s.d. sudhoff fall 2005 - purdue engineeringsudhoff/ee595s/reference frame... · ee595s: class...

EE595S: Class Lecture NotesChapter 3: Reference Frame Theory

S.D. Sudhoff

Fall 2005

Fall 2005 EE595S Electric Drive Systems 2

Reference Frame Theory

• Power of Reference Frame Theory:Eliminates Rotor Position Dependence Inductances and CapacitancesTransforms Nonlinear Systems to Linear Systems for Certain CasesFundamental Tool For Rigorous Development of Equivalent CircuitsCan Be Used to Make AC Quantities Become DC QuantitiesFramework of Most Controllers


History of Reference Frame Theory

• 1929: Park’s TransformationSynchronous 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


3.3 Equations of Transformation

• (3.3-1)

• (3.3-2)

• (3.3-3)

][)( 00 sdsqsT

sqd fff=f

abcsssqd fKf 0 =

][)( csbsasT

abcs fff=f



• Forward Transformation

• Inverse Transformation

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ −

=

21

21

21

32sin

32sinsin

32cos

32coscos

32 πθπθθ

πθπθθ

sK

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ −⎟

⎠⎞

⎜⎝⎛ −=−

13

2sin3

2cos

13

2sin3

2cos1sincos

)( 1

πθπθ

πθπθθθ

sK

(3.3-4)

(3.3-6)



• Geometrical Interpretation



• Transformation of PowerStarting Point

• (3.3-9)

Result• (3.3-10)

cscsbsbsasasabcs ivivivP ++=

)2(23

00

0

ssdsdsqsqs

abcssqd

iviviv

PP

++=

=


3.4 Stationary Circuit Variables Transformed to Arbitrary Reference Frame

• Resistive Circuits(3.4-1)abcssabcs irv =



• Inductive Circuits(3.4-4)abcsabcs pλ=v



• Inductive Circuits (Continued)


3.4 Stationary Circuit VariablesTransformed to Arbitrary Reference Frame

• Capacitive Circuits(3.4-19)abcsabcs pqi =



• Capacitive Circuits (Continued)



• Example 3B(3B-1)

(3B-2)

] [diag ssss rrr=r

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

s

s

s

sLMMMLMMML

L



• Example 3B (Continued)


• Example 3B: SummaryVoltage Equations

• (3B-13)• (3B-14)• (3B-15)

Flux Linkage Equations• (3B-16)• (3B-17)• (3B-18)


qsdsqssqs pirv λωλ ++=

dsqsdssds pirv λωλ +−=

ssss pirv 000 λ+=

qssqs iML )( −=λ

dssds iML )( −=λ

sss iML 00 )2( +=λ



• Equivalent Circuit


3.5 Commonly Used ReferenceFrames

Notation

Reference frame speed Interpretation Variables

Trans- formation

ω (unspecified) Stationary circuit variables referred to the

arbitrary reference frame sqd0f or

sdsqs fff 0 , , sK

0 Stationary circuit variables referred to the stationary reference frame

ssqd0f or

ssds

sqs fff 0 , ,

ssK

rω Stationary circuit variables referred to a reference frame fixed in the rotor

rsqd0f or

s

r

ds

r

qs fff 0 , ,

rsK

eω Stationary circuit variables referred to the synchronously rotating reference frame

esqd0f or

seds

eqs fff 0 , ,

esK


3.6 Transformation Between Reference Frames

• Consider Reference Frame x(3.6-1)

• Consider Reference Frame y(3.6-2)

• Thus

abcsys

ysqd fKf =0

abcsxs

xsqd fKf =0


3.6 Transformation Between Reference Frames

• Evaluating Yields

(3.6-7)

• It Can Be Shown That(3.6-8)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−−−−

=1000)(cos)(sin0)(sin)(cos

xyxy

xyxyyx θθθθ

θθθθK

Tyxyx )()( 1 KK =−


3.7 Transformation of a Balanced Set

• Consider 3-phase Variables(3.7-1)

(3.7-2)

(3.7-3)

• Where

(3.7-4)

efsas ff θcos2=

⎟⎠⎞

⎜⎝⎛ −=

32cos2 πθefsbs ff

⎟⎠⎞

⎜⎝⎛ +=

32cos2 πθefscs ff

dtd ef

eθ

ω =


3.7 Transformation of a Balanced Set

• It Can Be Shown(3.7-5)

(3.7-6)

)(cos2 θθ −= efsqs ff

)(sin2 θθ −−= efsds ff


3.7 Transformation of a Balance Set

• Some Special Cases …


3.8 Balanced Steady-State PhasorRelationships

• For Steady State Conditions…(3.8-1)

(3.8-2)

(3.8-3)

]2[Re

)]0([cos2)0( tjj

s

efesas

eef eeF

tFFωθ

θω

=

+=

]2[Re

32)0(cos2

]3/2)0([ tjjs

efesbs

eef eeF

tFF

ωπθ

πθω

−=

⎥⎦⎤

⎢⎣⎡ −+=

]2[Re

32)0(cos2

]3/2)0([ tjjs

efescs

eef eeF

tFF

ωπθ

πθω

+=

⎥⎦⎤

⎢⎣⎡ ++=



• From (3.7-5) and (3.7-6)(3.8-4)

(3.8-5)

]2[Re

)]0()0()[(cos2)()]0()0([ tjj

s

efesqs

eef eeF

tFFωωθθ

θθωω

−−=

−+−=

]2[Re

)]0()0()[(sin2)()]0()0([ tjj

s

efesds

eef eeFj

tFFωωθθ

θθωω

−−=

−+−−=



• From (3.8-1)(3.8-6)

• Consider Non-Synchronous Reference Frame

(3.8-7)(3.8-8)

• With Appropriate Choice of Reference Frame

(3.8-9)

)0(~ efjsas eFF θ=

)]0()0([~ θθ −= efjsqs eFF

qsds FjF ~~ =

qsas FF ~~ =



• For Synchronous Reference Frame

(3.8-10)

(3.8-11)

• With Appropriate Choice of Frame(3.8-12)

(3.8-13)

]2[Re)]0()0([ eefj

seqs eFF

θθ −=

]2[Re)]0()0([ eefj

seds eFjF

θθ −=

)0(cos2 efseqs FF θ=

)0(sin2 efseds FF θ−=



• Thus, Choosing Time Zero Position of Zero(3.8-14)e

dseqsas jFFF −=~2


3.9 Balanced Steady-State Voltage Equations

• Consider Non-Synchronous Reference Frame• Recall

(3.8-6)(3.8-7)

• Now Suppose(3.9-4)

• We Can Show That(3.9-9)

)0(~ efjsas eFF θ=

)]0()0([~ θθ −= efjsqs eFF

assas IZV ~~ =

qssqs IZV ~~ =


3.9 Balanced Steady-State Voltage Equations

• Proof


3.10 Variables Observed From Several Frames of Reference

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

(3.10-1)

(3.10-2)

(3.10-3)

tVv esas ωcos2=

⎟⎠⎞

⎜⎝⎛ −=

32cos2 πω tVv esbs

⎟⎠⎞

⎜⎝⎛ +=

32cos2 πω tVv escs



• Using Basic Circuit Analysis Techniques

(3.10-4)

(3.10-5)

(3.10-6)

)](coscos[2 / αωατ −+−= − teZ

Vi et

s

sas

⎥⎦⎤

⎢⎣⎡

⎟⎠⎞

⎜⎝⎛ −−+⎟

⎠⎞

⎜⎝⎛ +−= −

32cos

32cos2 / παωπατ te

ZVi e

t

s

sbs

⎥⎦⎤

⎢⎣⎡

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ −−= −

32cos

32cos2 / παωπατ te

ZVi e

t

s

scs



• In (3.10-4)-(3.10-8)

(3.10-7)

(3.10-8)

(3.10-9)

sess LjrZ ω+=

s

srL

=τ

sse

rLωα 1tan−=



• Transforming to the Synchronous Reference Frame

(3.10-10)

(3.10-11)

]})[(cos

)(cos{2 /

αωω

αωτ

−−+

−−= −

t

teZVi

e

ts

sqs

]})[(sin

)(sin{2 /

αωω

αωτ

−−−

−−= −

t

teZVi

e

ts

sds



• In Stationary Reference Frame



• In Synchronous Reference Frame



• In Strange Reference Frame


Transformation of Measured Quantities


Transformation of Measured Currents

• Suppose we are measuring 2 of 3 currents in a wye-connected load …


Transformation of Measured Currents

• Result

⎥⎦

⎤⎢⎣

⎡−−

−=

)cos()6/sin()sin()6/cos(

32

ee

eeei θπθ

θπθK


Transformation of Measured Line-to-Line Voltages

• Suppose we are measuring voltages in a delta-connected load …


Transformation of Line-to-Line Voltages

• Result

⎥⎦

⎤⎢⎣

⎡+−+−

=)3/2sin()sin()3/2cos()cos(

32

πθθπθθ

ee

eeevK

s.d. sudhoff fall 2005 - purdue engineeringsudhoff/ee595s/reference frame... · ee595s: class...

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