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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 salient-pole

or round-pole 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 1.14-1.16 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) q-axis current, d) d-axis 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 single-phase or three-phase utilised as matter of

study in the next subsections, windigs are constructed by copper conductors, terminals of

the trhee phase windings can be star-connected or delta-connected [2]. Rotor winding arein short circuit, this rotor windings are different from stator windings, they are constructed

with laminated iron forming a cage (squirrel-cage rotor).

2.1.1 INDUCTION MACHI NE MODEL

From (Krause, 2002) Voltage equations of a thre-phase 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: Three-phase 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 short-circuit and could be fed as the stator windings with

three-phase 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 steady-state, 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 2.4-2.9 could be replaced by 0 in the case of stationary

reference frame, = rfor the rotor reference frame, and naturally = efor the syn-chronous 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, open-loop scalar control and closed-loop

scalar control.Open-loop 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 volts-per-hertz 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 three-phase 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 open-loop) 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, idflux-linkage 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 2.41-2.44, 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-

to-back PWM voltage source converters in the rotor circuit [5]. The supply-side converter is

a voltage-fed PWM converter generally made with a matrix of IGBT, is conected to a DC bus

and tho the grid via three single-phase chokes. The rotor-side is conected to the DC bus and

to the three phase rotor slips. Also is capable to control the pitch angle of the turbine.

SUPPLY-SIDEPWM CONVERTER PWM converter connected to the grid allows to keep the

DC-link 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.

ROTOR-SIDE 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, C-M.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. Wiley-IEEE 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 back-to-

back PWM converters and its application to variable-speed wind-energy generation,"

Electric Power Applications, IEE Proceedings - , vol.143, no.3, pp.231,241, May 1996.

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