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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
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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
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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
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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
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Arbitrary Reference Frame
Consider stator winding of a 2-pole 3-phase symmetrical machine
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Arbitrary Reference Frame
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.
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Arbitrary Reference Frame
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
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“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.
Arbitrary Reference Frame
.
1)3
2sin()3
2cos(
1)3
2sin()3
2cos(1sincos
1
sK
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Arbitrary Reference Frame
=0: Stationary reference frame.
=e: synchronoulsy rotaingreference frame.
=r: rotor reference frame (i.e., the reference frame is fixed on the rotor).
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Arbitrary Reference Frame
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
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Arbitrary Reference Frame
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
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Arbitrary Reference Frame
Inductive elements: For a 3-phase inductive circuit,
cs
bs
as
s
s
s
abcssabcs
abcsabcs
iii
LL
Li
dtdpwhere
pV
000000
,,
,
L
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Arbitrary Reference Frame
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
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Arbitrary Reference Frame
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
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Arbitrary Reference Frame
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
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Arbitrary Reference Frame
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
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Arbitrary Reference Frame
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
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Arbitrary Reference Frame
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
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Arbitrary Reference Frame
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(
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Commonly Used Reference Frames
Our equivalent circuit in arbitrary reference frame can be represented as
Commonly used reference frame
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Commonly Used Reference Frames
=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.
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Commonly Used Reference Frames
= 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.
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Commonly Used Reference Frames
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
)(
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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
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Transformation of a Balanced Set
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
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Transformation of a Balanced Set
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
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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
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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
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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 ~~
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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
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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.
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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 ~)(~~
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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
)(,)(
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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
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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 ~)(~
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qdqd٠٠ Transformation to Shunt CapacitancesTransformation to Shunt Capacitances
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qdqd٠٠ Transformation to Shunt CapacitancesTransformation to Shunt Capacitances
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Suppose Following Voltages Applied to 3 Phase Wye- Connected RL Circuit:
Variables Observed From Several Frames of Reference
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Variables Observed From Several Frames of Reference
Using Basic Circuit Analysis Techniques
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Variables Observed From Several Frames of Reference Transforming to the Synchronous Reference
Frame:
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Variables Observed From Several Frames of Reference In Stationary Reference Frame
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Variables Observed From Several Frames of Reference In Synchronous Reference Frame
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Variables Observed From Several Frames of Reference
In Strange Reference Frame
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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 :
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Transformation Transformation BBetween Reference etween Reference FramesFrames
For example:For example:
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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