Fundamentals of Electrical Drivess - Appendix pps

Appendix AConcept of sinusoidal distributed windings Electrical machines are designed in such a manner that the flux density distribution in the airgap due to a single phase winding is a

Appendix A

Concept of sinusoidal distributed windings

Electrical machines are designed in such a manner that the flux density distribution in the airgap due to a single phase winding is approximately sinusoidal This appendix aims to make plausible the reason for this and the way in which this is realized In this context the so-called sinusoidally distributed winding concept will be discussed

Figure A.1 represents an ITF based transformer or IRTF based electrical machine with a

finite airgap g A two-phase representation is shown with two n1turn stator phase windings

The windings which carry the currents i 1α , i 1βrespectively, are shown symbolically This implies that the winding symbol shown on the airgap circumference represents the locations of the majority windings in each case, not the actual distribution, as will be discussed shortly If

we consider the α winding initially, i.e we only excite this winding with a current i 1α, then the aim is to arrange the winding distribution of this phase in such a manner that the flux density in

the airgap can be represented as B 1α = ˆB α cos ξ Similarly, if we only excite the β winding with a current i 1β, a sinusoidal variation of the flux density should appear which is of the form

B 1β= ˆB β sin ξ The relationship between phase currents and peak flux density values is of the form B 1α = Ci1α , B 1β = Ci1β where C is a constant to be defined shortly In space vector

terms the following relationships hold





Given that the current and flux density components are linked by a constant C, it is important to

ensure that the following relationship holds, namely



If for example the current is of the form i1 = ˆi1 e jρthen the flux density should be of the

form  B1 = C ˆi1 e jρ for any value of ρ and values of ˆi1which fall within the linear operating

range of the machine The space vector components are in this case of the form i 1α = ˆi1 cos ρ,

i 1β = ˆi1 sin ρ If we assume that the flux density distributions are indeed sinusoidal then the resultant flux density B res in the airgap will be the sum of the contributions of both phases namely

B res (ξ) = C ˆi1 cos ρ

  

ˆ

cos ξ + C ˆi1 sin ρ

  

ˆ

328 FUNDAMENTALS OF ELECTRICAL DRIVES

Figure A.1. Simplified ITF model, with finite airgap, no secondary winding shown

Expression (A.3) can also be written as B res = C ˆi1cos (ξ − ρ) which means that the resultant

airgap flux density is again a sinusoidal waveform with its peak amplitude (for this example)

at ξ = ρ, which is precisely the value which should appear in the event that expression (A.2)

is used directly It is instructive to consider the case where ρ = ω s t, which implies that the

currents i α , i β are sinusoidal waveforms with a frequency of ω s Under these circumstances the

location within the airgap where the resultant flux density is at its maximum is equal to ξ = ω s t.

A traveling wave exists in the airgap in this case, which has a rotational speed of ω srad/s Having established the importance of realizing a sinusoidal flux distribution in the airgap for each phase we will now examine how the distribution of the windings affects this goal For this purpose it is instructive to consider the relationship between the flux density in the

airgap at locations ξ, ξ + ∆ξ with the aid of figure A.2 If we consider a loop formed by the two ‘contour’ sections and the flux density values at locations ξ, ξ + ∆ξ, then it is instructive

to examine the sum of the magnetic potentials along the loop and the corresponding MMF

enclosed by this loop The MMF enclosed by the loop is taken to be of the form N ξ i, where N ξ represents all or part of the α phase winding and i the phase current The magnetic potentials

in the ‘red’ contour part of the loop are zero because the magnetic material is assumed to be magnetically ideal (zero magnetic potential) The remaining magnetic potential contributions when we traverse the loop in the anti-clockwise direction must be equal to the enclosed MMF which leads to

g

µ o

B (ξ) − g

µ o

Expression (A.4) can also be rewritten in a more convenient form by introducing the variable

n(ξ) = N ξ which represents the phase winding distribution per radian Use of this variable

Appendix A: Concept of sinusoidal distributed windings 329

Figure A.2. Sectional view of phase winding and enlarged airgap

with equation (A.4) gives

B (ξ + ∆ξ) − B (ξ)

µ o

which can be further developed by imposing the condition ∆ξ → 0 which allows equation (A.5)

to be written as

dB (ξ)

dξ =− g

µ o

The left hand side of equation (A.6) represents the gradient of the flux density with respect to

ξ An important observation of equation (A.6) is that a change in flux density in the airgap is

linked to the presence of a non-zero n(ξ)i term, hence we are able to construct the flux density

in the airgap if we know (or choose) the winding distribution n(ξ) and phase current Vice versa

we can determine the required winding distribution needed to arrive at for example a sinusoidal flux density distribution

A second condition must also be considered when constructing the flux density plot around the entire airgap namely

π

−π

Equation (A.7) basically states that the flux density versus angle ξ distribution along the

en-tire airgap of the machine cannot contain an non-zero average component Two examples are considered below which demonstrate the use of equations (A.6) and (A.7) The first example

as shown in figure A.3 shows the winding distribution n(ξ) which corresponds to a so-called

‘concentrated’ winding This means that the entire number of N turns of the phase winding are concentrated in a single slot (per winding half) with width ∆ξ, hence N ξ = N The

cor-responding flux density distribution is in this case trapezoidal and not sinusoidal as required The second example given by figure A.4 shows a distributed phase winding as often used in practical three-phase machines In this case the phase winding is split into three parts (and three

slots (per winding half), spaced λ rad apart) hence, N ξ = N

3 The total number of windings

of the phase is again equal to N The flux density plot which corresponds with the distributed

330 FUNDAMENTALS OF ELECTRICAL DRIVES

Figure A.3 Example: concentrated winding, N ξ = N

Figure A.4 Example: distributed winding,N ξ=N3

winding is a step forward in terms of representing a sinusoidal function The ideal case would

according to equation (A.6) require a n(ξ)i representation of the form

n (ξ) i = g

µ o

ˆ

in which ˆB represents the peak value of the desired flux density function B (ξ) = ˆ B cos (ξ).

Equation (A.8) shows that the winding distribution needs to be sinusoidal The practical imple-mentation of equation (A.8) would require a large number of slots with varying number of turns placed in each slot This is not realistic given the need to typically house three phase windings, hence in practice the three slot distribution shown in figure A.4 is normally used and provides a flux density versus angle distribution which is sufficiently sinusoidal

Appendix A: Concept of sinusoidal distributed windings 331

In conclusion it is important to consider the relationship between phase flux-linkage and

circuit flux values The phase circuit flux (for the α phase) is of the form

φ mα=

π

2

− π

2

which for a concentrated winding corresponds to a flux-linkage value ψ 1α = N φmα If a distributed winding is used then not all the circuit flux is linked with all the distributed winding

components in which case the flux-linkage is given as ψ 1α = Neff φ mα , where N eff represents the ‘effective’ number of turns

Appendix B

Generic module library

The generic modules used in this book are presented in this section In addition to the generic representation an example of a corresponding transfer function (for the module in question) is provided Transfer functions given, are in space vector and/or scalar format Some modules, such as for example the ITF module, can be used in scalar or space vector format However, some functions, such as for example the IRTF module, can only be used with space vectors

334 FUNDAMENTALS OF ELECTRICAL DRIVES

Appendix B: Generic module library 335

336 FUNDAMENTALS OF ELECTRICAL DRIVES

Appendix B: Generic module library 337

338 FUNDAMENTALS OF ELECTRICAL DRIVES

Appendix B: Generic module library 339

340 FUNDAMENTALS OF ELECTRICAL DRIVES

B¨odefeld, Th and Sequenz, H (1962) Elektrische Machinen Springer-Verlag, 6thedition

Holmes, D.G (1997) A generalised approach to modulation and control of hard switched

con-verters PhD thesis, Department of Electrical and Computer engineering, Monash University,

Australia

Hughes, A (1994) Electric Motors and Drives Newnes.

Leonhard, W (1990) Control of Electrical Drives Springer-Verlag, Berlin Heidelberg New York

Tokyo, 2 edition

Mathworks, The (2000) Matlab, simulink WWW.MATHWORKS.COM

Miller, T J E (1989) Brushless Permanent-Magnet and Reluctance Motor Drives Number 21

in Monographs in Electrical and Electronic Engineering Oxford Science Publications

Mohan, N (2001) Advanced Electrical Drives, Analysis, Control and Modeling using Simulink.

MNPERE, Minneapolis, USA

Svensson, T (1988) On modulation and control of electronic power converters Technical Report

186, Chalmers University of Technology, School of Electrical and Computer engineering van Duijsen, P.J (2005) Simulation research, caspoc 2005 WWW.CASPOC.COM

Veltman, A (1994) The Fish Method: interaction between AC-machines and Switching Power

Converters PhD thesis, Department of Electrical Engineering, Delft University of

Technol-ogy, the Netherlands

actuators, 2, 33

airgap, 18, 19, 22, 25, 50, 170, 175, 183, 239

algebraic loop, 57, 254

alternative differentiator module, 155

Amp` ere, A H., 265

amplitude invariant, 87, 88

armature, 269, 273

armature reaction, 271

asymmetrically sampled, 305

asynchronous, 169, 233, 237, 246, 266

bar magnet, 13, 14

bipolar, 296, 305, 314, 316, 318

Blondel diagram, 201, 205, 207

braking, 3

brushes, 194, 267

brushless machines, 195

building block, 9, 10, 169, 297, 302

Carter, 25

cartesian, 153

Caspoc, 7

classical, 5, 169, 179

commutator, 265, 267–270, 279

commutator segments, 271, 272, 284

comparator, 302, 311

compensation winding, 269

complex plane, 34, 36

control algorithm, 6, 7

convention, 8, 172, 201

conversion module, 186

converter, 6, 7, 75, 76, 104, 169, 193, 195, 296–

302, 304–306, 310, 314–316, 318,

323

converter switch, 299, 301, 302

Cumming, J S., 265

current control, 295, 318

current density, 22, 23

current sensor, 296

damper winding, 197 Davenport, T, 265

DC machine, 272 delta connected, 80, 84, 93, 95–97, 102, 103,

145, 146 discrete, 297, 308–310 double edged, 306 drives, 2–4, 6, 12, 30, 193, 243, 266 DSP, 6, 7, 296

efficiency, 2, 6 electro-magnetic interaction, 29 energy, 3, 5, 75, 122, 124, 125, 172, 173 falling edge, 300

Faraday, M., 265 feed-forward, 309, 321 field current, 195, 203, 218 finite-element, 7

flux density, 13–16, 18, 20, 23 flux lines, 13–16, 25 flux-linkage, 19–22, 32, 47, 50, 61, 152, 153,

156, 171 four parameter model, 185 four-quadrant, 2

fringing, 14, 17, 18, 25 generator, 5, 169, 203, 204, 209, 222, 244 generic model, 9, 295, 319

grid, 6, 75 half bridge converter, 306, 314 Heyland diagram, 242, 244–247 Holmes, G., 305

Hopkinson, 16, 20, 50 Hughes, A., 12, 193, 194 idle mode, 296 imaginary power, 124

incremental flux, 30, 298, 300, 304, 316

inductance, 20, 21, 31, 32, 273

induction machine, 231

inertia, 174, 211

iron losses, 67

IRTF, 169, 173, 197, 271

ITF, 45, 48, 49, 149

Kirchhoff, 76, 81

leakage inductance, 55, 56, 58

Leonhard, W., 10

linear-motors, 23

load, 3, 5, 211, 234, 276, 286, 287, 296

load angle, 195, 201

load torque, 211

logic signal, 297

Lorentz, 12, 173

m-file, 36, 41, 65

machine sizing, 22, 23

magnetic circuit, 14, 16, 20, 45, 46, 50

magnetic field, 12, 13, 17, 22, 182, 195, 231, 265

magnetic poles, 181

magnetizing inductance, 50, 51, 58, 183

MATLAB, 7

maximum output power point, 208

micro-processor, 6, 296, 298

Miller, T J E., 23

modulator, 6, 7, 296, 298, 301, 303, 304, 306,

307, 310, 311, 313, 314, 316, 318

motoring, 204, 208, 233, 240

multi-pole, 182, 183

mutual inductance, 60, 61

neutral, 76

no-load, 2, 67, 233, 276

non-linear, 21, 29, 33, 39, 42

non-salient, 197, 201

Oersted, H C., 265

permanent magnet, 195, 222, 267

permeability, 16, 17, 29, 45, 46, 50

phasor, 33–35

polar, 153

pole-pair, 181

power factor, 124, 203, 204, 208, 244

power invariant, 87–89

power supply, 4, 6, 14, 194

predictive dead-beat, 307, 321

primary, 45–50

primary referred, 51, 52, 61

proportional, 309, 319

proportional-integral, 309

pull-out slip, 241, 243

PWM, 300, 301

quasi-stationary, 268 quasi-steady-state, 179 reactive power, 124, 125, 127, 132–134, 137,

218 real power, 124, 131, 132 reference incremental flux, 298, 306, 320 regenerative, 3

reluctance, 16, 17, 20, 50, 51, 175 rising edge, 299, 306

robots, 1 rotating flux vector, 178, 195 rotating reference frame, 180 rotor, 170, 171, 195, 231 rotor angle, 173 rotor speed, 178, 195, 233 sampling interval, 298–300, 306, 307 saturation, 19, 21, 23, 32, 33, 38 saw tooth, 301, 311

Schweigger, J C S., 265 secondary, 45–48, 152 self inductance, 19, 29, 61 sensors, 5–7

separately excited DC, 277 series wound DC, 278 set-point, 307 shear-stress, 23 shoot-through mode, 296 shunt DC machine, 277 simplified model, 196, 235, 240, 268 Simpson, 308

Simulink, 36 sinusoidal, 33, 34, 53, 178, 200 sinusoidal distributed, 185, 327 slip, 233, 238–241

slipring machine, 232 slipring/brush, 194, 195, 267 space vector, 84

speed condition, 179 squirrel cage, 231 stable operation, 208 star connected, 76, 91, 102, 103, 109 star point, 76

stationary reference frame, 180 stator, 170, 171, 175, 193, 266 steady-state, 276

Sturgeon, S, 265 supply voltage, 84, 100, 296, 302–304, 310, 316 switching point, 299, 301

symbolic model, 32 synchronous, 169, 193–197, 201, 231, 267 synchronous speed, 211, 239

Tesla, N, 231 three inductance model, 57

FUNDAMENTALS OF ELECTRICAL DRIVES

344

three-phase, 75, 76, 84, 103, 121, 136, 149, 193,

195

toroidal, 29, 45

torque, 172, 173, 175

traction drives, 277

transformer, 45, 46, 49, 50, 149, 157, 169

triangular function, 304

two inductance model, 57, 59, 60, 157, 183

two-phase, 84, 183

uni-polar, 295, 296, 300, 301, 307

universal DC machine, 278

universal machine, 267 V/f drive, 243 Vector to RMS module, 136 Veltman, A., 169

Westinghouse, 231 winding ratio, 47, 63, 174 wye connected, 76 zero sequence, 81, 89, 91, 93, 96, 97, 99, 104 zero-order hold, 310

Index