TVE-MFE 18004

Examensarbete 30 hpJuni 2018

Field Current Control for the Damping of Rotor Oscillations and for the Alternative

Start of Synchronous Machines

Further Innovative Applications of FieldCurrent Active Control besides UMP-Compensation

Roberto Felicetti

Masterprogram i förnybar elgenereringMaster Programme in Renewable Electricity Production

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Field Current Control for the Damping of RotorOscillations and for the Alternative Start of Syn-chronous Machines __________________________________________Roberto Felicetti

The possibility to save energy in synchronous machines operation bydismissing d-axis damping bars and surrogating them with active excita-tion current control in sectored field winding is proved. In particular a way to recover the energy of rotor oscillations during power regulation is shown by means of a study-case generator whereas a self-starting machine is analytically and numerically designed in view of its next con-struction and test. Principal design requirements and limits for both ap-plications are presented and discussed.

TVE-MFE 18004Examinator: Juan de SantiagoÄmnesgranskare: Urban LundinHandledare: Johan Abrahamsson

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A Rita e Clara

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Contents

List of symbols ........................................................................................... vi

1 Introduction ........................................................................................ 11.1 Project aims ................................................................................... 31.2 Project background ........................................................................ 31.3 Outline, constraints and limits ........................................................ 6

2 Theory................................................................................................ 72.1 Synchronous machines ................................................................... 7

2.1.1 Types of synchronous machines ............................................ 72.1.2 Synchronous machine models ............................................... 8

2.2 Active current control .................................................................. 132.3 Rotor oscillations and energy recovery ......................................... 16

2.3.1 Rotor hunting and its description ......................................... 162.3.2 Energy repartition during the hunting .................................. 192.3.3 Rotor oscillations damping by active current forming .......... 21

2.4 Motor/Generator alternative start .................................................. 252.4.1 Stator-to-rotor voltage transformation ratio.......................... 252.4.2 Starting torque generation ................................................... 28

3 Method ............................................................................................. 323.1 Study-case synchronous generator ................................................ 323.2 Motor design and parameters ....................................................... 33

3.2.1 Motor design ....................................................................... 333.2.2 Motor parameters ................................................................ 393.2.3 Four sectors rotor winding arrangement............................... 40

3.3 Simulink models .......................................................................... 423.3.1 Simulink model for rotor oscillations active damping .......... 433.3.2 Simulink model for the alternative starting .......................... 45

4 Results and discussion ...................................................................... 464.1 Rotor oscillations and energy recovery ......................................... 464.2 Motor/Generator alternative starting ............................................. 54

5 Conclusions ...................................................................................... 62

6 Future work ...................................................................................... 66

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7 Bibliography .................................................................................... 67

Appendix .................................................................................................. 69A.1 ....................................................................................................... 69A.2 ....................................................................................................... 71A.3 ....................................................................................................... 73A.4 ....................................................................................................... 77A.5 ....................................................................................................... 79A.6 ....................................................................................................... 82

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List of symbols

A Electric load or linear current density [A/m]B1

max Air gap induction fundamental amplitude [T]c Number of parallel current paths per phase [-]CD Derivative controller gain [Vs2]CI Integrative controller gain [V]CP Proportional controller gain [Vs]cosιR Rated power factor [-]cosιOE Power factor in over-excitation [-]D Damping factor [kgm2/s]Di Machine bore diameter [m]E Single phase RMS electromotive force (EMF) [V]EOE RMS electromotive force in over-excitation [V]f0 Grid frequency [Hz]H Machine time constant [s]I RMS phase current [A]I Moment of inertia [kgm2]iA...B...C Instantaneous armature phase current [A]iad Instantaneous d-axis damping bars current [A]iaq Instantaneous q-axis damping bars current [A]Id d-axis armature current [A]id Instantaneous d-axis armature current [A]if,0 Excitation current for rated voltage at open circuit [A]if,R Rated excitation current [A]if,sc Excitation current for rated short circuit current [A]Iq q-axis armature current [A]iq Instantaneous q-axis armature current [A]iref Reference current [A]Isc,R Short circuit current at rated excitation current [A]Isc,0 Short circuit current at rated armature voltage [A]Iswitch

max Absolute maximum current for the switch [A]i0 Instantaneous homopolar current [A]JR Rated RMS current density [A/m2]K Rotor oscillations natural angular frequency [rad/s]kad Armature to d-axis damping bars transformation ratio [-]kaq Armature to q-axis damping bars transformation ratio [-]kd Winding distribution factor [-]

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Kd d-axis field form factor [-]kf Induction form factor [-]kfw Armature to field winding transformation ratio [-]kp Winding pitch-factor [-]Kq q-axis field form factor [-]kR Skin-effect correction factor for DC-resistance [-]kw Winding factor [-]kΕ Flux form factor [-]Lad d-axis damping bars self inductance [H]Laq q-axis damping bars self inductance [H]Ld d-axis armature inductance [H]Lq q-axis armature inductance [H]lFe Machine equivalent magnetic axial length [m]Lf Field winding self inductance [H]Lfρ Field winding stray inductance [H]Lf,ad Field winding to d-axis damping bars mutual inductance [H]Lf,aq Field winding to q-axis damping bars mutual inductance [H]Lm Synchronous magnetization inductance [H]Lm

avg Single phase armature average magnetization inductance [H]Lm

d Single phase d-axis armature magnetization inductance [H]Lm

q Single phase q-axis armature magnetization inductance [H]Ls Synchronous armature inductance [H]LXX Armature single phase self inductance of phase X [H]LXY Armature mutual inductance between phases X and Y [H]Lρ Armature stray inductance [H]Mad d-axis damping bars to armature mutual inductance [H]Maq q-axis damping bars to armature mutual inductance [H]Mf Field winding to armature mutual inductance [H]Nf Number of field winding turns per pole [-]Nm Armature equivalent number of turns per pole per phase [-]ns Number of conductors per slot [-]Nsectors Number of rotor winding sectors [-]N0 Stator to rotor voltage transformation ratio at rest [-]p Pole pairs [-]P Active power [W]PEM Electromagnetic synchronous power [W]POE Mechanical power in over-excitation [W]PR Rated mechanical power [W]q Number of slots per pole per phase [-]Q Armature slots number [-]Q Reactive power [VAR]R Armature phase resistance [ς]Rad d-axis damping bars resistance [ς]Radd Rotor winding additional phase resistance [ς]

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Raq q-axis damping bars resistance [ς]Rf Field winding resistance [ς]Rr Rotor windings phase resistance [ς]R75

AC Armature AC phase resistance at 75 °C [ς]R75

DC Armature DC phase resistance at 75 °C [ς]dℑ Magnetization reluctance of the d-axis magnetic path [H-1]

s Rotor slip [-]SCu Copper wire or bar cross section [m2]Ssector Apparent power per rotor sector [VA]SR Rated power [VA]SCR Short Circuit Ratio [-]TEM Electromagnetic torque [Nm]Tm Mechanical torque [Nm]Tsyn Synchronizing torque [Nm]Td’ Short-circuit d-axis transient time constant [s]Td’’ Short-circuit d-axis subtransient time constant [s]Td0’ Short-circuit d-axis transient time constant at open armature [s]Tq’’ Short-circuit q-axis subtransient time constant [s]UR Single-phase armature rated voltage [V]U DC-link voltage [V]V Single-phase RMS voltage [V]Vd d-axis armature voltage [V]vd Instantaneous d-axis armature voltage [V]Vf,0 Rotor winding open circuit voltage at rest [V]vf Instantaneous field winding voltage [V]Vq q-axis armature voltage [V]vq Instantaneous q-axis armature voltage [V]Vswitch

max Absolute maximum voltage for the switch [V]w0 Magnetization energy per machine unit length [W/m]Xd d-axis synchronous reactance [ς]Xd’ d-axis transient reactance at grid frequency [ς]Xd’’ d-axis subtransient reactance at grid frequency [ς]Xq q-axis synchronous reactance [ς]Xq’ q-axis transient reactance at grid frequency [ς]Xq’’ q-axis subtransient reactance at grid frequency [ς] Pole enclosure [-]χ Pole span factor [-]0 Armature current phase constant [rad]α Oscillation decay constant [s-1]φ Armature MMF-wave angular mech. position from the d-axis [rad]φFe Iron density at 25° C [kg/m3]χ Air gap [m]χ’ Magnetic equivalent air gap [m]

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χavg Average airgap [m]χm Tip air gap [m]χ0 Load angle at t=0 [rad]Χχ Load angle variation with respect to χ0 [rad]ΧLm Armature magnetizing inductance maximal excursion [H]ΧP Slip power [W]ΧTdam Electromagnetic damping torque [Nm]ΧTm Mechanical torque variation [Nm]ΧWuse Oscillations maximal theoretical recoverable kinetic energy [W]δ Relative power regulation [-]ψ Damping coefficient [-]γ Efficiency [-]π Electrical angular position of the d-axis [rad]Πr

+ Progressive rotor MMF wave [A]Πr

+ Regressive rotor MMF wave [A]λ0 Vacuum permeability [H/m]θ Short-pitching coefficient [-]θCu Copper resistivity [ςm]σ Pole pitch [m]ι0 Field current phase constant [rad]βf Relative electromagnetic synchronizing power [-]βr Relative reluctance synchronizing power [-]Ξad d-axis damping bars linked flux [Wb]Ξaq q-axis damping bars linked flux [Wb]Ξd d-axis armature linked flux [Wb]Ξq q-axis armature linked flux [Wb]Ξf Field winding linked flux [Wb]Ξ0 Armature homopolar linked flux [Wb]ϖ Electrical angular speed or frequency [rad/s]ϖ0 Grid angular frequency [rad/s]ς Rotor mechanical angular speed [rad/s]

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1 Introduction

Continuous improvements in construction and management of synchronousmachines make it nowadays possible to achieve very high efficiencies (over98%) for both generators and motors [6]. Nevertheless research for achiev-ing further marginal improvements is still ongoing since their impact on themaintenance and duty costs reduction is anything but negligible, especiallyfor big rate machines.One example about this effort in heavy vertical axis hydropower units re-search is the partial unloading of the Mitchell’s bearing obtained by meansof an electromagnetic thrust bearing [7]. This component allows reducing thefriction between bearing body and pads which results in lower energy wasteand favorable downsizing of oil cooling apparatus. Another notable exampleis the compensation of the Unbalanced Magnetic Pull (UMP) [1] performedby the system schematically represented in figure 1.

Figure 1. A digitally controlled drive performs the regulation of the excitation cur-rent in three magnetically independent rotor sectors [1].

It pursues both aims of reducing maintenance costs by eliminating mechani-cal stress on frames and guidance bearings and to reduce additional lossesdue to magnetic unbalance [8].The active current control in magnetically independent rotor pole groupsplays a key-role in this driving strategy. Figure 2 shows how a radial forceΧF can be generated and displaced around the rotor by controlling suitablecurrents1 in each sector.

1 The presented UMP compensating mechanism refers here to the dynamic unbalance. Thesystem in figure 1 is also capable to compensate a static unbalance by forcing different levelsof current bias in the three rotor sectors.

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Figure 2. Compensating pull generation mechanism graphically explained

For the dynamic unbalance compensation it is beneficial that the air gapinduction adjustments react promptly and unweakened to the commandedcontrol currents, especially when the machine is a fast rotating one.Unfortunately many authors [2][3][4] had proved that the magnetizing in-ductances strongly decrease during transient conditions due to the reaction ofdamping circuits, field winding and eddy currents.It is therefore advisable to dismiss the damping bars when it comes to a sec-tor-wise excitation current control. Nevertheless damping bars are part of thecurrent state of art for synchronous generators and motors. Traxler-Samek,Lugand and Schwery [5] recall the reasons for that by pointing out all per-formances they ensure:

a) damping of torque oscillationsb) reduction of parasitic air gap magnetic field harmonicsc) suppression of negative-sequence fieldd) protection of the excitation winding at faulte) transient stabilityf) asynchronous start.

However, the field current active control per rotor sectors proposed in [1]discloses new ways to perform the duties enlisted above, overcoming more-over the intrinsic energy wasting nature of damping circuits.In the present work two beneficial applications of the field current activecontrol technique are presented for alternatively achieving points a) and f).

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1.1 Project aimsThis thesis aims to prove how the field current active control can surrogateand even improve at least two duties performed by damping bars in synchro-nous machines: damping of rotor oscillations and asynchronous start.About the first application, by means of a 36-poles, 175 MVA study-casehydropower generator, a way to damp the rotor oscillations by recoveringtheir related kinetic energy is theoretically proved and simulated.A self-starting 4-poles, 35 kW synchronous motor without damping bars isanalytically and numerically designed in view of its next construction inorder to test the second application. At the same time a starting strategy byfield current active forming is studied and its validity checked by simulation.Principal design requirements, actual limits and future perspectives of bothapplications are presented and discussed in the following.

1.2 Project background

The damping bars are particularly interested by losses when a slip betweenthe rotating armature Magneto Motive Force (MMF) and the machine spin-ning rotor arises2.By that mechanism the kinetic energy of rotor oscillations is intentionallydissipated after few mechanical periods. In this kind of praxis, the more fre-quent the power adjustments producing the oscillations the grater theamount of wasted energy. An alternative but conservative oscillations damp-ing technique would be then of advantage, especially for big synchronousgenerators undergoing a large amount of power regulations per day and forsynchronous motors performing e.g. S3 and S6 duties according to the inter-national standard IEC 60034-1 (or equivalent German norm VDE 0530)[10]. This mechanism, which pretends to be conservative, must necessarilybe reversible. So if the UMP-compensation system is able to detect perturba-tions in the excitation current [9] being compensated in turn by perturbationsimpressed on the field current [1], the rotor oscillations, which induce har-monics in the field current [3], must be necessarily compensated by control-ling a suitable current in the rotor winding.R1)3 The excitation system shown in figure 1, which has revealed to be effec-tive for solving the UMP issue is therefore suitable to perform the conserva-tive rotor oscillations damping by actively controlling the field current.In the asynchronous behavior of synchronous motor and generators the loss-es produced in the damping bars are intentionally used for speeding up therotor close to the synchronism.

2 It is known that also at synchronism air gap induction harmonics and negative sequencearmature currents provoke some losses in the damping bars but they are disregarded here.3 R stands for remark

4

From the theory of the asynchronous machine [4] it is known that the elec-tromagnetic torque is proportional to the rotor copper losses according to theequation:

sRpI3

sPpT r

0

2r

0

Cur

em ϖ<

ϖ< . ( 1.1 )

In a wound rotor asynchronous machines a constant torque all over the startis classically achieved by connecting a three phase additional resistive loadRadd to the rotor winding, the value of which changes continuously or step-wise, so that:

∋ ( consts

sRR addr <∗

. ( 1.2 )

The rotor active power balance in this kind of drive is highlighted in figure 3by means of the equivalent single-phase circuit of an asynchronous machine.

Figure 3. Asynchronous speed control by slip power dissipation: the red boxesrepresent the dissipated energy, the green ones the kinetic energy.

Figure 4 offers correspondingly a graphical idea of the energy balance relat-ed to the start of an asynchronous machine at constant torque by help of con-tinuously varying additional rotor resistances: in a) at constant frequency; inb) by means of three subsequent frequency and armature voltage steps4.

Figure 4. Asynchronous start: a) at constant frequency, b) with three frequency steps

4 The magnetizing flux constancy is ensured by two conditions: 1) Volt/Hertz constant ratio atthe armature supply; 2) prevalent resistive nature of the rotor circuit (R >> sϖL). For high slipthe second requirement urges the reduction of the supply frequency and voltage. This perfor-mance would require the usage of an expensive cycloconverter.

5

Figure 5 shows the replacement of the additional external resistance by anexternal inverter connected to the rotor single phase circuit. It can be ob-served that for the sake of torque generation it is absolutely irrelevant if thevoltage drop ΧVadd in the rotor winding is produced by the insertion of anadditional resistance Radd or by an active component capable of generating it.

Figure 5.Asynchronous speed control by slip power recovery

However, this last driving strategy presents the advantage of recovering thatpart of slip power which is irremediably lost when the additional resistanceis used. Its beneficial effect can be observed in figure 6.

Figure 6.Asynchronous start at constantly variable frequency

The rotor current is held constant as far as the back Electro Motive Force(EMF) Er enables it and so are torque and rotor losses (red area). The restand major part of the slip power is recovered by the inverter and given backto the DC-link (blue area).All theoretical considerations presented in this paragraph about a three phaserotor winding can be valuably applied to the excitation winding of a syn-chronous machine even though it behaves as a single phase asynchronousrotor winding all over the start. The inverter proposed by [1] for activelysupplying the excitation circuit can conveniently be used as an active rectifi-er which enables the energy to flow from the rotor to the DC link.R2) The excitation system shown in figure 1 is therefore suitable to performthe asynchronous start of a synchronous machine by partially recovering itsslip power via the field current active control.

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1.3 Outline, constraints and limitsActive current control applications envisaged in the previous paragraph areduly examined from a theoretical point of view in chapter 2. Chapter 3 ex-plains how the Simulink models have been conceived and built, which madeit possible to perform the simulation work on both generator and motor. Fur-thermore, it illustrates the design steps for a 35 kW synchronous motor.The obtained results for the simulations are presented and discussed in chap-ter 4 . In chapter 5 some conclusions are drawn whereas chapter 6 deals withsuggestions for future work.Coming to the constraints posed from the beginning to the present MasterThesis, rated power lower than 100 kW and pole pairs equal to 2 were theonly requirements to be fulfilled for the motor design. They descend fromthe need to keep the handling of future experimental work as easy and inex-pensive as possible, considering all voltage- and power-level limitations tobe faced in a laboratory test-setting.The intrinsic limits of the present work are essentially two:

1) iron saturation effects are not included in the Simulink models;2) contributions of eddy currents, magnetic hysteresis and mechanical

friction to the damping of rotor oscillations are neglected.The simplicity in the model gained by that way pays a cost to the accuracy ofthe simulated currents profiles but ensures at the same time less calculationeffort.Anyway not having accounted for those second order effects, far fromchanging the nature of the electromagnetic phenomena under examination,has made them even more clear and understandable.

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2 Theory

2.1 Synchronous machines2.1.1 Types of synchronous machines

A classical synchronous machine (SM) is a two-windings5 alternating currentmachine in which a first winding called armature is connected to a supplyline operating at constant frequency f0 and a second one, called field wind-ing, provides the machine’s excitation being supplied by direct current.While the armature of SM shows more or less the same kind of structure -which usually means in slots distributed short-pitched three-phase lap- orwave- double layer winding with more parallel strands - the field windingexecution is strongly dependent on the chosen rotor type. Synchronous ma-chine can be essentially sorted out according to their rotor structure in:

∂ round or cylindrical rotor∂ salient pole rotor

The first sort shows a smooth rotor surface like the one depicted in figure 7and it is typical for critical steam high speed turbogenerators.

Figure 7.Cylindrical rotor structure6

The second type encompasses all those rotor wheel arrangements whichproduce an angular varying radial air gap height due either to rotor saliencies(figure 8) or to claw poles (figure 9).Hydropower and back-up generators as well as synchronous motor are al-most exclusively built with salient pole rotors even though turbogeneratorswhich are driven by subcritical low speed steam turbines (nuclear and geo-thermal power groups) are often manufactured in the same way.Claw poles rotor arrangements, especially in the brushless execution, areused when the machine must work under harsh service conditions for long

5 Reluctance and permanent magnet synchronous machines are not included in the study6 Figure from [3]

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time, without possibility of any maintenance but still granting its duty (minesand railways cars motors and generators [3], car and trucks alternators [11]).

Figure 8.Rotor with saliencies7

Figure 9.Claw pole rotor8

2.1.2 Synchronous machine modelsSynchronous machine models are used for describing and foreseeing mo-tors/generators steady state or transient behaviors. A simple inversion of thearmature current sign convention transforms a generator model into a motorone and vice versa. This is why only the generator current convention9 hasbeen considered in this review.

7 Figure from [3]8 Figure form [3]9 Armature positive current comes out from the phase terminal

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The most simple model which fits for cylindrical rotor unsaturated machinesin symmetrical steady state behavior is Behn-Eschenburg’s single phaseequivalent circuit shown in figure 10.

Figure 10.Behn-Eschenburg’s equivalent circuit for cylindrical rotor machine

It shows the internal motional electromotive force E proportional to the exci-tation current if and it takes in account the effects on the output voltage regu-lation due to load current I by means of the synchronous armature reactanceϖLs. The synchronous stator inductance Ls encompasses the equivalent mag-netizing inductance Lm seen from the armature winding during a symmetricload and the armature stray inductance Lρ:

ρ∗< LL23L ms . ( 2.1 )

The symbolic equation related to this model is:

∋ ( ILjREV s √ϖ∗,< ( 2.2 )

and it can be represented in phasorial way as shown in figure 11:

Figure 11. Behn-Eschenburg’s model phasors diagram.

As soon as an anisotropic rotor is used the magnetization inductance is notconstant anymore but, according to the relative angular position betweenrotor and chosen armature phase axis φ, it fluctuates periodically around anaverage value Lm

avg and between two extreme values separated by ΧLm.The armature self-inductance as well as the mutual inductances betweenphases become all functions of the relative rotor to stator angular position:

∋ ( ρ∗φΧ

∗<φ Lp2cos2LLL mavg

mAA , ( 2.3 )

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∋ ( ⌡

∑ ο∗φ

Χ∗,<φ

34p2cos

2LL

21L mavg

mAB , ( 2.4 )

∋ ( ⌡

∑ ο∗φ

Χ∗,<φ

32p2cos

2LL

21L mavg

mAC . ( 2.5 )

When the rotor saliency is aligned with the phase axis the magnetizing in-ductance presents its maximal value Lm

d:

2LLL mavg

mdm

Χ∗< . ( 2.6 )

Lays the rotor saliency orthogonal to the phase axis then it assumes itssmallest value Lm

q:

2LLL mavg

mqm

Χ,< . ( 2.7 )

In order to have a machine model dealing with constant parameters theBlondel’s two-axis armature reaction theory is needed. Figures 12 and 13show the application of Kirchhoff’s Voltage Law (KVL) to the EMFs gener-ated respectively by the magnetic flux linkages acting along the d- and the q-axis.

Figure 12.Blondel’s equivalent circuit for the d-axis armature reaction

The inductance Ld represented in figure 12 is the synchronous direct axisinductance:

ρ∗< LL23L d

md . ( 2.8 )

Figure 13.Blondel’s equivalent circuit for the q-axis armature reaction

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The inductance Lq represented in figure 13 is the synchronous quadratureaxis inductance:

ρ∗< LL23L q

mq . ( 2.9 )

The steady state equations related to the circuital models above are:

dqqd RIILsinVV ,ϖ,<χ< , ( 2.10 )

qddq RIEILcosVV ,∗ϖ,<χ< . ( 2.11 )

They can be represented in phasorial way as shown in figure 14:

Figure 14. Blondel’s model phasors diagram

As soon as the phase currents, the excitation current or the rotor speed un-dergo some transients, the previous models are not useful anymore. Thereasons for that are mainly that they do not give explicit and separated ac-count for two kinds of electromotive forces:

∂ the transformational ones, which are related to all possible magneticlinkages among circuits on the armature and on the rotor, under assump-tion of their relative immobility;

∂ the motional ones, which are produced by the relative movement be-tween rotor and stator.

Clarke et al. [12] and Park [13] have proposed two different changes of ref-erence frame called respectively α0-frame and dq0-frame in order to per-form, by means of only two orthogonal phases, the same armature MMFprovided by a set of time varying currents in three 120° spaced phases.These are shown in figure 15.Moreover, Park’s transformation shows the same advantage of the Blondel’stwo axis armature reaction theory since it results in circuital models withconstant parameters.

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Machine inductances in Park’s model are not functions of the electric angleπ=pφ anymore. This makes it possible to approach transient problems bysolving differential equations with constant coefficients.

Figure 15. Clarke’s (a) and Park’s (b) reference frames

The electric differential equations for a synchronous machine with dampingcircuits on both d- and q-axis are:

,<Ξ

,<Ξ

,<Ξ

∗<Ξ

∗Ξϖ,<Ξ

∗Ξϖ∗<Ξ

.iRdt

d

iRdt

d

iRvdt

d

Rivdt

d

Rivdt

d

Rivdt

d

aqaqaq

adadad

ffff

000

qdqq

dqdd

( 2.12 )

The following electromechanical differential equation is to be added:

∋ (

Ξ,Ξ,<

ϖdqqdm iip

23T

Ip

dtd

. ( 2.13 )

The several linked fluxes shown in the former equations are expressed aslinear combinations of the currents by means of constant inductance coeffi-cients:

√∗√∗√∗√,√,√,<Ξ√∗√∗√∗√,√,√,<Ξ

√∗√∗√∗√,√,√,<Ξ√∗√∗√∗√,√,√,<Ξ

√∗√∗√∗√,√,√,<Ξ√∗∗∗√,√,,<Ξ

.iLi0i0i0iMi0i0iLiLi0i0iM

i0iLiLi0i0iMi0i0i0iLi0i0

iMi0i0i0iLi0i0iMiMi0i0iL

aqaqadf0qaqdaq

aqadadff,ad0qdadad

aqadad,fff0qdff

aqadf00qd0

aqaqadf0qqdq

aqadadff0qddd

( 2.14 )

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In [2] it is shown how to obtain circuital representations of (2.12) and (2.14)after having referred all rotor parameters to the armature side by means ofthe following stator-to-rotor transformation ratios:

0d

ffw LL

M23k

,< ;

0d

adad LL

M23k

,< ;

0q

aqaq LL

M23k

,< . ( 2.15 )

All parameters which have been referred to the armature side are representedwith a prime sign in the Park’s circuits of figures 16 and 17.

Figure 16.Park’s equivalent circuit for the d-axis

Figure 17.Park’s equivalent circuit for the q-axis

2.2 Active current controlFrom the analysis of linear networks in time domain it is known that thecurrent response i(t) to a step-wise voltage of amplitude U applied over thetime interval t-t0 to an inductive-resistive load is equal to:

∋ (

⌡

∑,∗<

,,

RL

tt

0

0

e1RU)t(iti . ( 2.16 )

14

If the duration of the voltage step is way shorter than the inductor time con-stant the increment of current over the interval t-t0 can be expressed by:

∋ (00 ttLU)t(i)t(ii ,?,<Χ . ( 2.17 )

Is the impressed voltage U positive the current rises otherwise is U negativethe current drops (two points control). In some cases it is also possible toforce the impressed voltage to zero (three points control), circumstancewhich makes the current stationary. In figure 18 a current control by meansof a H-bridge is represented, which makes use of the presented strategy.

Figure 18.Two points current control by hysteresis comparator (bang-bang control)

A time varying reference current iref is compared with the phase currentsensed by a measurement device (green dot) in a Schmitt’s trigger. The ob-tained digital error signal is used to drive the switches in the bridge. As longas the phase current is smaller than the actual current value plus a given tol-erance ΧI the red wired switches are switched on and the phase current keepsgrowing. As soon as the phase current overcomes the upper tolerance limitthe trigger output changes state and forces the blue wired switches to turnon. The voltage on the load becomes then negative and the load current startsdropping until it reaches the lowest tolerance limit. At that point the triggeroutput changes state again and the process continues forth and forth. By thisway the actual value of the phase current is forced to follow the referenceone, being trapped in the tolerance band (bang-bang strategy).A further very widespread current control technique is based on the PulseWidth Modulation (PWM). In figure 19 it can be seen how the departure δ ofthe actual current i from the reference one iref (colored in blue in the figure)produces a control voltage vδ (shaded brown) by means of a PID controller.That control voltage, which represents the modulating signal, is comparedwith a triangular wave vc of given frequency called carrier (orange). Theoutput state of the comparator is then used to drive the switches in the bridgewhich establish the voltage vAB on the load (black). As a result of it the cur-rent in the load (green) reproaches the reference one.

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Usually, in order to decrease the harmonic distortion of the controlled cur-rent the frequency of the carrier is chosen way higher than the highest fre-quency the fundamental of iref can assume.

Figure 19.Two levels current control by pulse width modulation (PWM)

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2.3 Rotor oscillations and energy recovery2.3.1 Rotor hunting and its descriptionIt is known from the theory of synchronous machines that for each synchro-nous steady state there is a defined position between the armature magneto-motive force and the rotor, which is represented by the load angle χ10. Withreference to figure 20, any change that affects the power generated or ab-sorbed by the machine inevitably modifies the position of the rotor relativeto the armature rotating field.

Figure 20.Mechanical angles and speeds in a salient pole synchronous machine

This usually results in damped oscillatory perturbations which interest me-chanical rotor angle π, rotor angular speed ς, electromagnetic torque, arma-ture current and instantaneous electric power at the same time.Such a phenomenon is known as “hunting” [14]. It can be described in themost general case and under the assumption of small departures “Χ” in theneighborhood of the steady-state load angle χ0 by the following differentialequation called swing equation:

dtdDTT

dtd

pI

synm2

2 χΧ,χΧ√,Χ<

χΧ. ( 2.18 )

The variation of the torque at the shaft ΧTm in (2.18) can be regarded ascause of the hunting and the load angle departure Χχ as its effect, whereasthe synchronizing torque Tsyn and the damping factor D stand respectivelyfor the conservative and the dissipative torques opposing this kind of behav-ior.

10 The load angle is usually defined as an electrical angle.

17

The typical pseudo-periodical rotor response, which is also solution of(2.18), results in (see appendix A.2):

∋ (t1Kcose1 2tKmax ψ,,χΧ<χΧ ψ,

. ( 2.19)

Where Χχmax and K are respectively the amplitude of the load angle depar-ture and the pulsation during the free oscillations whereas ψ is the dampingcoefficient related to the damping factor D.Before investigating a way to recover the harmonic inertial work of the rotor,otherwise converted in heat by the damping bars, it is relevant to present anestimate of the maximal attainable energy from a given hunting behavior. Tothat end the analysis of free oscillations for D=0 reveals that the amount ofenergy exchanged in a period between rotor and power network is the samewhich decays exponentially in time in the damping circuits, in the field andarmature windings and even in the iron core due to the induced losses. Afirst estimation of it, for oscillations triggered by a step-wise relative powerregulation δ, has been found (see appendix A.2) to be

0r0f

0

0

Ruseful 2cos2cos

H2SWχ√β√∗χ√β

ϖ√ϖ

δ?Χ , ( 2.20 )

where βf and βr are respectively the coefficient of the maximal electromag-netic- and reluctance-related synchronizing powers expressed in p.u.:

dRf X~S

EU3 √<β , ( 2.21 )

⌡

∑,<β 1

X~X~

X~SU

23

q

d

dR

2

r . ( 2.22 )

It must be observed that the reactances used in (2.21) and (2.22) are smallerthan the Blondel’s synchronous reactances because of the partial magnetiz-ing flux capture exerted by the damping bars and the field winding when therotor hunts. The flux is forced to leave the main way and to flow in somemeasure through the magnetic stray paths. More precisely, according to thefrequency of the rotor oscillations those inductances can range between theirclassical transient and sub-transient values, being closer to the subtransientvalues the higher said frequency:

'XX~''X ddd ′′ , ( 2.23 )

'XX~''X qqq ′′ . ( 2.24 )

Since the pulsation of free oscillations and the maximal load angle departureare given respectively by

18

00r0f

H22cos2cosK ϖ√χ√β√∗χ√β<

( 2.25 )

and0r0f

max 2cos2cos χ√β∗χ√βδ

<χΧ , ( 2.26 )

considered that the maximal angular frequency departure is related to (2.25)and (2.26) by

maxmax K χΧ<ϖΧ , ( 2.27 )

(2.20) can finally be expressed as:

H2SW R0

maxuseful ϖ

ϖΧ?Χ . ( 2.28 )

By using (2.28) on the study-case generator presented in paragraph 3.1 it ispossible to find out the potentially recoverable energies respectively for thesynchronous, the transient and the subtransient behavior.

Table 1.Electromechanical oscillations parameters for synchronous behavior

βf βr Χχmax K f T ςmax ςmin ΧWmax ΧWuse

[p.u.] [p.u.] [°] [rad/s] [Hz] [s] [rad/s] [rad/s] [kWh] [kWh]1.804 0.171 2.61 9.19 1.46 0.68 17.476 17.430 0.907 0.453

Table 2.Electromechanical oscillations parameters for transient behavior

βf βr Χχmax K f T ςmax ςmin ΧWmax ΧWuse

[p.u.] [p.u.] [°] [rad/s] [Hz] [s] [rad/s] [rad/s] [kWh] [kWh]5.723 -1.010 1.27 13.19 2.10 0.48 17.469 17.437 0.632 0.316

Table 3.Electromechanical oscillations parameters for sub-transient behavior

βf βr Χχmax K f T ςmax ςmin ΧWmax ΧWuse

[p.u.] [p.u.] [°] [rad/s] [Hz] [s] [rad/s] [rad/s] [kWh] [kWh]8.736 -0.359 0.654 18.36 2.92 0.34 17.465 17.442 0.454 0.227

A relevant result can be concluded by observing (2.28):

R3) the maximal attainable recovery of kinetic energy from the rotor oscilla-tions triggered by a single step-wise relative power regulation δ in a givensynchronous machine, is proportional to the entity of the relative regulationitself, to the machine rated power and to its own inertial time constant.

19

2.3.2 Energy repartition during the huntingIn order to determine in what measure the kinetic energy of the rotor oscilla-tions supplies the losses in the armature winding and in the rotor dampingcircuits11, a closer insight into the electromechanical energy conversion dur-ing a transient is needed.Park’s model for a salient pole synchronous machine, which has been pre-sented in paragraph 2.1, is the tool used in the following. One of its expectedduties is that to establish numerical relationships between the magnitude ofthe angular deviation of the rotor from its steady state position and the inten-sities of all energy-wasting induced currents during the hunting occurrence.Since all quantities involved in the oscillating behavior have sinusoidalpseudo periodical nature the accomplishment of this task is convenientlyperformed in an approximated way by solving a succession of quasi-steadystates through the symbolic method.

2.3.2.1 Method of departuresAs long as the oscillations interesting the rotor are small all departures whichintervene in the machine quantities are also small and undergo the superposi-tion principle12. Park’s equations [2] can be then rewritten in term of super-posed steady state values and their related departures:

⌡

∑ ∗∗

⌡

∑∗<∗

⌡

∑ ∗∗

⌡

∑ ∗<∗

⌡

∑ ∗∗

⌡

∑ ∗<∗

⌡

∑,∗∗

⌡

∑,∗<∗

⌡

∑ ,,∗

⌡

∑ ,,<∗

.iRdtdΨΔiR

dtdΨ

Δv0

iRdtdΨΔiR

dtdΨΔv0

iRdtdΨΔiR

dtdΨΔvv

RiΨωdtdΨ

ΔRiΨωdtdΨ

Δvv

RiΨωdtdΨΔRiΨω

dtdΨΔvv

adadad

aqaqaq

aq

adadad

adadad

ad

fff

fff

ff

qdrq

qdrq

qq

dqrd

dqrd

dd

( 2.29 )

11 It has been here generally referred to rotor damping circuits in order to encompass all thoseclosed galvanic paths present in the rotor which are seat of induced currents during the hunt-ing. They include not only dumping bars and excitation winding but rotor iron core too.12 It states that the effect caused by two or more causes is the sum of the responses that wouldhave been produced by each cause acting individually. This principle requires and impliessystem linearity, which is always fulfilled for small amplitude stimuli in systems with C2

transfer functions.

20

By considering the impressed voltages on the rotor circuits constant and byrecalling the initial condition of equilibrium (2.29) reduces to:

Χ∗ΧΞ<

Χ∗ΧΞ<

Χ∗ΧΞ<

Χ,ΧΞϖ∗ΞϖΧ∗ΧΞ<Χ

Χ,ΧΞϖ,ΞϖΧ,ΧΞ<Χ

.iRdtd0

iRdtd0

iRdtd0

iRdtdv

iRdtdv

aqaqaq

adadad

fff

qdrdrqq

dqrqrdd

( 2.30 )

Stapleton [16] and Krause et al. [17], who make extensive use of the smalldepartures method, both suggest the following reasonable simplifications for(2.30):

a) voltages generated owning to changes in speed are negligible (thismeans voltage terms involving Χϖ);

b) voltages induced in the armature by rate of change of armature fluxlinkages are negligible (terms in d/dtΧi);

c) voltage drops on resistances are negligible in both armature andexcitation winding (terms in RΧi).

Under those assumptions the simplified system becomes:

Χ∗ΧΞ<

Χ∗ΧΞ<

ΧΞ?

ΧΞϖ∗?ΧΧΞϖ,?Χ

.iRdtd0

iRdtd0

dtd0

vv

aqaqaq

adadad

f

drq

qrd

( 2.31 )

Tables 1,2 and 3 show that rotor oscillations and all related quantities evolveat very low angular frequency K with a period which is several times multi-ple of the electric one. Since the energy repartition question aim only to de-termine the relative values of the energies involved in the different dampingcircuits, the absolute amplitudes of the departures do not matter here. Theanalysis can be then limited to the first half rotor oscillation. By taking inaccount (2.10) (2.11) (2.14) and (2.31) it is possible to solve the pseudo si-nusoidal currents departures in the frequency domain:

∋ (∋ (

∗,<∗∗,<

,,,<∗∗,<√,

,∗<√

.ΔijKLRΔijKM0ΔijKLRΔijKLΔijKM0

ΔijKLΔijKLΔijKM0ΔiMωΔiMωΔiLωΔδsinδV

ΔiMωΔiLωΔδcosδV

aqaqaqqaq

adadadfadf,dad

adadf,ffdf

adad0ff0dd00

aqaq0qq00 ( 2.32 )

21

(2.32) can be written in form of matrix as:

.

iiiii

jKLRjKMjKLRjKLjKM

jKLjKLjKMMM0L

ML0

000sinV

cosV

aq

ad

f

q

d

aqaqaq

adadad,fad

ad,fff

ad0f0d0

aq0q0

0

0

ΧΧΧΧΧ

√

∗,∗∗,

∗∗,ϖ∗ϖ∗ϖ,

ϖ,ϖ∗

<χΧ√

χ,χ

( 2.33 )

The first relevant result descending from (2.33) is thatR5) the intensities of current departures are all proportional to the ampli-tude of the oscillations. This fact lets foresee that all currents peaks decay inthe same way the rotor oscillations do.

The second relevant result is thatR6) the voltage sources which force the currents departures on the d-axisand on the q-axis are respectively proportional to the sine and to the cosineof the steady state load angle. This fact implies that for small load angles thecontribution of the d-axis current departures to the damping is less relevantthan that of the q-axis ones. For big load angles vice versa the q-axis cur-rents departures play a minor role than the d-axis ones.This result is very important because it highlights that, at least for small loadangles, the d-axis does not play a major role in damping the rotor oscilla-tions. The cause behind this weak damping effect via the d-axis current con-trol at small load angles is explained in the next paragraph. It is relevant toremark here how favorable it is, that the current departures on the d-axisdepends on the sinus of the load angle. That means in fact that the activefield current regulation for energy recovery gets more and more effective thehigher the load angle is. Fortunately the conclusion of paragraph 2.2.1 goesin the same direction since the stronger the power regulation effort the high-er the amount of energy which can be potentially recovered from the oscilla-tions.

2.3.3 Rotor oscillations damping by active current formingPark’s theory of the synchronous machine [2][3][4][18][19][20] offers a wayto express the electromagnetic torque in terms of d- and q-axis quantities:

∋ (dqqdem iip23T ξ,ξ< . ( 2.34 )

By specializing the expressions of linked fluxes (2.14) for a machine withoutdamping bars and by substituting them in (2.34), the following expressionfor the torque is obtained:

∋ ( qffqddqem iipM23iiLLp

23T ∗,< . ( 2.35 )

(2.35) shows that

22

R7) the control over the torque by means of the excitation current, in a ma-chine which performs the excitation flux only along the d-axis, it can be pur-sued exclusively by interacting with the stator quadrature current.According to this remark and having in mind the damping torque seen in(2.18) the solution of the active damping issue can be stated as it follows:R8) determining a suitable excitation current departure Χif which is able toproduce a damping torque proportional to the speed departure Χϖr bymeans of its only interaction with the q-axis component of the armature cur-rent.By putting remark n.8 in analytical formula it is obtained:

piipM

23T 0

rqffdampϖ,ϖ

<ϖΧ×Χ<Χ . ( 2.36 )

From Blondel’s equation (2.10) it can be deduced that the armature currenton the quadrature axis depends essentially from the armature voltage on thed-axis:

0q0q0

dq sin

LV

Lvi χ

ϖ<

ϖ? . ( 2.37 )

The excitation field current departure can be obtained instead by integratingthe control voltage applied to the excitation field and by neglecting the rotorresistance:

∋ ( σσ?Χ ⟩ dvL1i

t

0f

ff . ( 2.38 )

In particular, for compensating the delay introduced in the current controlchain by the high excitation field time constant a derivative controller isadvisable, which processes the error signal on the rotor speed according to:

∋ ( ∋ ( ∋ (0DDf dtdCt

dtdCtv ϖ,ϖ,<δ,< , ( 2.39 )

where CD is a constant to be determined.The expression for the field departure current becomes then:

∋ ( ∋ (0f

Dt

0f

ff L

CdvL1i ϖ,ϖ,<σσ<Χ ⟩ . ( 2.40 )

By recalling (2.27)

χΧ<ϖ,ϖdtd

0 ( 2.41 )

and by substituting it in (2.40) and then in (2.36) together with (2.37) theexpression for the breaking torque can be rewritten as:

23

dtdsin

LV

LM

p23CT 0

q0f

fDdam

χΧχ

ϖ,<Χ . ( 2.42 )

This kind of damping torque is qualified as viscose or Newtonian, since italways opposes the speed of the oscillations, exactly how a viscose mediumdoes on a pendulum oscillating in it.By comparing the damping torque term in (2.18) with (2.42) an expressionfor the damping factor D can be achieved:

0q0f

fD sin

LV

LM

23pCD χ

ϖ< . ( 2.43 )

(2.43) is the key for performing the conservative electromagnetic dumpingof the rotor oscillations:R9) the strength of the damping mechanism is set by the factor D which canbe tuned via the excitation field by means of the derivative controller gainCD.

In order to obtain the fastest settling time of the perturbations the systemmust be critically damped with damping coefficient ψ equal to one (figure22). Appendix A.1 shows that this condition is met when the decay constantof the oscillations α is equal to the rotor natural angular frequency K:

KDI2

p<<α . ( 2.44 )

Figure 21.Normalized indicial response of a 2nd order system for different values ofthe damping coefficient13

13(Wikipedia https://en.wikipedia.org/wiki/Damping_ratio#/media/File:2nd_Order_Damping_Ratios.svg)– Public domain source

24

By substituting (2.43) in (2.44) it is possible to obtain the critic value for theconstant CD:

0

R

f

fq

0cr,D sinV

SM

LLHK

38C

χϖ< . ( 2.45 )

Some important remarks can be done by observing (2.45):R10) CD,cr is inversely proportional to the load angle which suggests a highcontrol effort needed for damping the oscillations at low power levels;R11) CD,cr is directly proportional to q-axis inductance (the synchronous ortransient one) which means that strong rotor anisotropy and low resistivedamping circuit on the q-axis facilitate the active damping task;R12) control effort is proportional to the machine angular momentum(HS/ϖ0) and it is therefore inversely proportional to the number of pole-pairs;R13) high values of rotor field stray inductance would require an unneces-sarily heavier control effort.

25

2.4 Motor/Generator alternative start2.4.1 Stator-to-rotor voltage transformation ratioThe analysis of synchronous machines in asynchronous behavior reveals thatas soon as a rotational speed slip happens an EMF is induced in the excita-tion circuit [21]. This is maximal for a given nominal armature voltage whenthe rotor is blocked. With reference to the incipient asynchronous start [3]reports that the voltage existing across the open-circuited rotor circuit mayreach three to five times the armature phase voltage rising up to 20-50 kV.That is roughly a hundred times the rated excitation voltage. For that reasonduring the asynchronous start by damping bars the excitation winding isclosed through an external very low resistive (<5 R f) resistor. It establishes ashort-circuit-like current in the field winding, the strong reaction of whichsuppresses almost entirely the voltage induced by the slip.Since the rotor at rest is the recurring initial condition of every start proce-dure a deeper understanding of the back EMF generation mechanism is rele-vant for avoiding unbearable voltage level on the rotor winding insulation. Inthe following design some machine parameters are examined which influ-ence the rotor open circuit voltage.By observing that the magnetizing circuit presents the same reluctance ℑ forboth excitation winding and single phase of the stator, it is possible to writethe stator-to-rotor voltage transformation ratio as a ratio between inductancesaccording to:

dm

f

d

2m

d

mf

m

f0 L

MN

NN

NNN <

ℑ

ℑ<< . ( 2.46 )

From the expression for the inductances given in [3] it is easy to relate theopen circuit voltage ratio N0 to the machine geometry and design features by:

pds

ff0 kkqn

cNk2

N ο< . ( 2.47 )

(2.47) is a simple and valuable equation for a first estimation per excess ofthe open-circuit voltage ratio since it does not take in account the attenuatingeffect of the armature leakage inductance, which works actually as a voltagedivider by further reducing the rotor voltage. From steady state Park’s modelat ret and field winding open circuited it is possible to obtain (Appendix A.3)a more precise estimation of it by :

dm

0o

f

LL

321

NVVN

ρ∗?< . ( 2.48 )

A first relevant result emerges from (2.48):

26

R14) an increase of the armature stray inductance reduces the rotor open-circuit voltage ratio.This fact can be useful for decreasing the voltage ratio during the first in-stants of the start by increasing artificially the stray inductance through anexternal tie- or ballast-inductor joined to the armature.(2.47) and (2.48) do not offer anyway a direct estimation of how big therotor open-circuit voltage at rest could be. The reason for that is their reli-ance on the internal machine parameters which are usually not immediatelyavailable or even unknown. In Appendix A.3 a more useful equation isgained, which relates immediately to the rated machine parameters. In fact,when the rotor stands the synchronous machine behaves like a transformerenabling the transmission of the apparent power from the armature phases(primary windings) to the field winding (secondary winding). By that heuris-tic way it has been found that the peak value of the rotor open voltage de-pends on the machine typology according to:

for salient-pole machines

R,f

Rmaxo,f

R,f

R

iS9.2V

iS2.1 √′′√ ; ( 2.49 )

for non salient-pole machines

R,f

Rmaxo,f

R,f

R

iS9.1V

iS0.1 √′′√ . ( 2.50 )

By means of (2.49) it is easily proved that for a SR=175 MVA power hydro-generator with if,R=1500 A rated excitation current, the open circuit voltageat rest ranges between:

kV221VkV140 maxo,f ′′ . ( 2.51 )

These levels of voltage are not only harmful for the rotor insulation but theyare such to impair the current control mechanism used in the arrangement forthe UMP compensation of figure 1. In order to guarantee to the inverter therethe possibility to control the current supplied to the rotor all along the start-ing process, its DC-link voltage level must always exceed the maximal back-EMF induced in the rotor.Since the link voltage level is essentially limited by the technology of theswitches adopted in the bridge, an important relationship can be posed byusing (2.51):

R,ftorssec

R

torssec

maxo,fmax

switch iNS2.1

NV

V <= . ( 2.52 )

The admissible current in the switch must also be higher than the rated exci-tation current which is expected to be supplied. From (2.52) descends there-fore a first estimation of the maximal machine rated power that can be han-dled per rotor sector given a specific switches technology:

27

2.1IE

NSS

maxswitch

maxswitch

torssec

Rswitchtorsec ;<

. ( 2.53 )

Taking in consideration the actual maximal ratings for GTOs, IGBTs andMOSFETs according to the state of art presented in figure 23, the followinglimits for the sector rated power have been found by (2.53): GTO/IGCT

MVA102.120006000

2.1IE

NSS

maxGTO

maxGTO

torssec

RGTOtorsec ?

√<;< , ( 2.54)

IGBT

MVA25.22.110003000

2.1IE

NSS

maxIGBT

maxIGBT

torssec

RIGBTtorsec ?

√<;< , ( 2.55 )

MOSFET

kVA842.11001000

2.1IE

NSS

maxMOSFET

maxMOSFET

torssec

RMOSFETtorsec ?

√<;< .( 2.56 )

Figure 22.Absolute maximal ratings for the most common switches14

Since the usual configurations for the UMP compensation system make useof 3•6 rotor sectors [1]R15) according to actual maximal absolute ratings of the switches in thedifferent technologies, the maximal admissible rated power for a machine tobe started by the arrangement of figure 1 is respectively 60 MVA forGTOs/IGCTs, 13.5 MVA for IGBTs and 500 kVA for MOSFETs.

14Author Cyril Buttay, Wiki free-media repositoryhttps://commons.wikimedia.org/wiki/File:Switches_domain.svg#filelinks

28

2.4.2 Starting torque generationThe idea of controlling an alternate current Χif in the excitation circuit, hav-ing frequency sf0 equal to the slip-one, can have essentially two differenttargets or a combination of those:

a) contributing to the magnetization/demagnetization of the machine byexchanging reactive power;

b) contributing to the electromechanical energy conversion by provid-ing or subtracting active power.

In any of the enlisted performances the excitation circuit of a synchronousmachine is unable to generate a rotating MMF as a poly-phase system doessince it is a single-phased-like circuit.In Appendix A.4 it has been proved how a sinusoidal current with frequencysf0 and initial phase ι0 established in the field winding by the inverter offigure 1 builds up a pulsating MMF spatially anchored to the rotor. ThisMMF is perceived from the armature side as the superposition of two MMFsmoving in opposite directions with different angular speeds. Assuming α tobe the mechanical angular position measured from the A-phase axis inclockwise sense along the armature, the total rotor MMF results in:

∋ ( ∋ ( ∋ (Ζ ∴α∗ι∗ϖ,Π

∗α,ι∗ϖΠ

<αΠ pt1s2sin2

ˆptsin

2

ˆt, 00

r00

rr . ( 2.57 )

(2.57) makes evident that the FMM generated by a pulsating rotor currentconsists of two rotating terms, a progressive one turning clockwise with syn-chronous speed and a regressive one turning normally counter-clockwisewith variable speed according to the slip (figure 24).

Figure 23.Counter-rotating MMFs due to the sinusoidal field current.

In particular, when the rotor is at rest (s=1) its speed is exactly the synchro-nous one but in negative direction. For s=0.5 the regressive MMF is at restwith respect to the stator and for s=0 it rotates at synchronism.

29

Progressive and regressive MMFs are responsible for the induction of re-spectively positive and negative sequence EMFs in the armature.In appendix A.4 by considering a set of symmetric currents entering the sta-tor phases

∋ ( ∋ (00A tsinIti ∗ϖ< ,

∋ ( ⌡

∑ ο,∗ϖ<

32tsinIti 00B , ( 2.58 )

and ∋ ( ⌡

∑ ο,∗ϖ<

34tsinIti 00C ,

an expression for the instantaneous electromagnetic torque has been ob-tained:

∋ ( ∋ ( ∋ ( ∋ (000d00d

EM ts2sinI23p1s2sinI

223ptT ∗ι∗ϖΞ,,,ι

Ξ<

((((

.( 2.59 )

In (2.59) it is recognized that the average EM torque for s÷0 is given by:

∋ (00davg

EM sinI22

3pT ,ιΞ

<(

(

. ( 2.60 )

Since ι0 is the phase angle of the excitation current it tracks the position ofthe d-axis. Given the armature current phasor I with phase angle 0, the dif-ference ι0-0 shows the relative position of that phasor with respect to the d-axis. The sinus of the ι0-0 is then the cosine director of the q-axis and(2.60) can be rewritten as:

qdavg

EM I22

3pT Ξ<

(

. ( 2.61 )

Some important remarks arise from the analysis of (2.59) and (2.60):

R16) the single phase structure of the field winding is responsible for thepulsating nature of the electromagnetic torque;

R17) the smaller the slip the lower the torque ripple frequency but unfortu-nately the higher its intensity;

R18) for given intensities of armature q-current and field current the aver-age electromagnetic torque is maximized when the field current and the q-current are in phase (generator convention);

R19) for a given field current intensity only the half of the obtained excita-tion flux contributes to the electromagnetic torque generation.

30

The instantaneous composition of the armature and field currents whichachieve the maximal average value of the electromagnetic torque is repre-sented in figure 25.

Figure 24.Instantaneous stator and rotor currents for the maximal average torque

When the excitation current is maximal and the d-axis is aligned with the A-phase axis the current of that phase is zero. This confirms that the armaturecurrent does not need any direct axis component. It must also be observedthat, in spite of the difference between the armature and rotor frequencies,the progressive rotor field is always synchronized with the armature field.This must not be confused with the pulsating change of magnetic polarity ofthe rotor wheel which proceeds sub-synchronously and represents the rotorMMF standing wave.Coming to energetic considerations, the electromagnetic active power tres-passing the air gap, the so called synchronous power, which is responsiblefor the torque generation, is easily found from (2.61):

qr

00

EMEM I22

3p

TP Ξϖ<

ϖ<

(

. ( 2.62 )

This power must be equal to the mechanical power delivered to the rotorplus an additional term which ensures the energy balance in the rotor:

PPP mechEM Χ∗< . ( 2.63 )

ΧP is called slip power and relates to the energy which leaves the rotor in thetime unit. Since (2.63) can be expressed by means of the slip s

31

∋ ( Ps1p

Tp

T 0EM

0EM Χ∗,

ϖ<

ϖ, ( 2.64 )

the slip power results in

EMsPP <Χ . ( 2.65 )

It is easy to recognize by means of (2.65) that

R20) during the machine starting, when the slip decreases from 1 to 0, theslip power is always positive and represents a power extracted from therotor in form of heat (joule losses in the resistive asynchronous start) and inform of generated active power (generated power exiting the excitation cir-cuit in the start by active field current control).

This last theoretical remark confirms that the conclusions of the introductoryparagraph 1.1, which have been achieved with reference to poly-phase rotorwindings, can be definitely extended to a single phase rotor winding.Moreover (2.65) proves that the magnitude of the energy recovered throughthe excitation winding decays linearly with the slip factor s exactly how de-picted in figure 6 (blue area).

32

3 Method

3.1 Study-case synchronous generatorThe study-case generator considered in this paragraph refers to a mediumsize low speed salient pole synchronous machine, the technical data of whichhas been found in [22].Tables 4 and 5 show respectively its synchronous and transient parameters.

Table 4.Study-case generator synchronous parametersSR UR PR cosι If,R n H xd xq x0

[MVA] [kV] [MW] [-] [kA] [rpm] [s] [p.u.] [p.u.] [p.u.]175 14.4 150 0.857 1.5 }167 3.5 0.92 0.70 0.142

Table 5.Study-case generator transient and sub-transient parametersxd' xq' xd'' xq'' Td0' Td' Td'' Tq''

[p.u.] [p.u.] [p.u.] [p.u.] [s] [s] [s] [s]0.29 0.70 0.19 0.22 7.3 2.3 0.03 0.035

In order to build an Ordinary Differential Equation (ODE) -based model ofthis synchronous machine in Simulink the values of the resistances and in-ductances present in (2.12) and (2.14) has been needed. [2] shows how thetransient and sub-transient parameters can be related to the synchronous onesby means of the d- and q-axis equivalent circuits of figures 16 and 17. Thefinal parameters shown in tables 6 have been obtained by the calculationspresented in Appendix A.5.

Table 6.Absolute electrical parameters of the study-case generatorR Ld Lq Rf Lf Mf R’ad L’ad M’ad R’aq L’aq M’aq

[mς] [mH] [mH] [ς] [H] [mH] [ς] [H] [mH] [ς] [H] [mH]10 3.47 2.64 0.139 1.018 41.6 11.89 1.163 41.6 9.80 0.803 29.8

33

3.2 Motor design and parameters3.2.1 Motor designMotor design requirements have been posed for satisfying the followingneeds:∂ rated supply voltage, frequency and power that can be easily and afford-

ably managed in the test laboratory (20 •100 kW);∂ 4 poles rotor, since it represents the most widespread rotor type for syn-

chronous motors;∂ apparent power per rotor sector which admits the use of a MOSFETs

based H-bridge (<84 kVA);∂ rotor open-circuit voltage at rest lower than the maximal DC-link volt-

age achievable by rectifying 230 V AC (<325 V).All those features are summarized in table 7.

Table 7.Motor design requirements

UR PR cosιR f0 p Vf,0 (peak)

[V] [kW] [-] [Hz] [-] [V]400 35 0.85 (lag) 50 2 < 325

The design has been performed both analytically and numerically with theaid of a FEM-program (KALK) available at the Division for Electrical Ener-gy at the Uppsala University. All details of the design process are duly de-scribed in appendix A.6. In the present paragraph the most important resultsare reported and summarized, which are essential for performing the motorstart simulation.Figure 26 represents the transversal and longitudinal cross-sections of thedesigned motor.

Figure 25. Motor cross section with principal dimensions

The bore of the machine has been inspired by the armature of an availableasynchronous motor which fits for a rated power of 37 kW and which is soldby the company Kurt Meier Motor-press GmbH. In table 8 all armature

34

characteristics are summarized which influence the windings factor in orderto target low EMF- and torque-harmonic-distortion.

Table 8.Armature features and windings factorsQ q θ kd1 kp1 kw1 kd5 kp5 kw5 kd7 kp7 kw7

48 4 5/6 0.958 0.966 0.925 0.205 0.259 0.053 -0.158 0.259 -0.041

It can be observed that the assumption of short-pitching the armature wind-ings with a classical factor θ=5/6 has the effect to drastically attenuate thefifth and the seventh harmonic of the EMF according to the respective wind-ing factors kw.By assuming experienced values for the fundamental of the air gap inductionB1

max=1T and for the admissible linear electric load (or linear current link-age) A=25 kA/m, it has been possible to determine both the effective axiallength lFe of the machine and the number of conductors per slot per path byrespectively:

m214.0fADBk

2p

Sl0

2i

max1w

2R

Fe ?ο

< , ( 3.1 )

08.6flDqBk2

3U

cn

0Feimax1w

R

s ?ο

< . ( 3.2 )

For lFe=217 mm and without adopting parallel paths for the phase current inthe armature (c=1) the number of conductor per slot becomes exactly ns=6.The minimal air gap χ has been found by assuming that the average armaturereaction equals at least the open-circuit MMF due to the excitation field [4](design on load):

mm7.1pB2AD

p2D

BA1A2B2 max

1

i0

imax1

00max1 ?λ<

ο√

ολ<χ↑σ√

ολ<χ√ .

It has been assumed χ=1.8 mm in order to have a little bit more inductioninsensitivity to the mechanical tolerances of the electro steel sheets and toother potential air gap irregularities.Some considerations about the further reduction of the harmonic distortionon the EMF have suggested to adopt a cosecant profile for the pole shoe andto optimize its pole enclosure [3]. That has meant to choose the maximallateral air gaps χmax of the pole shoe (tip air gap) and its relative width to thepole-pitch so to target the best compromise between air gap inductionprofile and needed MMF. With reference to figure 27 the values summarizedin table 9 have been obtained (see Appendix A.6 for their derivation).

35

Figure 26. Pole shoe principal dimensions15

Table 9. Pole shoe geometry: equivalent air gap χavg and pole span factor χ

χ χ’ χm kf kΕ χ χavg

[mm] [mm] [mm] [-] [-] [-] [-] [mm]1.8 2.17 2.34 0.7 1.17 1.04 0.77 2.8

The pole span factor χ allows determining the equivalent air gap χavg whichis fundamental for the analytical calculation of the magnetizing inductancealong the d- and the q-axis:

mH59.6lDckqn

2L Feiav

2ws

0dm <

χ

⌡

∑

λο

< , ( 3.3 )

mH75.3lDckqn

2KK

L Feiav

2ws

0d

qdm <

χ

⌡

∑

λο

< , ( 3.4 )

where Kd=1.02 and Kq=0.58 are correction coefficients [3] related to therotor geometry and anisotropy.The magnitudes of the yoke radial height hy, the slot depth hz and width wz

have been determined by assuming at no load in the tooth facing the d-axisand in q-axis cross section of the yoke an induction level equal to Bmax=1.75T.For the armature has been chosen a double layer lap-winding short pitchedby 1/6 as already said. Each slot shows two strands of three turns super-posed. In order to achieve a high slot fill-factor, considered that the penetra-tion depth of the skin effect in copper at 50 Hz is roughly 7 mm, a massivecopper bar with squared cross section and rounded corners has been adopted.The copper bar cross section has been determined assuming that the currentdensity must not exceed 4 A/mm2 at rated power for thermal reasons.Figure 28 shows the winding arrangement with the coordination of the insu-lation and the mechanical fixing.

15 Figure from [3]

36

a) b)Figure 27.a) stator arrangement b) principal slot and strand dimensions

The armature phase resistance has been calculated moving from its DC-value at 75 °C and taking in account the skin effect according to [4]:

mH1.57RkR DC75R

AC75 << , ( 3.5 )

where kR=1.037 is the skin-effect resistive factor.

The motor, with rotor and stator made of non-grains-oriented electric steel,has been simulated in KALK at no load and on load, assuming Nf=18 turnsper rotor pole. The results of the simulation are shown in figures 29 a) and b)and are summarized in Table 10.

a) b)Figure 28. Motor induction FEM-calculation: a) no load; b) load-over-excited

Table 10.FEM calculation for excitation current, armature current and EMF in o.e.

UR POE cosιOE If I EOE

[V] [kW] [-] [A] [A] [V]400 27 0.7 (lead.) 212 57.7 398

Figure 30 shows the designed rotor winding arrangement.

37

Figure 29. Rotor winding arrangement for 18 turns per pole (pole cross-section)

The field winding resistance has been calculated at 75 °C resulting in:∋ (

ς<√

∗∗θ< m4.33N4

tddwl2R f

corecore75f . ( 3.6 )

The KALK calculation has also allowed achieving the magnetizing induct-ance in over excitation (o.e.)In table 11 a comparison between the analytically derived d- and q-axismagnetizing inductances and those obtained by KALK is presented.

Table 11.Comparison between analytical and numerical calculated inductances

Analytic KALK

Ldm Lqm Ldm Lqm

[mH] [mH] [mH] [mH]6.59 3.75 7.12 3.95

It is to observe that the inductances calculated in over excitation (in motorbehavior) in KALK do not resent the saturation whereas the inductancescalculated by means of experimental curves [3] include it.Coming to the stray inductances calculation, it has been chosen to trust theresults obtained by experienced formulas [3][4] more than those provided byunknown routines in KALK. In table 12 the values for the four principalarmature stray inductances are shown, which has been obtained by the for-mulas proposed by Smolensky [3] and Pyrhönen [4].

Table 12.Stray inductances values according to Smolensky and Pyrhönen

Smolensky Pyrhönen

Air gap leakage inductance Lρχ [λH] 52 42Slot leakage inductance Lρs [λH] 279 285Tooth tip leakage inductance Lρt [λH] 70 60End winding leakage inductance Lρe [λH] 32 49Total leakage inductance Lρ [λH] 433 436

38

According to (2.8) and (2.9) the Blondel’s motor synchronous inductancesare then:

Table 13. Blondel’s synchronous inductances

Ld Lq

[mH] [mH]11.12 6.36

The field winding mutual inductance Mf for Nf=18 turns can be calculatedfrom the values of table 7 by:

mH46.8IE2M

f

OEf <

ϖ< . ( 3.7 )

From the finite elements calculation performed in KALK a specific energyfor the exciting field at no load w0= 260.4 J/m per pole has resulted, whichleads to a rotor inductance Lf:

mH96.9I

lw2p2L 2f

Fe0f <

√√< . ( 3.8 )

The field winding leakage inductance is then:

mH50.1MLL fff <,<ρ . ( 3.9 )

By knowing all machine parameters it is possible to find out the rated valueswhich are represented in table 14.

Table 14. Rated values of the motor

UR PR cosιR IfR IR ER

[V] [kW] [-] [A] [A] [V]400 35 0.85 (lag.) 108.4 57.7 203.6

Figures 31 a) and b) show the phasor diagrams for the machine respectivelyover excited and in rated behavior.

a) b)Figure 30. Blondel’s diagrams: a) over-excited b) at rated power

39

3.2.2 Motor parametersIn this section all parameters needed for the machine simulation in Simulinkare calculated.The moment of inertia of the machine is determined by considering the rotoras a massive cylinder of iron with the bore dimensions:

2Fe

2i

Fe kgm34.0l4D

21I ?

οφ< . ( 3.10 )

The machine time constant results then:

s1.0S

pI

21

HR

20

?

⌡

∑ ϖ

< . ( 3.11 )

Since the motor does not have damping bars during the transient behavioronly the field winding can oppose the armature flux changes. Hence, thereare not sub-transient phenomena to be expected. In a short circuit for exam-ple the unipolar component of the current decays with the time constant Td0’whereas the alternated one decays with the time constant Td’.By referring the rotor stray inductance Lf

ρ to the stator side the transientinductances can be calculated according to [2]:

mH82.1'LL

23

'LL23

L'Lf

dm

fdm

d <∗

∗<ρ

ρ

ρ , ( 3.12 )

qqmq LL

23L'L <∗< ρ . ( 3.13 )

The spoken time constants are then:

s298.0RL'T

f

f0d ?< , ( 3.14 )

s051.0'R'L'T

f

dd ?< . ( 3.15 )

Table 15.18 turns/pole motor parameters

PR UR cosιR H ZR xd xq xd’ x0 r Td0’ Td’

[kW] [V] [-] [s] [ς] [p.u.] [p.u.] [p.u.] [p.u.] [p.u.] [s] [s]

35 400 0.85 0.1 4.0 0.876 0.499 0.143 0.034 0.0143 0.298 0.051

Table 16.18 turns/pole rotor parameters

If,R Nf Rf Mf Lf Lfρ

[A] [-/pole] [ς] [mH] [mH] [mH]108.4 18 33.4 8.46 9.96 1.50

40

3.2.3 Four sectors rotor winding arrangementIn view of the application of the driving arrangement of figure 1, which re-quires a maximal rotor back-EMF per sector lower than the adopted DC-linkvoltage according to (2.54), another version of the motor, equipped withsame stator (q=4, ns=6, kw=0.925, c=1, JCu= 3.74 A/mm2) but having differ-ent number of turns for the excitation winding, has been conceived. Thismachine shows of course the same Blondel’s diagrams of the previous one,since it has the same d- and q-reactance, but it requires definitely a smallerexcitation current. This fact does not affect the efficiency of the motor sincethe rotor MMF and the rotor current density (3.36 A/mm2) remain still thesame. The most important benefit of this change is its influence on the sta-tor-to-rotor transformation ratio and on the resulting rotor back EMF.For the machine with 18 turns the transformation ratio according to (2.49) is:

∋ ( V500E53.1k

cqn

Nk2U2

E18N max0

ws

ff

R

max0

0 ?↑<ο

<< .( 3.16 )

For the machine with 44 turns it results

∋ ( V1202E68.3k

cqn

Nk2U2

E44N max

0

ws

ff

R

max0

0 <↑<ο

<< ,( 3.17 )

where kf=1.17 (air gap induction form factor due to chosen pole shoe formand saturation). So, the motor with 18 turns can be used without tie imped-ance towards the bus bar with a 565 VDC-link (rectification of a three phase400V system) even without splitting the rotor in sectors. The other one canbe used without tie impedance even with a 326 VDC-link (rectification of asingle phase 230V system) once the rotor winding has been divided in fourparts.Whilst the machine parameters of table 15 remain the same, those of table 16need to be updated accordingly to Nf=44:

Table 17.44 turns/pole - 4 rotor sectors parameters

If,R Nf Rf Mf Lf Lfρ

[A] [-/pole] [ς/pole] [mH/pole] [mH/pole] [mH/pole]

45.0 44 49.9 5.17 14.87 9.70

In figure 31 the rotor winding cross section for a 44 turns is depicted.

41

Figure 31. Rotor winding arrangement for 44 turns per pole

42

3.3 Simulink modelsThe simulation of the control strategies presented in chapter 2 - for achievingrespectively the active damping of the rotor oscillations and the alternativestarting of a synchronous machine - have been performed by using self-madeMatlab-Simulink models. Simulink is a simulation tool which obtains thestates space describing ODEs for a given physical system directly from itsfunctional flow chart. The user must essentially establish interconnectionsbetween the system functional blocks and provide initial conditions for thestate variables. Simulink is moreover able to solve the self-obtained ODEsby using numerical integration routines. The ODE-system to be solved fordescribing the state-space of a synchronous machine with damping circuitson the d- and q-axis is represented by (2.12) and (2.13) according to Park’sreference frame. The implementation of (2.12) and (2.13) in Simulink isdepicted by the scheme of figure 33.

Figure 32.Simulink synchronous machine model

The state-space for the machine is defined by its five linked fluxes ξ plus theelectrical angular speed ϖ, since the time derivatives of all those variables -five electrical and one mechanical - show up in the ODEs.The currents are related to the linked fluxes by their linear combinationsthrough the inductances as shown by (2.14). They can be obtained from thelinked fluxes expressed in form of matrix by:

Ζ ∴ Ζ ∴ Ζ ∴ξ√< ,1Li . ( 3.18 )

This simple linear approach is possible since the effects of saturation andhysteresis on the inductances have been neglected in the present model.

43

The mechanical block of figure 33 is described in detail in figure 34. It ispossible to observe there how the unbalance between mechanical and elec-tromagnetic torque generates the electric angular acceleration through theangular moment of inertia I and the pole-pairs number p. The electrical an-gular speed ϖ and the electric load angle χ are then respectively obtained bymeans of two successive time integrations.

Figure 33.Simulink model (detailed) of the mechanical block

The Simulink model for the synchronous machine which has been just dis-cussed is the core of both driving systems proposed in chapter 2. It is repre-sented in the next two paragraphs by a functional box simply called “syn-chronous machine”.

3.3.1 Simulink model for rotor oscillations active dampingAccording to (2.39), in order to damp actively the rotor oscillations a controlvoltage Χvf is needed which is proportional to the derivative of the electricalangular speed departure. Figure 35 shows a control system capable to per-form that.

Figure 34.Speed control for oscillations active damping

44

In the scheme of figure 35 the actual rotor speed is sensed and transformedin the electrical angular speed ϖ. This last is compared with the constantelectric angular speed ϖ0 of the armature magnetic MMF and the differenceamong them is processed by an automatic controller for generating the con-trol voltage Χvf. It can be observed that the essential component of the auto-matic controller is the D-block performing the time-derivative of the error δ.However, since the regulator block acting on the field winding plays a cru-cial role in voltage and power regulation of synchronous machines, it hasbeen depicted including also the P- and I-blocks for more generality.Coming to the power electronics chain it consists of a PWM-generator and aH-bridge of switches like the one represented in figure 19. In the Simulinkmodel a bipolar PWM generator has been used and the H-bridge has beenassumed containing ideal switches, by providing an output voltage commu-tating between +VDC-link and –VDC-link instantaneously. That performance hasbeen obtained by the Simulink blocks chain shown in figure 36.

Figure 35.Bipolar PWM-generator and ideal H-bridge emulator

The PWM block has an internal triangular carrier at frequency 1 kHz whichis compared with the input voltage Uref. The resulting output is a digital sig-nal owing level 0 or 1. By subtracting the offset 0.5 to that it is possible toobtain a digital PWM signal commutating between -0.5 and +0.5. Finally, bymultiplying that signal by two times the DC-link voltage a bipolar PWMvoltage VAB with +VDC-link and –VDC-link is obtained.The VDC-link has been set to 565V which is the maximal DC output voltageachievable from a 400 V line-to-line three phase rectifying bridge.The rotor winding of the study-case generator has not been divided in moresectors in the simulation since it had produced an unnecessary complicationof the model without adding beneficial information. The simulation resultsobtained by that way can be easily extrapolated to an UMP-balancing suita-ble rotor arrangement. A possible division of the rotor in three sectors with 6poles each would require for example a three legs H-bridge as shown in fig-ure 1, the same control current achieved for the entire winding simulationimpressed in all three sectors at the same time and of course only one thirdof the simulated voltage control effort.Specific considerations about advantages and drawbacks of the sectoredrotor winding are presented in the conclusions.

45

3.3.2 Simulink model for the alternative startingAccording to the achievements of paragraph 2.4.3 in a synchronous motorthe average EM-torque, generated by actively controlling the rotor current i f

during the start, is maximal when it is in phase with the q-axis armature cur-rent. It is important to highlight that the field current if and the armature cur-rents iA, iB and iC have in general different frequencies. Nevertheless, oncethe armature currents are referred to the Park’s frame their resulting d- andq-axis components share the same frequency with the controlled current i f.This is one of the most important advantages of the Park’s transformations inthe quasi-steady-state behavior.It must be observed that the Simulink model of figure 33 for the synchronousmachine has been developed with reference to the generator currents con-vention. By using the same model, in order to achieve the maximization ofthe average torque in the motor behavior the control current i f and the iq mustbe then in opposition. The scheme of figure 37 presents a driving systemable to control a current in the field winding which has the desired constantamplitude Iref and which is in phase opposition with iq.

Figure 36.Current control for machine active self-starting

The hardcore of this drive is the signals sensing and processing unit. It con-sists of two main blocks. The first one is a system capable of sensing theangular slip speed and the position of the rotor relative to the armatureMMF. The second one is a block performing the Park’s transformation onthe armature currents after having sensed them. By this way a track signal ofthe q-axis current with unit amplitude <iq> is obtained which is a base tobuild the reference current signal iref. The rest of the control system is madeby the functional blocks explained in the previous paragraph.

46

4 Results and discussion

4.1 Rotor oscillations and energy recoveryIn this section the simulation results on the study-case generator which hasbeen obtained by the Simulink model of figure 35 are presented and dis-cussed.

Study-case generator characterization with and without damping bars

In figure 38 the simulated performance of the damping bars is shown, whichensures the quick attenuation of speed and load angle oscillations provokedby a step-wise power regulation of +10% in the study-case synchronousgenerator.

Figure 37. Study-case generator speed and load angle transient responses for a stepadjustment of the prime-mover power. The generator is equipped with damping barson the direct and quadrature axis.

In figures 39, 40 and 41 instead it can be observed how long the oscillatorytransient triggered by the power regulation lasts, once the damping bars areremoved.

Figure 38.Study-case generator speed transient response for a step adjustment of theprime-mover power. The generator direct and quadrature axis damping being re-moved.

47

Figure 39.Study-case generator load angle transient response for a step adjustmentof the prime-mover power. The generator direct and quadrature axis damping barsbeing removed.

Figure 40.Study-case generator power transient response for a step adjustment of theprime-mover power. The generator direct and quadrature axis damping being re-moved.

The weak residual damping recognizable in the last three pictures is due tothe inescapable armature and field windings losses induced by the rotorhunting. Since they are both low resistive circuits for the sake of the machineefficiency their time constants are usually bigger than the period of the rotoroscillations. This fact is confirmed by the comparison of the transient timeconstants Td0’=7.3 s and Td’=2.3 s of the study-case generator shown in table5 with the period T=0.6 s of the rotor oscillations shown in table 2 and cal-culated for the transient armature reactances.

Providing oscillations damping by means of field current active control

In figure 42, 43 and 44 the quick attenuation of the oscillations for the samepower regulation used before are obtained by active field current control.

Figure 41.Study-case generator speed transient response for a step adjustment of theprime-mover power. The generator direct and quadrature axis damping bars beingremoved and the field current being controlled by the arrangement of figure 35 withCD=4000 Vs2.

48

Figure 42.Study-case generator load angle transient response for a step adjustmentof the prime-mover power. The generator direct and quadrature axis damping barsbeing removed and the field current being controlled by the arrangement of figure 35with CD=4000 Vs2.

Figure 43.Study-case generator power transient response for a step adjustment of theprime-mover power. The generator direct and quadrature axis damping bars beingremoved and the field current being controlled by the arrangement of figure 35 withCD=4000 Vs2.

A comparison of figure 42 and 43 with figure 38 lets recognize that it is pos-sible to perform the rotor oscillation damping by means of the circuit of fig-ure 1 driven through the control system of figure 35. The gain of the deriva-tive controller is set at CD=4000 Vs2. In fact the suggested control strategyproves capable of attenuating the amplitude of the oscillations as well as thedamping bars do. A comparison of the same performances with the bar-lessmachine response of figures 39 and 40 shows how effective the proposedtechnique is. In order to generate the control current depicted in figure 45,which is necessary to the active damping of the spoken rotor oscillations, thecontrol voltage effort of figure 46 is needed.

Figure 44.Needed excitation current control for achieving the actively damped oscil-lations of figures 42, 43 and 44. The generator direct and quadrature axis dampingbars being removed and the field current being controlled by the arrangement offigure 35 with CD=4000 Vs2.

49

Figure 45.Needed voltage control effort for controlling the field current of figure 44and achieving the damped oscillations shown in figures 42, 43 and 44. The generatordirect and quadrature axis damping bars being removed and the field current beingcontrolled by the arrangement of figure 35 with CD=4000 Vs2.

It can be easily checked that the peak of that voltage (}140 V) is way smallerthan the rated excitation winding voltage (209 V). Even though the controlvoltage of figure 46 is indirectly applied to the rotor winding by its PWM-modulation on the base of a 565 V DC-link voltage (figures 47 and 48), thevoltage levels envisaged by this solution are such to not require an alterna-tive and more expensive insulation for the rotor winding. That means that theproposed control system behavior does not expose the rotor winding insula-tion to harmful voltage level. Anyway it would be always possible to reducethe number of turns per rotor pole in order to admit a lower voltage for theinverter link, as long as that the machine voltage regulation performancefulfills the grid-owner’s requirements. A reduction of the turns of -30% forexample would require half of the previously needed VDC-link16.Figure 47 and its magnification in figure 48 show from the top to the bottomrespectively the PWM-voltage applied to the field winding (green), the re-sulting controlled current (blue) and the comparison between the triangularcarrier (light blue) and the control voltage (brownish red).

Figure 46.PWM-control mechanism of the excitation current explained by the com-parison between the control voltage (red) and the 1 kHz triangular carrier (blue).

The carrier frequency is 1 kHz and only the magnification provided in figure48 enables to look at the time-work which triggers the commutation of the

16 It would of course also mean +30% of excitation current but dynamic and losses un-changed.

50

PWM voltage. In particular in figure 48 the duty cycle bigger than 50%forces the standing increase of the field current.

Figure 47.PWM-control mechanism of the excitation current explained by the com-parison between the control voltage (red) and the 1 kHz triangular carrier (blue) –magnification of figure 47 at t=10 s.

Providing the non-periodic critical oscillations damping by means offield current active control with the aim to recover some kinetic energy

A relevant result of the field current active control applied to the damping ofrotor oscillations is shown in figures 49, 50 and 51.

Figure 48.Study-case generator speed transient for a step adjustment of the prime-mover power. The generator direct and quadrature axis damping bars being removedand the field current being controlled by the arrangement of figure 35 withCD=23000 Vs2.

Figure 49.Study-case generator load angle transient response for a step adjustmentof the prime-mover power. The generator direct and quadrature axis damping barshave been being removed and the field current being controlled by the arrangementof figure 35 with CD=23000 Vs2.

51

Figure 50.Study-case generator power response for a step adjustment of the prime-mover power. The generator direct and quadrature axis damping bars have beenbeing removed and the field current being controlled by the arrangement of figure 35with CD=23000 Vs2.

At cost of a higher control effort and just by increasing the derivative con-troller gain CD from 4000 to 23000 Vs2 it is possible to achieve the shortestsettling time of the rotor speed for that given level of power generation.Whereas it is sure that the damping bars perform different damping accord-ing to the amount of power generated at the moment the perturbation inter-venes, by the system proposed in figure 35 it is always possible to suppressthe oscillations with the same strength and within the same settling time, forall possible load angles (at least all those between the rated one and the un-der-excitation limit).Moreover figure 52, which magnifies the power overshoot of figure 51,shows a beneficial aspect of practicing the active damping of rotor oscilla-tions: the yellow shaded area represents the quota of rotor kinetic energy,which has been recovered and converted into available electric power for thegrid.

Figure 51.Area (yellow shaded surface) representing the quota (}0.16 kWh) of rotorkinetic energy (}0.23•0.32 kWh) involved in the rotor oscillations recovered by thenon-periodic-critical active dumping via the field current control.

In a normal passive damping system all that energy is wasted in the dampingbars. Analytical considerations performed in tables 1,2 and 3 show that themaximal theoretical recoverable energy for the study-case generator is about0.23•0.32 kWh per each regulation of that kind, whereas the recovered ener-gy calculated via integration of the yellow shaded area accounts for 0.16kWh. That means that at least 50% of the otherwise lost kinetic energy hasbeen recovered by the proposed system.

52

It is worth to observe that the recovered energy is of the same order of mag-nitude of the energy needed for magnetizing the study-case generator. Infact, by using the values in tables 4 and 6 it is possible to estimate the open-circuit magnetizing energy expressed in kWh by:

kWh318.0IL21

106.31W 2

R,ff60,f ?√√

< ( 4.1 )

The field current regulation for a critical response requires of course astronger driving effort than the one needed for the damping-bars-like per-formance seen before. Figure 53 shows the current pulse of 130 A neededfor achieving the critical settling time.

Figure 52.Needed excitation current control for achieving the non-periodic criticaldamped oscillations of figures 49, 50 and 51. The generator direct and quadratureaxis damping bars being removed and the field current being controlled by the ar-rangement of figure 35 with CD=23000 Vs2.

That is witnessed also by the control voltage in figure 54, which reaches inthis case almost three times the excitation winding rated voltage.

Figure 53.Needed voltage control effort for controlling the field current of figure 52and for achieving the non-periodic critical damped oscillations of figures 49, 50 and51. The generator direct and quadrature axis damping bars being removed and thefield current being controlled by the arrangement of figure 35 with CD=23000 Vs2.

Even though this control case envisages the worst case scenario the involvedvoltage levels do not seem so severe to require a drastic change in the insula-tion design criteria and costs.Figures 55 and 56 show from the top to the bottom respectively the PWM-voltage applied to the field winding (green), the resulting controlled current

53

(blue) and the comparison between the triangular carrier (light blue) and thecontrol voltage (brownish red) for the critical current control given byCD=23000 Vs2.

Figure 54.PWM-control mechanism of the excitation current explained by the com-parison between the control voltage (red) and the 1 kHz triangular carrier (blue).Over-modulation occurrence.

In spite of the PWM-over-modulation occurring in figure 56 in correspond-ence of the heaviest control effort, the control action has not been impairedas it is witnessed in figure 55 by the achieved smooth field current.

Figure 55.PWM-control mechanism of the excitation current explained by the com-parison between the control voltage (red) and the 1 kHz triangular carrier (blue) –Over-modulation occurrence. Magnification of figure 54 at t=10 s.

This circumstance, which shows no need for the control voltage to remainwithin the limits of the carrier amplitude, could represent a rational criterionfor targeting an economical design of the control system and drive.

54

4.2 Motor/Generator alternative startingIn this section the simulation results for the designed motor, which has beenobtained by the Simulink model of figure 37, are presented and discussed.The timely starting profile of the designed motor at no load is depicted infigure 57.

Figure 56.Electrical speed transient during the alternative starting of the designedmotor by field current active control. The start is performed by the driving arrange-ment of figure 37.

The used driving criterion keeps the EM-torque constant against the inertialreacting torque up to the synchronism. From that point on the control voltageis shut-off and the motor free-wheels as a reluctance machine thanks to itsrotor magnetic anisotropy.In figure 58 and in its magnification in figure 59 it can be observed that thecontrolled rotor current frequency drops according to the rotor slip by main-taining the current amplitude constant.

Figure 57.Excitation current (pink) and q-axis armature current (light brown) duringthe active starting of the designed motor by field current active control. The start isperformed by the driving arrangement of figure 37.

Figure 58. Magnification of picture 58 at t= 1 s. It proves the phase oppositionbetween field current and q-axis armature current in the proposed driving strategy.

55

This last is chosen intentionally equal to 45 A, which is the intensity of therated excitation current shown in table 14. In figure 59 it is shown that thecontrol system conceived in figure 37 succeeds in controlling the field cur-rent. In fact it keeps both field current amplitude and phase opposition to-wards the q-axis armature current component constant over the frequency.According to (2.61) those conditions pursue the constant average torquegeneration foreseen by the theoretical considerations of this work. In figures60 and 61 it is possible to observe that also the d- and q-axis currents holdtheir amplitudes almost constant all over the start procedure. Since the ma-chine keeps going through an infinite sequence of quasi-steady state condi-tions during the acceleration, the q- and d-axis currents lack to be perfectlyin phase quadrature.

Figure 59.d-axis (green) and q-axis armature current (light brown) during the activestarting of the designed motor by field current active control. The start is performedby the driving arrangement of figure 37.

Figure 60.d-axis (green) and q-axis armature current (light brown) during the activestarting of the designed motor by field current active control. The start is performedby the driving arrangement of figure 37.Magnified picture at t= 1 s. It proves that thetwo currents are not in phase quadrature outside of a steady-state condition.

That explains why the rotor back-EMF and the field current are not exactlyin phase in figures 62 and 63. In figure 62 it can be observed that the designrequirement posed in (3.17) - about the back-EMF to be smaller than 325 V -has been fulfilled. The rotor back-EMF at s=1 is lower than 300 V (onefourth of 1202 V achieved in (3.17)) because of the attenuating effect playedby the stray inductance disregarded in (3.17).

56

Figure 61.Field current (pink) and rotor back-EMF (black) during the active startingof the designed motor by field current active control. The start is performed by thedriving arrangement of figure 37.

Figure 62.Field current (pink) and rotor back-EMF (black) during the active startingof the designed motor by field current active control. The start is performed by thedriving arrangement of figure 37. Magnified picture at t= 1 s. It proves that the twoquantities are almost in phase but not perfectly.

The armature currents profiles depicted in figure 64 are worth a couple ofremarks. It seems that they undergo a kind of amplitude modulation whilethe rotor keeps accelerating.

Figure 63.Armature phase currents during the active starting of the designed motorby field current active control. The start is performed by the driving arrangement offigure 37.

In particular the changes of amplitude do not affect all the three phases sim-ultaneously when the rotor speed is low. This fact can be observed by com-paring figure 65 with figure 66 where a strong armature currents asymmetryis to be recognized. On the contrary the more the rotor reproaches the syn-chronism the smaller is the unevenness among the three phase currents.It must be considered that as soon as an asymmetry in the armature currentarises a negative sequence armature magnetic field is to be taken in account.That is explained by the natural reaction of the armature windings to thecounter-rotating magnetic field produced by the pulsating rotor MMF.

57

Figure 64.Armature phase currents during the active starting of the designed motorby field current active control. The start is performed by the driving arrangement offigure 37. Magnified picture at t= 3.8 s. It proves that the armature current asym-metry is negligible close to the synchronism.

Figure 65.Armature phase currents during the active starting of the designed motorby field current active control. The start is performed by the driving arrangement offigure 37. Magnified picture at t= 0.5 s. It proves that the armature current asym-metry is strong at low rotor speed.

In appendix A.4 it is proved that the speed of said counter rotating magneticfield relative to the armature is proportional to 2s-1. It becomes then clearwhy the EMFs induced in the armature change their nature along the startprocedure, belonging to a negative sequence for 1>s>0.5 and to a positiveone for 0.5>s>0. So in both cases the occurrence of a beat phenomenon mustbe foreseen. However in the case of a negative EMFs sequence, even if theperiod of the EMFs is close to that of the supplying network, the deformingeffects of the beat arise due to the discrepancy in the phases sequence order.Another effect produced by the rotor counter rotating magnetic field can berecognized in figure 57. A slight change in the speed slope (acceleration) canbe observed at circa 2 s when the rotor crosses the slip s=0.5. At that pointthe counter rotating magnetic field speed relative to the armature changessign and keeps spinning in same sense of the rotor. It can be easily provedthat the asynchronous drag produced by this field component for 1>s>0.5 issuch to help the work of the EM-torque. This is why in that slip interval theacceleration is higher than elsewhere. As soon as the slip falls down 0.5 thegenerated asynchronous torque starts working against the EM-torque. Thatexplains why in the slip interval 0.5>s>0 the rotor acceleration is lower thanelsewhere. As explained in the theoretical part of this work both the occur-rences of armature currents asymmetry and different actions of the asyn-

58

chronous torque depend on the single phase nature of the proposed drivingsystem. Another occurrence related to the same cause is the pulsating char-acter of the EM-torque. In the green profile of figure 67 the instantaneousmechanical power can be recognized, which is proportional to the EM-torque by means of the angular rotor speed. Its shape confirms that the gen-erated accelerating torque has a pulsating nature as envisaged by the formula(2.59).

Figure 67.Comparison between the instantaneous powers during the suggested star-ing process: absorbed armature power (black), generated rotor power (blue) andEM-power (green).

It must be also observed that the starting time at rated power should be equalto two times the motor time constant. Assumed that in the simulation themoment of inertia is ten times that of the motor (for accounting all inertiarelated to a possible load), the motor time constant becomes H=1 s. Thestarting at rated power should be then performed in 2 s. In figure 57 a start-ing time of roughly 4 s is instead recorded. That means that the acceleratingtorque is half as big as the motor nominal one. This remark is confirmed by(2.62) which states that only half of the rotor flux contributes actually to thegeneration of the EM-torque.Coming to the efficiency of the proposed drive in figure 67 a comparisonbetween all powers involved in the motor start is possible.The picture confirms that the EM-power grows almost linearly with the timeor with the complement of the slip 1-s. The same figure confirms that theprincipal supply of the rotor is connected to the armature (black profile)which absorbs power all over the start. Because of the PWM high frequencyit is not possible to perceive from the figure if the energy is extracted fromthe rotor (positive) or provided to it (negative). The representation of thesame behavior in terms of energies in figure 68 reveals that the energy isactually generated by the rotor winding. In other words it is extracted fromthe field winding all over the start. The theoretical previsions about the be-havior of the driving strategy implemented by the scheme of figure 37 findcomplete confirmation here. Moreover by means of figure 68 it is possible toperform an estimate of all the energy involved in the balance of figure 6.Since the absorbed energy at the synchronism corresponds to 85000 J the

59

stored to 40000 J and the recovered through the rotor to 40000 J, the overallenergy efficiency of the start performed by means of the suggested fieldcurrent active forming strategy amounts to roughly 89%.

Figure 68.Comparison between the energies respectively absorbed (black), recov-ered (blue) and stored (green) during the suggested staring process.

Figures 69 shows the PWM voltage acting on one of the four rotor windingsectors and the resulting controlled current.

Figure 69.PWM-control mechanism of the excitation current during the startingperformed by the driving arrangement of figure 37.

In figure 70 the mechanism of generation of a bipolar PWM by comparisonof a modulating control voltage and a 25 kHz triangular wave carrier is pre-sented.

Figure 70.PWM-control mechanism of the excitation current explained by the com-parison between the control voltage (blue) and the 25 kHz triangular carrier (green).

60

The control voltage generated by the derivative controller (blue) is comparedwith the triangular shaped carrier (green) and the PWM voltage (red) is gen-erated. Up till the synchronism the modulation is linear since the controlsignal ranges between the carrier limits. As soon as the control system isdisconnected from the rotor winding at the synchronism the control signalincreases enormously since it senses that the field current does not follow thereference anymore. In figure 69 it is possible to observe that after the open-ing of the control loop the field current start dropping to zero by followingthe rotor oscillations.In figure 71 and its magnification of figure 72 a sample of the power elec-tronics behavior in linear region is shown.

Figure 71.PWM-control mechanism of the excitation current explained by the com-parison between the control voltage (blue) and the 25 kHz triangular carrier (green).Magnification of figure 68 in linear modulation region.

The mechanism of generation of the PWM voltage by comparison betweenthe control voltage and the carrier is clearly depicted in figure 72.

Figure 72.PWM-control mechanism of the excitation current explained by the com-parison between the control voltage (blue) and the 25 kHz triangular carrier (green).Magnification of figure 69.

It is interesting to observe the 25 kHz disturbance superposed to the controlsignal (blue) which represents the propagation of the PWM fundamentalthrough the control chain. In the Simulink model a low pass filter before thecontroller has been used to reduce that kind of undesired noise. The systemmodel performed in Simulink by the author has therefore highlighted a realproblem in the current control strategies using a derivative controller.

61

The high-pass nature of the D-controller is responsible for the lack of attenu-ation of all noises produced in the system, in particular the ones related tothe PWM modulation.Figures 73 and 74 catch two moments of the PWM control where the currentis almost constant.

Figure 73.Particular of the PWM-voltage and resulting stationary excitation current.

Figure 74.Particular of the PWM-voltage and resulting stationary excitation current.

It is interesting to observe that at different rotor speeds the needed averagePWM voltage required for obtaining a stationary current is different. It mustbe recognized that the duty cycle in the first case is less than 50% whichmeans negative average PWM voltage. In the second case the duty cycle isclose to 50% which means average voltage close to zero. The reason forthose behaviors is the different value of rotor back-EMF acting in the fieldwinding in the two examples. In fact in figure 73 the rotor is close to the restwhereas in figure 74 it is close to the synchronism.

62

5 Conclusions

Conclusion about active damping of rotor oscillations

The preliminary theoretical study conducted on the rotor hunting in a medi-um-large salient pole synchronous machine together with the results of theSimulink simulation presented in chapter 4 and discussed above let concludethat it is definitively possible to damp the rotor oscillations by means of fieldcurrent active control. The same system proposed by [1] and already used forthe UMP-compensation proved also to be able to recover more than 50 % ofthe rotor kinetic energy involved in the rotor oscillations. It established bythat a new performance not otherwise achievable by using traditional damp-ing bars. The energy amount attainable by that way for each power regula-tion event showed to be proportional to the entity of the power regulationitself and to the machine inertial time constant. Even though systematic stud-ies and experimentations about technical exploitation of conservative damp-ing mechanisms in synchronous machines have been lacking till now, thecontrol strategy proposed in here could be of considerable advantage forsynchronous generators undergoing a large amount of power regulations perday and for synchronous motors performing heavy highly frequent intermit-tent duties (e.g. stone mills and ice-breakers).The active damping of oscillations via the excitation current forming provedalso that the needed control effort is inversely proportional to the sinus of theload angle17, which also means inversely proportional to the generated pow-er. This circumstance appears to be very favorable since the damping mech-anism results more effective and efficient when the amounts of energy to berecovered become more relevant and when the amplitudes of the angularrotor departures are more intense.Some valuable knowledge was gained about how efficient the traditionaldamping circuits work: they accomplish their function differently accordingto the entity of the load angle. For small load angles the quadrature dampingbars contribute more than the direct axis ones to the oscillations extinction.At big load angles the opposite is true.It must be therefore concluded that in a bars-less synchronous machine, inabsence of a control winding along the q-axis, the active damping rotor os-

17at low power angles the quadrature current is very low and the mechanism of torque generation via theinteraction with the excitation flux becomes not efficient.

63

cillations at low power angle is not effective, even useless for synchronouscapacitors/inductors and fly-wheels where χ ?0.

That important result suggest to forego only the damping bars laying on thed-axis for the following reasons:

∂ they affect and undermine the dynamic of the air gap induction con-trol during the UMP compensation;

∂ they waste the energy of the rotor oscillations, which can be conven-iently and efficiently recovered by the active excitation current con-trol for load angles χ> 20°;

∂ they do not damp effectively the rotor oscillations for small load an-gles where the q-axis damping bar instead perform their best;

∂ they reduce in some measure the cross section of the magnetizingpath by causing higher exciting MMF need.

Coming to other advantages that the proposed damping technique showstowards the classical amortisseurs, the results showed the possibility to ad-just the strength of the damping action simply by tuning the gain of the de-rivative controller. This feature could prove useful even in presence ofdamping bars, when a wrong design of them could not be solved without anextremely expensive substitution or reprocessing of the entire rotor. In thatcase the system suggested by this work could cheaply empower, compensateor more generally correct the damping bars performance providing at thesame time further valuable benefits such as voltage- and reactive-power-regulation, UMP-compensation, and static and dynamic stability improve-ment.Finally it must not be forgotten that the proposed control system proved tobe able of fast dynamic corrections of the electromagnetic torque accordingto a given control signal. This property could reveal extremely interesting insuppressing the Sub Synchronous Resonance (SSR) in those power plantssupplying power through transmission lines serially compensated by capaci-tors. It is in fact well known [23] [24] that SSR occurrence is accompaniedby prejudicial torsional stress on shafts connections between generator, tur-bine and exciter. The proposed control system would be able to favorablycompensate those harmful torque oscillations (typically 10•40 Hz) and theirrelated voltage fluctuations (flicker) even in cylindrical rotor synchronousmachines. Even though it is ascertained that hydropower plants are moreimmune than turbogenerators toward SSR [25], it is undisputed that by dis-missing the damping bars the asynchronous energizing mechanism at thebase of SSR insurgence would be directly stricken.

64

Conclusion about start by active current control

A 35 kW-400V synchronous three phase motor was analytically designedand cross-checked by a FEM program. Its design was performed in order toobtain a rotor back-EMF voltage lower than the usual DC-link voltage levelsall over the start procedure. The design procedure itself showed that by di-viding the field winding in more sectors it is possible to increase the statorEMF several times at all advantage of a smaller rated excitation current,without overcoming anyway the DC-link voltage level by that.The motor start simulation performed by a self-made Simulink model provedthat the alternative starting by means of field current active control is defi-nitely possible. The possibility of impressing a constant average acceleratingtorque to the rotor by recovering active power through its rotor winding wassuccessfully confirmed. The conceived control and driving system succeededin maintaining the excitation current constant and constantly in phase oppo-sition toward the q-axis armature current, all over the start procedure. Thesimulation proved also the exactness of several theoretical conclusions andprevisions related to the intrinsic restraint of controlling the rotor currentonly along the d-axis, and precisely:

∂ very high torque ripple due to the pulsating nature of the instantane-ous EM-power ;

∂ presence of remarkable torque beat close to the synchronism. The re-lated very low frequency torque ripple excites the proper-resonancefrequency of the rotor, fact that becomes evident on the speed rippleoccurrence;

∂ single phase asynchronous disturbances related to the regressive ro-tor-MMF, which appear in the speed start characteristic. The firsthalf of it shows higher slope than the remaining part, since the re-gressive rotor-field helps accelerating the rotor for n < 750 rpm andbreaking it elsewhere;

∂ only one half of the rotor field helps generating the EM-torque eventhough the rotor thermal stress accounts for the rated full one.

In spite of all the enlisted drawbacks the proposed starting system achievedan outstanding energetic starting efficiency (}89%) if compared with theefficiencies pursued by analog classical driving strategies.The theoretical work helped also to gain a deeper insight in the design re-quirements and limits for the proposed driving system. According to actualmaximal absolute ratings of the power-electronics switches belonging to thedifferent available technologies, the maximal admissible motor rated powershas been found, which can be handled by the suggested starting techniquewithout facing whatever kind of technical prejudice. They are respectively60 MVA for GTOs/IGCTs, 13.5 MVA for IGBTs and 480 kVA forMOSFETs.

65

The limitation of the back-EMF levels to values which do not require specialand expensive coordination of the rotor winding insulation appears to be thebiggest challenge that this kind of technique needs to face. Under this pointof view the application of the proposed starting strategy to big-size powergenerators appears quite problematic, unless helping means for reducing therotor back-EMF are admitted (tie-reactance, voltage lowering and specialarmature winding connections).Given the ranges of power envisaged above the proposed starting techniqueby field current active control seems to target preferably synchronous motorsin the range of power up till 10 MVA. This fact, if also confirmed by thefuture experimental work and practical executions, would contend the su-premacy of asynchronous machines in their traditional field of application.

66

6 Future work

Future work about active damping oscillations requires primarily an im-provement of the simulation model aimed to include saturation effects, non-linear phenomena and all losses mechanisms contributing to the mechanicaldamping of the oscillations. In order to explore systematically all potentialreceptions for the proposed technique a survey of all applications of syn-chronous generators and motors should be planned. In particular those oneswhere the recurrence and the intermittence of the power regulation makesthe rotor energy recovery more attractive. Since the presented work dis-closed the possibility to alternatively perform just two main duties amongthose performed by the damping bars, the remaining unexplored ones shouldbe also inquired for disclosing other potential applications or limits of thefield current active control. Some experimental tests about rotor oscillationsand energy recovery, possibly conducted on the designed motor, wouldcomplete the validation of all theoretical considerations performed in thepresent work.Coming to the motor active starting, the drive arrangement, which hasproved to be successful in the simulation needs to be built and tested. Somefurther work on the design side is also required for optimizing the EM be-havior and mainly for checking the mechanical design (best cost-benefitsmaterial choice, tolerances, inertial and electromechanical tensions andstresses, thermal analysis). Finally, since the work done on the proposedstarting technique highlighted some drawbacks, improving solutions for thesuppression or the reduction of the rotor counter rotating field component areneeded.

67

7 Bibliography

[1] J.J. Pérez-Loya, C.J.D. Abrahamsson, U. Lundin – Demonstration of activecompensation of unbalanced magnetic pull in synchronous machines – CigreScience & Engineering – n. 8 June 2017.

[2] C. Concordia – Synchronous machines – Theory and performance – JohnWiley & Sons Inc. – New York 1951.

[3] A. Ivanov-Smolensky – Electrical Machines – vol. 3, MIR Publishers, Moscow1983.

[4] J. Pyrhönen, T. Jokinen and V. Hrabovcova - Design of Rotating ElectricalMachines - John Wiley & Sons Ltd. - New York 2008.

[5] G. Traxler-Samek , T. Lugand, A. Schwery - Additional Losses in the DamperWinding of Large Hydrogenerators at Open-Circuit and Load Conditions -IEEE Transactions on Industrial Electronics, vol. 57, n. 1, January 2010.

[6] ABB Group Press Release – ABB motor sets world record in energy efficiency– saves half a million dollars – Zurich, 5 July 2017.

[7] J.J. Pérez-Loya, F. Evestedt, U. Lundin– Magnetic thrust bearing for hydro-power unit with a Kaplan turbine – DiVA-portalen, November 2017..

[8] J.J. Pérez-Loya, C.J.D. Abrahamsson, U. Lundin – Electromagnetic losses inSynchronous Machines during Active Compensation of Unbalanced MagneticPull – IEEE Transactions on Industrial Electronics, PP(99):1-1, April 2018.

[9] C. Bruzzese and G. Joksimovic – Harmonic signatures of stator eccentricitiesin the stator voltages and in the rotor current of no-load salient-pole synchro-nous generators - IEEE Transactions on Industrial Electronics,vol.58, no.5, pp.1606-1624, May 2011.

[10] D. Schröder – Elektrische Antriebe 1 – Grundlagen - p. 40, 48, Springer-Verlag Berlin Heidelberg New York, 1994

[11] Robert Bosch GmbH (ed.) – Bosch Automotive Electrics and Automotive Elec-tronics– 5th edition – Springer Vieweg – 2007.

[12] W.C. Duesterhoeft, M.W. Schulz; E. Clarke - Determination of InstantaneousCurrents and Voltages by Means of Alpha, Beta, and Zero Components -Transactions of the American Institute of Electrical Engineers, Volume 70 (2),pp. 1248–1255, July 1951.

[13] R.H. Park - Two Reaction Theory of Synchronous Machines - Transactions ofthe American Institute of Electrical Engineers, Volume 48, pp. 716-730, 1929.

[14] C. Concordia - Hunting of a salient-pole synchronous machine - ElectricalEngineering, Volume 75, Issue 4, April 1956.

68

[15] N.V. Balasubramanian, J.W. Lynn, D.P. Sen Gupta – Differential forms onelectromagnetic networks – Buttersworth & Co. Ltd. - London 1970.

[16] C.A. Stapleton – Root locus study of synchronous machine regulation – Proc.IEEE London, vol. 111, p. 751-767, April 1964.

[17] P.C. Krause, James N. Towle – Synchronous Machine Damping by ExcitationControl with Direct and Quadrature Axis Field Windings – IEEE Transactionson Power Apparatus and Systems, Volume PAS-93, Issue 8, 1969.

[18] S.N. Vukosavic - Electrical Machines - Springer Science+Business MediaNew York 2013.

[19] E. Johansson - Detailed Description of Synchronous Machine Models Used inSimpow - Master Thesis B-EES-0201 – KTH.

[20] J. Bladh - Hydropower generator and power system interaction – PhD Disser-tation presented at Uppsala University November 16, 2012.

[21] C.K. Seetharaman, S.P. Verma, A.M. El-serafi – Operation of SynchronousGenerators in the Asynchronous Mode - IEEE Transactions on Power Appa-ratus and Systems, Volume PAS-93, Issue 3, May 1974.

[22] F. Iliceto - Impianti elettrici – vol. 1, Patron Editore, Bologna 1981.[23] D.N. Walker, C.E.J. Bowler, R.L. Jackson – Results of subsynchronous reso-

nance tests at Mohave – IEEE Transactions on Power Apparatus and Systems,Volume PAS-94, Issue 5, September/October 1975.

[24] E.H. Allen, J.W. Chapman, M.D. Ilić - Eigenvalue Analysis of the StabilizingEffects of Feedback Linearizing Control on Subsynchronous Resonance –Proceedings of the 4th IEEE Conference on Control Applications, pp. 395 -402, 28-29 September 1995.

[25] J. Bladh, P. Sundqvist, U. Lundin - Torsional Stability of Hydropower UnitsUnder Influence of Subsynchronous Oscillations - IEEE Transactions on PowerSystems, Volume 28, Issue 4, 2013.

Reprints of figures 7, 8 and 9 could not get the permission of the publishersince MIR-publisher unfortunately does not exist anymore.

69

Appendix

A.1Under the hypothesis of small angular rotor departures around the rest posi-tion, in presence of damping effects, the swing equation becomes:

∋ (dt

dT2cosP2cosPPdt

dp

I dam0r,eM0f,eMm2

2

20 χΧ

,χΧ√χ√∗χ√,Χ<χΧϖ

,(0.1 )

dtdTpTT

dtd

pI

dam0

synm2

2 χΧϖ

,χΧ√,Χ<χΧ

( 0.2 )

or

dtdDTT

dtd

pI

synm2

2 χΧ,χΧ√,Χ<χΧ, ( 0.3 )

where

H2II

21

pI

21

SS

'RU3

p'R

U3pTD 20

n

n

dam

2

2

0dam

2

0dam

φ<

⌡

∑ ϖ√<

⌡

∑ϖ

<ϖ

< .( 0.4 )

The differential equation (0.3) can be written as:

⌡

∑ Χ<χΧ√

⌡

∑∗χΧ

⌡

∑∗χΧ

msyn2

2

TIpT

Ip

dtdD

Ip

dtd

. ( 0.5 )

The solution of (0.5) consists of two terms:

steadytrans χΧ∗χΧ<χΧ , ( 0.6 )

where

syn

m

tsteady TTlim Χ

<χΧ<χΧ⁄↑

( 0.7 )

andtK

2tK

1trans21 eCeC √∗√<χΧ , ( 0.8 )

70

with

∋ ( .KTIpD

I2pD

I2p

2

TIp4D

IpD

Ip

K

22

2

syn

2

syn

2

2,1

,αα,<

⌡

∑,

⌡

∑,<

<,

⌡

∑,

<

λλ

λ

( 0.9 )

With α≥K there is the over-damped case and the transient response is a-periodical:

⌡

∑ ∗√<χΧ ,α,,αα, tK

2tK

1t

trans

2222

eCeCe . ( 0.10 )

Since

∋ ( steady0 χΧ<χΧ and ∋ ( 00dt

d<

χΧ, ( 0.11 )

the final a-periodical response is:

⌡

∑,ψ

,ψ

ψ∗,ψ,χΧ<χΧ α, t1Ksinh

1t1Kcoshe1 2

2

2tsteady

( 0.12 )

where

∋ ( 00r0fsyn

H22cos2cos

p21

TIp

DI2

p

K ϖχ√β√∗χ√β

φ<<

α<ψ . ( 0.13 )

With α< K there is the under-damped case and the transient response will bepseudo-periodical:

∋ (tjK2

tjK1

ttKj2

tKj1

ttrans

pp2222

eCeCeeCeCe ∗,α,α,∗α,,α, ∗√<⌡

∑ ∗√<χΧ .( 0.14 )

Given the same initial conditions (0.11) the final pseudo-periodical responsebecomes:

∋ (t1Kcose1 2tsteady ψ,,χΧ<χΧ α, . ( 0.15 )

71

A.2With reference to figure 20 the following kinematic equations can be writ-ten:

pt

ppt

p000 χΧ∗χ

∗ϖ

<χ

∗ϖ

<π , ( 0.16 )

dtd

p1

pdtd

p1

pdtd 00 χΧ

∗ϖ

<χ

∗ϖ

<π

, ( 0.17 )

2

2

2

2

2

2

dtd

p1

dtd

p1

dtd χΧ<χ<π

. ( 0.18 )

Under the hypothesis of small angular rotor departures around the rest posi-tion, in absence of damping effects, the swing equation becomes:

∋ ( χΧ√χ√∗χ√,Χ<χΧϖ0r,eM0f,eMm2

2

20 2cosP2cosPP

dtd

pI . ( 0.19 )

The solution of (0.19) for β=ζ=0 has the form:

∋ ( ∋ (Ζ ∴Ktcos1t max ,χΧ<χΧ , ( 0.20 )

where

0r0f0

n

r,eM0

n

f,eM

n

m

max 2cos2cos2cosS

P2cos

SP

SP

χ√β∗χ√βδ

<χ√∗χ√

Χ

<χΧ ( 0.21 )

with

dnf X~S

EU3 √<β and

⌡

∑,<β 1

X~X~

X~SU

23

q

d

dn

2

r ( 0.22 )

and

.H2

2cos2cos

Ip

2cosP2cosPK

00r0f

20

0r,eM0f,eM

ϖ√χ√β√∗χ√β

<

<ϖ

χ√∗χ√<

( 0.23 )

The maximal variation of the rotor speed can be found by:

.2cos2cos

1H2p

Kp1

dtd

dtd

21

0r0f

0

maxminmax

r

χ√β√∗χ√βϖδ

<

<χΧ√<

⌡

∑ π,

π<ϖΧ

( 0.24 )

72

The maximal energy variation is then:

∋ ( .2cos2cos

H2S2p

Kp

I2

dtd

dtdI

21W

00r0fn

max0

2

min

2

maxmax

ϖ√χ√β√∗χ√βδ<

χΧ√√

ϖ<

<

⌡

∑ π,

⌡

∑ π<Χ

( 0.25 )

The maximal theoretically recoverable energy results instead:

\

.H2S2cos2cos

H2S

pK

p1

pI

21

pI

21

dtdI

21W

0n

0r0f

0

0

n

20

2

max0

20

2

maxuseful

ϖΧϖ

<χ√β√∗χ√β

ϖ√ϖ

δ?

<

⌡

∑ ϖ,

⌡

∑χΧ√∗

ϖ<

⌡

∑ ϖ,

⌡

∑ π<Χ

( 0.26 )

73

A.3The maximal flux generated by the excitation field under a pole and linkingan armature coil with N turn is:

Fec

fff0pd

max1pdpole l

kiNk2knNkknNk σχο

λ<Ε<ξ , ( 0.27 )

where n is the number of coils per pole per phase.The flux linked with an entire phase is:

Feic

ffpdsf0

c

ffpdf0polephase lD

kiN

ckkqn

k'lp2k

iNc

knNkk2

cp2

χλ<σ

χλ

ο<ξ<ξ ( 0.28 )

with ns=2n.The mutual inductance between the field winding and the armature is then:

Fei0

c

fpds

ff

phasef

lDk

Nc

kkqn

ki

M

λχ

⌡

∑

<ξ

< . ( 0.29 )

Since the single phase magnetizing inductance is:

Fei0

c

2dps

dm

lDk

ckkqn

2L

λχ√

⌡

∑

ο< , ( 0.30 )

the stator-to-rotor transformation ratio becomes:

.kkqn

cNk2

ckkqn

2

lDk

lDk

Nc

kkqn

kLMN

pds

ff2

dps

Fei0

c

Fei0

c

fpds

fdm

f0

ο<

⌡

∑

λχ√

ο

λχ

⌡

∑

<< ( 0.31 )

In a machine without damping bars using the Park’s equations:

∗ξ<

,ξϖ∗ξ

<

,ξϖ,ξ

<

.iRdt

dv

Ridt

dv

Ridt

dv

fff

f

qdrq

q

dqrd

d

( 0.32 )

With the excitation field open-circuited and the rotor turning at slip s insteady-state:

ϖ,<,ϖ,<

,ϖ<

dfof

qddq

dqqd

I'MjsVIRILV

IRILV⇓

ϖ,<,ϖ,<

,ϖ?

.I'MjsVIRILV2j

IRILV2

dfof

qdd

dqq

( 0.33 )

74

By solving for Id, Iq and the open circuit voltage:

ϖ∗ϖ∗

ϖ<

ϖ∗ϖ∗

,<

ϖ∗ϖ∗

,<

.V2LLR

LjR'MjsV

LLRLjRV2jI

LLRLjR

V2I

dq22

qf

0f

dq22

dq

dq22

qd

( 0.34 )

Since ϖLq>> R the expression for the rotor open circuit voltage becomes:

V2L

'MsVd

f0f ,? . ( 0.35 )

By remembering that the direct axis inductance is equal to

ρ∗< LL23L d

md and ff M23'M < , ( 0.36 )

the transformation ratio becomes:

dm

0

dm

dm

f0f

LL

321

Ns

LL

321

1LMs

V2VN

ρρ ∗,<

∗,?< . ( 0.37 )

At rotor at rest, when s=1:

dm

0

LL

321

NNρ∗

? . ( 0.38 )

Heuristic calculation of the stator-to-rotor transformation ratio at restFrom the stator-to-rotor transformation ratio definition :

Rd

fR

d

fR

maxo,f U2

LM

23U2

L'MU2NE

ϖϖ

<ϖϖ

?< . ( 0.39 )

By multiplying both sides by the excitation current at rated conditions thefollowing formula is obtained:

Rd

R,ffR,f

maxo,f U2

LiM

23iE

ϖ

ϖ? . ( 0.40 )

By having a look at the synchronous machine short circuit characteristic infigure 75

75

Figure 66.Synchronous machine short-circuit characteristic

it is recognized that:

RRsc.f

o,f

d

Ro,sc ISCRI

ii

LUI √<<ϖ

< . ( 0.41 )

where SCR stands for the Short Circuit Ratio.(0.40) becomes then:

RR,ffR,fmax

o,f ISCRiM223iE √ϖ< . ( 0.42 )

From the same figure it is possible to obtain:

o,sc

R,sco,fR,f I

Iii < . ( 0.43 )

By substituting (0.43) in (0.42):

.SSCRII

IUSCRII

223

ISCRII

iM223iE

Ro,sc

R,scRR

o,sc

R,sc

Ro,sc

R,sco,ffR,f

maxo,f

√<√<

<√ϖ<

( 0.44 )

From [3] the following typical values for the SCR are obtained :

dx06.1SCR ? for salient-pole machines ( 0.45 )

and

dx15.1SCR < for non-salient-pole machines.( 0.46 )

76

Still from [3] it is obtained:

5.25.1II

o,sc

R,sc •< for salient-pole machines ( 0.47 )

and

32II

o,sc

R,sc •< for non-salient-pole machines.( 0.48 )

Taking in account that for [22]:

3.19.0xd •< for salient-pole machines ( 0.49 )

and

3.28.1xd •< for non-salient-pole machines,( 0.50 )

(0.44) gives at the end

R,f

Rmaxo,f

R,f

R

iS9.2E

iS2.1 √′′√ for salient-pole machines ( 0.51 )

and

R,f

Rmaxo,f

R,f

R

iS9.1E

iS0.1 √′′√ for non-salient-pole machines.( 0.52 )

77

A.4Taking reference on the d-axis for accounting the mechanic angle φ, the fun-damental MMF generated by a rotor with 2p-poles and an excitation currentof frequency sf0 and initial phase ι0 is given by:

∋ ( ∋ ( ∋ (00rr tssinpcosˆt, ι∗ϖ√φΠ<φΠ . ( 0.53 )It represents a standing MMF with respect to the rotor which shows anti-nodal and nodal points respectively on the d-axis and on the q-axis. Thisstanding MMF rotates solidly with the rotor at the speed (1-s)ϖ0/p with re-spect to the stator in the clock-wise direction.By means of a trigonometric transformation the standing MMF can be re-garded as the sum of two MMFs, which rotate at the same speed but in op-posite directions with respect to the rotor:

∋ ( ∋ ( ∋ (φ∗ι∗ϖΠ

∗φ,ι∗ϖΠ

<φΠ ptssin2

ˆptssin

2

ˆt, 00

r00

rr . ( 0.54 )

The same MMFs can be referred to the armature by considering that theposition of the d-axis seen from the stator is:

∋ ( tp

s1 0ϖ,∗φ<χ . ( 0.55 )

By substituting χ in the previous formula the following expression for theMMF is obtained:

∋ ( ∋ ( ∋ (Ζ ∴χ∗ι∗ϖ,Π

∗χ,ι∗ϖΠ

<χΠ pt1s2sin2

ˆptsin

2

ˆt, 00

r00

rr . ( 0.56 )

It is now evident that the MMF generated by the pulsating rotor current con-sists of two rotating terms, a progressive one turning clockwise with syn-chronous speed and a regressive one turning counter-clockwise with variablespeed according to the slip. In particular, when the rotor is resting s=1 itsspeed is exactly the synchronous one but in opposite direction. For s=0.5 theregressive MMF is at rest with respect to the stator and for s=0 it rotates atsynchronism.By considering now the position of the stator phases A at χ=0, B at χ=2/3ο/pand C at χ=4/3ο/p, it is possible to find the following expressions of the rotorflux linkage with the armature phases, just by integrating (0.56) over a polepitch between -ο/2p and +ο/2p:

∋ ( ∋ ( ∋ (Ζ ∴00r

00r

A,r t1s2sin2

tsin2

t,0 ι∗ϖ,Ξ

∗ι∗ϖΞ

∗<Ξ((

,

∋ (

ο∗ι∗ϖ,

Ξ∗

⌡

∑ ο,ι∗ϖ

Ξ∗<

⌡

∑ οΞ

32t1s2sin

232tsin

2t,

32

00r

00r

B,r

((, ( 0.57 )

∋ (

ο∗ι∗ϖ,

Ξ∗

⌡

∑ ο,ι∗ϖ

Ξ∗<

⌡

∑ οΞ

34t1s2sin

234tsin

2t,

34

00r

00r

C,r

((.

78

From (0.57), via derivation, the expressions for the induced electromotiveforces in the single phases are obtained:

∋ ( ∋ ( ∋ ( ∋ (Ζ ∴00r

000r

0A,r t1s2cos2

1s2tcos2

t,0e ι∗ϖ,Ξ

ϖ,,ι∗ϖΞ

ϖ,<((

,

∋ ( ∋ (

ο∗ι∗ϖ,

Ξϖ,,

⌡

∑ ο,ι∗ϖ

Ξϖ,<

⌡

∑ ο

32t1s2cos

21s2

32tcos

2t,

32e 00

r000

r0B,r

((

, ( 0.58 )

∋ ( ∋ (

ο∗ι∗ϖ,

Ξϖ,,

⌡

∑ ο,ι∗ϖ

Ξϖ,<

⌡

∑ ο

34t1s2cos

21s2

34tcos

2t,

34e 00

r000

r0C,r

((

.

By considering a set of symmetric currents entering the stator phases∋ ( ∋ (00A tsinIti ∗ϖ< ,

∋ ( ⌡

∑ ο,∗ϖ<

32tsinIti 00B

,( 0.59 )

∋ ( ⌡

∑ ο,∗ϖ<

34tsinIti 00C

.and interacting with the EMFs, it is easy to obtain an expression for the in-stantaneous electromagnetic torque by:

∋ ( ∋ (

∋ ( ∋ ( ∋ (.ts2sinI22

3p1s2sinI22

3p

ieieieptT

000r

00r

CC,rBB,rAA,r0

avgEM

∗ι∗ϖΞ

,,,ιΞ

<

<√∗√∗√ϖ

<

((

((

( 0.60 )

It can be recognized that the average EM torque for s=0 is given by:

∋ ( ∋ (

.I22

3p

sinI22

3pdieieieT1pT

qr

00r

Tt

tCC,rBB,rAA,r

0

avgEM

Ξ<

<,ιΞ

<σ√∗√∗√ϖ

< ⟩∗

(

((

( 0.61 )

79

A.5

Study-case machine parameters from [22] are:

Table 18.Study-case generator given data

S V P cosι n H Xd Xd' Xd'' Xq=Xq' Xq'' X0 Td0' Td' Td'' Tq''

[MVA] [kV] [MW] [-] [rpm] [s] [p.u.] [p.u.] [p.u.] [p.u.] [p.u.] [p.u.] [s] [s] [s] [s]

175 14.4 150 0.857 166.667 3.5 0.92 0.29 0.19 0.70 0.22 0.142 7.3 2.3 0.03 0.035

Rated current

kA016.74.143

175V3

SI

n

nR <

√<< . ( 0.62 )

Nominal impedance

ς?√

<< 185.1016.734.14

I3V

Zn

nR . ( 0.63 )

Inductances and resistances (all referred to the armature).

.LL23

L2

LL2

LL23L

2L

L23

2L

23L

21LLL

dm

qm

dm

qm

dmmavg

m

mavgm

avgmd

ρ

ρρ

ρ

∗<

<∗

⌡

∑ ,∗

∗<∗

⌡

∑ Χ∗<

<Χ

∗∗∗<

( 0.64 )

.LL23

L2

LL2

LL23L

2LL

23

2L

23L

21LLL

qm

qm

dm

qm

dmmavg

m

mavgm

avgmq

ρ

ρρ

ρ

∗<

<∗

⌡

∑ ,,

∗<∗

⌡

∑ Χ,<

<Χ

,∗∗<

( 0.65 )

Thus

∋ (

∗<

⌡

∑,

∗

,<,

ρ .LLL2

LL34

LLLL32

qm

dm

qd

qm

dmqd

( 0.66 )

80

At the synchronism in p.u.:

∋ (

⌡

∑,

∗<∗

,<,

ρ .x2

xx34xx

xx32xx

qdqm

dm

qdqm

dm

( 0.67 )

Since xρ=x0:

∋ (

<

⌡

∑,

∗<∗

<,<,

8907.0x2

xx34xx

1467.0xx32xx

0qdq

mdm

qdqm

dm

⇓

<<

.3720.0x5187.0x

qm

dm ( 0.68 )

From [2] and with reference to the circuits of figures 16 and 17:1

qmq

aq

L1

L''L1L

,

ρρ

⌡

∑,

,< ⇓ ,0987.0

x1

x''x1x

1

qmq

aq <

⌡

∑,

,<

,

ρρ

( 0.69 )

1

dmd

f

L1

L'L1L

,

ρρ

⌡

∑,

,< ⇓ ,0783.0

x1

x'x1x

1

dmd

f <

⌡

∑,

,<

,

ρρ

( 0.70 )

1

fdmd

ad

L1

L1

L''L1L

,

ρρρ

⌡

∑,,

,< ⇓ ,1632.0

x1

x1

x''x1x

1

fdmd

ad <

⌡

∑,,

,<

,

ρρρ

( 0.71 )

'TLL

'TLR

0d

fdm

0d

ff

ρ∗<< ⇓ 4

0d

fdm

0d

ff 106.2

'Txx1

'Tx1r ,ρ √<

∗ϖ

<ϖ

< , ( 0.72 )

''T

LLLLLLL

LLL

Rd

addm

ffdm

fdm

ad

ρρρρρ

ρρ ∗∗∗

<⇓ .022.0

''T

xxxxxxx

xxx1r

d

addm

ffdm

fdm

ad <∗

∗∗ϖ

<ρ

ρρρρ

ρρ

( 0.73 )

''T

LLL

LL

Rq

aqqm

qm

aq

ρρ

ρ ∗∗

<⇓ .0183.0

''T

xxx

xx1r

q

aqqm

qm

aq <∗

∗ϖ

<ρ

ρ

ρ

( 0.74 )

Rotor inductances and resistances referred to the excitation circuitThe nominal RMS-value of the electromotive force corresponds to 13.856kV at an excitation current of 1.5 kA. The magnetization inductance referredto the excitation circuit is then:

.mH6.41IE2

I'M

fff <

ϖ<

Ξ< ( 0.75 )

81

The transformation ratio between excitation circuit and one phase armatureallows referring all quantities to the armature side:

262.21Zx

'ML

'MN

NN

NN

ndm

fdm

f2m

mf

m

f <ϖ

<<

ℑ

ℑ< , ( 0.76 )

H8845.0NN'M'L

m

ff

fm <√< , ( 0.77 )

mH8.29NNZ

xNN

L'Mm

fnqm

eq

fqmaq <

ϖ<< , ( 0.78 )

H6343.0NN

'M'Lm

faq

qm <√< , ( 0.79 )

H1335.0NNZx

NNL'L

2

m

fnf2

m

fff <

⌡

∑ϖ

<

⌡

∑< ρρρ , ( 0.80 )

H018.1'L'L'L ffmf <∗< ρ , ( 0.81 )

H2783.0NNZx

NNL'L

2

m

fnad2

m

fadad <

⌡

∑ϖ

<

⌡

∑< ρρρ , ( 0.82 )

H163.1'L'L'L adfmad <∗< ρ , ( 0.83 )

H1683.0NNZx

NNL'L

2

m

fnaq2

m

faqaq <

⌡

∑ϖ

<

⌡

∑< ρρρ , ( 0.84 )

H8026.0'L'L'L aqqmaq <∗< ρ , ( 0.85 )

ς<

⌡

∑< 1393.0

NN

Zr'R2

m

fnff

, ( 0.86 )

ς<

⌡

∑< 8927.11

NN

Zr'R2

m

fnfad

, ( 0.87 )

ς<

⌡

∑< 8034.9

NN

Zr'R2

m

fnaqaq

. ( 0.88 )

82

A.6Motor sizing

The available starting data about the motor design are the type of electricalmotor, a synchronous one, the number of pole-pairs p=2, the required powerP=35 kW and the rated line voltage U=400 V.The first step consists in finding a standard steel armature type for a commonasynchronous machine in the same rated power. In the catalogue of KurtMeier Motor-press GmbH it is possible to find out the stator item SSA225 40240 080 related to a 37 kW 4-poles asynchronous motor, with the followingdimensions and features:

Figure 67.Reference dimensions for a 37 kW asynchronous machine

48 slots, outer diameter Da=340 mm, inner diameter Di=215 mm and statorlength lFe= 240 mm.Since the induced electromotive force in the stator phase is

fDlBc

qnk2E iFemax1

sw ο< , ( 0.89 )

by multiplying both sides of the equation by three times the rated stator cur-rent an expression for the rated power can be obtained:

RiFemax1

swRR fIDlB

cqnk23EI3S ο<< . ( 0.90 )

It is possible then to define the machine linear current density A as:

Rs

ii

Rs

z

slot Ic

qnDp6

q3p2D

cInIA

ο<

√ο

<σ

< ( 0.91 )

83

and to substitute it in the expression of the rated power:

.fDAlBk2p

S 2iFe

max1w

2

Rο

< ( 0.92 )

By assuming a classical 5/6 short-pitching ratio – which reduces convenient-ly the 5th and the 7th EMF harmonics - the winding factor kw results in:

925.0966.0958.026

5sin

2q3sinq

2q3qsin

kkk pdw ?√<⌡

∑ ο

⌡

∑ ο

⌡

∑ ο

<< ( 0.93 )

being q=4 slots per pole per phase in a 48 slots three phase machine.If the losses are momentarily disregarded, at the rated power SR=40 kVA andwith a mechanical power PR= 35 kW, the power factor must be:

875.04035

SPcos

R

R <<<ι . ( 0.94 )

By assuming a linear current density A=25 kA/m and a maximal value of theair gap fundamental induction B1

maxequal to 1 T, a first estimate of the ma-chine length is obtained:

m214.0fADBk

2p

Sl2i

max1w

2R

Fe ?ο

< . ( 0.95 )

Armature windings and slots

By coming back to the expression for the induced EMF it is possible to findout the number of conductor per parallel path per slot:

08.6flDqBk2

3U

cn

Feimax1w

s ?ο

< . ( 0.96 )

It is possible to adjust the machine length for obtaining ns as an integer num-ber. For lFe=0.217 m ns/c becomes exactly 6.By choosing c=1 there are not parallel paths for the phase current to go. Itresults then ns=6 conductors per slots, which means two strands with 3 con-

84

ductors each, since the short-pitching requires the armature winding to be adouble layer one.Coming to the stator structure, since the pole pitch is

mm9.168p2

D i <ο

<σ , ( 0.97 )

by considering that the pole-shoe coverage factor for a sinusoidal air gapinduction profile is close to 2/ο, the needed yoke height hy must be:

mm31BB

B1B2

21

B1

2h sat

Fe

max1

satFe

max1sat

Fe

poley ?

οσ

<σο

<Ε

< , ( 0.98 )

where BFe=1.75 T. The yoke height is assumed equal to hy= 31.5 mm so thatthe slot depth (or tooth height) results:

mm31h2

DDh yia

z <,,

< . ( 0.99 )

If the same level of saturation BFe=1.75 T is chosen for the most saturatedtooth, the tooth width is then:

mm8BB

q3BBw sat

Fe

max1

satFe

max1

sz <σ

<σ< . ( 0.100 )

In this design a constant slot opening is assumed:

mm6wq3

ww zzss <,σ

<,σ< . ( 0.101 )

Considering now the conductors as bare copper bars, since the current densi-ty must not exceed Jadm=4 A/mm2 the copper cross section must be biggerthan:

2

adm

RCu mm5.14

J1

cIS <= . ( 0.102 )

By choosing a squared cross section with side s=4 mm and rounded cornerswith r=0.8 mm, the active copper cross section becomes

Ζ ∴ 2

adm

22Cu mm45.15

J1r4sS <<ο,,< , ( 0.103 )

which results in a rated armature current density:

2Cu

RR mm

A74.3S1

cIJ << . ( 0.104 )

Both turn to turn insulation and strand to strand insulation are chosen 0.5mm thick. The bottom of the slot shows also a 0.5 mm thick dielectric cush-

85

ion. The opening of the slot presents on both sides triangular shaped troughswhere a 2.5 mm thick insulating wedge can be slit.Figure 77 a) and b) show the stator windings cross-section with the coordi-nation of the insulation.

a) b)Figure 68. a) stator windings arrangement b) principal slot and strand dimensions

Pole-shoe design: real and equivalent air gap

With the aim to obtain an EMF as sinusoidal as possible the pole shoes ofsalient pole synchronous machines are usually shaped so to obtain a cosecantprofile of the mechanical air gap. In figure 78 a typical air gap pole-shoestructure is shown with its resulting air gap induction.

Figure 6918.Air gap saliencies and air gap induction profile

18Figure from [3]

86

There are four kinds of air gap involved in the FEM generation:∂ the mechanical air gap χ measured against the stator tooth along the d-

axis (it is the smallest air gap);∂ the maximal air gap χm measured in radial direction against the stator

tooth at the pole-shoe sides (tip air gap);∂ the equivalent air gap χ’ which would admit the same average magnet-

izing flux in a smooth stator machine given the same pole enclosure and the same MMF;

∂ the average air gap χav which would admit the same average magnetiz-ing flux in a smooth air gap isotropic machine given the same MMF.

A pole-shoe-stator system is geometrically characterized by the followingparameters, which are responsible for the average magnetizing flux but alsofor the harmonic content of the air gap induction profile:

pole enclosureσ

<,

,< pb

pitchpolewidthshoepole

, ( 0.105 )

pole tipχ

χ<φ m , ( 0.106 )

relative air gapσχ

<δ , ( 0.107 )

the Carter’s coefficientχχ

<'kc . ( 0.108 )

The first step in the air gap design is to estimate the minimal air gap χ byassuming that the peak value of the fundamental armature reaction equals atleast the maximal MMF due to the excitation field [4] (design on load):

.mm7.1pB2AD

p2D

BA1

2A4B2 max

1

i0

imax1

00max1 ?λ<

ο√

ολ<χ↑

σ√ο

λ<χ√ ( 0.109 )

For the present design a minimal mechanical air gap χ0= 1.8 mm is assumed.The second step requires the calculation of the Carter’s coefficient kc via theair gap factor cs, which takes in account the saliencies due to the stator teeth:

33.1w5

w

2w

1ln2w

arctanw2c

s

2s

2sss

s <

χ∗

⌡

∑

χ?

⌡

∑⌡

∑

χ∗,

⌡

∑

χχο< , ( 0.110 )

20.1c

kss

sc <

χ,σσ

< . ( 0.111 )

The equivalent air gap along the d-axis is then:

87

mm17.2k' c <χ<χ . ( 0.112 )

It is possible to define [9] the field form factor for the v-th air gap inductionharmonic as:

χ

<B

Bkmaxv

fv . ( 0.113 )

There are special curves [3] which relates the form factors for the fundamen-tal, the third and the fifth harmonics to the pole-shoe geometry through theparameters , φ and δ. In particular the field form factors are obtained as sumof two terms:

fvfvfv ''k'kk ∗< . ( 0.114 )

This curves are depicted in figure 79 a) for the fundamental component andin figure79 b) for the third and the fifth harmonics.

a) b)Figure 7019. a) fundamental field form factor; b) 3rd& 5th harmonics field form fac-tors

In the present design can be recognized that δ=χ/σ? 0.01. If a pole enclosure in the range 0.69•0.72 is chosen together with a pole tip between 1.3 and1.5 the field form factor for the fundamental approximates 1 and that for thethird harmonic roughly zero. The fifth harmonics is less important because itwill account for 1/5th of its relative value on the pole pitch with reference tothe total flux.

19Figure from [3]

88

By choosing the tip φ=1.3 and the pole enclosure =0.7 it is possible to findout a first estimation of the induction form factor:

07.102.005.1''k'kk fff ?∗<∗< ( 0.115 )which is a very good indicator for a sinusoidal EMF.The maximal air gap of the pole shoe is chosen then equal to:

mm34.2m ?φχ<χ . ( 0.116 )For the following calculations it is very important to define the excitationflux form factor kΕ and the pole span factor χ [3]:

max1

mean

max1

mean

m,1f

fm

BB

2Bl2Blk χ

χ

χχΕ

ο<

σο

σ<

ΕΕ

< , ( 0.117 )

χ

χχ <

BBmean

. ( 0.118 )

It results at the end that:

f

max1

max1

meanmean

kk2B

BBB

BB

Εχ

χ

χ

χχ ο

<<< . ( 0.119 )

When the induction in the air gap is perfectly sinusoidal kΕ=kf=1and χ=2/ο.In figure 80 the pole span factor χ can be obtained by taking in accountboth the effects of saturation and that of armature saliencies according to thepole-shoe tips. The ratio between χm and χ’is equal to:

08.1'

m <χ

χ( 0.120 )

Figure 7120.Most important air gap parameters in function of the modified air gap tip

20 Figure from [3]

89

It is possible to estimate now kf=1.17 and kΕ=1.04 which lead to a pole spanfactor:

77.0kk2f <

ο< Εχ . ( 0.121 )

It must be observed that the saturation and the effect of rotor saliencies onthe tips make the air gap more uniform by worsening the induction formfactor. It is now sensibly bigger than one. The effects of the pole shoe tipshave to be observed in the pole span factor which is bigger than the poleenclosure. This will increase the magnetizing flux at cost of a less sinusoidalFEM. This is anyway not a problem since the armature winding is conven-iently short pitched. Once the pole span factor is given the average air gap χav is known:

mm8.2k' c

av <χ

<χ

<χχχ

. ( 0.122 )

This last air gap value is of capital importance for the calculation of the ma-chine inductances related to the magnetizing flux path.

Machine inductances: magnetizing inductances

The magnetizing inductance along the d-axis for a phase is equal to:

mH59.6lDckqn

2L Feiav

2ws

0dm <

χ

⌡

∑

λο

< . ( 0.123 )

Since the inductances are proportional to the linked flux given the sameMMF according to

xFe0wxm Kl2nk

⌡

∑ σ

χΠ

λο

<ξ , ( 0.124 )

it follows that:

d

qdm

qm

dm

qm

KK

LL

<ΞΞ

< ( 0.125 )

where Kq= q-axis field form factor and Kd= d-axis field form factor.In figure 81 the field form factors are represented in function of the modifiedrelative air gap and tip ratio.

90

Figure 72.Direct- and quadrature-axis field form factors

It can be observed that for χm/χ’=1.08 and χ’/σ≡0.02 the following valuescan be obtained: Kq=0.58 and Kd=1.02.The magnetization inductance along the q-axis for a phase is then equal to:

mH75.3LKK

L dm

d

qqm << . ( 0.126 )

Machine inductances: stray inductances

Coming to the leakage inductances they are classically distinguished in fourmain classes [3] [4] and their calculation is partially based on electrical ma-chines theory and partially on experienced factors:

∂ air gap leakage inductance or differential leakage inductance Lρχ;∂ slot leakage inductance Lρs;∂ tooth tip leakage inductance Lρt;∂ end winding leakage inductance Lρe.

In the present design all the above enlisted inductances are calculated ac-cording to the formulas proposed by Pyrhönen [4] and Smolensky [3]. Theobtained values are indicated respectively by the suffixes P and S.

Differential inductance (harmonics related leakage inductance)According to Pyrhönen

dm

pP Ly

,q,mL √

⌡

∑σ

ρ< χρχ ( 0.127 )

where the leakage factor ρχ depends on the number of phases m, on thenumber of slots per pole per phase q and on the relative short pitch. For m=3,q=4 and the adopted 5/6 short-pitching the ρχ is ?0.007.Thus:

91

H42L007.0L dm

P λ?√<ρχ . ( 0.128 )

According to Smolensky

∋ (

⌡

∑χ

σ•√√

⌡

∑λ<κλ< ρχρχ

av

sFe2

s0

Fe2eq0

S

120.17.0

pql

cpqn2

pqln2L .( 0.129 )

By assuming the average value 0.85 for the experienced coefficient in theformula the following value is obtained:

H52LS λ?ρχ . ( 0.130 )

Slot leakage inductanceAccording to Pyrhönen and to figure 82, for a double layer winding in rec-tangular slots:

PsFe

2s

0P

s lc

nmQL ρρ κ

⌡

∑λ< ( 0.131 )

where

44

12

4

41

Ps b4

'hbhk

b3'hhk ∗∗

,<κρ , ( 0.132 )

Figure 7321.Stator slot arrangement according to Pyrhönen

⌡

∑σ

,,< p1

y1

1691k and

⌡

∑σ

,,< p2

y1

431k . ( 0.133 )

For a 5/6 short pitching k1=0.906 and k2=0.875. With the dimensions fromfigure 77 b) the slot specific leakage permeance results in:

817.1Ps <κρ . ( 0.134 )

21Figure from [4]

92

The slot leakage inductance is then:

H285LPs λ<ρ . ( 0.135 )

According to Smolensky, for a double layer winding in rectangular slots:

H2794

1y

3

b3h

bh

pql

cpqn

2pql

n2L

p

4

4

4

1Fe2

s0

Ss

Fe2eq0

Ss λ<

∗σ

⌡

∑∗√√

⌡

∑λ<κλ< ρρ. ( 0.136 )

Tooth tip leakage inductanceAccording to Pyrhönen:

PtFe

2s

0P

t lc

qpnQm4L ρρ κ

⌡

∑λ< ( 0.137 )

where

5b

4

b5

k

4

av

4

av

2P

t

∗χ

χ

<κ ρ . ( 0.138 )

Thus:

H49LPt λ?ρ . ( 0.139 )

According to Smolensky:

.H324

1y

326.0

b35.0

b1.1

pql

cpqn

2

pql

n2L

p2

4

av

4

avFe2

s0

St

Fe2eq0

St

λ?∗

σ

⌡

∑,

⌡

∑ χ,

χ√√

⌡

∑λ<

<κλ< ρρ

( 0.140 )

End winding leakage inductanceAccording to Pyrhönen:

Pee

2s

0P

e lc

qpnqQm4L ρρ κ

⌡

∑λ< ( 0.141 )

where

wplewewP

ee yl2l κ∗κ<κρ . ( 0.142 )

withlew= axial length of the end-winding [m]yp = shortened winding pitch [m]κlew = specific linear permeance for the axial end-winding [-]κw = specific linear permeance for the transversal end-winding [-]

93

With yp =5/6, σ=0.141 m , lew=0.3*yp=0.042 m, κlew =0.55 and κw=0.35

H60lc

qpnqQm4L P

ee

2s

0P

e λ?κ⌡

∑λ< ρρ . ( 0.143 )

According to Smolensky:

.H70l

qy1.0

yl34.0

pql

cpqn2

pqln2L

Fe

p

p

ewFe2

s0

Se

Fe2eq0

Se

λ?

⌡

∑∗√

⌡

∑λ<

<κλ< ρρ

( 0.144 )

Total leakage inductanceAccording to Pyrhönen:

H436604928542LLLLL Pe

Pt

Ps

PP λ?∗∗∗<∗∗∗< ρρρρχρ . ( 0.145 )

According to Smolensky:

H433703227952LLLLL Se

St

Ss

SS λ?∗∗∗<∗∗∗< ρρρρχρ .( 0.146 )

Machine inductances: synchronous d-axis and q-axis reactances

By assuming the highest stray inductance from Blondel’s theory it is finallyobtained:

mH32.10LL23L d

md <∗< ρand mH06.6LL

23L q

mq <∗< ρ. ( 0.147 )

By expressing these inductances in p.u.:

81.0Z

LxR

d0d <

ϖ< , ( 0.148 )

48.0Z

Lx

R

q0q <

ϖ< , ( 0.149 )

034.0Z

LxxR

00 <

ϖ<< ρ

ρ . ( 0.150 )

Machine inductances: KALK 2D simulation

The same machine has been simulated in KALK in over-excitation withcosι=0.7 ( leading ) with field current If=212 A and Nf=18 turns/pole. Thevalues of the obtained Blondel’s inductances in p.u. are:

94

876.0xd < and 499.0xq < . ( 0.151 )

These values are very close to the analytic ones presented above and it mustbe expected that the magnetization inductances calculated via the finite ele-ments are more reliable than the analytic ones. It is however to observe thatthe calculation of the stray inductances in KALK must be considered verypoor in comparison to the two methods used in the present work for calculat-ing them. This hybrid way is followed in the present design by assuming asvalid the q- and d-axis inductances obtained by KALK and considering atthe same time the leakage inductances calculated analytically.Since

ρ∗< LL23L d

md and ρ∗< LL23L q

mq , ( 0.152 )

it follows from Blondel’s theory that:

∋ (

⌡

∑,

∗<∗

,<,

ρ .L2

LL34LL

LL32LL

qdqm

dm

qdqm

dm

( 0.153 )

The same expressed at the synchronous speed in p.u. is:

∋ (

⌡

∑,

∗<∗

,<,

ρ .x2

xx34xx

xx32xx

qdqm

dm

qdqm

dm

( 0.154 )

Since xρ=x0:

∋ (

<

⌡

∑,

∗<∗

<,<,

8693.0x2

xx34xx

2493.0xx32xx

0qdq

mdm

qdqm

dm

⇓

<<

3100.0x5593.0x

qm

dm ( 0.155 )

which means in absolute values:

<ϖ

<

<ϖ

<

mH95.3ZxL

mH12.7ZxL

0

Rqm

qm

0

Rdm

dm

( 0.156 )

and

mH12.11Ld < and mH36.6Lq < . ( 0.157 )

95

Machine inductances: Blondel’s diagram and rotor inductances

Since the motor is simulated for cosι=0.7 (leading) the absorbed activepower and the generated reactive one must be:

,<ι√<<ι√<

.kVA4.20sinSQkW28cosSP

R

R( 0.158 )

Those powers can be expressed according to Blondel’s theory as:

⌡

∑ χ,

χ∗χ<

⌡

∑,∗χ<

.X

cosX

sinV3cosXEV3Q

1XX

X2V3sin

XEV3P

d

2

q

22

d

q

d

d

2

d( 0.159 )

Given P, Q, V, Xd and Xq it is possible to solve the previous system in E andχ by getting:

↓,<χ<

.44.14V3.398E

.e.o

.e.o ( 0.160 )

The nominal RMS-value of the electromotive force corresponds to 398.3 Vat an excitation current of 212 A. The magnetization inductance referred tothe excitation circuit is then:

mH46.8IE2

IM

fff <

ϖ<

Ξ< . ( 0.161 )

From the finite elements calculation performed in KALK it resulted a specif-ic energy for the exciting field w0= 260.4 J/m per pole, which means a rotorinductance including the leakage one:

mH96.9212

215.04.260222I

lw2p2IW2L 22

f

e02f

0f <

√√√√<

√√<

√< . ( 0.162 )

The excitation field leakage inductance is then:

mH50.146.896.9MLL fff <,<,<ρ . ( 0.163 )

For cosι=0.875 (lagging) the absorbed active and reactive power must bethe rated ones:

∗<ι√<<ι√<

.kVA36.19sinSQkW35cosSP

R

R( 0.164 )

96

Given P, Q, V, Xd and Xq it is possible to solve the previous system in E andχ by getting:

↓,<χ<

.93.29V6.203E

R

R ( 0.165 )

By neglecting the effects of saturation the rated excitation current can beoverestimated as:

A4.108IEEI .e.o,f

R

.e.oR,f << . ( 0.166 )

The current density in the rotor will result in:

A37.3SI

Jr,Cu

R,fr << . ( 0.167 )

In figures 83 and 84 the Blondel’s diagrams for respectively the overexcitedand the rated machine are shown:

Figure 74.Blondel’s diagrams for overexcited machine

Figure 75. Blondel’s diagrams at rated power

97

Machine resistances

The DC stator resistance at 20° C is equal to:

∋ (ς<

∗θ< m2.45qpn

Sy6.1l2

R sCu

ps2020 . ( 0.168 )

Since

2075 22.1 θ√<θ ( 0.169 )

the DC stator resistance at 75° C is equal to:∋ (

ς<∗

θ< m1.55qpnS

y6.1l2R s

Cu

ps7575 . ( 0.170 )

In order to consider the influence of the skin effect in the slot on the ACstator resistance [4] suggests

DC75R

AC75 RkR < , ( 0.171 )

where the skin-effect resistance factor is

037.19

2.0n1k 4

R

2s

R <ω,

∗< ( 0.172 )

and the relative penetration depth is equal to:

31.0ww

21h

s

c

75

00cR <

θλ

ϖ<ω . ( 0.173 )

The AC stator resistance at 75° C is equal then to:

ς< m1.57R75 . ( 0.174 )

In p.u.:

0143.0ZR

rR

75 << . ( 0.175 )

Coming to the rotor winding, it consists of 4 x 18.5 turns in series around thefour poles, which are w=70 mm wide and ls=215 mm long. The copper wirecross section is d=23 mm x t=1.4 mm. Thus

∋ (ς<

√∗∗

θ< m4.33N4td

d2wl2R fs

75f . ( 0.176 )

98

Transient parameters

Since the stator-to-rotor transformation ratio is

188.1LMN d

m

f0 << ( 0.177 )

the leakage rotor inductance referred to the armature side is:

mH59.1NL

23'L 2

0

ff <<

ρρ . ( 0.178 )

The transient direct inductance Ld’ according to [2] results in:

mH82.1'LL

23

'LL23

L'Lf

dm

fdm

d <∗

∗<ρ

ρ

ρ . ( 0.179 )

In p.u.:

143.0Z

'L'x

R

d0d <

ϖ< ( 0.180 )

According to [2] it is possible to define the following time constants:

s298.010321096.9

RL'T 3

3

f

f0d ?

√√

<< ,

,

, ( 0.181 )

s051.0N32Z

R'x

'R'L'T 2

00

n

f

d

f

dd ?

ϖ<< . ( 0.182 )