EE595S: Class Lecture Notes
Chapter 3: Reference Frame Theory
S.D. Sudhoff
Fall 2005
Reference Frame Theory
• Power of Reference Frame Theory:
¾Eliminates Rotor Position Dependence
Inductances and Capacitances
¾Transforms Nonlinear Systems to Linear
Systems for Certain Cases
¾Fundamental Tool For Rigorous Development
of Equivalent Circuits
¾Can Be Used to Make AC Quantities Become
DC Quantities
¾Framework of Most Controllers
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History of Reference Frame Theory
• 1929: Park’s Transformation
¾ Synchronous Machine; Rotor Reference Frame
• 1938: Stanley
¾ Induction Machine; Stationary Reference Frame
• 1951: Kron
¾ Induction Machine; Synchronous Reference Frame
• 1957: Brereton
¾ Induction Machine; Rotor Reference Frame
• 1965: Krause
¾ Arbitrary Reference Frame
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3.3 Equations of Transformation
• f qd 0 s = K s f abcs
(3.3-1)
• (f qd 0 s )T = [ f qs
f ds
f0s ]
(3.3-2)
• (f abcs ) = [ f as
f bs
f cs ]
(3.3-3)
T
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3.3 Equations of Transformation
• Forward
Transformation
• Inverse
Transformation
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⎡
⎢cos θ
⎢
2⎢
K s = sin θ
3⎢
⎢ 1
⎢
⎢⎣ 2
2π ⎞
2π ⎞⎤
⎛
⎛
cos ⎜θ −
⎟ cos ⎜θ +
⎟
3 ⎠
3 ⎠⎥⎥
⎝
⎝
2π ⎞
2π ⎞ ⎥
⎛
⎛
sin ⎜θ −
⎟ sin ⎜θ +
⎟
3 ⎠
3 ⎠⎥
⎝
⎝
⎥
1
1
⎥
2
2
⎥⎦
⎡
⎤
⎢
cosθ
sin θ
1⎥
⎢
2π ⎞
2π ⎞ ⎥
⎛
⎛
−
1
(K s ) = ⎢cos ⎜θ − ⎟ sin ⎜θ − ⎟ 1⎥
3 ⎠
3 ⎠ ⎥
⎝
⎢ ⎝
⎢ ⎛
2π ⎞
⎛θ + 2π ⎞ 1⎥
+
cos
sin
θ
⎟
⎜
⎟
⎢ ⎜
3 ⎠
3 ⎠ ⎥⎦
⎝
⎣ ⎝
EE595S Electric Drive Systems
(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 Power
¾ Starting Point
•
Pabcs = vas ias + vbs ibs + vcsics
(3.3-9)
¾ Result
• Pqd 0 s = Pabcs
3
= (vqsiqs + vdsids + 2v0 si0 s )
2
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(3.3-10)
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3.4 Stationary Circuit Variables
Transformed to Arbitrary Reference Frame
• Resistive Circuits
¾ v abcs = rs i abcs
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(3.4-1)
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3.4 Stationary Circuit Variables
Transformed to Arbitrary Reference Frame
• Inductive Circuits
¾ v abcs = pλ abcs
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(3.4-4)
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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 Variables
Transformed to Arbitrary Reference Frame
• Capacitive Circuits
¾i
abcs = pq abcs
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(3.4-19)
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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
¾ rs = diag [ rs
⎡ Ls
¾ Ls = ⎢M
⎢
⎢⎣ M
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M
Ls
M
rs
rs ]
M⎤
M⎥
⎥
Ls ⎥⎦
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(3B-1)
(3B-2)
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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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3.4 Stationary Circuit Variables
Transformed to Arbitrary Reference Frame
• Example 3B: Summary
¾ Voltage Equations
• vqs = rs iqs + ωλds + pλqs
• vds = rs ids − ωλqs + pλds
• v0s = rsi0s + pλ0s
(3B-13)
(3B-14)
(3B-15)
¾ Flux Linkage Equations
• λqs = ( Ls − M )iqs
• λ ds = ( Ls − M )ids
• λ0 s = ( Ls + 2 M )i0 s
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(3B-16)
(3B-17)
(3B-18)
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3.4 Stationary Circuit Variables
Transformed to Arbitrary Reference Frame
• Equivalent Circuit
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3.5 Commonly Used Reference
Frames
Notation
Reference
frame speed
ω
(unspecified)
0
Interpretation
Stationary circuit variables referred to the
arbitrary reference frame
Stationary circuit variables referred to the
stationary reference frame
ωr
Stationary circuit variables referred to a
reference frame fixed in the rotor
ωe
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Stationary circuit variables referred to
the synchronously rotating reference frame
Variables
f qd0s or
Transformation
Ks
fqs, fds, f0s
s
fqd0
s or
K ss
s
s
fqs
, fds
, f0s
r
fqd0
s or
r
K rs
r
fqs , fds , f0 s
e
fqd
0s or
Kes
e e
fqs
, fds, f0s
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3.6 Transformation Between
Reference Frames
• Consider Reference Frame x
y
y
¾ f qd 0 s = K s f abcs
• Consider Reference Frame y
x
¾ fx
=
K
s f abcs
qd 0 s
(3.6-1)
(3.6-2)
• Thus
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3.6 Transformation Between
Reference Frames
• Evaluating Yields
¾
⎡cos (θ y − θ x ) − sin (θ y − θ x ) 0⎤
x y ⎢
K = sin (θ y − θ x ) cos (θ y − θ x ) 0⎥
⎢
⎥
0
0
1⎥⎦
⎢⎣
(3.6-7)
• It Can Be Shown That
¾ ( x K y ) −1 =( x K y )T
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(3.6-8)
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3.7 Transformation of a Balanced Set
• Consider 3-phase Variables
¾ f as = 2 f s cosθef
(3.7-1)
¾ fbs = 2 f s cos ⎛⎜θ ef − 2π ⎞⎟
(3.7-2)
¾
(3.7-3)
3 ⎠
⎝
2π ⎞
⎛
f cs = 2 f s cos ⎜θ ef +
⎟
3 ⎠
⎝
• Where
¾ ωe =
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dθef
(3.7-4)
dt
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3.7 Transformation of a Balanced Set
• It Can Be Shown
¾
f qs = 2 f s cos (θ ef − θ )
¾ f ds = − 2 f s sin (θ ef − θ )
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(3.7-5)
(3.7-6)
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3.7 Transformation of a Balance Set
• Some Special Cases …
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3.8 Balanced Steady-State Phasor
Relationships
• For Steady State Conditions…
¾
Fas = 2Fs cos [ωet + θ ef (0)]
= Re [ 2Fs e
¾
¾
(3.8-2)
j[θ ef ( 0) − 2π / 3] jωet
e
]
2π ⎤
⎡
Fcs = 2Fs cos ⎢ωet + θ ef (0) + ⎥
3⎦
⎣
= Re [ 2Fs e
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jθef ( 0) jωet
e
]
2π ⎤
⎡
Fbs = 2Fs cos ⎢ωet + θ ef (0) − ⎥
3⎦
⎣
= Re [ 2Fs e
(3.8-1)
(3.8-3)
j[θ ef ( 0 ) + 2π / 3] jωet
e
]
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3.8 Balanced Steady-State Phasor
Relationships
• From (3.7-5) and (3.7-6)
¾ Fqs = 2Fs cos [(ωe − ω )t + θ ef (0) − θ (0)]
= Re [ 2Fs e
j[θef ( 0) −θ ( 0)] j (ωe −ω )t
e
(3.8-4)
]
¾ Fds = − 2Fs sin [(ωe − ω )t + θ ef (0) − θ (0)] (3.8-5)
= Re [ j 2Fs e
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j[θef ( 0) −θ ( 0)] j (ωe −ω )t
e
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]
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3.8 Balanced Steady-State Phasor
Relationships
• From (3.8-1)
¾ F~as = Fs e jθef (0)
(3.8-6)
• Consider Non-Synchronous Reference
Frame
¾
¾
j[θ ( 0 ) −θ ( 0 )]
~
Fqs = Fs e ef
~
~
Fds = jFqs
(3.8-7)
(3.8-8)
• With Appropriate Choice of Reference
Frame
¾ F~as = F~qs
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(3.8-9)
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3.8 Balanced Steady-State Phasor
Relationships
• For Synchronous Reference Frame
j[θ ef (0) −θ e (0)]
e
¾ Fqs = Re [ 2Fs e
]
(3.8-10)
j[θef (0) −θe (0)]
e
¾ F = Re [ j 2Fs e
]
ds
(3.8-11)
• With Appropriate Choice of Frame
e = 2F cosθ (0)
¾ Fqs
s
ef
(3.8-12)
e = − 2F sin θ (0)
¾ Fds
s
ef
(3.8-13)
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3.8 Balanced Steady-State Phasor
Relationships
• Thus, Choosing Time Zero Position of Zero
e
e
¾ 2F~as = Fqs
− jFds
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(3.8-14)
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3.9 Balanced Steady-State Voltage
Equations
• Consider Non-Synchronous Reference Frame
• Recall
¾ F~ = F e jθef (0)
as
s
¾ F~qs = Fs e j[θef (0)−θ (0)]
(3.8-6)
(3.8-7)
• Now Suppose
~
~
V
=
Z
I
¾ as
s as
(3.9-4)
• We Can Show That
¾V~qs = Z s I~qs
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(3.9-9)
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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 3Phase Wye-Connected RL Circuit
¾ vas = 2Vs cos ωet
(3.10-1)
2π ⎞
⎛
= 2Vs cos ⎜ ωet −
⎟
3
⎠
⎝
(3.10-2)
¾ vcs = 2Vs cos ⎛⎜ ωet + 2π ⎞⎟
(3.10-3)
¾ vbs
⎝
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3 ⎠
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3.10 Variables Observed From
Several Frames of Reference
• Using Basic Circuit Analysis Techniques
¾
¾
2Vs
ias =
[−e −t /τ cos α + cos (ωet − α )]
Zs
ibs =
¾ ics =
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(3.10-4)
2Vs ⎡ −t / τ
2π ⎞
2π ⎞⎤
⎛
⎛
e
α
ω
t
α
cos
cos
−
+
+
−
−
⎜ e
⎟ (3.10-5)
⎜
⎟
Z s ⎢⎣
3 ⎠
3 ⎠⎥⎦
⎝
⎝
2Vs
Zs
2π ⎞
2π ⎞ ⎤
⎡ −t / τ
⎛
⎛
e
α
ω
t
α
cos
cos
−
−
+
−
+
⎜ e
⎟ ⎥(3.10-6)
⎜
⎟
⎢⎣
3 ⎠
3 ⎠⎦
⎝
⎝
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3.10 Variables Observed From
Several Frames of Reference
• In (3.10-4)-(3.10-8)
¾ Z s = rs + jωe Ls
(3.10-7)
Ls
¾ τ=
rs
(3.10-8)
−1 ωe Ls
α
=
tan
¾
r
(3.10-9)
s
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3.10 Variables Observed From
Several Frames of Reference
• Transforming to the Synchronous Reference
Frame
2Vs
{−e −t / τ cos (ωt − α )
¾ iqs =
Zs
(3.10-10)
+ cos [(ωe − ω )t − α ]}
2Vs
−t / τ
¾ ids =
{−e
sin (ωt − α )
Zs
(3.10-11)
− sin [(ωe − ω )t − α ]}
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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
K ie
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2 ⎡cos(θ e − π / 6) sin(θ e ) ⎤
=
⎢ sin(θ − π / 6) − cos(θ )⎥
3⎣
e
e ⎦
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Transformation of Measured Lineto-Line Voltages
• Suppose we are measuring voltages in a
delta-connected load …
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Transformation of Line-to-Line
Voltages
• Result
K ev
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2 ⎡cos(θ e ) − cos(θ e + 2π / 3)⎤
= ⎢
3 ⎣ sin(θ e ) − sin(θ e + 2π / 3) ⎥⎦
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