Chapter 3 reference frame theory

Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek

3.1 INTRODUCTION

We have found that some of the machine inductances are functions of rotor position, whereupon the coeffi cients of the differential equations (voltage equations) that describe the behavior of these machines are rotor position dependent A change of variables is often used to reduce the complexity of these differential equations There are several changes of variables that are used, and it was originally thought that each change of variables was unique and therefore they were treated separately [1–4] It was later learned that all changes of variables used to transform actual variables are contained

in one [5, 6] This general transformation refers machine variables to a frame of ence that rotates at an arbitrary angular velocity All known real transformations are obtained from this transformation by simply assigning the speed of the rotation of the reference frame

In this chapter, this transformation is set forth and, since many of its properties can

be studied without the complexities of the machine equations, it is applied to the tions that describe resistive, inductive, and capacitive circuit elements Using this approach, many of the basic concepts and interpretations of this general transformation

equa-Analysis of Electric Machinery and Drive Systems, Third Edition Paul Krause, Oleg Wasynczuk,

Scott Sudhoff, and Steven Pekarek.

© 2013 Institute of Electrical and Electronics Engineers, Inc Published 2013 by John Wiley & Sons, Inc.

REFERENCE-FRAME THEORY

3

BACKGROUND 87

are readily and concisely established Extending the material presented in this chapter

to the analysis of ac machines is straightforward, involving a minimum of trigonometric manipulations

3.2 BACKGROUND

In the late 1920s, R.H Park [1] introduced a new approach to electric machine analysis

He formulated a change of variables that in effect replaced the variables (voltages, currents, and fl ux linkages) associated with the stator windings of a synchronous machine with variables associated with fi ctitious windings rotating at the electrical angular velocity of the rotor This change of variables is often described as transforming

or referring the stator variables to a frame of reference fi xed in the rotor Park ’ s formation, which revolutionized electric machine analysis, has the unique property of eliminating all rotor position-dependent inductances from the voltage equations of the synchronous machine that occur due to (1) electric circuits in relative motion and (2) electric circuits with varying magnetic reluctance

In the late 1930s, H.C Stanley [2] employed a change of variables in the analysis

of induction machines He showed that the varying mutual inductances in the voltage equations of an induction machine due to electric circuits in relative motion could be eliminated by transforming the variables associated with the rotor windings (rotor variables) to variables associated with fi ctitious stationary windings In this case, the rotor variables are transformed to a frame of reference fi xed in the stator

G Kron [3] introduced a change of variables that eliminated the position-dependent mutual inductances of a symmetrical induction machine by transforming both the stator variables and the rotor variables to a reference frame rotating in synchronism with the fundamental angular velocity of the stator variables This reference frame is commonly referred to as the synchronously rotating reference frame

D.S Brereton et al [4] employed a change of variables that also eliminated the varying mutual inductances of a symmetrical induction machine by transforming the stator variables to a reference frame rotating at the electrical angular velocity of the rotor This is essentially Park ’ s transformation applied to induction machines

Park, Stanley, Kron, and Brereton et al developed changes of variables, each of which appeared to be uniquely suited for a particular application Consequently, each transformation was derived and treated separately in literature until it was noted in 1965 [5] that all known real transformations used in induction machine analysis are contained

in one general transformation that eliminates all rotor position-dependent mutual tances by referring the stator and the rotor variables to a frame of reference that may rotate at any angular velocity or remain stationary All known real transformations may then be obtained by simply assigning the appropriate speed of rotation, which may in

induc-fact be zero, to this so-called arbitrary reference frame Later, it was noted that the

stator variables of a synchronous machine could also be referred to the arbitrary ence frame [6] However, we will fi nd that the varying inductances of a synchronous machine are eliminated only if the reference frame is rotating at the electrical angular velocity of the rotor (Park ’ s transformation); consequently, the arbitrary reference frame

refer-does not offer the advantages in the analysis of the synchronous machines that it refer-does

in the case of induction machines

3.3 EQUATIONS OF TRANSFORMATION: CHANGE OF VARIABLES

Although changes of variables are used in the analysis of ac machines to eliminate time-varying inductances, changes of variables are also employed in the analysis of various static, constant-parameter power-system components and control systems asso-ciated with electric drives For example, in many of the computer programs used for transient and dynamic stability studies of large power systems, the variables of all power system components, except for the synchronous machines, are represented in a reference frame rotating at synchronous speed, wherein the electric transients are often neglected Hence, the variables associated with the transformers, transmission lines, loads, capacitor banks, and static var units, for example, must be transformed to the synchronous rotating reference frame by a change of variables Similarly, the “average value” of the variables associated with the conversion process in electric drive systems and in high-voltage ac–dc systems are often expressed in the synchronously rotating reference frame

Fortunately, all known real transformations for these components and controls are also contained in the transformation to the arbitrary reference frame, the same trans-formation used for the stator variables of the induction and synchronous machines and for the rotor variables of induction machines Although we could formulate one trans-formation to the arbitrary reference frame that could be applied to all variables, it is preferable to consider only the variables associated with stationary circuits in this chapter and then modify this analysis for the variables associated with the rotor wind-ings of the induction machine at the time it is analyzed

A change of variables that formulates a transformation of the three-phase variables

of stationary circuit elements to the arbitrary reference frame may by expressed as

fqd s0 =K fs abcs (3.3-1) where

23

232

3

2

12

12

EQUATIONS OF TRANSFORMATION: CHANGE OF VARIABLES 89

where the angular position and velocity of the arbitrary reference frame are related as

3

2

23

the variables, parameters, and transformation associated with stationary circuits The

angular displacement θ must be continuous; however, the angular velocity associated

with the change of variables is unspecifi ed The frame of reference may rotate at any constant or varying angular velocity, or it may remain stationary The connotation of arbitrary stems from the fact that the angular velocity of the transformation is unspeci-

fi ed and can be selected arbitrarily to expedite the solution of the system equations or

to satisfy the system constraints The change of variables may be applied to variables

of any waveform and time sequence; however, we will fi nd that, for a three-phase

electrical system, the transformation given above is particularly appropriate for an abc

sequence

Although the transformation to the arbitrary reference frame is a change of ables and needs no physical connotation, it is often convenient to visualize the trans-formation equations as trigonometric relationships between variables as shown in Figure 3.3-1 In particular, the equations of transformation may be thought of as if the

f qs and f ds variables are “directed” along paths orthogonal to each other and rotating at

an angular velocity of ω , whereupon f as , f bs , and f cs may be considered as variables

directed along stationary paths each displaced by 120° If f as , f bs , and f cs are resolved

into f qs , the fi rst row of (3.3-1) is obtained, and if f as , f bs , and f cs are resolved into f ds , the

second row is obtained It is important to note that f 0 s variables are not associated with

the arbitrary reference frame Instead, the zero variables are related arithmetically to

the abc variables, independent of ω and θ Portraying the transformation as shown in

Figure 3.3-1 is particularly convenient when applying it to ac machines where the

direction of f as , f bs , and f cs may also be thought of as the direction of the magnetic axes

of the stator windings We will fi nd that the direction of f qs and f ds can be considered

as the direction of the magnetic axes of the “new” windings created by the change of

variables It is also important not to confuse f as , f bs , and f cs or f qs and f ds with phasors

The total instantaneous power of a three-phase system may be expressed in abc

variables as

P abcs =v i as as+v i bs bs+v i cs cs (3.3-7)

The total power expressed in the qd 0 variables must equal the total power expressed

in the abc variables, hence using 1) to replace actual currents and voltages in

=

The 3/2 factor comes about due to the choice of the constant used in the transformation

Although the waveforms of the qs and ds voltages, currents, fl ux linkages, and electric

charges are dependent upon the angular velocity of the frame of reference, the waveform

of total power is independent of the frame of reference In other words, the waveform of the total power is the same regardless of the reference frame in which it is evaluated

3.4 STATIONARY CIRCUIT VARIABLES TRANSFORMED TO

THE ARBITRARY REFERENCE FRAME

It is convenient to treat resistive, inductive, and capacitive circuit elements separately

STATIONARY CIRCUIT VARIABLES TRANSFORMED 91

From (3.3-1)

vqd s0 =K r Ks s( s)−1iqd s0 (3.4-2)

It is necessary to specify the resistance matrix r s before proceeding All stator phase

windings of either a synchronous or a symmetrical induction machine are designed to have the same resistance Similarly, transformers, capacitor banks, transmission lines and, in fact, all power-system components are designed so that all phases have equal

or near-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 matrix

r s are equal, then

Thus, the resistance matrix associated with the arbitrary reference variables ( f qs , f ds , and

f 0 s ) is equal to the resistance matrix associated with the actual variables if each phase

of the actual circuit has the same resistance If the phase resistances are unequal anced or unsymmetrical), then the resistance matrix associated with the arbitrary

(unbal-reference-frame variables contains sinusoidal functions of θ except when ω = 0,

where-upon K s is algebraic In other words, if the phase resistances are unbalanced, the

transformation yields constant resistances only if the reference frame is fi xed where the unbalance physically exists This feature is quite easily illustrated by substituting

r s = diag[ r as r bs r cs ] into K s r s ( K s ) − 1

Inductive Elements

For a three-phase inductive circuit, we have

where p is the operator d / dt In the case of the magnetically linear system, it has been

customary to express the fl ux linkages as a product of inductance and current matrices before performing a change of variables However, the transformation is valid for fl ux linkages and an extensive amount of work can be avoided by transforming the fl ux linkages directly This is especially true in the analysis of ac machines, where the inductance matrix is a function of rotor position Thus, in terms of the substitute vari-ables, (3.4-4) becomes

vqd s0 =Ks p[(Ks)−1lqd s0 ] (3.4-5) which can be written as

vqd s0 =Ks p[(Ks)−1]lqd s0 +K Ks( s)−1plqd s0 (3.4-6)

It is easy to show that

that the speed voltage terms are zero if ω is zero, which, of course, is when the

refer-ence frame is stationary Clearly, the voltage equations for the three-phase inductive circuit become the familiar time rate of change of fl ux linkages if the reference frame

is fi xed where the circuit physically exists Also, since (3.4-4) is valid in general, it follows that (3.4-11) – (3.4-13) are valid regardless if the system is magnetically linear

or nonlinear and regardless of the form of the inductance matrix if the system is netically linear

For a linear system, the fl ux linkages may be expressed

Whereupon, the fl ux linkages in the arbitrary reference frame may be written as

STATIONARY CIRCUIT VARIABLES TRANSFORMED 93

As is the case of the resistive circuit, it is necessary to specify the inductance matrix before proceeding with the evaluation of (3.4-15) However, once the inductance matrix

is specifi ed, the procedure for expressing any three-phase inductive circuit in the trary reference frame reduces to one of evaluating (3.4-15) and substituting the resulting

λ qs , λ ds , and λ 0 s into the voltage equations (3.4-11) – (3.4-13) This procedure is

straight-forward, with a minimum of matrix manipulations compared with the work involved

if, for a linear system, the fl ux linkage matrix λ abcs is replaced by L s i abcs before

perform-ing the transformation

If, for example, L s is a diagonal matrix with all nonzero terms equal, then

A matrix of this form could describe the inductance of a balanced three-phase inductive load, a three-phase set of line reactors used in high-voltage transmission systems or any symmetrical three-phase inductive network without coupling between phases It is clear that the comments regarding unbalanced or unsymmetrical phase resistances also apply in the case of unsymmetrical inductances

An inductance matrix that is common is of the form

Ls s s s

where L s is a self inductance and M is a mutual inductance This general form can be

used to describe the stator self- and mutual inductance relationships of the stator phases

of symmetrical induction machines, and round-rotor synchronous machines with trary winding arrangement, including double-layer and integer and noninteger slot/pole/phase windings From our work in Chapter 2 , we realize that this inductance matrix is

arbi-of a form that describes the self- and mutual inductances relationships arbi-of the stator phases of a symmetrical induction machine and the stator phases of a round-rotor syn-chronous machine with or without mutual leakage paths between stator windings It can also describe the coupling of a symmetrical transmission line Example diagrams that portray such coupling are shown in Figure 3.4-1 It is left to the reader to show

that for L s given by (3.4-17)

K s L s ( K s ) − 1 yields a diagonal matrix that, in effect, magnetically decouples the substitute variables in all reference frames This is a very important feature of the transformation

On the other hand, we have seen in Section 1.4 and Chapter 2 that the self- and mutual inductances between the stator phases of the salient-pole synchronous machine form a magnetically unsymmetrical system It will be shown that for this case, there is only one reference frame, the reference frame rotating at the electrical angular velocity of the rotor, wherein the substitute variables are not magnetically coupled

+ (b)

– – –

(a)

STATIONARY CIRCUIT VARIABLES TRANSFORMED 95

Utilizing (3.4-8) yields

iqd s0 =ωqdqs+pqqd s0 (3.4-22) where

Once the capacitance matrix is specifi ed, q qs , q ds , and q 0 s can be determined and

sub-stituted into (3.4-24) – (3.4-26) The procedure and limitations are analogous to those

in the case of the inductive circuits A diagonal capacitance matrix with equal nonzero elements describes, for example, a three-phase capacitor bank used for power factor correction and the series capacitance used for transmission line compensation or any three-phase electrostatic system without coupling between phases A three-phase trans-mission system is often approximated as a symmetrical system, whereupon the induc-tance and capacitance matrices may be written in a form similar to (3.4-17)

EXAMPLE 3A For the purpose of demonstrating the transformation of variables to

the arbitrary reference frame, let us consider a three-phase RL circuit defi ned by

2

121

Here we have broken up the self-inductance into a leakage, L ls , and magnetizing

induc-tance, L ms Also, the mutual inductance M is equal to − (1/2) L ms The voltage equations in

the arbitrary reference frame can be written from (3.4-2) and (3.4-9) , in expanded form as

v qs =r i s qs+ωλds+pλ qs (3A-3)

v ds =r i s ds−ωλqs+pλ ds (3A-4)

Since the example inductance matrix given by (3A-2) is in the same form as (3.4-17) ,

we can use (3.4-18) as a guide to evaluate K s L s ( K s ) − 1 Thus

words, the above voltage equations are most easily solved with ω = 0 However, our

purpose is to set forth the basic concepts and the interpretations of this general formation; its advantages in machine analysis will be demonstrated in later chapters

trans-COMMONLY USED REFERENCE FRAMES 97

3.5 COMMONLY USED REFERENCE FRAMES

It is instructive to take a preliminary look at the reference frames commonly used in the analysis of electric machines and power system components; namely, the arbitrary, stationary, rotor, and synchronous reference frames Information regarding each of these reference frames as applied to stationary circuits is given in the following table For purposes at hand, it is suffi cient for us to defi ne the synchronously rotating or the synchronous reference frame as the reference frame rotating at the electrical angular velocity corresponding to the fundamental frequency of the variables associated with

stationary circuits, herein denoted as ω e In the case of ac machines, ω e is the electrical

angular velocity of the air-gap rotating magnetic fi eld established by stator currents of fundamental frequency

ω (unspecifi ed) Stationary circuit variables referred to an

arbitrary reference frame

f qd 0 s or f qs , f ds ,

f 0 s

K s

stationary reference frame

fqd s s0 or

s ds s s

Ks

reference frame fi xed in the rotor

fqd s r0 or

r ds r s

Ks

synchronously rotating reference frame

fqd s e or

e ds e s

Ks e

The notation requires some explanation We have previously established that the s

sub-script denotes variables and transformations associated with circuits that are stationary

-in “real life” as opposed to rotor circuits that are free to rotate Later, when consider-ing

the induction machine, we will use the subscript r to denote variables and the mation associated with rotor circuits The raised index is used to denote the qs and ds

transfor-variables and transformation associated with a specifi c reference frame except in the

case of the arbitrary reference frame that does not have a raised index Since the 0 s

variables are not associated with a reference frame, a raised index is not assigned to

f 0 s The transformation of stationary circuits to a stationary reference frame was

devel-oped by E Clarke [7] , who used the notation f α , f β , and f 0 rather than fqs s , f ds s , f 0 s In

Park ’ s transformation to the rotor reference frame, he denoted the variables f q , f d , and

f 0 rather than fqs r , f ds r , and f 0 s There appears to be no established notation for the variables

in the synchronously rotating reference frame We will use the e superscript as indicated

in the table As mentioned previously, the voltage equations for all reference frames may be obtained from those in the arbitrary reference frame by assigning the speed of

the desired reference frame to ω

3.6 TRANSFORMATION OF A BALANCED SET

Although the transformation equations are valid regardless of the waveform of the variables, it is instructive to consider the characteristics of the transformation when the three-phase system is symmetrical and the voltages and currents form a balanced three-

phase set of abc sequence as given by (3.6-1) – (3.6-4) A balanced three-phase set is

generally defi ned as a set of equal-amplitude sinusoidal quantities that are displaced by

120° Since the sum of this set is zero, the 0 s variables are zero.

θω

BALANCED STEADY-STATE PHASOR RELATIONSHIPS 99

With the three-phase variables as given in (3.6-1) – (3.6-3) , the qs and ds variables form

a balanced two-phase set in all reference frames except when ω = ω e In this, the

syn-chronously rotating reference frame, the qs and ds quantities become

f qs e = 2f scos(θef −θ e) (3.6-8)

f ds e = − 2f ssin(θef −θ e) (3.6-9)

where θ e is the angular position of the synchronously rotating reference frame It is

important to note that θ e and θ ef both have an angular velocity of ω e Hence, θ ef − θ e is

a constant depending upon the initial values of the variable being transformed, θ ef (0),

and the initial position of the synchronously rotating reference frame, θ e (0) Equations

(3.6-8) and (3.6-9) reveal a property that is noteworthy; there is one reference frame

where a balanced set will appear as constants if the amplitude f s is constant In other

words, if a constant amplitude balanced set appears in any reference frame, then there

is another reference frame where this balanced set appears as constants Obviously, the converse is true

3.7 BALANCED STEADY-STATE PHASOR RELATIONSHIPS

For balanced steady-state conditions, the amplitude and frequency are constants and θ ef

becomes ω e t + θ ef (0), whereupon (3.6-1) – (3.6-3) may be expressed as

where θ ef (0) corresponds to the time-zero value of the three-phase variables The

upper-case are used to denote steady-state quantities If the speed of the arbitrary reference

frame is an unspecifi ed constant, then θ = ω t + θ (0), and for the balanced steady-state

conditions (3.6-5) and (3.6-6) may be expressed as

From (3.7-1) , the phasor representing the as variables is

It is necessary to consider negative frequencies since ω can be greater than ω e The

phasors rotate in the counterclockwise direction for ω < ω e and in the clockwise

direc-tion for ω > ω e

In the analysis of steady-state operation, we are free to select time zero It is often

convenient to set θ (0) = 0 Then from (3.7-6) and (3.7-7)

Thus, in all asynchronously rotating reference frames ( ω ≠ ω e ) with θ (0) = 0, the

phasor representing the as quantities is equal to the phasor representing the qs

quantities For balanced steady-state conditions, the phasor representing the variables

of one phase need only be shifted in order to represent the variables in the other phases

In the synchronously rotating reference frame, ω = ω e and θ e (0) is the zero position

of the arbitrary reference frame Recall θ ef (0) is the zero position of the abc quantities

If we continue to use uppercase letters to denote the steady-state quantities, then from (3.7-4) and (3.7-5) , we obtain

If we select the time-zero position of the synchronously rotating reference frame to be

zero, then, θ e (0) = 0 in (3.7-10) and (3.7-11) and

BALANCED STEADY-STATE PHASOR RELATIONSHIPS 101

represents a sinusoidal quantity; however, F qs e and F ds e are not phasors They are real quantities representing the constant steady-state variables of the synchronously rotating reference frame

EXAMPLE 3B It is helpful to discuss the difference between the directions of f as ,

f bs , and f cs , as shown in Figure 3.3-1 and phasors The relationships shown in Figure

3.3-1 trigonometrically illustrate the transformation defi ned by (3.3-1) – (3.3-6) Figure 3.3-1 is not a phasor diagram and should not be interpreted as such It simply depicts

the relationships between the directions of f as , f bs , f cs , f qs , and f ds as dictated by the

equa-tions of transformation regardless of the instantaneous values of these variables On the other hand, phasors provide an analysis tool for steady-state, sinusoidal variables The magnitude and phase angle of the phasor are directly related to the amplitude of the sinusoidal variation and its phase position relative to a reference The balanced set given by (3.6-1) – (3.6-3) may be written as (3.7-1) – (3.7-3) for steady-state conditions

The phasor representation for as variables is given by (3.7-6) The phasor representation

for the balanced set is

The phasor diagram is shown in Figure 3B-1 For balanced conditions, the phasors that

form an abc sequence are displaced from each other by 120° and each with a phase angle

of θ ef (0) The directions of f as , f bs , and f cs in Figure 3.3-1 , that are fi xed by the

transforma-tion, are such that f cs is directed − 120° from f as However,  F cs is + 120° from F as for anced conditions (Fig 3B-1 ) Another important difference is that the phasor diagram

bal-must be rotated at ω e in the counterclockwise direction and the real part of the phasors

is related to the instantaneous values of the three-phase set However, the diagram of f as ,

f bs , and f cs shown in Figure 3.3-1 is always stationary for stationary circuits

3.8 BALANCED STEADY-STATE VOLTAGE EQUATIONS

If the three-phase system is symmetrical and if the applied voltages form a balance set

as given by (3.6-1) – (3.6-3) , then the steady-state currents will also form a balanced set

For equal resistance in each phase, the steady-state voltage equation in terms of the as

variables is

For linear, symmetrical inductive elements and since p = j ω e , the steady-state voltage

equation may be written as

where Z s is the impedance of each phase of the symmetrical three-phase system

For equal resistance in each phase of the circuit, the balanced steady-state voltage

equation for the qs variables in all asynchronously rotating reference frames can be

written from (3.4-2) as

For linear symmetrical inductive elements, the steady-state qs voltage equation in all

asynchronously rotating reference frames may be written from (3.4-11) as

V qs =ωΛds+ j(ω ωe− )Λqs (3.8-6)

the ( ω e − ω ) factor comes about due to the fact that the steady-state variables in all

asynchronously rotating reference frames vary at the frequency corresponding to

( ω e − ω ) From (3.7-8) , Λds = jΛqs, thus (3.8-6) becomes

Similarly, for a linear symmetrical capacitive circuit, the steady-state qs current

phasor equation in all asynchronously rotating reference frames may be written from (3.4-24) as