding to maximum electromagnetic torque Tel for a given stator current value is, are obtained by substituting (2.51) in (2.50) and setting the derivative dTel/did zero:

22dsq iii −= (2.51)

( )( ) 0!

22 =

−−Ψ−= ddqMdds

del

d

iLLiipdidT

did (2.52)

⇒ )(4

)(8 222

dq

sdqMdMdd LL

iLL−

−+Ψ−Ψ=i (2.53)

⇒ ddq

Mddq i

LLi

−Ψ

−= 22i (2.54)

Figure 2.12 highlights the differences of the torque producing mechanism. At one design extreme, the reluctance term naturally disappears in a non-salient surface mounted PMSM (Ld = Lq). Torque-per-amp in the non-salient PMSM is maximized by setting id zero in all operation points.

Tel1

Tel2

Tel3

i1

i2

i3

-Tel1

-Tel3

max T/A trajectory

i1

-i1

i2

-i2

i3

-i3

ΨMdi1

i2

i3

-Tel2

id

iqTel3

Tel2

Tel1

i2i3

i1

-i2-i3

-i1id

ΨMd

max T/A trajectory iq

-Tel1

-Tel3

-Tel2

Figure 2.12: Trajectories of stator current phasors in the d/q reference frame for maximum torque-per-

amp control. Left: Non-salient pole with Ld = Lq. Right: Salient pole with Ld < Lq. The implementation of the optimum torque control is shown in figure (1.139). Note that the torque is no longer proportional to the stator current amplitude in the presence of reluctance torque. Due to lack of a closed algebraic expression, the torque command Tel

* is mapped into a current command is* via a look-up table,

predetermined from the known motor parameters. In speed control mode, the current value is

* is usually approximated by:

60 Chapter 2

Md

els p

TiΨ

≈*

* (2.55)

sign(Tel*)

Tel*

Look-up table

is*

Eq. (2.49)id*

2*2*dS ii −

iq*

Figure 2.13: Optimum torque control with look-up table. The error of the incorrect torque-current mapping is automatically corrected by the integral-acting part of the speed controller. A modification of the torque-current mapping, avoiding the look-up table, is presented in figure 2.14. The introduced approach uses the instantaneous d-axis current quantity and motor parameter variations are compensated by real-time flux adaptation.

id*

pΨMd

Lq-Ld 2

iq*

x1

x2 2

221 xx +

Tel*

id

Torque-current mapping

Motor parameter

Motor parameter

Figure 2.14: Modified torque-current mapping (Ld ≤ Lq). Figure (3.31) presents the optimum torque control at a step of the speed reference from standstill to 1000 rpm using a 3 kW PMSM. The torque command T*

el, calculated by an overlaid speed controller, is mapped into current references according to figure 2.14. As illustrated, the optimal control of the motor takes advantage of the reluctance torque by introducing a negative (Ld < Lq) direct axis current component. The corresponding speed signal is shown in same figure below, marked optimum torque control. In the same figure, a comparison is given of motor control with optimum d-axis current and no d-axis current, respectively. In spite of identical maximum current amplitude, the maximum torque using optimum torque control is higher, yielding a faster acceleration of the motor.


0 0.2 0.4 0.6 0.8 1 1.20

5

10

15

20

t [s]

i q [A]

0 0.2 0.4 0.6 0.8 1 1.2-10

-5

0

5

t [s]

i d [A]

0 0.2 0.4 0.6 0.8 1 1.2 1.40

200

400

600

800

1000

1200

t [s]

n [rp

m]

nref id = 0

Optimum torque control

Figure 2.15: Comparison of torque control with optimum d-axis current and no d-axis current at a step of the speed reference using a 3 kW PMSM with Ld ≤ Lq. Top: Optimal distribution of d- and q-axis current.

Bottom: Speed response with and without optimum torque control.

2.5.2 Flux Weakening

Since the maximum inverter output voltage is limited, a speed controlled ac motor cannot operate in speed-regions where the back-EMF, almost proportional to the field and the motor speed, is higher than the maximum output voltage of the inverter. Thus, the field must be decreased to enable higher motor speed. However, flux weakening for PMSM is not as straightforward as in the case of induction motor drives. The maximum fundamental motor voltage is limited by the power electronic inverter due to the limited dc bus voltage Udc. Furthermore, the maximum fundamental motor voltage depends on the PWM method used. Using space vector modulation, the maximum fundamental output voltage without over-modulation is, according to the later described relation (2.87)-(2.88), defined by:

dcphaseqd UUuu21ˆ

23 *2*2* ≤=+ (2.56)

This implies that for a given rotor speed, the current state phasor must lie within a corresponding boundary. When stator resistance is neglected, these limiting curves are ellipses decreasing in size when rotor speed increases [Jah 86].

22

2

2

21

Ψ+

+≥

d

Mdd

q

dq

qr

dc

Li

LLi

LU

ω (2.57)

62 Chapter 2

Consequently, rated torque cannot be maintained above the speed at which the voltage limit ellipse intersects with the maximum torque-per-amp trajectory. This speed is usually the base speed of the PMSM.

22

221

Ψ+

+

=

d

MddN

q

dqNq

dcrb

Li

LL

iL

Uω

(2.58)

Reducing the current along with the maximum torque-per-ampere trajectory above base speed, results in a fast decrease of output torque. Alternatively, the operating point can be forced to leave the maximum torque-per-amp trajectory resulting in further flux weakening at maximum current (figure 2.16). The high-speed operating region is considerably extended using the flux-weakening algorithm. The maximum speed is reached when all stator current is in the direct axis and is:

d

MdNd

dcr

LIL

UΨ

+−=

321

3ω (2.59)

Note that the presentation (2.57)-(2.59) differs from the equations in [Jah 86] and [Van 98]. They assume sinusoidal PWM and allow strong over-modulation. In fact, pure six-step block modulation is applied over the whole flux weakening range. In real-time implementations, this results in high current harmonics, significant motor losses and acoustic noise and may lead to instabilities. The maximum speed becomes theoretically infinite when:

Nd

Md IL

3=Ψ (2.60)

Introducing the voltage limited maximum power curve, without regarding the current limit [Mor 90], the current can be reduced at very high speeds when:

Nd

Md IL

3<Ψ (2.61)

In this case, the control trajectory yielding maximum output at all speeds is indicated in figure 2.16 on the right.


iq

id

ωr2

ωr3

−ΨMd d/L

imax

ω ωr1 rb=

Tel3

Tel2

Tel1

max. trajectory T/A

imax

iq

id

−ΨMd d/L

ωrbωr2

ωr3

T/Amax-trajectoryP -max trajectory

Figure 2.16: Constant torque hyperbolas, maximum torque-per-amp trajectory, current limit circle, voltage limit ellipses and optimum current trajectory. Left: ΨMd/Ld > imax. Right: ΨMd/Ld < imax.

Suitable algorithms calculating reference values for the d-axis current in the flux weakening range are described in literature, e.g. [Bose 97], [Jah 87], [Mor 90], [Van 98]. However, the algorithms presented, are based on pre-calculations postulating a constant dc bus voltage Udc. Frequently, the dc bus voltage is not constant but variable over a wide range. Then, an alternative approach is required to allow flux weakening. An automatic flux adaptation scheme may be based on following approach: The flux is automatically reduced by the d-axis current, whenever the dc bus voltage is too low to reach the reference speed. A stop criterion of the flux reduction, i.e. the reduction of the d-axis current, is given, when the q-axis current reaches its maximum allowed value. The proposed algorithm can be easily implemented using the anti-windup system of the current controller described in paragraph 2.6.3.

64 Chapter 2

2.6 Torque and Current Control

Torque control, constituting the most important motor basic control function, maps very directly into current control because of the close association between current and torque generation in any PMSM and induction motor drive. According to the basic FOC-scheme (figure 2.1), the current controller calculates the reference voltages used as input of a voltage source inverter. As will be shown, current control of PMSM and induction motor drives is very similar. The presented controller design is based on the voltage equations in the rotor flux reference frame.

2.6.1 Current Control of the Induction Motor

Considering FOC of induction motor drives, current control as described in literature, e.g. [Bose 97], [Hen 92], [Leo 85], is usually based on the stator voltage equation (2.22)-(2.23) neglecting a change of the magnetizing current. In many applications, e.g. using flux weakening and flux optimization, the rotor flux is varied very quickly and over a wide range. Therefore, the stator voltage equations are rewritten in a slightly different form. Substitutions are made by applying (2.17) to (2.22) and (2.19) to (2.23), respectively. As will be shown, this approach eliminates the flux derivative, increases the performance of the decoupling network and decreases the dominant time constant in the stator voltage equations:

µµ ττσωστστ i

Ki

KRKui

dtdi

K rq

rsr

dd

d

r 2

111 )1( −++=+ (2.62)

µµ ωτσωστστ iK

iKRK

ui

dtdi

K rr

drsr

qq

q

r

111 )1( −−−=+ (2.63)

with:

1)1(12

1 >

−+=

ττσ

rK (2.64)

Equations (2.62)-(2.63) describe a nonlinear-coupled system. Postulating the current control loop much faster than a change of the rotor speed and rotor field, decoupling of the two current controllers can be achieved by adding voltages ∆ud and ∆uq at the output of the current controller compensating the cross coupling within the motor:

sr

d

sr

dd

d

r RKu

RKui

dtdi

K∆

−=+1στ (2.65)


sr

q

sr

qq

q

r RKu

RKu

idtdi

K∆

−=+1στ (2.66)

with:

µµ τωσ i

LLiLu

r

hqsd

2

21−−=∆ (2.67)

µµ ωωσ iLLiLu

r

hrdsq

21+=∆ (2.68)

The structure of the decoupled current control loop is shown in figure 2.17. The realization of the discrete current controller for real-time applications considering voltage limitation is shown in figure 2.22.

PI-controller

ud*

σLs

id*

2

2 τ L

L r

h

r

h

LL2

ωµ

ωr

id

∆ud

iµ

iq

∆uq

PI-controller

uq*iq

*

1 11

1 +sK

RKr

srστ

11

+sτ σ

uq iq

11

+sτ σ

ud id

1 11

1 +sK

RKr

srστ

Inverter model

Induction motor model

Figure 2.17: Current control and decoupling for the FOC of induction motor drives. The decoupled current control loop contains a dominant time constant and a time constant τσ. The latter is the sum of equivalent time constants representing measurement filter τfilt and time required for the conversion of the reference voltage to the inverter output voltage, which is mainly depending on sample time Ts and PWM frequency fPWM = 1/TPWM:

filtPWMs TT ττσ ++= (2.69)

Considering the proposed decoupling and employing a PI controller (figure 2.18), the open-loop current transfer function of both d-axis and q-axis is:

66 Chapter 2

11

1

1111

0 ++

+=

ssK

RKssKG

r

sri

ii

στστττ (2.70)

PI-controller

ud*id

*

id

∆ud

iq ∆uq

PI-controller

uq*iq

*

1

111 +s

KRK

r

srστ

11

+sτ σ

uq iq

11

+sτ σ

ud id

1

111 +s

KRK

r

srστ

Inverter model

Induction motor model

∆ud

∆uq

Figure 2.18: Current control loop for of induction motor drives. The time constant τi of the PI controller within such systems is optimally chosen to neutralize the largest time constant in the loop:

ri K

1σττ = (2.71)

This yields following closed-loop transfer function:

σσ

σ

τττττ

isr

iisr

i

qd

qd

RKKssRK

Kii

++=

2*

,

, 1 (2.72)

Equation (2.72) describes a second order system. The damping factor ζ of the transfer function can be adjusted by the controller gain Ki. Optimum value of damping factor and resulting closed-loop gain are [Mayr 91]:

21

=ς (2.73)

⇒σττ

2isr

iRKK = (2.74)


For control purpose of overlaid control loops, e.g. the speed or flux control loop, the closed-loop transfer function (2.72), employing the calculated controller gain Ki given in (2.74), is often simplified by a first order system with an equivalent time constant τeq:

11

1221

22*,

,

+≈

++=

sssii

eqqd

qd

τττ σσ

(2.75)

with:

σττ 2=eq (2.76)

Figure 2.19 shows a typical step response of both d-and q-axis current using a 1,5 kW induction motor. Additionally, the current control loop simplified by a first order system with the equivalent time constant (τeq ≈ 1ms) according to (2.76) is drawn in the same figure. As indicated by the current signals marked every sample time-step (Ts = 200µs), only a few samples are required to reach the current reference.

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

2

4

6

t [s]

i q [A]

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

1

2

3

4

t [s]

i d [A]

i*q

i*d

iq,eq iq

id id,eq

Figure 2.19: Step response of the current control loop and equivalent time constant (τeq ≈ 1ms).

2.6.2 Current Control of the PMSM

The current control of the PMSM is based on the stator equations (2.45)-(2.46) in a rotor flux reference frame employing PI-controllers and a decoupling network according to figure (1.71):

68 Chapter 2

s

d

s

dd

dd R

uRui

dtdi ∆

−=+τ (2.77)

s

q

s

qq

qq R

uRu

idtdi ∆

−=+τ (2.78)

with:

qqrd iLu ω−=∆ (2.79)

Mdrddrq iLu Ψ+=∆ ωω (2.80)

Due to the similarity of the voltage equations (2.77)-(2.78) compared to the equations valid for the induction motor (2.65)-(2.66), the current control loop and the parameters of the PI controllers are obtained in the same way. The calculation method is explained in detail in the previous paragraph. Considering a possible rotor asymmetry, the different parameters of d-axis and q-axis current controller are:

ddi ττ =, (2.81)

σττ

2,ds

diRK = (2.82)

qqi ττ =, (2.83)

σττ

2,qs

qi

RK = (2.84)

PI-controller

ud*id

*

ωr

id

∆ud

iq

∆uq

PI-controller

uq*iq

*

11

+sτ σ

uq iq

11

+sτ σ

ud id

Inverter model

PMSM model

111

+sτ R qs

111

+sτR ds

Ld

-Lq

ΨMd

Figure 2.20: Current control and decoupling for the FOC of PMSM.


The resulting closed-loop transfer function and its first order equivalent are equal to the transfer functions of the induction motor:

11

1221

22*,

,

+≈

++=

sssii

eqqd

qd

τττ σσ

(2.85)

with:

σττ 2=eq (2.86)

This property reflects the ambivalence of the FOC and the large influence of delays within the current control loop as e.g.: measurement filter, sample time, PWM frequency and signal lag in data transmission.

2.6.3 Discrete current controller with anti-windup

Anti-windup systems for the speed control loop, providing current and torque limitation of the inner control loops without overrun of the integrator within the overlaid speed controller, are widely known in literature [Bose 97], [Van 98]. Less appreciated is an anti-windup system within the current controller since the maximum voltage is limited by the power inverter itself. Furthermore, the maximum output voltage can be controlled by the earlier mentioned flux weakening and flux optimization techniques. However, a variable dc bus voltage or (involuntary) voltage drops of the power supply may result in instabilities without such a system. Especially the integrator of the q-axis current control loop will overrun when the system experiences voltage limitation by the inverter. Then, the supplied voltage is insufficient to maintain the q-axis current and the motor will accelerate uncontrolled if the load decreases or the dc bus voltage returns to higher values. This is obviously since the over-saturated integrator value must be decreased first before a suitable voltage, maintaining the q-axis current, can be applied. Compared to the speed control loop, the anti-windup of the current control loop is more complicated due to the interaction of the two current controllers. Furthermore, the voltage limitation is physically located outside of the digital motion control. Considering space vector modulation, the controllable range of linear modulation terminates when the reference voltage phasor Uref in the α/β reference frame touches the hexagon (figure 2.21), opened up by the six active switching state vectors. Control in the intermediate range, limited by the outer circle around the hexagon, can be achieved by over-modulation.

70 Chapter 2

Uβ

u1

Over- modulation

Uref

Uα

u4

u3 u2

u5 u6

Outer circle

Inner circle

Figure 2.21: Hexagon formed by the basic space phasors and over-modulation range. Since the maximum output voltage is limited by the inverter, also the reference voltages should be limited to achieve anti-windup of the current controller. This limit depends on the dc bus voltage Udc and the acceptable range of over-modulation indicated by the factor Kc:

dccphaseqd UKUuu ≤=+ *2*2* ˆ23 (2.87)

Suitable values for Kc are given by the inner and outer circle around the hexagon shown in figure 2.21. A geometrical calculation yields:

32

21

≤≤ cK (2.88)

For sinusoidal PWM techniques, the factor Kc must be multiplied by 21 resulting in a much lower maximum fundamental output voltage of the inverter. However, the maximum voltage should not be exploited since over-modulation results in a firmly increased spectrum of lower current harmonics especially of the 5th, 7th and 11th harmonics. A limitation to the inner circle rejects over-modulation but the maximum fundamental output voltage is then 15% lower. Usually, a compromise between current harmonics and voltage exploitation is made (e.g.: Kc = 0,75) and a limiting factor is chosen according to the given application, respectively. The entire current control loop with anti-windup of the reference voltage can be considered as a master-slave system. The controller is always forced to maintain the flux at the reference value. The d-axis voltage, controlling the flux of the motor, is not limited, since the back-EMF of the motor is predominantly active in the q-axis. Thus, an anti-windup system is only necessary in the q-axis of the current control


loop. In contrast to the speed control loop, the maximum allowed value of the q-axis voltage is not constant but depends on the momentary d-axis voltage as well as on the dc bus voltage level. Figure 2.22 presents the real-time implementation of the entire current control loop with a variable limit of the q-axis voltage. The proposed control scheme is valid for both PMSM and induction motor drives. Reaching the voltage limit yields a forced reduction of the q-axis current. Consequently, the speed cannot reach the reference value. In order to return to a regular operating mode, an adaptation of the flux must be applied. Suitable (predetermined) adaptation schemes are flux weakening and flux optimization. Figure 2.23 illustrates the different operating conditions of an induction motor drive during voltage limitation, flux weakening and flux optimization. In the beginning of the experiment shown in figure 2.23, flux weakening is disabled and the speed cannot reach the reference value due to the limited maximum output voltage. However, the implemented anti-windup system guarantees stable operation. Flux weakening at t = 0,55 s reduces the required motor voltage and the drive returns to a regular operation mode. Finally at t = 1,2 s, the system switches in flux optimization mode, increasing the efficiency of the drive. Among these predetermined approaches, an automatic flux adaptation scheme is possible. The flux may be automatically reduced by the d-axis current, whenever the dc bus voltage is too low to reach the reference speed. However, if the load exceeds the maximum possible electromagnetic torque, which can be generated at the reduced flux, the reference speed cannot be reached. Thus, a stop criterion of the flux reduction is given, when the q-axis current reaches its maximum allowed value. Then, an expert must decide either to increase the dc bus voltage or to reduce the load in order to reach the speed reference.

z-1

Ts /τq Kq

sign(u2)

u1 > |u2|

iq

u1

iq*

Ts /τd Kd

id

id*

2*2dc uu −Kc

u2

Udc uc

ud*

ud*

uq*

∆ud

∆uq

z-1

Figure 2.22: Current control loop with voltage limitation (variable anti-windup system).

72 Chapter 2

1.20 0.2 0.4 0.6 0.8 1 1.4 1.6 1.8 20

1000

2000

t [s]

n [r

pm

]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1

2

3

t [s]

i µ [A

]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

100

200

300

t [s]

|Ure

f| [V

]

Voltage limit

Flux weakening Flux optimization

Figure 2.23: Experimental results during current control with voltage limit, flux weakening and flux optimization of an induction motor (Udc ≈ 400 V). Top: Speed reference and motor speed. Middle:

Magnetizing current. Bottom: Amplitude of the reference voltage in the α/β reference frame.


2.7 Speed Control

The motion control algorithms are based on the assumption of nearly ideal electromagnetic torque control. This assumes ideal field orientation and a current controller with bandwidth considerably beyond required motion control bandwidth. The speed control output feeds a cascaded current regulated field oriented drive, creating electromagnetic torque of the motor. In speed control mode, the torque reference is calculated by the speed controller. This reference can be easily mapped into a current command by (2.26) and (2.48) for induction motor and PMSM (see also figure 2.14), respectively. The easiest controller type is a proportional gain Kn converting the speed error in a torque reference value Tel

*. This structure is shown in figure 2.24. Keeping torque and current within predetermined boundaries is achieved by limiting the output of the speed controller.

ω* Tel*

Speed reference

ω Kn

Tel

Tload

ω

eqsτ+11

sJ1

|Tel|<Tmax

ac motor Digital motion control

e

Figure 2.24: Speed control loop with proportional gain. The mechanical property of both PMSM and induction motor drive is:

dtd

pJ

dtdJTT r

loadelωω

==− (2.89)

Due to the linear relation between torque and q-axis current, the time constant τeq of the torque and current control loop are identical. According to (2.85), the transfer function of the electromagnetic torque is:

eqq

q

el

el

sii

TT

τ+==

11

** (2.90)

Neglecting load torque and the current/torque-limiter, the closed-loop transfer function of the speed is easily obtained by multiplying the open-loop transfer functions in figure 2.24 with the speed error e:

74 Chapter 2

( )Jss

KJss

Keeq

neq

n1

111

11 *

τωω

τω

+−=

+= (2.91)

⇒ 1

11

111

11

1

2*

++=

++

+=

sKJs

KJ

JssK

JssK

nn

eq

eqn

eqn

ττ

τωω (2.92)

⇒ 200

2

20

2* 2

1ωως

ω

τττω

ω++

=++

=ss

JKssJ

K

eq

n

eq

eq

n (2.93)

Equation (2.93) describes a simple second order system. The damping ζ of the transfer function can be adjusted by the controller gain Kn. Optimal values of damping and resulting closed-loop gain are [Mayr 91]:

21

=ς (2.94)

⇒ eq

nJK

τ21

= (2.95)

Note: If the control loop (figure 2.24) is based on speed signals in rpm-unit, the controller Kn gain (2.95) has to be multiplied by the factor 2π/60. Assuming a reference speed higher than the measured speed, the speed error is positive (e > 0) and the (very fast) current/torque control loop generates a positive electromagnetic torque Tel = Kn e > 0. Without load applied to the motor, also the input of the integral acting part 1/(J s) in figure 2.24 is positive and the electromagnetic torque accelerates the drive until the motor speed equals the reference signal, i.e. the speed error disappears. The same applies for a negative speed error. Thus, a speed controller consisting of a simple proportional is gain sufficient if no load is applied. However, in the presence of load Tload ≠ 0, this simple control approach yields a steady state error. In steady state, the load torque equals the electromagnetic torque

nelload KeTT == (steady state) (2.96)

and the input of the integral acting part 1/(J s) in figure 2.24 is zero, i.e. the motor speed stays constant (no acceleration). According to (2.96), this simple control


approach yields a steady state error proportional to the load torque and the inverse of the loop gain:

⇒ ( )n

loadt K

Te =−=∞→

ωω * (2.97)

However, this problem can be overcome by load torque rejection. As shown in figure 2.25, the (estimated) load torque, compensating the real load, is added to the output of the speed controller. This very robust control approach, requiring the exact information on the load torque but vastly enlarging the dynamic stiffness of the drive, is explained in detail in a subsequent chapter.

Tel*

ω*

ω |Tel|<Tmax

α̂J Load torque rejection

Proportional gain

Tel Tload

Kn iq*iq

* = f(Tel*)

Current mapping

qr

hel ii

LL

pT µ

21=

( )[ ]ddqMdqel iLLipT −−Ψ=

Induction motor:

PMSM:

Figure 2.25: Speed control with proportional gain, load torque rejection and torque-current mapping. Accurate information of the load torque is not always available. Then, an integrator must be added to the speed control loop compensating for the error due to the load. As long as the speed error is not zero, the integrator adds up the error and varies the electromagnetic torque. Consequently, the speed error vanishes in steady state. The discrete structure of the PI controller is shown in figure 2.26.

Ts /τn Kn

z-1

Tel*ω*

|Tel|<Tmax

Integrator with anti-windup

|Tel|<Tmax ω

Figure 2.26: Discrete structure of the PI speed controller with a simple anti-windup system. The PI velocity loop is the “de facto” industry standard. Suitable values concerning controller gain Kn and time constant τn are obtained according to symmetrical optimization [May 91]. Due to the current/torque limitation, the integrator within the speed controller must be equipped with an anti-windup system avoiding uncontrolled overrun of the integrator. A simple system is shown in figure 2.26 limiting the integrator value to the maximum predetermined electromagnetic torque. More advanced anti-windup systems restrict the value of the integrator to the load

76 Chapter 2

torque, e.g. estimated by an observer, whenever the limit of the torque command is reached. This results in a very smooth step response with almost no speed overshoot. To improve the dynamic performance, the speed controller can be equipped with a derivative part. However, pure PID controllers are rarely used because calculating acceleration from a speed/position signal suffers from resolution limitations so severe as to make it impractical: Measurement and quantization errors are vastly amplified which may result in a strong increased spectrum of current/torque harmonics. Thus, a PID-speed control requires a very accurate measurement device. Alternatively, the acceleration signal calculated by an observer can be used as input of a modified PID controller (figure 2.27). Note that PID-position control is quite different from PID-speed control. The D term of a PID speed controller is proportional to acceleration whereas the D term of a PID position controller is proportional to speed. Other controller types frequently used are PDF and PI+, also termed PDFF controller [Ell 99]. Their structures, similar to the PI controller, are shown in figure 2.27. PI+ is a general controller including PDF and PI control (when KF is 0 and 1, respectively). A PI controlled system is more responsive to speed commands. However, the change in structure allows PDF to have higher integral gains while avoiding overshoot. While KF can be set to these extremes, it can also be set anywhere in between. A comparative study and optimizing KF are treated in [Ell 99].

ω*

ω

Kd

derivative part

PI with anti windup

sTz 11 −−

Tel*

|Tel|<Tmax

αe

Ts /τn

KF

z-1

Tel* ω*

|Tel|<Tmax

PDF-controller

|Tel|<Tmax

ω

Kn

Figure 2.27: Different speed controller types. Left: Modified PID controller. Right: PI+ controller (PDF controller indicated by the gray box).

Fuzzy logic applications in power electronics and drives are relatively new. A comparison of the fuzzy-controlled system performance with that of a PID control is given in [Li 89] demonstrating the superiority of the former. The different speed controller types (PI, PID, PI+ and PDF) have been implemented and tested in [Ter 02]. Experimental results have shown a clear inferior performance compared to that, which has been achieved with a simple proportional gain together with the load torque rejection approach. Usually, speed controllers are optimized regarding an optimum speed response. To the contrary, the proportional gain (2.95) is optimized still regarding optimum speed response while the observer calculating the load of the load rejection approach is optimized with respect to an optimum disturbance rejection. The proportional gain (2.95) clearly exhibits the only drive parameter possibly taken into account for adaptation of Kn: a variation of the inertia J; With varying inertia,


e.g. rolling mills, conveyors, winding machines, Kn can be tuned in real-time, e.g. if the inertia is estimated by an observer. However, only an underrated inertia is critical regarding drive stability. An overrated inertia yields no instabilities, but the speed control loop is not optimally tuned, i.e. resulting in a slow response to change of the speed reference. Figure 2.28 presents a typical speed response to a step of both speed reference and load torque using a common PI-controller and a simple proportional gain with load torque rejection, respectively. Avoiding an integral-acting part within the speed controller, the overshoot at steps of both speed reference and load torque vastly decreases or even vanishes.

0 0.2 0.4 0.6 0.8 10

500

1000

1500

t [s]

n [r

pm]

0 0.2 0.4 0.61400

1450

1500

1550

t [s]

n [r

pm]

Reference

Load torque rejection

r

Figure 2.28: PI-controller versus load torq

load step (90% rated torque) using a 1,5 k

2.8 Conclusions

Fundamentals of high-performance motion contSuitable motor models are derived and different cthis chapter. Squirrel-cage induction motor and due to the similarity of the electrical and mliterature, the model of the induction motor is rewith major consequences. This approach eliminaperformance of the decoupling network and decrthe stator voltage equations. This chapter prcontroller design, taking care for the viability of t The basic control scheme is refined systematicallstep. Different flux weakening and flux optimizweakening in the high-speed range, providing coin literature. Less appreciated is the ability to opto enhance the torque per amp and thus bette

PI-controlle
0.8 1
ue rejection at a speed and W induction motor drive

rol employing FOC are explained. ontrol approaches discussed within

PMSM are handled simultaneously echanical behavior. Compared to written in a slightly different form

tes the flux derivative, increases the eases the dominant time constant in ovides a clear description of the he real-time implementation.

y adding additional features step by ation schemes are described. Flux-nstant power mode is widely known erate above rated flux at low speed r use the available power supply

78 Chapter 2

current. The approach is further refined by flux optimization and optimum torque control. Widely neglected in literature is an anti-windup system within the current controller since the maximum voltage is limited by the power inverter itself. However, the presented approach is essential considering the dc bus voltage variable over a wide range. Furthermore, the proposed system is refined by an automatic flux adaptation scheme. Different speed controller types have been implemented and tested and an introduction in advanced speed control is given within this chapter. Experimental results have shown a clear superior performance of a speed controller with load torque rejection.

3. Advanced Speed Control & Speed Measurement

3.1 Introduction

The de facto industry standard for motion control is to use a PI velocity loop and a proportional position loop. One input of the motion controller is usually an optical position feedback device. The optical incremental encoder together with a digital counter interface is preferred over the less accurate resolver feedback. The encoder produces a certain number of sine or square wave pulses for each shaft revolution. The higher the number of pulses, the better the resolution becomes. Subsequently, the cost of the unit increases. Incremental encoders as position sensors are not able to generate simultaneously a velocity signal. The speed of the drive is obtained by signal processing in hardware or software. The speed can be determined by numerical differentiation of the position information, counting the inner clock pulse time between encoder pulses or by applying observer based estimation approaches. However, the first approaches are used in combination with a filter to smooth the speed signal resulting in phase lags, while the latter method described in literature [Bose 97], [Lor 99] is motor parameter depending and clearly sensitive to the accuracy and dynamic of the torque estimation. This chapter reviews former methods that have become commonplace and explains a new approach of speed estimation. Along with the speed, also rotor position and acceleration are estimated. The implemented algorithm is based on a linear Kalman Filter. This approach is shown to offer a significant improvement of the drive performance. The noise reduction is especially relevant for servo drives due to the high current/torque loop bandwidths required for highly dynamic operation. The proposed observer is vastly insensitive to parameter variations and avoids numerical differentiation as well as signal lags. Furthermore, the proposed algorithm can be implemented easily in software with a negligible requirement of computation time. As will be shown, the entire drive performance is improved by adding acceleration feedback. Then, the speed controller, consisting of a simple proportional gain mapping the speed error into a torque command, is optimized with respect to an optimum speed response. The feedback of the estimated load torque yields an optimized system regarding disturbance sensitivity.

80 Chapter 3

The discussion extends to the design and implementation of a linear Kalman filter for position, speed and acceleration estimation using an incremental encoder, the implementation of an advanced speed control loop and ends by presenting representative results simplifying a interpretation of the observer function. This chapter can be also regarded as a smart introduction into observer theory.

3.2 Speed Measurement Devices

Continuous and precise control of speed with long-term stability and good transient performance is an important feature of machine control. The speed resolution of feedback devices is the most important property, since erroneous speed-signals are directly reflected in torque ripples through the speed controller mapping the speed error into a torque command. The most popular speed measurement device is the mechanical dc tachogenerator, which has the advantages of small size, simple connections and good linearity. In fact, the tachogenerator is usually a dc motor separately exited by permanent magnets. Since the resistive voltage drop is negligible when connected to a highly resistive measurement device, the output voltage is equal to the back-EMF and proportional to the rotor speed. The common analog tachometer is the only feedback device providing directly a speed signal. All other devices are in fact position measurement devices indirectly providing velocity information. The analog tachometer has no specific limit of speed resolution and owns still a superior performance in ultra-low-speed applications. However, its mechanical commutator is often undesired because of the regular maintenance required. Mechanical tachogenerator normally use silver-graphite brushes for commutation. Approximately, they have quoted life time of 109 revolutions in industrial conditions, equivalent to 347 days long-term function with a motor running on average at n = 2000rpm. When interruptions of the operating cannot be tolerated or when the transducer is used in inaccessible locations, constant maintenance is difficult. Contrary to the analog tachogenerator, brushless resolvers have a robust rotor construction providing reliable maintenance-free operation at a wide speed range. Because the resolver is an absolute position device, it intrinsically provides velocity information. A resolver is, in principle, a rotating transformer fed with a high frequency voltage ue (figure 3.1). The resolver is an absolute position feedback device, even able to determine directly the position after a power supply interruption. Within each electrical cycle, the voltages ua1 and ua2 maintain a constant (fixed) relationship determining the absolute rotor position Θ. The excitation voltage ue may be coupled to the rotating winding by slip rings and brushes, though this arrangement is a drawback. In maintenance-

Advanced Speed Control & Speed Measurement 81

free applications, a brushless resolver may be used so that the excitation voltage is inductively coupled to the rotor windings.

ua1

Θ = ∫ω dt

ua1

ua2

ue

Θsin~2 ea uu Θcos~1 ea uu

)2sin(0 ee fuu π=

=

ΘΘ

=Θ2

1arctancossinarctan

a

a

uu

Stator:

Rotor:

Figure 3.1: General operating principle of a resolver. In former times, resolvers were the dominant high-accurate position feedback devices (resolution: 1,5-5mrad). Presently, resolvers are largely replaced by less expensive incremental encoders providing equal or even better accuracy. For highly accurate applications, sine encoders, similar to incremental encoders with sine-wave interpolation (e.g. interpolation factor 256), are used.

3.3 Incremental Rotary Encoders

The basic operation principle of incremental optical encoder has changed little over the past 50 years. Conventional incremental encoders consist of a light source, a stationary mask with a spoke pattern of clear slots that shutters the light through an identical pattern on a rotating disc, a photoelectric diode on the side of the disc opposite to the light source and a signal processor. As the disc rotates, light either passes through one of the slots or it falls between two slots and is blocked. The flashes of light passing through are detected by the photo detector and interpreted by the signal processor. The simplest type of incremental encoder is a single-channel tachometer encoder that produces a certain number of sine or square wave pulses for each shaft revolution. The pulses are fed into an up/down counter to produce a digital word for the rotor position. These relatively inexpensive devices are well suited as velocity feedback sensors in medium to high-speed control systems, but run into noise and stability problems at extremely slow velocities due to quantization errors. In addition to low speed instabilities, single-channel tachometer encoders are also incapable of determining the rotation direction and thus cannot be used as position sensors. Phase-quadrature incremental encoders overcome these problems by adding a second channel with a displaced arrangement of the detectors. The resulting pulse trains are 90° out of phase (figure 3.2).

82 Chapter 3

1 cycleChannel A

Channel B Detector Mask Grating Light source

Figure 3.2: Digital incremental encoder and signals.

This technique allows the decoding electronics to determine which channel is leading the other and hence ascertain the direction of rotation, with the added benefit of increased resolution. Since each full cycle contains four transitions, or edges, an encoder provides four edges or counts per encoder line. The resolution of the incremental encoder depends on the number of encoder lines:

linesencoderofnumberres ×=Θ

42 π (3.1)

The typical incremental position resolution used for industrial servo drives is Θres = 1,5 mrad (1024 lines). Lower and higher resolutions are chosen for special applications.

3.4 Velocity Estimation via Numerical Differentiation

The least complicated and often applied technique of speed calculation using an incremental encoder interface is obtained by numerical differentiation of the position information

s

kk

Tdtdk 1)( −Θ−Θ

≈Θ

=ω (3.2)

where Ts is the sample time and k is the sample number index. Problems related to this approach are its lagging nature (small for high sample rates) of the velocity estimation and the limited velocity resolution:

s

res

Tperiodsampleresolutionpositiondigitalresolutionspeed Θ

== (3.3)

The typical incremental position resolution used for industrial servo drives is 1024 lines. For this encoder type and a sample time Ts = 200 µs, the resolution of the speed signal is:


1min24.73min1sec60

sec0002.01

102441 −=⋅

=revresolutionspeed (3.4)

A representative shape of the speed signal calculated by numerical differentiation for a real motor speed of 50 rpm and 100 rpm respectively is shown in figure 3.3. The resolution deteriorates even at higher sample rates, whereas a high sample rate would be advantageous to get high bandwidth current/torque control loops.

0 0.002 0.004 0.006 0.008 0.01 0.0120

50

100

150

t [s]

n [rp

m]

real speed measurement

Figure 3.3: Quantization effect by numerical differentiation (Ts = 200 µs). A speed reference together with such a noisy speed signal forms the input of a speed controller calculating the torque commands. Therefore, this quantization error results in faulty torque commands, increased torque ripple, acoustic noise and motor heating by current harmonics. Especially a speed controller with a derivative part, e.g. a PID controller, should be avoided. Then, mainly the higher harmonics of this quantization noise are amplified. The speed estimation is often improved by digitally filtering the past values to smooth the signal. Both finite impulse response (FIR) and infinite impulse response (IIR) digital filters may be used. Such filtering can produce a very accurate, high-resolution speed signal at steady state, but the instantaneous accuracy and phase fidelity is compromised. The gain of the speed controller has to be reduced directly or indirectly due to stability considerations when applying an additional digital filter in the speed control loop. Thus, the transient performance during acceleration and the dynamic stiffness of the drive are reduced. It must be noted, that filtering of a signal generally is an unacceptable ripple reduction solution for servo drives, where the dynamic performance cannot be compromised.

84 Chapter 3

3.5 Velocity Estimation via Clock Pulse Counting Time

Especially for low motor-speed operation, an estimation technique based on clock pulse counting time is often preferred and implemented. This approach uses the relatively high temporal resolution of the microprocessor clock Tclk to measure the time between two encoder pulses [Bose 97]. The speed estimation is obtained by counting the number ∆N of clock pulses between the most recent pulses from the digital encoder interface:

clk

res

TNtdtd

∆Θ

=∆∆Θ

≈Θ

=ω (3.5)

This clock time measurement improves the resolution especially for low-speed operation. The incremental velocity resolution is:

( ) res

clk

clkres

clk

clk

res

clk

res TT

TTNTN

resolutionspeedΘ

≈+Θ

=+∆

Θ−

∆Θ

=22

1ω

ωω (3.6)

In contrast to numerical differentiation, this estimation technique does not depend on the sample period Ts. The number of clock pulses appearing between two encoder pulses is almost proportional to the square of the motor speed. Considering (3.6), the resolution is vastly improved at low velocities. Problems related to this approach are inherently large word lengths including possible register overflow at zero speed and problems caused by the manufacturing process inaccuracy. The individual line spacing directly affects the performance of this method and any lack of uniformity shows up as a speed jitter on the drive. The jitter can be softened by implementing an additional digital filter with the same consequences as mentioned in the previous subsection. Furthermore, the used development platform does not support this approach.

3.6 Observer based Velocity Estimation

In many servo drives, the dynamic performance of the velocity loop cannot be compromised. The problems due to quantization errors and estimation lag of formerly described approaches to obtain velocity feedback can be greatly reduced by applying an observer-based velocity estimation method. Steps in that direction have been taken, though, e.g. [Bose 97], [Lor 91] and [Lor 99]. However, they are motor parameter depending and clearly sensitive to torque estimation accuracy and torque dynamics. Torque errors will lead to velocity estimation errors since the observer


model is being fed the torque command whereas the real motor experiences the actual torque. Subsequently, a new method of speed estimation is proposed. Simultaneously, rotor position and the acceleration of the drive are estimated. This approach is shown to offer a significant improvement of the drive performance. The observer is hardly sensitive to parameter variations and avoids numerical differentiation as well as signal lags. The calculation of the observer parameter is based on the linear Kalman Filter algorithm considering both measurement error and model inaccuracy. These disturbances are explicitly evaluated. The discussion starts with the selection of a system model and the design of the affiliated linear Kalman filter using an incremental encoder interface as input and ends with the implementation of the proposed observer, influence of motor parameters and presentation of representative results.

3.6.1 System model

The mechanical motor speed ω is equal to the time-derivative of the mechanical rotor position Θ, whereas the acceleration α is obtained by the time-derivative of the speed. The following mechanical equations are valid in any case and without any inaccuracy:

dtdΘ

=ω (3.7)

dtdωα = (3.8)

In addition, the acceleration equals the difference between the electromagnetic Tel and load torque Tload related to the moment of drive inertia J.

αω JdtdJTT loadel ==− (3.9)

The load torque is generally an unknown variable, by definition constant at steady state. It creates a disturbance of the speed control loop, compensated by the affiliated controller. Therefore, a variation of the load torque (not the load torque itself) is handled as model inaccuracy. This inaccuracy is neglected at the moment, but is taken into account afterwards at the calculation of the observer feedback matrix. The acceleration (3.9) rewritten in state form as required by the system model yields the time-derivative of the acceleration:

noisemodel+

=

−=⇒

JT

dtd

JTT

dtd

dtd elloadelα (3.10)

86 Chapter 3

Contrary to many other observer designs, the differential equation of the acceleration (3.10) requires not the absolute torque value as input, but the time-derivative of the electromagnetic torque. Therefore, the influence of both an incorrect estimation of the electromagnetic torque due to electrical parameter variations and an incorrect identification of the drive inertia vanishes in steady state. Furthermore, the influence of incorrect parameter estimation is small compared to a potential load variation. Thus, the erroneous electromagnetic torque calculation and inertia identification are handled as model noise. Postulating a very small sample time (Ts ≈ 100-200 µs), the following transformation from continuous time (3.7)-(3.10) to the discrete time state space causes a negligible error. Nevertheless, the discretization error can be considered at the calculation of the feedback matrix through the noise covariance matrix of the system model as well as the influence of the load variations neglected in (3.10).

kskk T ω̂ˆˆ1 +Θ=Θ + (3.11)

kskk T αωω ˆˆˆ 1 +=+ (3.12)

( 1,,11ˆˆ −+ −+= kelkelkk TT )J

αα (3.13)

In (3.7)-(3.13), a stiff mechanical coupling between incremental encoder and the rotor of the drive is assumed. If this is not the case, the mechanical differential equations of motion have to be considered in more detail. The selection of the system model is completed by choosing the mechanical rotor position, motor speed and acceleration as state variable xk and the inertia related electromagnetic torque variation as input uk. The output vector yk consists of the encoder position. The structure of the system model, implemented in discrete form, is presented in figure 3.4.

kkk uxx BA +=+1 ; JTT kelkel

k1,, −−

=u ;

k

kx

Θ=

αωˆˆ

ˆ

(3.14)

with: ; (3.15)

=

10010

01

s

s

TT

A

=

100

B

kkkxy C=Θ= ˆ ; with: ( )001=C (3.16)


overflow protection

Θk+1→Θk

ωk+1→ωk

αk+1→αk

Electromagnetic torque Tel

Output Matrix C

Model matrix AInput matrix B

Θ ^ Ts

|Θ|<π z-1

Ts

z-1

z-1

z-1

1/J

(1,0,0)

Figure 3.4: Structure of the system model for velocity estimation. The rotor position is limited in speed control mode to |Θk| < π avoiding an overflow of a register occurring due to a potential rotation of the rotor in one direction over a long period of time. This non-linearity reflects no negative influence on the proposed observer. The overflow protection has to be dropped in position control mode, requiring the absolute position. As shown in (3.17), the determinant of the observable matrix QB is unequal to zero. Therefore, the system is observable and it is possible to identify the system state unambiguously by the output of the system [Mayr 92].

( ) 3

22101001

det s

ss

s TTT

T =

=

=

2B

CACAC

Q (3.17)

It should be noted, that the observation (3.17) is independent of any parameter variations. The sample time is pre-determined depending on the required computation time and should be as small as possible. Unfortunately, Figure 3.4 clarifies, that without feedback of a measured signal, the open-loop observer fails at any system disturbance. Such disturbances are caused by applying a load torque or an erroneous electromagnetic torque calculation. For instance setting the input Tel to zero, all estimated values never change and remain at the initial values. Therefore, the output vector has to be compared to all available measured signals and the difference must be used to correct the state vector of the system model. This results in a closed loop observer as shown in the next subsection.

88 Chapter 3

The closed loop observer proposed is based on the feedback of the measured encoder position. If additionally speed information is available from a sensor, the error between estimation and measurement of both speed and position are used by the observer to get information on the drive acceleration. This can improve vastly the drive performance and dynamic stiffness. However, a digital incremental encoder does not inherently produce an instantaneous velocity signal.

3.6.2 Closed loop observer and parameter calculation

The closed loop observer is obtained by feedback of the unaccompanied position information given by the incremental encoder. This information is compared with the estimation of the encoder position. As illustrated in the signal flow graph of the closed loop observer (figure 3.5), the difference is used by the feedback matrix (K1, K2, K3) to correct the state vector xk of the system model.

Ts

|Θ|<π z-1

overflow protection

Ts

z-1

z-1

K1

|∆|<π

K2

K3

overflow protection

Θk+1→Θk

ωk+1→ωk

αk+1→αk z-1

1/J

Electromagnetic torque Tel

αω ˆ,ˆ,Θ̂

Measured encoder position

Output

Kalman gains Model matrix AInput matrix B

Θ index reset

|Θ|<π

Figure 3.5: Structure of the closed loop acceleration, velocity and position observer. Generally, the feedback matrix can be time- or state-dependent and adaptive. As will be shown, the matrix elements are constant. They can be pre-determined reducing precious computation time. Calculating the elements can be done by different approaches, e.g., by pole placement. Pole placement guarantees a stable system, but results usually in an inferior observer performance, as additional information on the system is dropped. Due to the noisy characteristic of the position measurement, the elements are calculated by a linear Kalman filter algorithm considering both measurement error and model inaccuracy.


As mentioned above, the overflow protections of both estimation and measurement of the position are only required in speed control mode. The overflow protection at the input of the Kalman filter rejects erroneous calculations at phase jumps of the angle. The index reset is only used in position control mode, where the absolute rotor position is required, e.g. rotor position of a synchronous motor drive. The estimation and measurement registers are set to zero if the index line is found. The Kalman filter is sometimes referred to as being slowly and having a lagging characteristic due to the error driven nature of the observer. This might be true, if only position and speed are chosen as states. Here, the involuntary lag of the speed signal is avoided by the additional estimation of the acceleration. In fact, the acceleration is insignificantly lagging at continuous load torque variation. Nevertheless, apart an initial change, the acceleration is nearly constant during both changing the speed reference and applying a load torque. This special drive property is caused by current/torque limitation in the speed control loop. Thus, the acceleration is almost constant and can be estimated accurately. The performance of acceleration estimation might be increased by estimating additionally the derivative of the acceleration. This has been disregarded due to mentioned remarks and the difficulty of calculating the model inaccuracy of such an approach. If the measurement and model inaccuracy are known, the Kalman filter algorithm yields an optimum feedback matrix, minimizing automatically the RMS-error between measured quantities and estimated states. A more complete introduction to the general idea of the Kalman filter can be found in literature [Bram 94], [May 79]. Table 3.1 briefly repeats the algorithm of the linear Kalman filter calculating the gains of the feedback matrix [May 79].

Table 3.1: Calculation of the feedback matrix according to the linear discrete Kalman filter algorithm.

( )[ ]

), R() ,P(),(P);,(P);,(P kkkkkk

1111131211 1|1|1|

+=

+=

−−−

−−−

1T1k|k

T1k|kk RCPCCPK

(3.18)

1k|kk1k|kk|k PCKPP −− −= (3.19)

QAPAP k|kk|1k +=+ (3.20)

All model matrices (A, B and C), initial matrix P0|0 and noise covariance matrices (Q and R) of the algorithm given are constant. Thus, also the Kalman matrix Kk with the elements (K1, K2, K3) settles to a constant after a number of iterations indicated by the arrows in table 3.1. Practically, almost 100 iterations are sufficient to reach the settle point (figure 3.6).

90 Chapter 3

0 10 20 30 40 50 60 70 80 90 1000.242

0.244

0.246

k [ ]

K1 [

]

0 10 20 30 40 50 60 70 80 90 100170

172

174

176

k [ ]

K2 [1

/s]

0 10 20 30 40 50 60 70 80 90 1003.2

3.3

3.4

3.5

k [ ]

K3 [1

04 /s2 ]

Figure 3.6: Exemplary iterative calculation of the Kalman gains with respect to the algorithm given in

table 3.1 with iteration number k and subsequently described noise covariance matrices (Q33 = 0,1 Q33,max). In table 3.1, the state update and state projection are dropped, because they do not affect the calculation of the Kalman gains. The presented equations are used for the pre-determination of the feedback matrix, saving computing time with respect to the subsequent real-time implementation. As illustrated in figure 3.5, the final observer with the pre-determined and constant feedback matrix is very computing-time friendly. Only state xk and system model (A, B, C) updates with the pre-determined constant Kalman-gains remain in real-time. A is the model matrix, B the input matrix and C the output matrix of the system model. Still remaining variables of the algorithm calculating the feedback matrix are the noise covariance matrices Q and R and the initial matrix P0|0 representing the covariance in knowledge of the initial conditions. They consist only of diagonal elements due to the lack of sufficient statistical information to evaluate their off-diagonal terms. Varying P0|0 does not affect the settle point of the Kalman gains and can be chosen at random. The covariance matrices Q and R have to be set-up based on the stochastic properties of the corresponding noise. The variance of a variable x with the mean value µ and the distribution p(x) is defined by:

( ) dxxpx )(22 ∫∞

∞−

−= µσ (3.21)

In literature, the initial entries of the covariance matrices are often set to the unity matrix. In order to achieve the optimal filter performance, the filter parameters R and Q can be obtained by tuning based on experimental investigations. This describes an iterative process of searching the best values. Increasing R reflects a stronger disturbance of the measurement. The noise is weighted less by the filter, causing a more filtered position signal but also a slower transient performance of the


system. The noise covariance Q describes the system model inaccuracy. Q has to be increased at stronger noise levels driving the system, entailing a more heavily weighting of the measured signal and a faster transient performance. Thus, changing the covariance matrices R and Q affects both the transient duration and the steady state operation of the observer. Without any previous knowledge of the matrices, tuning is very arduously or can even lead to an unstable behavior of the observer. Obviously, it is preferable to have a rational basis for choosing the required parameters. The following approach reduces the set of unknown parameters to the only parameter Q33 = Q(3,3), adjusted depending on the given application. The noise covariance R accounts for the measurement noise introduced by the incremental encoder and can be easily pre-determined. Assuming a near-normal distribution p(∆Θ) and a zero mean value µ of the position error, the approximate distribution of the position error depends only on the digital encoder resolution Θres:

p(∆Θ)

∆Θ

1/Θres

Θres/2Θres/2

Figure 3.7: Approximate distribution of the position error. According to (3.21), the variance of a signal with a zero mean value µ and equally distributed as in figure 3.7 is:

121

23121 232

2

22 res

res

res

res

dxxres

res

Θ=

Θ

Θ

=Θ

= ∫Θ

Θ−

σ (3.22)

Thus, the noise covariance R, describing the measurement error of the position, depends only on the digital encoder resolution Θres:

12

2resΘ

=R (3.23)

Since the position is the only measured signal, R is a 1×1 matrix being constant for a given installation. In simulations, the measurement noise variance is speed dependent, e.g. the variance at constant speed, being a multiple of the speed resolution, equals zero. However, the real motor speed is never absolute constant due to torque ripple, the encoder pulse counting is mostly asynchronous to the fixed sample time and the encoder lines are individually spaced due to manufacturing process inaccuracy. Thus, the assumption of a near-normal distribution of the position error is reasonable. Furthermore, the given assumption of a near-normal distribution of the position measurement error is backed-up by experiments [Ter 02].

92 Chapter 3

The covariance matrices Q is a 3×3 matrix consisting only of diagonal elements. Q11 = Q(1,1) Q22 = Q(2,2) and Q33 = Q(3,3) describe the model inaccuracy of the position, speed and acceleration estimation, respectively. The model inaccuracy of speed and position estimation is only caused by the discretization of the continuous equations. Considering a very small sample time, the error is negligible and may usually be disregarded. Nevertheless, the worst of all approximations is to set the model inaccuracy to zero. White noise is a much better approximation than zero [May 79]. Thus, this discretization error is considered by a very small value in the noise covariance matrix Q. The maximum discretization error is appraised by the maximum possible acceleration of a drive. An estimation of the acceleration top-limit can be obtained by assuming a maximum torque-inertia relation of the drive. According to catalogues of motor manufacturers, this relation is approximately proportional to the inverse rotor diameter of the motor and varies between αmax ≈ 500 s-2 for large machines (~500 kW) and αmax ≈ 2000 s-2 for small motor drives (~500 W). Here, a constant relation of αmax = Tmax/J = 1000 s-2 is assumed. According to (3.11), the next position value is estimated by the current position plus the current speed multiplied by the sample time Ts. Since the real rotor position is the time-integral of the speed, the discretization error of the position within one computing cycle is the difference between (3.11) and the total area below the speed signal as illustrated in figure 3.8.

t kTs

(k+1)Ts

(k+2)Ts

(k+3)Ts

maximum discretization error:0,5 ∆ωmax Ts = 0,5 αmax Ts

2

∆ωmax

Ts

ω

t kTs

(k+1)Ts

(k+2)Ts

(k+3)Ts

no discretization error at constant speed: ∆ω = 0

Ts

ω(t) ωk ωk = ω(t)

ω

Figure 3.8: Maximum position error due to discretization. The estimated position is indicated by the area of the shadowed rectangle, while the real position equals the total area below the real speed signal. Thus, the discretization error is approximately equal to the area of the textured triangle:

∫+

−=∆Θs

s

Tk

kTks dttT

)1(

)(ωω (3.24)

⇒ 2maxmaxmax 2

121

ss TT αω =∆≈∆Θ (3.25)


According to (3.22) and assuming a maximum drive acceleration of α = 1000 s-2 and a sample time Ts = 200 µs, the maximum variance of the estimated position due to the discretization error is:

112

2max

)1(

103,321

121)(var)1,1( −

+

⋅≈

<<

−= ∫ s

Tk

kTks TdttTQ

s

s

αωω (3.26)

Similarly, the variance of the estimated speed due to the discretization error is appraised. According to (3.12), the discretization error is at its maximum in the case of a maximally changing acceleration within one computing cycle

∫+

−=∆s

s

Tk

kTks dttT

)1(

)(ααω (3.27)

⇒ sTmaxmax αω ≈∆ (3.28)

as illustrated in figure 3.9 by the area of the textured rectangle. The variation of the acceleration can be caused by applying electromagnetic torque as well as by applying a load torque to the motor.

maximum discretization error: ∆ωmax = αmax Ts

α

t kTs

(k+1)Ts

(k+2)Ts

(k+3)Ts

no discretization error at constant acceleration: ∆α = 0

α

t kTs

Ts

αmax

Ts

(k+1)Ts

(k+2)Ts

(k+3)Ts

α(t)

αk

αk = α(t)

Figure 3.9: Maximum speed error due to discretization.

According to (3.22) and assuming again a maximum drive acceleration of α = 1000 s-2 and a sample time Ts = 200 µs, the maximum variance of the estimated speed due to the discretization error is approximately:

( )2

32

max)1( 1103,3

12)(var)2,2(

sTdttTQ s

Tk

kTks

s

s

−+

⋅≈<<

−= ∫

ααα (3.29)

The trickiest part of the proposed observer gain calculation is the variance determination of the acceleration. The load torque is generally unknown. It creates a disturbance of the speed control loop compensated by the affiliated controller. Therefore, the load torque is handled as model inaccuracy. Additionally, erroneous electromagnetic torque calculation and inertia identification are handled as (a small)

94 Chapter 3

part of the model noise. However, the influence of both an incorrect estimation of the electromagnetic torque due to electrical parameter variations and an incorrect identification of the drive inertia is small compared to a potential load variation and subsequently dropped. An estimation of the variance top-limit can be obtained considering the inner current control loop. Since the electromagnetic torque is varied with the equivalent time constant of the earlier derived current control loop (figure 3.10), faster load variations are not compensated anyway and consequently disregarded.

Ts

Tel

t

τeq

Tel

T*el

∆Tmax = Tmax Ts / τeq

Figure 3.10: Approximate step function response and maximum variation

of the electromagnetic torque within one computing cycle. Taking an equivalent time constant τeq = 1ms, the maximum process variance of the acceleration can be estimated by:

storque

kloadkload TJ

TJTT

τmax

max

1,,inaccuracymodel ≈

−< − (3.30)

43

2

max33

11033,3121)3,3(

sJTT

QQtorque

s ⋅≈

<=⇒

τ (3.31)

Equation (3.31) delivers a maximum value for the calculation of the Kalman gains. All other parameters, only slightly affecting the resulting Kalman gains, are fixed and, considering very small discretization errors due to the short sample time, set to the maximum values given in (3.26) and (3.29). Thus, only the acceleration inaccuracy has to be chosen and adapted depending on the given application. It should be noted, that the calculation of all process covariance matrices is proportional to the square of the sample time. The given constants should be adapted accordingly, if a different sample time is chosen. Table 3.2 gives a general idea of the observer performance as a function of the variance Q33. The spread of position, speed and acceleration are indicated by ∆Θ, ∆n and ∆α. The acceleration at an abrupt load torque variation can be estimated according to the given time constant τα. Compared to the encoder resolution Θres = 0,0015 rad, the estimation of the position is for all pre-determined parameters even more accurate than the measurement. The speed signal is vastly improved


considering the numerical differentiation approach (∆n = 73,34 rpm). For almost all applications where the load torque variation is totally unknown, even a ratio Q33/Q33,max = 10% is fast enough. Of course, this ratio should be chosen much smaller, if the load torque is known very well or a smooth speed signal is more important than the dynamic behavior of the loop.

Table 3.2: Kalman gains and observer performance as a function of Q33 (Ts = 200 ms).

Q33/Q33,max [%]

K1 [ ]

K2 [1/s]

K3 [104/s2]

∆Θ [10-4 rad]

∆n [1/min]

∆α [1/s2]

τα [ms]

10 0,245 175 3,4 ~4 ~2 ~150 ~5 50 0,284 244 7,7 ~5,5 ~5 ~250 ~0,4 100 0,307 290 10,8 ~8 ~7 ~400 ~2,5

3.6.3 Stability of the observer

A linear time-invariant system is asymptotically stable, if all eigenvalues are placed inside the unity-circle. Thus, the stability of the observer is proven by calculating the observer eigenvalues λ1, λ2, ..., λn, being solutions of the characteristic polynomial:

( )[ ] ))...((det)( 1 nAKCA λλλλλλχ −−=−−=− CKAAI (3.32)

The stability boundary and the eigenvalues with the variance of the acceleration as parameter are given in figure 3.11 proving the stability of the system. Q33 is varied in steps of 1% of the maximum variance.

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Real [ ]

Imag

inar

y [ ]

Stability boundary

Q33 = max

IncreasingQ33

Q33 = 0

Figure 3.11: Eigenvalues of the observer as a function of Q33 (Ts = 200 µs).

Increasing Q33 results in smaller eigenvalues and a faster observer response. However, the noise suppression is simultaneously reduced. The variance Q33 = 0 can be disregarded, since a zero variance is equivalent to the unrealistic assumption of a

96 Chapter 3

never changing load torque resulting in an estimator observing only position and speed with a phase lag during transients.

3.7 Implementation of the Observer

The presented observer achieves the objectives of avoiding numerical differentiation as well as eliminating lag of the estimated motor speed. Since the model matrices of the observer are independent on motor parameters, the closed-loop observer is vastly insensitive to both electrical and mechanical parameter variations. Furthermore, the proposed algorithm can be easily implemented in software with a negligible requirement of computation time. Figure 3.12 shows the closed-loop observer integrated into the speed/position control loop of a highly dynamic torque controlled ac motor drive.

SVM Inverter

Torque Control

& Torque

Estimation

ia

ub*

uc*

AC motor

ib

ua*

ω

ω*

ω*

Tel*

Udc

Θ

Speed controller

Position controller

Incremental encoder

Θ Load

AccelerationVelocity Position Observer

Tel

Θ*

mode

Position reference

Speed reference

ω

α

Θ

Digital motion control

Kpp

pns KTz

/1 1−− ω*

Figure 3.12: Velocity observer integrated into the speed/position control loop.

It should be noted, that the velocity estimation described here can be extended easily to allow further improvement of the entire drive performance especially at load torque variations by adding acceleration feedback. The information on load acceleration can be used either as derivative input part of a modified PID speed controller (figure 3.13) or directly by compensating the load torque (figure 3.14).

Tel*

ω*

ω

Tdn

pnKJα̂ derivative part

PI with anti windup

pns KTz

/1 1−−

|Tel| < Tmax

Figure 3.13: Modified PID speed controller.


Tel*

ω*

ω |Tel| < Tmax


PI with anti-windup

Tel Tload

Tel*

ω*

ω |Tel|<Tmax


Proportional gain

Tel Tload

Kn

Figure 3.14: P- and PI-speed controller with load torque rejection. Both approaches reject immediately load disturbances improving the dynamic stiffness of the drive. Using this extra information is in any case superior when compared to applying again a numerical differentiation of the (now smoothed) velocity signal. Figure 3.15 presents the simulation results of a response to a load step using a common PI controller and the same controller with additional load torque rejection according to figure 3.14. The applied load amounts to 90% of the rated torque. Thus, this feedback causes the disturbance to perceive a more robust system less sensitive to disturbances.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

1450

1500

1550

t [s]

n [rp

m]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

5

10

15

t [s]

T load

[Nm

]

Load torque rejection

PI-controller

90% of the rated torque

Figure 3.15: PI-controller versus load torque rejection at a load step (Simulation).

Top: Motor speed. Bottom: Load torque The proposed observer-based speed estimation is even applicable using a high-resolution encoder or a resolver. If additional speed information is directly available from a sensor, the algorithm has to be changed slightly. Then, the error between estimation and measurement of both speed and position are used by the Kalman filter to get information on the drive acceleration, additionally improving the drive performance and dynamic stiffness.

98 Chapter 3

3.8 Experimental Results

A drive set-up with a 1,5 kW induction motor has been used to clarify the practical operation and performance of the introduced observer. The inertia of the drive is about J = 0,008 kgm2. Generally, such a small inertia causes more problems for an estimator to track the real speed than a larger one, as the acceleration is much faster. All presented experimental results are obtained, if not explicitly stated differently, using the proposed observer with a sample time Ts = 200 µs and 10% of the maximum acceleration variance as defined in (3.31). Figure 3.16-3.18 shows experimental results during a step of both speed reference and load torque. The given speed, acceleration and position estimations are measurements corresponding to the same time span. The speed and acceleration estimation are used as feedback to control the motor. The speed reference changes from zero to 2000 rpm at t =0.1 s. According to common flux weakening algorithms, the flux is decreased inversely proportionally to the speed above n = 1500 rpm. A step of the load torque is applied after t =0,7 s. The applied load amounts to 65 % of the rated torque. Due to the flux weakening, a maximum load of 75 % of the rated torque can be generated by the motor in the given speed range. Figure 3.16 shows the vast improvement of the speed estimation during transients compared to a simultaneously offline measurement with the numerical differentiation approach using an additional filter to smooth the signal. The implemented filter is chosen to match the same speed smoothness in steady state. Nevertheless, the speed oscillates if, instead of using the observer, this filtered speed signal is used as feedback in the same control loop. The filter causes an extra large time delay in the speed loop. Due to lagging during transients, the decoupling of the current control loop is extremely deteriorated. With the filtered speed signal in the loop, the gain of the speed controller has to be decreased. To the contrary, the speed signal estimated by the observer requires no supplementary filter.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

t [s]

n [rp

m]

0 0.25 0.5 0.75 1-10

-5

0

5

10

t [s]

n real

- n es

t [rpm

]

Kalman Filter

0 0.25 0.5 0.75 1-25

0

25

50

75

t [s]

n real

- n fil

t [rpm

]

Num. Differentiation + Filter

Flux weakening Load

step

Figure 3.16: Estimation of the speed during start-up and at a step of the load torque. Top: Motor speed.

Bottom: Speed error of the observer (nest) and speed error of a filtered signal (nfilt).


Figure 3.17 shows the ability of the observer to track the real motor acceleration during the same step of the speed reference and at variable load torque. Without load torque, the estimation of the acceleration matches the acceleration due to the electromagnetic torque. Of course, the total acceleration is zero in steady state and the electromagnetic torque equals the load. The load can be calculated by the difference between the total acceleration and the acceleration due to the electromagnetic torque. The information on the load acceleration is used as part of the speed controller directly compensating the load torque (figure 3.14). If the speed-load torque relation of a given installation is known, this information can be added to the electromagnetic torque input of the observer. Consequently, the pre-determined variance of the acceleration can be reduced. This results in a further improvement of the drive performance with a very smooth speed and torque signal.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-400

-200

0

200

400

600

800

1000

1200

1400

1600

t [s]

α [1

/s2 ]

Acceleration due to electromagnetic

torque

Loadstep

Total acceleration (estimation)

Figure 3.17: Estimation of the acceleration during start-up and at a step of the load torque.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5x 10-3

t [s]

Θre

al -

Θm

eas [r

ad]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5x 10-3

t [s]

Θre

al -

Θes

t [rad

]

Figure 3.18:Position error during start-up and at a step of the load torque. Top: Error between real and measured position. Bottom: Error between real and estimated position.

The variance of the estimated position is even lower than the variance of the measured position (figure 3.18). Therefore, the estimated signal is preferred in

100 Chapter 3

position control mode or when using a synchronous motor requiring the information on the absolute rotor position. As mentioned earlier, the observer is not motor type dependent. Only the maximum torque-inertia ratio of the given machine can be used to pre-determine the observer parameters.

3.9 Optimal Speed Controller & Parameter Variation

In speed control loops using a common PI controller, the load torque is compensated by the integral-acting part, also if the output of the integrator does not match the real load torque. However, an additional integral-acting part of the speed controller is not required, if the estimation of the load torque is used as feedback (figure 3.19). In any case, neglecting the integral-acting part of the speed controller and applying load torque rejection is superior regarding speed response as well as disturbance sensitivity. The overshoot at steps of both speed reference and load torque vastly decreases or even vanishes. In common speed control loops the parameter of the controller, e.g. a PI-controller, are calculated regarding either optimum speed response or optimum disturbance rejection. To the contrary, the proportional gain Kn is optimized with respect to an optimum speed response while the feedback of the estimated load torque yields an optimized system regarding disturbance sensitivity. Furthermore, the determination of the speed controller parameter Kn is less complicated (2.95).

Tel*

ω*

ω |Tel|<Tmax


Proportional gain

Tel Tload

Kn iq*iq

* = f(Tel*)

Current mapping

qr

hel ii

LL

pT µ

21=

( )[ ]ddqMdqel iLLipT −−Ψ=

Induction motor:

PMSM:

Figure 3.19: P-controller with load torque rejection. In steady state, both real motor acceleration and its estimation is zero. Otherwise, the acceleration of either the real state or its estimation would not be zero and the observer obtains an error. This error is equalized by the observer independent on the inertia or motor parameter estimation. Thus, following equations are valid in steady state:

0ˆ == realαα ; (3.33)

⇒ T (3.34) 0!==− αJTloadel

⇒ T loadel T−= ; & (3.35) loadel TT ˆˆ −=


Since the estimated and real motor acceleration is zero in steady state, the generated electromagnetic torque equalizes the load. In fact, the torque calculation might be incorrect, but the real load is compensated by the real torque. Consequently, the steady state speed error remains zero, even if the torque estimation is incorrect due to a parameter mismatch. Another validation of motor parameter independence is the fact, that the estimated electromagnetic torque in figure 3.19 is calculated by (possibly incorrect) motor parameters, whereas the current mapping is calculated by the inverse motor parameters. Thus, a parameter mismatch is directly compensated. Consequently, the implemented algorithm together with the speed control loop in figure 3.19 yields no steady state speed error, is insensitive to a correct calculation of motor parameters and parameter variations are negligible considering the proposed method. Figure 3.20 presents the estimated and real speed using this controller type with a mismatch of motor parameters. The electromagnetic torque is 20%, the motor inertia 50% over calculated. The speed reference changes from zero to 1500 rpm at t =0.1 s. A step of the load torque is applied after t =0,7 s. The applied load amounts to 80 % of the rated torque.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

t [s]

n [rp

m]

0.2 0.3 0.4 0.5 0.6 0.7 0.81450

1500

1550

t [s]

n [rp

m]

Box 1 nest and nm

nref

Load step

Figure 3.20: Speed control using a P-controller with load torque rejection and mismatch of motor

parameters. Top: Real, estimated and reference speed. Bottom: Details indicated by “Box 1”.

3.10 Conclusions

The quality of the measured speed signal has a dramatic impact on the drive performance. A noisy speed signal results in erroneous torque commands, increased torque ripple, acoustic noise and motor heating by current harmonics. Filtering of the signal is not a generally acceptable solution to reduce the ripple of the speed measurement. The lag associated with the filter can substantially degrade the closed-loop drive performance.

102 Chapter 3

Alternative methods for velocity estimation described in literature are motor parameter depending and clearly sensitive to torque accuracy. This chapter presents the design and the implementation of a novel observer for position and speed estimation using an incremental encoder as input offering a significant improvement of the drive performance. The presented observer achieves the objectives of avoiding numerical differentiation as well as eliminating lag of the estimated motor speed. The lagging characteristic of the speed signal due to the error driven nature of the observer is avoided by the additional estimation of the motor acceleration. The observer is motor type independent. The variance of the estimated position is even lower than the variance of the measured position. Therefore, the estimated signal should be preferred in position control mode or when using a synchronous motor requiring information on the absolute rotor position. The entire drive performance is improved by adding acceleration feedback. The information on load acceleration is used as input of a speed controller rejecting directly load variations. Independent of motor parameters, a speed controller with load torque rejection requires no additional integrator part compensating for the load. This topology provides a system with extremely high stiffness to disturbance inputs as well as an improved dynamic performance. This closed loop observer is very much insensitive to both electrical and mechanical parameter variations. Furthermore, the proposed algorithm can be easily implemented in software with a negligible requirement of computing time.

4. Control

4.1 Discretization

As the drive is controlled by a DSP, it is to be considered as a sampled data system. There are different approaches to model these systems [Fra 90]. One possibility is to use a discrete model [Lab 96], using the z-transform and to carry out the controller design in the z-domain. To enable the use of frequency response methods, a second transformation is necessary (bilinear transformation). Alternatively, carrying out the initial design using continuous methods can serve as a guide for a direct discrete design. The performance of a system when it could be realized with continuous hardware is a target for how well the digital system should perform and assists in selecting the sample rate. As shown in [Büh 97], under certain conditions a sampled data or discrete system can be treated as a pseudo-continuous system. This has the advantages with respect to the applicability of optimal control formulations. The effect of A/D and D/A converters, as well as the execution time of the control algorithm itself are taken into account by using equivalent continuous transfer functions. The designed optimal controllers can directly be converted to their discrete equivalents. However, the control algorithms implemented here contain no fixed discrete transfer functions. They are rather constructed manually (user-defined library) by unit delays, sums and gains. This has the advantage of all variables being accessible and avoiding involuntary signal delays. Only where it is necessary to avoid algebraic loops, an additional delay is added. To design the controllers and simulate the entire drive, it is necessary to take also the power converter into account. A power electronic system can be modeled in different ways, or, as often encountered, not modeled at all. The level of modeling to be considered depends on the purpose of the simulation. For control design purposes, the inverter can be simply modeled by a delay. It has been shown that for the subharmonic PWM and SVM a general delay time of TPWM / 3 is adequate, where TPWM is the PWM period [Büh 97].

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