Reference Frame Theory

UNIT- II

Reference frame theory

16 November 2012 1PRB/Dept.of EEE/SCE

INTRODUCTION

Generally, the machine model can be described by differential equations withtime varying mutual inductances,

For example

( for a synchronous machine) (for a induction machine)but such a model tends to be very complex under dynamic analysis due tothe following reasons

(1) electric circuit in relative motion(2) electric circuits with varying magnetic reluctance

So that, Several change of variables are used to reduce the complexity ofthese differential equations.


TRANSFORMATION

The process of replacing one set of variables by another related set of variablesis called transformation. In the study of power systems and electrical machine

analysis, mathematical transformations are often used to decouple variables, tofacilitate the solution of difficult equations with time-varying coefficients, or torefer all variables to a common reference frame.

Evolution of transformation technique

R. H. Park - Transformation – 1920 ( for a synchronous machine)

He transformed a change of variables (voltages ,currents and flux linkages)associated with the stator windings of a synchronous machine to a variablesassociated with fictious windings rotating with the rotor at synchronousspeed.

(The stator variables are transformed to a synchronously rotating referenceframe fixed in the rotor)


H. C. Stanley - Transformation – 1930 (for a induction machine)

He transformed a change of variables associated with the rotor windings(rotorvariables) of a induction machine to a variables associated with fictiousstationary windings (stationary reference frame) fixed in the stator

G. Kron - Transformation (for a induction machine)

He transformed a change of variables associated with both stator and rotorwindings of a symmetrical induction machine to a variable associated withreference frame rotating in synchronism with the rotating magnetic field. Thisreference frame is commonly referred to as the synchronously rotatingreference frame.

D. S . Brereton -Transformation ( for a induction machine)

He proposed a change of variables associated with the stator windings of ainduction machine to a variables associated with the reference frame fixed inthe rotor.


Krause and Thomas- Transformation (after 1965)

Both were shown that time varying inductances can be eliminated by referringthe stator and rotor variables to a common reference frame which may rotate atany speed. This common reference frame is also called arbitrary referenceframe.

E. Clarke - Transformation

He transformed a change of variables associated with the stationary circuits to a variables associated with stationary reference frame. The stationary two-phase variables of Clarke’s transformation are denoted as α and β. (Both are orthogonal).


Advantages of reference frame transformation

1. The number of voltage equations are reduced.2. The time – varying voltage equations become time – invariant

ones.3. Performance of power systems and electric machines can be

analyzed without complexities in the voltage equations.4. Transformations make it possible for control algorithms to be

implemented on the DSP5. With aid of this technique , many of the basic concepts and

interpretations of this general transformations are concisely established.
































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Reference Frame Theory