Fundamentals of Electrical Drivess - Appendix pps

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 n 1 turn stator phase windings. The windings which carry the currents i 1α ,i 1β respectively, 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 = ˆ B 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 = ˆ B β sin ξ. The relationship between phase currents and peak flux density values is of the form B 1α = Ci 1α ,B 1β = Ci 1β where C is a constant to be defined shortly. In space vector terms the following relationships hold  i 1 = i 1α + ji 1β (A.1a)  B 1 = ˆ B 1α + j ˆ B 1β (A.1b) 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  B 1 = C  i 1 (A.2) If for example the current is of the form  i 1 = ˆ i 1 e jρ then the flux density should be of the form  B 1 = C ˆ i 1 e jρ for any value of ρ and values of ˆ i 1 which fall within the linear operating range of the machine. The space vector components are in this case of the form i 1α = ˆ i 1 cos ρ, i 1β = ˆ i 1 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 ˆ i 1 cos ρ    ˆ B 1α cos ξ + C ˆ i 1 sin ρ    ˆ B 1β sin ξ (A.3) 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 ˆ i 1 cos (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 s rad/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,whereN 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 B (ξ +∆ξ)=N ξ i (A.4) Expression (A.4) can also be rewritten in a more convenient form by introducing the variable n(ξ)= N ξ ∆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) ∆ξ = − g µ o n(ξ)i (A.5) 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 n (ξ) i (A.6) 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 B (ξ) dξ =0 (A.7) 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.Thecor- 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 = N 3 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 B sin (ξ) (A.8) in which ˆ B 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 B (ξ) dξ (A.9) which for a concentrated winding corresponds to a flux-linkage value ψ 1α = Nφ mα .Ifa 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 = N eff φ mα ,whereN 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 References B¨defeld, Th and Sequenz, H (1962) Elektrische Machinen Springer-Verlag, 6th edition o Holmes, D.G (1997) A generalised approach to modulation and control of hard switched converters PhD thesis, Department of Electrical and Computer engineering,... 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 FUNDAMENTALS OF ELECTRICAL DRIVES quasi-stationary, 268 quasi-steady-state, 179 reactive power, 124, 125,... 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 Technology,... 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... 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,... 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,... 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,... 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... 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 universal machine, 267 uni-polar, 295, 296, 300, 301, 307 universal DC machine, 278 zero sequence, 81, 89, 91, 93, 96, 97, 99, 104 zero-order hold, 310 V/f drive, 243 Vector to RMS module, 136 Veltman, A., 169 Westinghouse, 231 winding ratio, 47, 63, 174 wye connected,... 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, 57 345 Index three-phase, 75, 76, 84, 103, 121, 136, 149, 193, 195 toroidal, 29, 45 torque, 172, . 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. 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:. current is of the form  i 1 = ˆ i 1 e jρ then the flux density should be of the form  B 1 = C ˆ i 1 e jρ for any value of ρ and values of ˆ i 1 which fall within the linear operating range of the