Higher Harmonics of Selected
384 Bit Unfiltered Magic Sinwaves
by Jim Fitzsimons and Don Lancaster

Copyright c 1996 by Don Lancaster and Synergetics, Box 809, Thatcher, AZ, 85552
(520) 428-4073 https://www.tinaja.com email: synergetics@tinaja.com All commercial
rights and all electronic media rights are **FULLY** reserved.

Royalty and consulting fees **required** for any and all commercial uses.

v3.4 October 10, 1996

Magic sinewaves are ultra long repeating binary sequences highly useful for induction motor speed controls, electric autos, solar inverters, and home energy efficiency improvers. Compared to traditional PWM, magic sinewaves offer lower costs, higher efficieny, elegant simplicity, simplified filtering, and involve significantly less high frequency energy.

Additional details on magic sinewaves appear on www.tinaja.com

Copyright c 1996 by Don Lancaster and Synergetics, Box 809, Thatcher, AZ, 85552
(520) 428-4073 https://www.tinaja.com email: synergetics@tinaja.com All commercial
rights and all electronic media rights are **FULLY** reserved. Royalty and consulting
fees **required** for any and all commercial use.

v3.4 October 18, 1996

In this requested file, Jim Fitzsimons answers the question: Just where
do those square corners come from if a magic sinewave has no low harmonics?

The first 1001 harmonics for a few typical "minimum distortion" sinewaves
are analyzed. These are UNFILTERED results.

Note that mid-range harmonics can be dramatically reduced by filtering. Strong
filtering is often inherent in a motor's inductance, but effective filtering may present
problems over wider speed ranges.

Harmonics can also be selectively reduced by going to a longer word length or by
by making inclusive selections that allow somewhat higher lower harmonic values.

Note also that forcing magic sinewaves to be delta friendly dramatically increases
the harmonic amplitudes in the twenties and thirties. Any time that all ones are
restricted from giving their all to the fundamental, the harmonics have to increase.

Keywords:
magic,sinewave,Fourier,analysis,PWM,power,electronics,auto,solar,inverter

v3.4 October 18, 1996

Note that mid-range harmonics can be dramatically reduced by filtering. Strong
filtering is often inherent in a motor's inductance, but may present problems over
wider speed ranges.

Harmonics can also be selectively reduced by going to a longer word length or by
by making inclusive selections that allow somewhat higher lower harmonic values.

All harmonics are shown as ABSOLUTE amplitudes.

Fundamental Amplitude = 0.198

000000000000000000010000000000001000000010000000 100000000110000000000110000000001100000000110000 Hex: $00001000808080600600C030 Ones: 12 Transitions: 16 1 0.198527 3 0.000002 5 0.000242 7 -0.002810 9 0.000055 11 -0.007458 13 -0.006863 15 -0.001644 17 -0.018171 19 -0.024118 21 0.002725 23 0.033343 25 0.003250 27 0.015343 29 0.063783 31 -0.059382 33 -0.063294 35 0.095320 37 -0.164663 39 0.098434 41 0.024246 43 0.027821 45 -0.068840 47 0.012863 49 0.068337 51 0.010975 53 -0.057029 55 -0.035908 57 -0.006245 59 -0.021912 61 0.059277 63 0.034040 65 -0.050487 67 0.064270 69 -0.001419 71 -0.091690 73 0.085262 75 -0.069895 77 0.028794 79 -0.054063 81 0.055814 83 -0.008040 85 -0.001336 87 0.043382 89 0.050592 91 -0.068753 93 -0.080488 95 0.040593 97 0.005977 99 0.011267 101 0.026264 103 -0.056542 105 0.033712 107 0.057334 109 -0.036216 111 0.014609 113 -0.059596 115 0.030015 117 -0.057945 119 0.019110 121 0.036959 123 0.028748 125 -0.024321 127 0.026292 129 0.001318 131 -0.060182 133 0.014256 135 0.017861 137 -0.014419 139 -0.011448 141 -0.026590 143 0.040982 145 0.060089 147 -0.006996 149 -0.022956 151 -0.059920 153 0.025151 155 0.001001 157 -0.007012 159 -0.006098 161 0.031778 163 -0.005517 165 0.000751 167 0.022762 169 -0.003139 171 -0.014730 173 -0.020307 175 -0.012266 177 -0.002145 179 -0.002691 181 0.025294 183 0.036934 185 0.003015 187 -0.031247 189 -0.024431 191 0.018745 193 0.018550 195 -0.023679 197 -0.029661 199 0.002803 201 0.033626 203 0.022553 205 -0.002349 207 -0.001834 209 -0.010271 211 -0.016650 213 -0.011826 215 -0.002468 217 0.017517 219 0.000565 221 -0.004069 223 0.022943 225 -0.004309 227 -0.004850 229 0.000678 231 0.016658 233 -0.038832 235 -0.014555 237 -0.004339 239 0.036455 241 0.024317 243 -0.015429 245 -0.006495 247 -0.007997 249 0.009683 251 0.007554 253 -0.031162 255 0.000667 257 0.012992 259 -0.011738 261 0.013548 263 0.017004 265 0.008581 267 -0.025392 269 0.012832 271 -0.024850 273 0.005940 275 -0.014355 277 0.022147 279 0.012687 281 -0.020725 283 0.009373 285 0.003914 287 0.002020 289 0.013344 291 -0.025723 293 -0.021353 295 0.015263 297 0.012708 299 -0.000380 301 -0.002217 303 0.014921 305 -0.014003 307 0.007222 309 -0.016965 311 0.020013 313 -0.020799 315 -0.000311 317 0.013584 319 -0.010287 321 0.006681 323 0.011195 325 -0.003978 327 -0.001089 329 -0.006003 331 -0.009131 333 0.001681 335 0.009995 337 0.001794 339 -0.009138 341 0.003508 343 0.002898 345 0.011127 347 -0.017558 349 0.009559 351 -0.005951 353 -0.005215 355 0.005210 357 0.001160 359 0.000226 361 0.002124 363 0.000158 365 -0.001255 367 -0.000842 369 -0.000067 371 -0.000240 373 -0.000220 375 0.000001 377 -0.000052 379 0.000003 381 0.000000 383 0.000518 385 0.000516 387 0.000000 389 0.000003 391 -0.000050 393 0.000001 395 -0.000208 397 -0.000225 399 -0.000062 401 -0.000770 403 -0.001137 405 0.000141 407 0.001884 409 0.000199 411 0.001008 413 0.004479 415 -0.004436 417 -0.005009 419 0.007962 421 -0.014472 423 0.009075 425 0.002339 427 0.002802 429 -0.007221 431 0.001403 433 0.007733 435 0.001287 437 -0.006917 439 -0.004499 441 -0.000807 443 -0.002918 445 0.008126 447 0.004798 449 -0.007309 451 0.009548 453 -0.000216 455 -0.014308 457 0.013619 459 -0.011421 461 0.004809 463 -0.009225 465 0.009722 467 -0.001429 469 -0.000242 471 0.008013 473 0.009519 475 -0.013172 477 -0.015693 479 0.008051 481 0.001205 483 0.002309 485 0.005469 487 -0.011959 489 0.007239 491 0.012494 493 -0.008007 495 0.003276 497 -0.013550 499 0.006917 501 -0.013532 503 0.004521 505 0.008856 507 0.006974 509 -0.005973 511 0.006534 513 0.000331 515 -0.015309 517 0.003667 519 0.004646 521 -0.003791 523 -0.003043 525 -0.007141 527 0.011120 529 0.016470 531 -0.001937 533 -0.006417 535 -0.016912 537 0.007166 539 0.000288 541 -0.002035 543 -0.001786 545 0.009388 547 -0.001644 549 0.000226 551 0.006899 553 -0.000959 555 -0.004538 557 -0.006307 559 -0.003840 561 -0.000677 563 -0.000855 565 0.008103 567 0.011920 569 0.000980 571 -0.010233 573 -0.008058 575 0.006226 577 0.006205 579 -0.007975 581 -0.010057 583 0.000957 585 0.011554 587 0.007799 589 -0.000818 591 -0.000642 593 -0.003620 595 -0.005904 597 -0.004219 599 -0.000886 601 0.006325 603 0.000205 605 -0.001487 607 0.008429 609 -0.001592 611 -0.001802 613 0.000253 615 0.006257 617 -0.014664 619 -0.005526 621 -0.001656 623 0.013985 625 0.009377 627 -0.005980 629 -0.002530 631 -0.003131 633 0.003809 635 0.002986 637 -0.012377 639 0.000266 641 0.005209 643 -0.004728 645 0.005482 647 0.006912 649 0.003504 651 -0.010414 653 0.005286 655 -0.010281 657 0.002468 659 -0.005990 661 0.009281 663 0.005339 665 -0.008758 667 0.003977 669 0.001667 671 0.000864 673 0.005730 675 -0.011090 677 -0.009242 679 0.006631 681 0.005542 683 -0.000166 685 -0.000974 687 0.006581 689 -0.006199 691 0.003209 693 -0.007564 695 0.008956 697 -0.009340 699 -0.000140 701 0.006143 703 -0.004668 705 0.003042 707 0.005114 709 -0.001823 711 -0.000501 713 -0.002770 715 -0.004227 717 0.000781 719 0.004657 721 0.000839 723 -0.004285 725 0.001650 727 0.001367 729 0.005266 731 -0.008335 733 0.004551 735 -0.002842 737 -0.002498 739 0.002503 741 0.000559 743 0.000109 745 0.001029 747 0.000077 749 -0.000612 751 -0.000411 753 -0.000033 755 -0.000118 757 -0.000108 759 0.000001 761 -0.000026 763 0.000002 765 0.000000 767 0.000259 769 0.000258 771 0.000000 773 0.000002 775 -0.000025 777 0.000001 779 -0.000105 781 -0.000114 783 -0.000031 785 -0.000394 787 -0.000582 789 0.000073 791 0.000970 793 0.000102 795 0.000521 797 0.002321 799 -0.002304 801 -0.002608 803 0.004155 805 -0.007568 807 0.004757 809 0.001229 811 0.001475 813 -0.003810 815 0.000742 817 0.004099 819 0.000683 821 -0.003682 823 -0.002400 825 -0.000431 827 -0.001563 829 0.004362 831 0.002581 833 -0.003940 835 0.005157 837 -0.000117 839 -0.007759 841 0.007401 843 -0.006218 845 0.002624 847 -0.005043 849 0.005325 851 -0.000784 853 -0.000133 855 0.004414 857 0.005254 859 -0.007283 861 -0.008694 863 0.004468 865 0.000670 867 0.001287 869 0.003053 871 -0.006686 873 0.004055 875 0.007011 877 -0.004501 879 0.001845 881 -0.007644 883 0.003909 885 -0.007661 887 0.002564 889 0.005030 891 0.003969 893 -0.003404 895 0.003731 897 0.000190 899 -0.008770 901 0.002104 903 0.002670 905 -0.002183 907 -0.001754 909 -0.004125 911 0.006433 913 0.009543 915 -0.001124 917 -0.003730 919 -0.009845 921 0.004178 923 0.000168 925 -0.001190 927 -0.001046 929 0.005507 931 -0.000966 933 0.000133 935 0.004065 937 -0.000566 939 -0.002682 941 -0.003733 943 -0.002276 945 -0.000402 947 -0.000509 949 0.004824 951 0.007107 953 0.000585 955 -0.006119 957 -0.004825 959 0.003733 961 0.003726 963 -0.004795 965 -0.006055 967 0.000577 969 0.006975 971 0.004715 973 -0.000495 975 -0.000389 977 -0.002197 979 -0.003588 981 -0.002568 983 -0.000540 985 0.003859 987 0.000125 989 -0.000909 991 0.005163 993 -0.000976 995 -0.001106 997 0.000156 999 0.003852

Fundamental Amplitude = 0.395

000000000000010000000000011000000001100000000111 000000011100000001111000000011110000000111110000 Hex: $00040060180701C0780F01F0 Ones: 24 Transitions: 16 1 0.395282 3 -0.000372 5 -0.000705 7 -0.001547 9 -0.002207 11 -0.005611 13 -0.005409 15 -0.001677 17 0.003680 19 0.033235 21 -0.039268 23 -0.010072 25 -0.046552 27 0.015374 29 -0.093435 31 0.116247 33 -0.169864 35 0.330776 37 0.014365 39 -0.206773 41 0.018120 43 0.030003 45 0.021579 47 -0.011221 49 0.012177 51 -0.023428 53 -0.031218 55 -0.045754 57 0.003010 59 0.001981 61 0.046650 63 -0.010112 65 0.089197 67 -0.021844 69 0.060219 71 -0.039629 73 -0.153398 75 0.062617 77 0.091249 79 -0.023198 81 -0.051760 83 -0.038884 85 -0.002145 87 -0.027811 89 0.047743 91 0.035494 93 0.035808 95 -0.004810 97 0.026933 99 -0.018362 101 -0.010221 103 -0.020178 105 -0.046814 107 0.018998 109 0.013359 111 0.022700 113 -0.049993 115 -0.037075 117 -0.006220 119 0.064708 121 0.023153 123 0.043162 125 -0.009990 127 -0.003827 129 -0.033377 131 0.021522 133 -0.030889 135 -0.023202 137 0.016903 139 -0.040227 141 0.039293 143 -0.015502 145 -0.028110 147 0.034878 149 0.008006 151 0.015580 153 0.036982 155 0.017862 157 -0.025006 159 -0.037007 161 0.002040 163 -0.006454 165 -0.003097 167 -0.023969 169 -0.005507 171 0.011071 173 0.002564 175 0.004638 177 0.021046 179 -0.009139 181 0.048295 183 -0.009199 185 -0.013072 187 -0.000360 189 0.012964 191 -0.032284 193 -0.031949 195 0.012565 197 -0.000342 199 -0.012152 201 -0.008376 203 0.043061 205 -0.007980 207 0.017996 209 0.003884 211 0.002102 213 0.008888 215 -0.004328 217 -0.018446 219 -0.002333 221 -0.004760 223 0.001473 225 -0.026151 227 -0.017295 229 0.012090 231 0.024494 233 0.010097 235 0.005076 237 0.021633 239 -0.017054 241 -0.009198 243 0.022800 245 -0.022823 247 0.009375 249 -0.012579 251 -0.016367 253 0.011144 255 -0.016885 257 -0.001891 259 -0.004821 261 0.020341 263 0.010652 265 0.029058 267 -0.002726 269 -0.015850 271 -0.020846 273 0.009230 275 0.005295 277 0.007338 279 -0.017618 281 -0.007396 283 -0.003648 285 -0.006378 287 0.009103 289 -0.001581 291 0.011444 293 0.011024 295 0.014404 297 -0.008147 299 -0.000610 301 -0.010722 303 -0.013837 305 -0.006009 307 0.022886 309 0.015198 311 -0.036007 313 -0.008989 315 0.013191 317 -0.004617 319 0.018175 321 -0.001985 323 0.008810 325 0.000360 327 0.000525 329 -0.007649 331 -0.004999 333 -0.003588 335 0.001781 337 -0.001565 339 0.002864 341 0.003783 343 0.002166 345 -0.023374 347 0.001532 349 0.033172 351 -0.015970 353 0.010209 355 -0.007633 357 0.001163 359 -0.003242 361 -0.000642 363 -0.002272 365 0.001730 367 0.000170 369 -0.000068 371 -0.000190 373 -0.000165 375 -0.000053 377 -0.000029 379 -0.000009 381 -0.000003 383 0.001032 385 0.001027 387 -0.000003 389 -0.000009 391 -0.000028 393 -0.000051 395 -0.000156 397 -0.000177 399 -0.000063 401 0.000156 403 0.001567 405 -0.002036 407 -0.000569 409 -0.002845 411 0.001010 413 -0.006561 415 0.008684 417 -0.013442 419 0.027630 421 0.001262 423 -0.019064 425 0.001748 427 0.003021 429 0.002264 431 -0.001224 433 0.001378 435 -0.002747 437 -0.003786 439 -0.005732 441 0.000389 443 0.000264 445 0.006395 447 -0.001425 449 0.012913 451 -0.003245 453 0.009172 455 -0.006184 457 -0.024503 459 0.010231 461 0.015241 463 -0.003958 465 -0.009016 467 -0.006911 469 -0.000389 471 -0.005137 473 0.008983 475 0.006800 477 0.006982 479 -0.000954 481 0.005431 483 -0.003764 485 -0.002128 487 -0.004268 489 -0.010052 491 0.004140 493 0.002954 495 0.005090 497 -0.011367 499 -0.008544 501 -0.001453 503 0.015309 505 0.005548 507 0.010471 509 -0.002453 511 -0.000951 513 -0.008393 515 0.005475 517 -0.007946 519 -0.006035 521 0.004445 523 -0.010691 525 0.010553 527 -0.004206 529 -0.007705 531 0.009655 533 0.002238 535 0.004397 537 0.010537 539 0.005137 541 -0.007257 543 -0.010836 545 0.000603 547 -0.001923 549 -0.000931 551 -0.007265 553 -0.001683 555 0.003411 557 0.000796 559 0.001452 561 0.006640 563 -0.002906 565 0.015471 567 -0.002969 569 -0.004250 571 -0.000118 573 0.004276 575 -0.010724 577 -0.010687 579 0.004232 581 -0.000116 583 -0.004148 585 -0.002878 587 0.014892 589 -0.002777 591 0.006303 593 0.001369 595 0.000746 597 0.003171 599 -0.001554 601 -0.006660 603 -0.000847 605 -0.001739 607 0.000541 609 -0.009662 611 -0.006426 613 0.004517 615 0.009200 617 0.003813 619 0.001927 621 0.008256 623 -0.006542 625 -0.003547 627 0.008836 629 -0.008890 631 0.003670 633 -0.004948 635 -0.006470 637 0.004426 639 -0.006738 641 -0.000758 643 -0.001942 645 0.008231 647 0.004330 649 0.011865 651 -0.001118 653 -0.006529 655 -0.008625 657 0.003835 659 0.002210 661 0.003075 663 -0.007414 665 -0.003125 667 -0.001548 669 -0.002717 671 0.003893 673 -0.000679 675 0.004934 677 0.004771 679 0.006258 681 -0.003553 683 -0.000267 685 -0.004711 687 -0.006103 689 -0.002660 691 0.010168 693 0.006777 695 -0.016112 697 -0.004037 699 0.005944 701 -0.002088 703 0.008247 705 -0.000904 707 0.004025 709 0.000165 711 0.000241 713 -0.003529 715 -0.002314 717 -0.001666 719 0.000830 721 -0.000731 723 0.001343 725 0.001779 727 0.001022 729 -0.011062 731 0.000727 733 0.015794 735 -0.007627 737 0.004890 739 -0.003667 741 0.000560 743 -0.001566 745 -0.000311 747 -0.001104 749 0.000843 751 0.000083 753 -0.000033 755 -0.000093 757 -0.000082 759 -0.000026 761 -0.000014 763 -0.000005 765 -0.000001 767 0.000515 769 0.000514 771 -0.000001 773 -0.000005 775 -0.000014 777 -0.000026 779 -0.000079 781 -0.000090 783 -0.000032 785 0.000080 787 0.000802 789 -0.001045 791 -0.000293 793 -0.001468 795 0.000522 797 -0.003400 799 0.004510 801 -0.006998 803 0.014417 805 0.000660 807 -0.009993 809 0.000918 811 0.001591 813 0.001194 815 -0.000647 817 0.000730 819 -0.001459 821 -0.002015 823 -0.003058 825 0.000208 827 0.000141 829 0.003433 831 -0.000767 833 0.006960 835 -0.001753 837 0.004964 839 -0.003354 841 -0.013315 843 0.005571 845 0.008315 847 -0.002164 849 -0.004938 851 -0.003792 853 -0.000214 855 -0.002830 857 0.004958 859 0.003760 861 0.003868 863 -0.000530 865 0.003020 867 -0.002097 869 -0.001188 871 -0.002386 873 -0.005631 875 0.002323 877 0.001660 879 0.002867 881 -0.006412 883 -0.004829 885 -0.000822 887 0.008681 889 0.003151 891 0.005958 893 -0.001398 895 -0.000543 897 -0.004800 899 0.003136 901 -0.004560 903 -0.003469 905 0.002559 907 -0.006165 909 0.006095 911 -0.002433 913 -0.004464 915 0.005603 917 0.001301 919 0.002560 921 0.006144 923 0.003000 925 -0.004244 927 -0.006347 929 0.000354 931 -0.001130 933 -0.000548 935 -0.004281 937 -0.000993 939 0.002016 941 0.000471 943 0.000861 945 0.003942 947 -0.001727 949 0.009211 951 -0.001770 953 -0.002538 955 -0.000071 957 0.002560 959 -0.006430 961 -0.006416 963 0.002544 965 -0.000070 967 -0.002501 969 -0.001737 971 0.009002 973 -0.001681 975 0.003821 977 0.000831 979 0.000453 981 0.001930 983 -0.000947 985 -0.004064 987 -0.000518 989 -0.001064 991 0.000331 993 -0.005926 995 -0.003946 997 0.002777 999 0.005664

Fundamental Amplitude = 0.591

000000000000011000000000001110000000011110000001 111000000111110000111110000011111100001111111000 Hex: $000600380781E07C3E0FC3F8 Ones: 36 Transitions: 16 1 0.591106 3 0.000124 5 -0.001931 7 0.005047 9 -0.005376 11 0.010440 13 0.006151 15 0.024378 17 -0.057223 19 0.031317 21 -0.030245 23 -0.045201 25 -0.081483 27 -0.015674 29 -0.081467 31 0.169454 33 0.057833 35 0.257464 37 -0.332899 39 -0.020905 41 0.205998 43 -0.080426 45 -0.012276 47 -0.012271 49 -0.042651 51 -0.030566 53 -0.035100 55 -0.053674 57 0.019206 59 0.097999 61 0.038369 63 0.076233 65 -0.016290 67 -0.035526 69 -0.063531 71 0.044405 73 0.041643 75 -0.043052 77 -0.039507 79 -0.078190 81 0.032827 83 -0.029477 85 0.022516 87 0.035818 89 0.076059 91 0.003272 93 -0.001324 95 0.003263 97 -0.015497 99 -0.031803 101 0.032974 103 -0.052382 105 -0.008946 107 -0.011787 109 -0.014892 111 -0.003072 113 -0.017276 115 0.076717 117 0.043542 119 -0.045411 121 0.033884 123 -0.003405 125 -0.003275 127 0.001351 129 -0.024377 131 -0.047256 133 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Fundamental Amplitude = 0.983

000000001000000011100000011100000111111100000111 111110011111111001111111111111110111111111111111 Hex: $0080E0707F07F9FE7FFF7FFF Ones: 60 Transitions: 15 1 0.983712 3 0.000128 5 0.000152 7 -0.000489 9 0.005720 11 -0.002576 13 0.006001 15 -0.004978 17 0.018369 19 -0.077944 21 -0.054319 23 0.079027 25 0.045292 27 0.043834 29 -0.080636 31 -0.065600 33 0.155470 35 0.099542 37 -0.161262 39 -0.219143 41 -0.028726 43 -0.027509 45 0.030362 47 0.143623 49 0.047040 51 0.053201 53 0.017845 55 -0.039186 57 -0.032614 59 0.004651 61 0.047286 63 0.009990 65 -0.014759 67 -0.069400 69 0.042018 71 0.071602 73 -0.058957 75 0.039028 77 0.036011 79 -0.066366 81 -0.063741 83 -0.011962 85 -0.050231 87 -0.084354 89 0.046436 91 0.047382 93 0.054247 95 0.038960 97 -0.037939 99 0.014007 101 0.001105 103 0.020305 105 0.035944 107 0.004044 109 -0.018361 111 0.007411 113 0.044198 115 -0.062642 117 -0.001328 119 0.056524 121 -0.040610 123 -0.041648 125 -0.028883 127 0.008220 129 -0.005699 131 -0.008061 133 -0.018604 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v3.4 October 18, 1996