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Electric Machines
Tuesman Castillo
March 13, 2014
1 PERMANENT MAG NETSYNCHRO NOUSMAC HINES
A permanent magnet synchronous machine is essentially a synchronous machine, with
the difference that PMSM produce the rotor magnetic field with a solid magnetic material,
the advantages are the elimination of copper looses, simpler construction, weight and sizedisminution and higher efficiency [2]. There are two main kinds of permanents magnets
arrangement in the rotor, surface mount, and interior mount, and they can form salientpole
or roundpole rotors, round rotors leads an uniform airgap wich is an uniform magnetizing
inductance in the direct and quadrature axis (LqandLd) [1].
1.1 MATH EM ATI CA LM ODEL OF APMSM
Accordingly to (Krause, pag. 264, 2002), voltage PMSM equations in the dq0sreference frame
can be written as:
vqs=rsiqs+wrd+dq
dt, (1.1)
vds=rsids+wrq+dd
dt, (1.2)
v0s=rsi0s+d0s
dt, (1.3)
Where flux linkages are:
q=Lqiq, (1.4)d=
Ldiq+m, (1.5)
0 = Ll si0, (1.6)Direct, quadrature inductances and magnetizing inductances are.
Lq=Ll s+Lmq, (1.7)
Ld=Ll s+Lmd, (1.8)
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Lmq=3
2(LALB), (1.9)
Lmd=3
2(LA+LB), (1.10)
Replacing the flux linkages equations into 1.1 1.3, voltages equations yields.
vq= rsiq+Lqdiqdt
+rLdid+rm, (1.11)
vd= rsid+Lddid
dtrLqi q, (1.12)
v0 = rsi0+Ll sdi0
dt, (1.13)
mis the amplitude of the flux linkages established by the permanent magnet as viewed
from the stator phase windings [1]. LB is zero for a round rotor machine,Lq=Ld.
In order to perform the simulation its useful to determine the current equationsidq0s.
id=1
Ld
t0
(vd rsid+rLqiq)dt (1.14)
iq=1
Lq
t0
(vq rsiq+rLdidrm)dt (1.15)
i0 =1
Ll s
t0
(v0 rsi0)dt (1.16)
The electromechanic torque is:
Te=3
2P[miq+ (LdLq)iqid], (1.17)Torque is negative for generator action. The relationship between rand r is:
r=dr
dtd t (1.18)
The total equation of various torque interactions is:
Te=J
2
p
d
d tr+B
2
p
r+TL (1.19)
WhereTLis the mechanic load,pis the number of poles. For a rotor with 2 poles we have:
r=1
J
t
0(TeTLBr)dt (1.20)
1.2 PMSM SIMULATION
The model viewed above descrives the behavior of a PMSM, the model has to be the same for
motor o generator opearation, this model is simulated using Simulink and Matlab software.
All the equations are implemented in blocks and agruped in subsystems.
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Figure 1.1: Simulink subsystem Vabc to Vdq
Figure 1.2: Simulink subsystem idqtoiabc
1.2.1 THREE PHASE VOLTAJE CONVERTION TOdqPARAMETERS
Figure1.1 shows the frame convertion of three phase voltage signals in the dqrotor reference
frame, the transformation is performed in two steps [2]. Also the block that transforms
currents in rotor frame back into 3 phase currents is show in the Figure1.2
1.2.2 DIRECT AXIS EQUATION AND QUADRATURE AXIS EQUATION BLOCKS
Equations 1.141.16 are used to obtain the blocks shown in Figure 1.3and Figure 1.4
1.2.3 MECHANIC EQUATIONB LOCK
The Figure1.5shows the subsystem that simulate the mechanic interaction of electromag
netic torqueTeand Mechanic TorqueTL, viscous coefficient are also included.
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Figure 1.3: Simulink subsystem quadrature axis
Figure 1.4: Simulink subsystem direct axis
Figure 1.5: Simulink subsystem mov
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Figure 1.6: Simulink subsystem PMSM
1.2.4 COMPLETE SIMULINK REPRESENTATION OF MODELEDPMSM
Figure 2.1shows the complete scheme of the PMSM in Simulink, including all the above
subsystems.
SIMULATION OF A COMMERCIALPMSM In order to validate the moldel presented above,
parameters of a commercial PMSM are used to obtain the curves in Fig 1.8, this parameters
where extracted from (Krause, Chp 6, 2002). Fig 1.7
Figure 1.7: parameters in matlab file .m
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Figure 1.8:a) Voltage, b)phase current, c) qaxis current, d) daxis current, e) electromagnetic
torqueTed) rotor speedr ad/s
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1.2.5 CONTROL OFPMSM
2 INDUCTIONMAC HINES
2.1 INTRODUCTION
Induction machines are rotating machins that operate in an asynchronous way, this is, for
motoring mode, rotor run under synchronous speed, and, for generator mode operation,
roor runs above synchronous speed [2]. Induction machines are widely used as industrial
motor because the easely maintain, high efficiency, and their robustness, they are the
principal electric to mechanic energy converters.
Induction machines are formed by two winding systems, one is installed in the rotor and
the other in the stator, between stator and rotor an airgap, that generally is between 0.1 and
0.9mm.
The stator winding arragement could be singlephase or threephase utilised as matter of
study in the next subsections, windigs are constructed by copper conductors, terminals of
the trhee phase windings can be starconnected or deltaconnected [2]. Rotor winding arein short circuit, this rotor windings are different from stator windings, they are constructed
with laminated iron forming a cage (squirrelcage rotor).
2.1.1 INDUCTION MACHI NE MODEL
From (Krause, 2002) Voltage equations of a threphase induction machine can be written as
follows:
Vabcs= rsiabcs+d
d tabcs (2.1)
Vabcr= rriabcr+d
dtabcr (2.2)
where: abcs
abcr
=Lss Lsr
Lr s Lr r
iabcs
iabcr
(2.3)
VOLTAGE EQUATIONS IN ARBITARYqd0REFERENCE FRAME Voltage equations in expanded
form according to (Krause, 2002) are:
vqs= rsiqs+ds+d
dtqs (2.4)
vds= rsids+qs+d
dtds (2.5)
v0s= rsi0s+d
dt0s (2.6)
vqr= r
si
qr+ (r)
dr+
d
dtqr (2.7)
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Figure 2.1: Threephase induction machine
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vdr= r
si
dr (r)
qr+
d
dtdr (2.8)
v0r= r
si
0r
d
dt0r (2.9)
Where flux linkages in expanded form are:
qs= Ll siqs+LM(iqs+ iqr) (2.10)
ds= Ll sids+LM(ids+ idr) (2.11)
0s= Ll si0s (2.12)
qr= L
l ri
qr+LM(iqr+ i
qr) (2.13)
dr= L
l ri
dr+LM(ids+ i
dr) (2.14)
0r= L
l ri
0r (2.15)
LM= 32Lms (2.16)
Ll s andLmsare, respectively the leakeage and magnetizing inductances of the stator
windings,Ll randLmrare for rotor windings[1].
TORQUE E QUATIONS developed torque by an induction machine in rotor speed funtion is:
Te=j
2
P
d
d tr+TL (2.17)
jis the inertia,TLis the load torque, positive for motoring and negative for generator
mode. Torque expression in arbitrary reference frame in terms of currents is:
Te= 1.5P
2
LM
iqsi
dr idsi
qr
(2.18)
Again,Teis positive for motor action, another equations for the torque could be:
Te= 1.5P
2
LM
qri
dr
dri
qr
(2.19)
Te= 1.5P
2
LM
dsiqsqsids
(2.20)
In steady state operation torque equation is shown in Fig 2.2:
where s is the slip defined by:
s= ere
(2.21)
The stator input current is:
 Ias =Vas
Z (2.22)
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Figure 2.2: Steady state torque equation
2.2 DOUBLYF ED I NDUCTION MAC HI NES
2.2.1 INTRODUCTION
A doubly fed induction machine (DFIM) is an induction machine with with the difference
that its rotor windings are not in shortcircuit and could be fed as the stator windings with
threephase voltages via slip rings[3]. This machine types are specially utilized as Adjustable
Speed Generators (ASG), to improve the operation of wind turbines, offering important
advantages as: Reduce of mechanical streses, dinamical compensation for torque a powerpulsations, etc [4]. The mathematical model of a DFIM its equal to a induction machine
(IM). Equations would be repeated but using the simbol instead of, accordingly to most
of the literature review[2][3][4].
VOLTAGE AND TORQUE EQUATIONS IN STATIONARY REFERENCE FRAME In an idealized model
show in fig 3.1, voltages and currents in fluxes function can be described by the following
equations:
Vas
=rsias
+
das
dt
(2.23)
Vbs= rsibs+dbs
dt(2.24)
Vcs= rsics+dcs
dt(2.25)
rsis the stator resistance,iabcs,vabcsare the currents and voltages applied,abcsare the
stator fluxes[3]. Rotor equations are described as.
Var= rriar+dar
dt(2.26)
Vbr= rribr+dbr
dt(2.27)
Vcr= rricr+dcr
dt(2.28)
Under steadystate, eis the angular frecuency imposed by the grid in stator windings,
r is the the angular frecuency in rotor windings. This two magnitudes, are related by:
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r+m=s (2.29)mis the electrical angular frecuency of the machine.
MATHEMATIACL MODEL INdqSYNCHRONOUS REFERENCE FRAME Voltages equations in
synchronous reference frame are denoted by:
Vds= rsids+dds
dteqs (2.30)
Vqs= rsiqs+dqs
dt+eds (2.31)
Vdr= rsidr+ddr
dt (er)qr (2.32)
Vqr= rsiqr+dqr
dt + (er)dr (2.33)Flux linkages are:
ds= Ll sids+LMidr (2.34)qs= Ll siqs+LMiqr (2.35)
dr= Ll ridr+LMids (2.36)qr= Ll riqr+LMiqs (2.37)
Torque equations:Electromagnetic torque is given by:
Te=3
2
P
2
LM
Ll s(qsidrdsiqr) (2.38)
The mechanic dynamic equation is expressed as:
Te=J2
P
dm
dt+TL (2.39)
where:d
dtm=m (2.40)According to the literature review [3], [4] its useful to know the expressions in thed q
reference frame for electric powers on rotor and stator side:
Ps=3
2(Vdsids+ vqsi qs) (2.41)
Pr=3
2(Vdridr+ vqri qr) (2.42)
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Qs=3
2(Vqsids+ vdsi qs) (2.43)
Qr=3
2(Vqridr+ vdri qr) (2.44)
REFERENCE FRAMES The equations described above are in the synchronous rotating refer
ence frame, as the case of equations 2.42.9 could be replaced by 0 in the case of stationary
reference frame, = rfor the rotor reference frame, and naturally = efor the synchronous rotating reference frame. Conditions of operation will determine the reference
frame most convenient to use:
In case of unbalanced stator voltages and balanced or zero rotor voltages its better to use
stationary reference frame. For unbalanced external applied rotor voltages and balanced
stator voltages, a rotor reference frame should be used. Synchronous reference frame is used
to analyze balanced rotor and stator conditions [1].
SLIP AND OPERATING MODES Depending of the slip as seen on equation 2.21, we can
distiguish three operating modes for the machine [3].
s> 0, r> 0 Subsynchronous operation.s< 0, r< 0 Hypersynchronous operation.s= 0,r= 0 Synchronous operation.
MODES OF OPERATION OF ADFI M Next table, shows the four modes of operation for a
DFIM.
Figure 2.3: Four Modes of operation of a DFIM, extracted from [3]
2.2.2 INDUCTIONMAC HI NE CONTROLTECHNIQUES
According to the literature review, induction machines control techniques are divided into
two branches: Scalar based controllers, and Vector Based controllers [3].
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SCALARCONTROL: In some applications, induction machines could vary the speed without
changing the torque, the simplest method is changing the number of pole pairs, also a
variation of the supplied voltage magnitude or its frequency can be performed. Scalar control
techniques are based in the simultaneous variation of voltage magnitude and frequency. It
can be differenced two types of scalar controllers, openloop scalar control and closedloop
scalar control.Openloop scalar control is based in the premise that by variyng frecuency one can control
the speed, this results in a change of impedances, and in an increase or decrease of the
current, causing a change in the torque. If the voltage increases can cause saturation, so, its
important to vary frequency and voltage at the same time. The control technique is to set
a reference angular velocity in the input of the control, this angular velocity is multiplied
by the number of pair of poles to obtain the electrical rotor speed, the electrical frecuency
is multiplied by the voltsperhertz ratio Vbb
, both rated values, from this mutiplicacion we
obtain theVsvoltage wich is multiplied by
2resulting in theVqvalue, this quadrature
voltage is used as variable for a VSI (Voltage Source Inverter) to vary the threephase voltage
source of the machine [1].Closed loop method permits to control the torque, using a slip control loop, slip is pro
portional to the torque. in this control method speed of the machine is sense, this signal is
compared with the desired speed, the difference between the two signals is decreased to
zero by a PI controller, and the motor will have a regulated speed [1].
This Scalar control methods are used especially in the industry, due its implementation
simplicity, low cost implementation, and good power quality[3]. A simplificated scalar
control is show in fig 2.4.
Figure 2.4: Scalar control (closed and openloop) block diagram [3]
VECTORBAS ED CO NT RO L This methods controls the machine in a synchronously rotating
dqreference frame, with thedaxis oriented along the rotor flux space vector position. The
dcurrent is thus proportional to the rotor fluxrand theqcurrent is proportional to the
electromagnetic torqueTe [3]. Controlling both currentsiq, idfluxlinkage in rotor and
electromagnetic torque can be controlled independently. This method is used in the same
way for DFIM, but, in this case a reference frame orientated with the stator flux could be
used to control independently the active and reactive power flows withiqandid. Fig. 2.5
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shows a general block diagram of a vector control system. This control method over scalar
control has the advantage that torque can be controlled and a better dynamic response, but
requires to tunning controllers and constant switching frecuency.[3].
Figure 2.5: Vector control block diagram for IM and DFIM [3]
Figure 2.6: Direct torque control (DTC) block diagram [3].
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DIRECTC ONTROLT ECHNIQUES : Are alternative solutions to control AC machines, there
are many variants of his type of control[3], escentially we can find two clasic direct control
techniques named as: direct torque control (DTC) and direct power control (DPC), both
controls are basically equal in structure but oriented to control different magnitudes of the
machine.
DTC controls torque and rotor flux, the classic DTC is based on a space representation ofthe source voltages for the machine, in the control input are present two variables: Torque
and rotor flux amplitude. The rotor flux and stator flux in a space vector representation
are separated by an angle denoted by, if we can drive the distance between those space
vector, it is posible to control the machine torque, rotor flux amplitude and direction in the
machine can be influenced by the VSC, a complete DTC block diagram is presented en fig.
2.6[3].
DPC Direct power control is based on the same control principlesas DTC, but in DPC the
stator active and reactive powers are directly controlled.
Stator active and reactive powers of the machine can be calculated directly from stator
voltage and currets, just like equations 2.412.44, as in DTC, injecting directly rotor voltage
vectors,Pand Qare controlled, the schem of a simple DPC could be draw modifing inputs
and outputs on fig 2.6, changingTembyQsandrbyPs.
CONTROLSYS TE M FO R ADFIMACTING AS P OWER GENERATOR DFIM acting as generator are
present in modern wind powered generator, they generate power from its stator windings
and its rotor widings too, the power generated in the rotor has an important aplication, a
control system can generare active power or reactive power, PandQ, to and from the grid.
While the stator circuit delivers active power to the grid, and it is conected directly to the
system.
The most common way to control the Power Pand Q, torque, and pitch is to use two back
toback PWM voltage source converters in the rotor circuit [5]. The supplyside converter is
a voltagefed PWM converter generally made with a matrix of IGBT, is conected to a DC bus
and tho the grid via three singlephase chokes. The rotorside is conected to the DC bus and
to the three phase rotor slips. Also is capable to control the pitch angle of the turbine.
SUPPLYSIDEPWM CONVERTER PWM converter connected to the grid allows to keep the
DClink voltage constant regardless of the magnitude and direction of the rotor power [5]. A
vector control method is used to drive this side, it is necesary to obtain two currents vector
IqandId, PWM converter is current regulated,Idis used to control the voltage level of the
DC bus,Iqis used to control the reactive power libered or extrated from the grid.
ROTORSIDE P WM CONVERTER Rotor converter has the objective of control the output
active and reactive power, a Vrsignal has to be send to the PWM converter, this signal has
two components Vdrand Vqr.Vqr, is produced by a PI regulator comparing the actual Iqrsignal with a reference quadrature current signal. Signal Iqris the vector that controls the
Tem, and controls the active power produced by the machine [6].
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Figure 2.7: Wind turbine and DFIG System [Matlab Help]
Reactive power is controlled by theVdrsignal, this signal is generated by comparing actual
Idrvector and a reference direct current vector obtained from a AC voltage regulator and a
Var regulator modules [6].
Figure 2.8: Power Flow in DFIG system [Matlab Help]
REFERENCES
[1] Krause, P.C., O. Wasynczuk, and S.D. Sudhoff. Analysis of Electric Machinery and Drive
Systems. IEEE Press, 2002.
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[2] Ong, CM.Dynamic Simulations of Electric Machinery: Using MATLAB/SIMULINK.
Prentice Hall, 1997
[3] G.Abad, J.Lopez, M.A. Rodriguez, L. Marroyo. Doubly Fed Induction Machine: Modeling
and Control for Wind Energy Generation Applications. WileyIEEE Press, 2011.
[4] Muller, S.; Deicke, M.; De Doncker, R.W., "Doubly fed induction generator systems forwind turbines," Industry Applications Magazine, IEEE , vol.8, no.3, pp.26,33, May/Jun
2002.
[5] Pena, R.; Clare, J.C.; Asher, G.M., "Doubly fed induction generator using backto
back PWM converters and its application to variablespeed windenergy generation,"
Electric Power Applications, IEE Proceedings  , vol.143, no.3, pp.231,241, May 1996.
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