Chapter 3 reference frame theory

86

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 refer￾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 equa￾tions that describe resistive, inductive, and capacitive circuit elements. Using this

approach, many of the basic concepts and interpretations of this general transformation

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 trans￾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 induc￾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

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 refer￾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

88 REFERENCE-FRAME THEORY

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

f Kf qd s s abcs 0 = (3.3-1)

where

( )[ ] fqd s T qs ds s fff 0 0 = (3.3-2)

()[ ] fabcs T as bs cs = fff (3.3-3)

Ks =

− ⎛

⎝

⎜ ⎞

⎠

⎟ + ⎛

⎝

⎜ ⎞

⎠

⎟

− ⎛

⎝

⎜ ⎞

⎠

⎟ 2

3

2

3

2

3

2

3

cos cos cos

sin sin sin

θ θ π θ π

θ θ π θ π

+ ⎛

⎝

⎜ ⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

2

3

1

2

1

2

1

2

(3.3-4)