EE595S: Class Lecture NotesChapter 3: Reference Frame Theory
S.D. Sudhoff
Fall 2005
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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
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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
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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
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3.3 Equations of Transformation
• 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)
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3.3 Equations of Transformation
• Geometrical Interpretation
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3.3 Equations of Transformation
• Transformation of PowerStarting Point
• (3.3-9)
Result• (3.3-10)
cscsbsbsasasabcs ivivivP ++=
)2(23
00
0
ssdsdsqsqs
abcssqd
iviviv
PP
++=
=
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3.4 Stationary Circuit Variables Transformed to Arbitrary Reference Frame
• Resistive Circuits(3.4-1)abcssabcs irv =
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3.4 Stationary Circuit Variables Transformed to Arbitrary Reference Frame
• Inductive Circuits(3.4-4)abcsabcs pλ=v
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3.4 Stationary Circuit Variables Transformed to Arbitrary Reference Frame
• Inductive Circuits (Continued)
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3.4 Stationary Circuit VariablesTransformed to Arbitrary Reference Frame
• Capacitive Circuits(3.4-19)abcsabcs pqi =
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3.4 Stationary Circuit Variables Transformed to Arbitrary Reference Frame
• Capacitive Circuits (Continued)
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3.4 Stationary Circuit Variables Transformed to Arbitrary Reference Frame
• Example 3B(3B-1)
(3B-2)
] [diag ssss rrr=r
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
s
s
s
sLMMMLMMML
L
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3.4 Stationary Circuit Variables Transformed to Arbitrary Reference Frame
• Example 3B (Continued)
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3.4 Stationary Circuit Variables Transformed to Arbitrary Reference Frame
• Example 3B (Continued)
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• Example 3B: SummaryVoltage Equations
• (3B-13)• (3B-14)• (3B-15)
Flux Linkage Equations• (3B-16)• (3B-17)• (3B-18)
3.4 Stationary Circuit Variables Transformed to Arbitrary Reference Frame
qsdsqssqs pirv λωλ ++=
dsqsdssds pirv λωλ +−=
ssss pirv 000 λ+=
qssqs iML )( −=λ
dssds iML )( −=λ
sss iML 00 )2( +=λ
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3.4 Stationary Circuit Variables Transformed to Arbitrary Reference Frame
• Equivalent Circuit
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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
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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
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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 =−
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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 =
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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
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3.7 Transformation of a Balance Set
• Some Special Cases …
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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
+=
⎥⎦⎤
⎢⎣⎡ ++=
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3.8 Balanced Steady-State PhasorRelationships
• 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
−−=
−+−−=
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3.8 Balanced Steady-State PhasorRelationships
• 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 ~~ =
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3.8 Balanced Steady-State PhasorRelationships
• 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−=
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3.8 Balanced Steady-State PhasorRelationships
• Thus, Choosing Time Zero Position of Zero(3.8-14)e
dseqsas jFFF −=~2
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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 ~~ =
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3.9 Balanced Steady-State Voltage Equations
• Proof
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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
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3.10 Variables Observed From Several Frames of Reference
• 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
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3.10 Variables Observed From Several Frames of Reference
• 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−=
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3.10 Variables Observed From Several Frames of Reference
• 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
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3.10 Variables Observed From Several Frames of Reference
• In Stationary Reference Frame
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3.10 Variables Observed From Several Frames of Reference
• In Synchronous Reference Frame
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3.10 Variables Observed From Several Frames of Reference
• In Strange Reference Frame
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Transformation of Measured Quantities
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Transformation of Measured Currents
• Suppose we are measuring 2 of 3 currents in a wye-connected load …
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Transformation of Measured Currents
• Result
⎥⎦
⎤⎢⎣
⎡−−
−=
)cos()6/sin()sin()6/cos(
32
ee
eeei θπθ
θπθK
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Transformation of Measured Line-to-Line Voltages
• Suppose we are measuring voltages in a delta-connected load …
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Transformation of Line-to-Line Voltages
• Result
⎥⎦
⎤⎢⎣
⎡+−+−
=)3/2sin()sin()3/2cos()cos(
32
πθθπθθ
ee
eeevK