ﻢﯿﺣﺮﻟا ﻦﻤﺣﺮﻟا ﷲا ﻢﺴﺑ1).pdf · 15 ﯽﺑﺎﺘﮐ سﺎﺒﻋ ﺮﺘﮐد:سرد دﺎﺘﺳا Arbitrary Reference Frame When the reference frame is

تئوري جامع ماشینهاي الکتریکی تئوري جامع ماشینهاي الکتریکی دکتر عباس کتابی دکتر عباس کتابی : مدرس

بسم اهللا الرحمن الرحیمبسم اهللا الرحمن الرحیم

: : مراجع درسمراجع درس“Analysis of Electric Machinery and Drive Systems”, Third Edition, By: P.

C. Krause, O. Wasynczuk and S. D. Sudhoff, IEEE Press, 2013.“Electromechanical Motion Devices”, Second Edition, By: P. C. Krause,

O. Wasynczuk and S. Pekarek, Wiley-IEEE Press, 2012. “Dynamic simulation of Electric Machinery using MATLAB”, by: Chee-

Mun Ong, Prentice Hall PTR, 1997 نحوه ارزشیابینحوه ارزشیابی: :

نمره 2: تکالیف نمره 13: امتحان پایانترم

نمره 5: پروژه

دکتر عباس کتابی دکتر عباس کتابی : : استاد درساستاد درس2

ReferenceReference FrameFrame TheoryTheory 33--Phase Phase Induction MachinesInduction Machines Synchronous Synchronous MachinesMachines Machine Equations in Operational Machine Equations in Operational

Impedances and Time ConstantsImpedances and Time Constants LinearizedLinearized Machine EquationsMachine Equations Reduced Order Machine EquationsReduced Order Machine Equations

::مطالب درس مطالب درس


ReferenceReference FrameFrame TheoryTheory

Power of Reference Frame Theory:Power of Reference Frame Theory: Eliminates Rotor Position Dependence

inductances Transforms Nonlinear Systems to Linear Systems

for Certain Cases Fundamental Tool For Development of Equivalent

Circuits Can Be Used to Make AC Quantities Become DC

Quantities Framework of Most Controllers


ReferenceReference FrameFrame TheoryTheory

History of Reference Frame Theory•1929: Park’s Transformation Synchronous Machine; Rotor Reference Frame

•1938: StanleyInduction Machine; Stationary Reference Frame

•1951:KronInduction Machine; Synchronous Reference Frame

•1957: BreretonInduction Machine; Rotor Reference Frame

•1965: KrauseArbitrary Reference Frame


Arbitrary Reference Frame

Consider stator winding of a 2-pole 3-phase symmetrical machine



Synchronous and induction machine inductances are functions of the rotor speed, therefore the coefficients of the differential equations (voltage equations) which describe the behavior of these machines are time-varying.

A change of variables can be used to reduce the complexity of machine differential equations, and represent these equations in another reference frame with constant coefficients.



A change of variables which formulates a transformation of the 3-phase variables of stationary circuit elements to the arbitrary reference frame may be expressed

abcsssqd fKf 0

,)(, 00 sdsqsT

sqd fffwhere f

,)( csbsasT

abcs ffff

,

21

21

21

)3

2sin()3

2sin(sin

)3

2cos()3

2cos(cos

32

sK

).0()(0

t

dtt


“f” can represent either voltage, current, or flux linkage.

“s” indicates the variables, parameters and transformation associated with stationary circuits.

“” represent the speed of reference frame.


.

1)3

2sin()3

2cos(

1)3

2sin()3

2cos(1sincos

1

sK



=0: Stationary reference frame.

=e: synchronoulsy rotaingreference frame.

=r: rotor reference frame (i.e., the reference frame is fixed on the rotor).



fas, fbs and fcs may be thought of as the direction of the magnetic axes of the stator windings.

fqs and fds can be considered as the direction of the magnetic axes of the “new” fictious windings located on qsand ds axis which are created by the change of variables.

Power Equations:

cscsbsbsasasabcs iViViVP

ssdsdsqsqsabcssqd iViViVPP 000 223



Stationary circuit variables transformed to the arbitrary reference frame.

Resistive elements: For a 3-phase resistive circuit,

abcssabcs irV

sqdsabcs ii 01 K sqdsabcs VV 0

1 K

sqdsssqds irV 01

01 KK

sqdssssqd irV 01

0 KK

sqdssqd irV 00 ssss rr 1, KK

s

s

s

s

rr

rr

000000



Inductive elements: For a 3-phase inductive circuit,

cs

bs

as

s

s

s

abcssabcs

abcsabcs

iii

LL

Li

dtdpwhere

pV

000000

,,

,

L



In terms of the substitute variables, we have

After some work, we can show that

sqdsssqdsssqdsssqd ppp 01

01

01

0 KKKKKKV

0)3

2cos()3

2sin(

0)3

2cos()3

2sin(0cossin

, 1

spwhere K

000001010

1 ss p KK



Vector equation Vqd0s can be expressed as

where “ds” term and “qs” term are referred to as a “speed voltage” with the speed being the angular velocity of the arbitrary reference frame.

sqdsssqdsssqd ppV 01

01

0 )(])[( KKKK

sqddqssqd pV 00

0)(, qsdsT

dqswhere

ss

dsqsds

qsdsqs

pV

pV

pV

00



When the reference frame is fixed in the stator, that is, the stationary reference frame (=0), the voltage equations for the three-phase circuit become the familiar time rate of change of flux linkage in abcs reference frame

For the three-phase circuit shown, Ls is a diagonal matrix, and

abcssabcs iL

sqdssqdssssqd ii 001

0 LKLK



For the three-phase induction or synchronous machine, Lsmatrix is expressed as

where, Lls: leakage inductance, Lms: magnetizing inductance

mslsmsms

msmslsms

msmsmsls

s

LLLL

LLLL

LLLL

21

21

21

21

21

21

L

ls

msls

msls

sss

L

LL

LL

KLK

00

0230

0023

)( 1



Consider the stator windings of a symmetrical induction or round rotor synchronous machine shown below

ssss rrrdiagr

s

s

s

s

LMMMLMMML

Lmslss LLL

msLM21



For each phase voltage, we write the following equations,

In vector form,

Multiplying by Ks

cscsscs

bsbssbs

asassas

pirVpirVpirV

,

abcssabcs

abcsssqd

abcsssqd

abcsssqd

i

ii

VV

L

K

K

K

0

0

0

,abcsabcssabcs piV r

abcssabcsssabcss piV KrKK



Replace iabcs and abcs using the transformation equations,

or

)()( 01

01

sqdsssqdsssabcss piV KKKrKKsqdsqdssqd iV 000 r

ssss

dsqsdssds

qsdsqssqs

pirV

pirV

pirV

000

000001010

where, qssqs iML )(

dssds iML )(

sss iML 00 )2(


Commonly Used Reference Frames

Our equivalent circuit in arbitrary reference frame can be represented as

Commonly used reference frame



=unspecified: stationary circuit variables referred to the arbitrary reference frame. The variables are referred to as fqd0s or fqs, fds and f0s and transformation matrix is designated as Ks.

=0: stationary circuit variables referred to the stationary reference frame. The variables are referred to as fs

qd0s or fs

qs, fsds and fs

0s and transformation matrix is designated as Ks

s.



= r: stationary circuit variables referred to the reference frame fixed in the rotor. The variables are referred to as fr

qd0s or frqs, fr

ds and fr0s and transformation matrix is

designated as Krs.

= e: stationary circuit variables referred to the synchronously rotating reference frame. The variables are referred to as fe

qd0s or feqs, fe

ds and fe0s and transformation

matrix is designated as Kes.



Representation

ssqdf 0

rsqdf 0

esqdf 0

Stationary reference frame

q-d axes of stator variables

q-d axes of stator variables,

q-d axes of stator variables,

Reference frame fixed on the rotor with speed of r

Synchronously rotating reference frame

t

rr dtt0

)(

t

ee dtt0

)(


Transformation of a Balanced Set

Consider a 3-phase circuit which is excited by a balanced 3-phase voltage set. Assume the balanced set is a set of equal amplitude sinusoidal quantities which are displaced by 120.

ef: Angular position of each electrical variable (voltage, current, and flux linkage) is ef with the f subscript used to denote the specific electrical variable.

)3

2cos(2

)3

2cos(2

cos2

efscs

efsbs

efsas

ff

ff

ff set) (balanced 0 csbsas fff

)0()(0 eft

eef dtt



e: Angular position of the synchronously rotating reference frame is e.

e and ef differ only in the zero position e(0) and ef(0), since each has the same angular velocity of e.

fas, fbs and fcs can be transformed to the arbitrary reference frame,

abcsssqd ff K0



After transformation, we will have,

qs and ds variables form a balanced 2-phase set in all reference frames except when =e,

In qse and dse reference frame, sinusoidal quantities appear as constant dc quantities.

0

)sin(2

)cos(2

0

s

efsds

efsqs

f

ff

ff

)0()0(sin2

)0()0(cos2

eefsdse

eefsqse

ff

ff


Balanced Steady-State PhasorRelationships For balanced steady-state conditions e is constant and

sinusoidal quantities can be represented as phasorvariables.

tejefj

sefesbs

tejefj

sefesbs

tejωefjθsefesas

eeFtFF

eeFtFF

eeFθtωFF

32)0(

32)0(

0

2Re3

2)0(cos2

2Re3

2)0(cos2

2Re)0(cos2


Balanced Steady-State PhasorRelationships Balanced steady-state qs-ds variables are,

fas phasor can be expressed as

tejefj

s

efesds

tejefjs

efesqs

eeFj

tFF

eeF

tFF

)0()0(

)0()0(

2Re

)0()0(sin2

2Re

)0()0(cos2

)0(~ efjsas eFF


Balanced Steady-State PhasorRelationships For arbitrary reference frame (e),

Selecting (0)=0,

Thus, in all asynchronously rotating reference frame (e) with (0)=0, the phasor representing the asvariables is equal to the phasor representing the qsvariables.

qsds

efjsqs FjFeFF ~~,~ )0()0(

qsas FF ~~


Balanced Steady-State PhasorRelationships In the synchronously rotating reference frame =e, Fe

qsand Fe

ds can be expressed as

Let e(0)=0, then

)0()0(

)0()0(

2Re

2Re

θefθjsds

e

θefθjsqs

e

eFjF

eFF

))0(sin(2)),0(cos(2 efsdse

efsqse θFFθFF

dse

qse

as jFFF ~2

))0(sin())0(cos(~,since ))0((efsefs

efjsas jFFeFF


Balanced Steady-State PhasorRelationships Consider the stator winding of a symmetrical induction

machine.

Assume the stator winding is excited by a balanced 3-phase sinusoidal voltage set.


Balanced Steady-State PhasorRelationships For phase as, we will have

For balanced conditions

For steady-state conditions, p = je

dtdp

csbsassassas MpiMpipiLirV

assassascsbsas

csbsascsbsas

piMLirViiMpMpiiiiVVV

)(),(0,0

asesassas IjMLIrV ~)(~~


Balanced Steady-State PhasorRelationships qs and ds voltage equations in the arbitrary reference frame

can be written as

Let =e, then

dssdsqssqs

dsqsdssds

qsdsqssqs

iMLiML

pirV

pirV

)(,)(

dse

sdse

qse

sqse

dse

qse

edse

sdse

qse

dse

eqse

sqse

iMLiML

pirV

pirV

)(,)(


Balanced Steady-State PhasorRelationships For balanced steady-state conditions, the variables in the

synchronously rotating reference frame are constants, therefore pe

qs and peds are zero. Therefore, the above can

be expressed as

Recall

Thus,

qse

sedse

sdse

dse

seqse

sqse

IMLIrV

IMLIrV

)(

)(

dse

qse

as jFFF ~2

dse

qse

as jVVV ~2


Balanced Steady-State PhasorRelationships

Now

Substituting in the above equation, we will have

qse

sedse

sdse

seqse

sas IMLIrjIMLIrV )()(~2

qse

dse

as

dse

qse

as

jIIIj

jIII

~2

~2

assesas IMLjrV ~)(~


qdqd٠٠ Transformation to Shunt CapacitancesTransformation to Shunt Capacitances


qdqd٠٠ Transformation to Shunt CapacitancesTransformation to Shunt Capacitances


Suppose Following Voltages Applied to 3 Phase Wye- Connected RL Circuit:

Variables Observed From Several Frames of Reference



Using Basic Circuit Analysis Techniques


Variables Observed From Several Frames of Reference Transforming to the Synchronous Reference

Frame:


Variables Observed From Several Frames of Reference In Stationary Reference Frame


Variables Observed From Several Frames of Reference In Synchronous Reference Frame



In Strange Reference Frame


Transformation Transformation BBetween Reference etween Reference FramesFrames

To transform the variables from To transform the variables from xx to to yy reference reference frame :frame :


Transformation Transformation BBetween Reference etween Reference FramesFrames

For example:For example:


Space Vectors

• Three-phase voltages

0)()()( tvtvtv CnBnAn• Two-phase voltages

• Space vector representation

)()()( tvjtvtV

3/43/20 )()()(32)( j

Cnj

Bnj

An etvetvetvtV

where

)()()(

23

230

21

211

32

)()()(

34sin

32sin0sin

34cos

32cos0cos

32

)()(

tvtvtv

tvtvtv

tvtv

Cn

Bn

An

Cn

Bn

An

axisV

V

axis

)(tV

frequency)lfundamentaf(where,

2tω)VV

(tanα

VV

s

s1

22

tf

V

s

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ﻢﯿﺣﺮﻟا ﻦﻤﺣﺮﻟا ﷲا ﻢﺴﺑ1).pdf · 15 ﯽﺑﺎﺘﮐ سﺎﺒﻋ ﺮﺘﮐد:سرد دﺎﺘﺳا Arbitrary Reference Frame When the reference frame is