Comparison of Three Analytical Methods for the Precise Calculation of Cogging Torque and Torque Ripple in Axial Flux PM Machines

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left">Analytical model type </td><td class="align_center"><span style="width: 82.4424ptpx;"><svg height="14.8173pt" id="M104" style="vertical-align:-5.52897pt" version="1.1" viewbox="-0.0498162 -9.28833 82.4424 14.8173" width="82.4424pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="411" vert-adv-y="411"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,5.343,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g 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271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.841,0)"><path d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z" id="g113-67"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="601" vert-adv-y="601"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,17.589,3.132)"><path d="M545 403C545 425 527 451 498 451C448 451 400 404 312 286L290 341C262 412 246 451 219 451C182 451 138 404 92 345L112 323C152 368 169 378 181 378C192 378 201 359 216 321L257 214C184 115 142 69 113 69C102 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163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 162 151 181 190 198C211 208 259 229 281 240V87Z" id="g58-95"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="449" vert-adv-y="449"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,36.057,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,46.594,0)"><path d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z" id="g113-67"></path><glyph.data 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ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,72.295,0)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="411" vert-adv-y="411"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,77.638,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg></span></td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left"> SC model</td><td class="align_center"> 1.2</td><td class="align_center"> 1.36</td></tr><tr><td class="align_left">SD model</td><td class="align_center"> 0.24</td><td class="align_center"> 0.24</td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Comparison of the full load flux density between the analytical and the 2D FE models considering only armature current.

Mathematical Problems in Engineering

tab4

Table 4

Table 4: Comparison of Three Analytical Methods for the Precise Calculation of Cogging Torque and Torque Ripple in Axial Flux PM Machines 