653:
641:
665:
3684:
151:, such as a 230/400 V system which provides 230 V between the neutral (centre hub) and any one of the phases, and 400 V across any two phases. A delta system arrangement provides only one voltage, but it has a greater redundancy as it may continue to operate normally with one of the three supply windings offline, albeit at 57.7% of total capacity. Harmonic current in the neutral may become very large if nonlinear loads are connected.
122:
3418:
25:
4021:
3679:{\displaystyle {\begin{aligned}i&=\sin \left(\theta \right)+\sin \left(\theta -{\frac {2\pi }{3}}\right)+\sin \left(\theta +{\frac {2\pi }{3}}\right)\\&=\sin \left(\theta \right)+2\sin \left(\theta \right)\cos \left({\frac {2\pi }{3}}\right)\\&=\sin \left(\theta \right)-\sin \left(\theta \right)\\&=0\end{aligned}}}
3355:
4350:
With linear loads, the neutral only carries the current due to imbalance between the phases. Devices that utilize rectifier-capacitor front ends (such as switch-mode power supplies for computers, office equipment and the like) introduce third order harmonics. Third harmonic currents are in-phase on
3689:
Since we have shown that the neutral current is zero we can see that removing the neutral core will have no effect on the circuit, provided the system is balanced. Such connections are generally used only when the load on the three phases is part of the same piece of equipment (for example a
1084:
2144:
1983:
4389:. Such arrays will evenly balance the polyphase load between the phases of the source system. For example, balanced two-phase power can be obtained from a three-phase network by using two specially constructed transformers, with taps at 50% and 86.6% of the primary voltage. This
146:
that are offset in time by one-third of the period. A three-phase system may be arranged in delta (∆) or star (Y) (also denoted as wye in some areas, as symbolically it is similar to the letter 'Y'). A wye system allows the use of two different voltages from all three
498:
363:
3810:
4166:
680:
Generally, in electric power systems, the loads are distributed as evenly as is practical among the phases. It is usual practice to discuss a balanced system first and then describe the effects of unbalanced systems as deviations from the elementary case.
865:
1822:
3128:
4372:
A rotating magnetic field of steady amplitude requires that all three phase currents be equal in magnitude, and accurately displaced one-third of a cycle in phase. Unbalanced operation results in undesirable effects on motors and generators.
652:
4340:
640:
4364:
possible. Indeed, where induction motors must run on single-phase power (such as is usually distributed in homes), the motor must contain some mechanism to produce a revolving field, otherwise the motor cannot generate any stand-still
4359:
Any polyphase system, by virtue of the time displacement of the currents in the phases, makes it possible to easily generate a magnetic field that revolves at the line frequency. Such a revolving magnetic field makes polyphase
2780:
1692:
1605:
125:
One voltage cycle of a three-phase system, labeled 0 to 360° (2π radians) along the time axis. The plotted line represents the variation of instantaneous voltage (or current) with respect to time. This cycle repeats with a
947:
1991:
1830:
3709:
When specifying wiring sizes in a three-phase system, we only need to know the magnitude of the phase and neutral currents. The neutral current can be determined by adding the three phase currents together as
940:
4554:
2956:
1518:
2863:
742:
228:
3698:
In practice, systems rarely have perfectly balanced loads, currents, voltages and impedances in all three phases. The analysis of unbalanced cases is greatly simplified by the use of the techniques of
3107:
4588:
4016:{\displaystyle I_{L1}+I_{L2}\cos \left({\frac {2}{3}}\pi \right)+jI_{L2}\sin \left({\frac {2}{3}}\pi \right)+I_{L3}\cos \left({\frac {4}{3}}\pi \right)+jI_{L3}\sin \left({\frac {4}{3}}\pi \right)}
3423:
3133:
2552:
2413:
2274:
755:
371:
236:
1161:
42:
4369:
and will not start. The field produced by a single-phase winding can provide energy to a motor already rotating, but without auxiliary mechanisms the motor will not accelerate from a stop.
3004:
1229:
1323:
1444:
4032:
3410:
750:
664:
4734:
570:
1379:
3350:{\displaystyle {\begin{aligned}I_{L1}&={\frac {V_{L1-N}}{R}},\;I_{L2}={\frac {V_{L2-N}}{R}},\;I_{L3}={\frac {V_{L3-N}}{R}}\\-I_{N}&=I_{L1}+I_{L2}+I_{L3}\end{aligned}}}
1703:
159:
In a star (wye) connected topology, with rotation sequence L1 - L2 - L3, the time-varying instantaneous voltages can be calculated for each phase A,C,B respectively by:
4177:
3037:
1262:
1184:
3802:
3772:
3742:
4630:
4558:
531:
3118:
For the case of equal loads on each of three phases, no net current flows in the neutral. The neutral current is the inverted vector sum of the line currents. See
614:
592:
1329:
The load need not be resistive for achieving a constant instantaneous power since, as long as it is balanced or the same for all phases, it may be written as
4351:
each of the supply phases and therefore will add together in the neutral which can cause the neutral current in a wye system to exceed the phase currents.
4581:
4427:
In a three-phase system, at least two transducers are required to measure power when there is no neutral, or three transducers when there is a neutral.
4396:
produces a true two-phase system with 90° time difference between the phases. Another example is the generation of higher-phase-order systems for large
2563:
89:
1613:
1526:
4715:
61:
4503:
4692:
1079:{\displaystyle p=\sin ^{2}\theta +\sin ^{2}\left(\theta -{\frac {2}{3}}\pi \right)+\sin ^{2}\left(\theta -{\frac {4}{3}}\pi \right)={\frac {3}{2}}}
4725:
68:
2139:{\displaystyle P_{L3}=V_{L3}I_{L3}=V_{P}I_{P}\sin \left(\theta -{\frac {4}{3}}\pi \right)\sin \left(\theta -{\frac {4}{3}}\pi -\varphi \right)}
1978:{\displaystyle P_{L2}=V_{L2}I_{L2}=V_{P}I_{P}\sin \left(\theta -{\frac {2}{3}}\pi \right)\sin \left(\theta -{\frac {2}{3}}\pi -\varphi \right)}
4381:
Provided two voltage waveforms have at least some relative displacement on the time axis, other than a multiple of a half-cycle, any other
75:
877:
4537:
57:
2874:
1186:
we can see that the total power does not vary with time. This is essential for keeping large generators and motors running smoothly.
4730:
4661:
1455:
108:
2791:
692:
165:
3045:
46:
2785:
Since the three terms enclosed in square brackets are a three-phase system, they add up to zero and the total power becomes
493:{\displaystyle V_{L3-N}=V_{P}\sin \left(\theta -{\frac {4}{3}}\pi \right)=V_{P}\sin \left(\theta +{\frac {2}{3}}\pi \right)}
358:{\displaystyle V_{L2-N}=V_{P}\sin \left(\theta -{\frac {2}{3}}\pi \right)=V_{P}\sin \left(\theta +{\frac {4}{3}}\pi \right)}
4171:
The polar magnitude of this is the square root of the sum of the squares of the real and imaginary parts, which reduces to
4781:
4637:
2421:
2282:
2159:
4740:
4710:
4685:
3702:. An unbalanced system is analysed as the superposition of three balanced systems, each with the positive, negative or
1095:
82:
3119:
4431:
states that the number of measurement elements required is one less than the number of current-carrying conductors.
4471:
4456:
3714:
and then converting from rectangular to polar co-ordinates. If the three-phase root mean square (RMS) currents are
139:
4161:{\displaystyle I_{L1}-I_{L2}{\frac {1}{2}}-I_{L3}{\frac {1}{2}}+j{\frac {\sqrt {3}}{2}}\left(I_{L2}-I_{L3}\right)}
2967:
1192:
35:
4720:
4441:
1270:
1390:
3690:
three-phase motor), as otherwise switching loads and slight imbalances would cause large voltage fluctuations.
3363:
860:{\displaystyle {\begin{aligned}P_{Li}&={\frac {V_{Li}^{2}}{R}}\\P_{TOT}&=\sum _{i}P_{Li}\end{aligned}}}
658:
Elementary three-wire three-phase alternator, showing how the phases can share only three transmission wires.
4771:
4678:
4613:
4496:
3699:
2150:
871:
689:
An important property of three-phase power is that the instantaneous power available to a resistive load,
135:
1817:{\displaystyle P_{L1}=V_{L1}I_{L1}=V_{P}I_{P}\sin \left(\theta \right)\sin \left(\theta -\varphi \right)}
538:
4776:
4745:
1335:
646:
Elementary six-wire three-phase alternator, with each phase using a separate pair of transmission wires.
4411:
When three-phase is needed but only single-phase is readily available from the electricity supplier, a
4582:"Harmonics in Low Voltage Three-Phase Four-Wire Electric Distribution Systems and Filtering Solutions"
4428:
4416:
4390:
143:
4656:. McGraw-Hill Electrical and Electronic Engineering Series (3rd ed.). New York: McGraw-Hill.
4335:{\displaystyle {\sqrt {I_{L1}^{2}+I_{L2}^{2}+I_{L3}^{2}-I_{L1}I_{L2}-I_{L1}I_{L3}-I_{L2}I_{L3}}}}
4670:
4657:
4555:"The 3rd Harmonic Blocking Filter: A Well Established Approach to Harmonic Current Mitigation"
4533:
127:
3009:
1234:
1169:
4476:
4466:
4446:
4382:
3777:
3747:
3717:
509:
4412:
4361:
670:
Each phase of a three-phase transformer has its own pair of windings, with a shared core.
4451:
4401:
3711:
630:
The below images demonstrate how a system of six wires delivering three phases from an
599:
577:
4765:
3703:
4461:
121:
2775:{\displaystyle P_{TOT}={\frac {V_{P}I_{P}}{2}}\left\{3\cos \varphi -\left\right\}}
1687:{\displaystyle I_{L3}=I_{P}\sin \left(\theta -{\frac {4}{3}}\pi -\varphi \right)}
1600:{\displaystyle I_{L2}=I_{P}\sin \left(\theta -{\frac {2}{3}}\pi -\varphi \right)}
4386:
24:
631:
4397:
4587:. Texas A&M University Power Electronics and Power Quality Laboratory.
4415:
can be used to generate three-phase power from the single phase supply. A
4405:
621:
voltages L1-N, L2-N and L3-N are referenced to the star connection point.
148:
634:
may be replaced by just three. A three-phase transformer is also shown.
4366:
120:
935:{\displaystyle \scriptstyle p\,=\,{\frac {1}{V_{P}^{2}}}P_{TOT}R}
4674:
2951:{\displaystyle P_{TOT}={\frac {3V_{P}^{2}}{2|Z|}}\cos \varphi }
4614:"Measurement of three-phase power with the 2-wattmeter method"
1513:{\displaystyle I_{L1}=I_{P}\sin \left(\theta -\varphi \right)}
18:
16:
Mathematics and basic principles of three-phase electric power
2858:{\displaystyle P_{TOT}={\frac {3V_{P}I_{P}}{2}}\cos \varphi }
737:{\displaystyle \scriptstyle P\,=\,VI\,=\,{\frac {V^{2}}{R}}}
223:{\displaystyle V_{L1-N}=V_{P}\sin \left(\theta \right)\,\!}
4532:. Clifton Park, NY: Cengage Learning/Delmar. p. 49.
3102:{\displaystyle P_{TOT}={\frac {3V^{2}}{Z}}\cos \varphi }
4385:
set of voltages can be obtained by an array of passive
881:
696:
4180:
4035:
3813:
3780:
3750:
3720:
3421:
3366:
3131:
3048:
3012:
2970:
2877:
2794:
2566:
2424:
2285:
2162:
1994:
1833:
1706:
1616:
1529:
1458:
1393:
1338:
1273:
1237:
1195:
1172:
1098:
950:
880:
753:
695:
602:
580:
541:
512:
374:
239:
168:
1189:
Notice also that using the root mean square voltage
2547:{\displaystyle P_{L3}={\frac {V_{P}I_{P}}{2}}\left}
2408:{\displaystyle P_{L2}={\frac {V_{P}I_{P}}{2}}\left}
2269:{\displaystyle P_{L1}={\frac {V_{P}I_{P}}{2}}\left}
49:. Unsourced material may be challenged and removed.
4419:is often used in factory industrial applications.
4334:
4160:
4015:
3796:
3766:
3736:
3678:
3404:
3349:
3101:
3031:
2998:
2950:
2857:
2774:
2546:
2407:
2268:
2138:
1977:
1816:
1686:
1599:
1512:
1449:for all phases and the instantaneous currents are
1438:
1373:
1317:
1256:
1223:
1178:
1155:
1078:
934:
859:
736:
608:
586:
564:
525:
492:
357:
222:
1156:{\displaystyle P_{TOT}={\frac {3V_{P}^{2}}{2R}}.}
561:
219:
142:systems have at least three conductors carrying
1697:Now the instantaneous powers in the phases are
4686:
2557:which add up for a total instantaneous power
1264:above takes the following more classic form:
8:
4530:Electricity 3: Power Generation and Delivery
2999:{\displaystyle V={\frac {V_{p}}{\sqrt {2}}}}
1224:{\displaystyle V={\frac {V_{p}}{\sqrt {2}}}}
1318:{\displaystyle P_{TOT}={\frac {3V^{2}}{R}}}
58:"Mathematics of three-phase electric power"
4693:
4679:
4671:
3231:
3185:
2964:Again, using the root mean square voltage
1439:{\displaystyle I_{P}={\frac {V_{P}}{|Z|}}}
4321:
4308:
4292:
4279:
4263:
4250:
4237:
4229:
4216:
4208:
4195:
4187:
4181:
4179:
4144:
4128:
4107:
4091:
4082:
4065:
4056:
4040:
4034:
3995:
3975:
3947:
3927:
3902:
3882:
3854:
3834:
3818:
3812:
3785:
3779:
3755:
3749:
3725:
3719:
3593:
3513:
3473:
3422:
3420:
3405:{\displaystyle i={\frac {I_{N}R}{V_{P}}}}
3394:
3380:
3373:
3365:
3360:We define a non-dimensionalized current,
3334:
3318:
3302:
3285:
3254:
3248:
3236:
3208:
3202:
3190:
3162:
3156:
3140:
3132:
3130:
3078:
3068:
3053:
3047:
3017:
3011:
2983:
2977:
2969:
2931:
2923:
2912:
2907:
2897:
2882:
2876:
2834:
2824:
2814:
2799:
2793:
2738:
2691:
2603:
2593:
2586:
2571:
2565:
2515:
2458:
2448:
2441:
2429:
2423:
2376:
2319:
2309:
2302:
2290:
2284:
2196:
2186:
2179:
2167:
2161:
2112:
2077:
2054:
2044:
2028:
2015:
1999:
1993:
1951:
1916:
1893:
1883:
1867:
1854:
1838:
1832:
1766:
1756:
1740:
1727:
1711:
1705:
1660:
1637:
1621:
1615:
1573:
1550:
1534:
1528:
1479:
1463:
1457:
1428:
1420:
1413:
1407:
1398:
1392:
1362:
1353:
1345:
1337:
1303:
1293:
1278:
1272:
1242:
1236:
1208:
1202:
1194:
1171:
1133:
1128:
1118:
1103:
1097:
1066:
1045:
1025:
1000:
980:
961:
949:
916:
904:
899:
890:
889:
885:
879:
870:To simplify the mathematics, we define a
844:
834:
811:
792:
784:
778:
762:
754:
752:
744:, is constant at all times. Indeed, let
722:
716:
715:
711:
704:
700:
694:
616:is the frequency in cycles per second and
601:
579:
560:
540:
517:
511:
472:
449:
424:
401:
379:
373:
337:
314:
289:
266:
244:
238:
218:
195:
173:
167:
109:Learn how and when to remove this message
4488:
636:
874:power for intermediate calculations,
7:
47:adding citations to reliable sources
565:{\displaystyle \theta =2\pi ft\,\!}
4654:Elements of Power Systems Analysis
4652:Stevenson, William D. Jr. (1975).
1374:{\displaystyle Z=|Z|e^{j\varphi }}
14:
4594:from the original on 13 June 2010
4377:Conversion to other phase systems
3039:can be written in the usual form
130:that depends on the power system.
4631:"THE TWO-METER WATTMETER METHOD"
4497:"Delta and Wye 3-phase circuits"
663:
651:
639:
23:
4557:. IAEI Magazine. Archived from
4509:from the original on 2013-05-13
4400:systems, to produce a smoother
34:needs additional citations for
3804:, the neutral RMS current is:
2961:showing the above contention.
2932:
2924:
1429:
1421:
1354:
1346:
572:is the phase angle in radians
1:
1384:so that the peak current is
1089:Hence (substituting back):
4798:
4751:three-phase electric power
4716:artificial neural networks
4472:Three-phase electric power
4457:Mikhail Dolivo-Dobrovolsky
2151:angle subtraction formulae
140:three-phase electric power
4706:
4701:"Mathematics of" articles
4442:Charles Proteus Steinmetz
4404:output and to reduce the
1166:Since we have eliminated
4528:Keljik, Jeffrey (2008).
4408:currents in the supply.
4355:Revolving magnetic field
3120:Kirchhoff's circuit laws
4726:cyclic redundacy checks
3032:{\displaystyle P_{TOT}}
1257:{\displaystyle P_{TOT}}
1179:{\displaystyle \theta }
685:Constant power transfer
4336:
4162:
4017:
3798:
3797:{\displaystyle I_{L3}}
3768:
3767:{\displaystyle I_{L2}}
3738:
3737:{\displaystyle I_{L1}}
3706:of balanced voltages.
3700:symmetrical components
3680:
3406:
3351:
3103:
3033:
3000:
2952:
2859:
2776:
2548:
2409:
2270:
2140:
1979:
1818:
1688:
1601:
1514:
1440:
1375:
1319:
1258:
1225:
1180:
1157:
1080:
936:
861:
738:
610:
594:is the time in seconds
588:
566:
527:
494:
359:
224:
136:electrical engineering
131:
4553:Lowenstein, Michael.
4337:
4163:
4018:
3799:
3769:
3739:
3681:
3407:
3352:
3104:
3034:
3001:
2953:
2860:
2777:
2549:
2410:
2271:
2141:
1980:
1819:
1689:
1602:
1515:
1441:
1376:
1320:
1259:
1231:, the expression for
1226:
1181:
1158:
1081:
937:
862:
739:
611:
589:
567:
528:
526:{\displaystyle V_{P}}
495:
360:
225:
124:
4782:Three-phase AC power
4178:
4033:
3811:
3778:
3748:
3718:
3419:
3364:
3129:
3046:
3010:
2968:
2875:
2792:
2564:
2422:
2283:
2160:
1992:
1831:
1704:
1614:
1527:
1456:
1391:
1336:
1271:
1235:
1193:
1170:
1096:
948:
878:
751:
693:
600:
578:
539:
533:is the peak voltage,
510:
372:
237:
166:
144:alternating voltages
43:improve this article
4423:System measurements
4242:
4221:
4200:
2917:
1138:
909:
797:
4731:general relativity
4332:
4225:
4204:
4183:
4158:
4026:which resolves to
4013:
3794:
3764:
3734:
3694:Unbalanced systems
3676:
3674:
3402:
3347:
3345:
3114:No neutral current
3099:
3029:
2996:
2948:
2903:
2855:
2772:
2544:
2405:
2266:
2136:
1975:
1814:
1684:
1597:
1510:
1436:
1371:
1315:
1254:
1221:
1176:
1153:
1124:
1076:
932:
931:
895:
872:nondimensionalized
857:
855:
839:
780:
734:
733:
606:
584:
562:
523:
490:
355:
220:
132:
4759:
4758:
4429:Blondel's theorem
4330:
4117:
4113:
4099:
4073:
4003:
3955:
3910:
3862:
3606:
3526:
3486:
3400:
3272:
3226:
3180:
3088:
2994:
2993:
2937:
2844:
2746:
2699:
2613:
2523:
2468:
2384:
2329:
2206:
2120:
2085:
1959:
1924:
1668:
1581:
1434:
1313:
1219:
1218:
1148:
1074:
1053:
1008:
910:
830:
801:
731:
609:{\displaystyle f}
587:{\displaystyle t}
480:
432:
345:
297:
119:
118:
111:
93:
4789:
4695:
4688:
4681:
4672:
4667:
4645:
4644:
4642:
4636:. Archived from
4635:
4627:
4621:
4620:
4618:
4610:
4604:
4603:
4601:
4599:
4593:
4586:
4580:Enjeti, Prasad.
4577:
4571:
4570:
4568:
4566:
4561:on 27 March 2011
4550:
4544:
4543:
4525:
4519:
4517:
4515:
4514:
4508:
4501:
4493:
4467:Polyphase system
4447:Galileo Ferraris
4362:induction motors
4346:Non-linear loads
4341:
4339:
4338:
4333:
4331:
4329:
4328:
4316:
4315:
4300:
4299:
4287:
4286:
4271:
4270:
4258:
4257:
4241:
4236:
4220:
4215:
4199:
4194:
4182:
4167:
4165:
4164:
4159:
4157:
4153:
4152:
4151:
4136:
4135:
4118:
4109:
4108:
4100:
4092:
4090:
4089:
4074:
4066:
4064:
4063:
4048:
4047:
4022:
4020:
4019:
4014:
4012:
4008:
4004:
3996:
3983:
3982:
3964:
3960:
3956:
3948:
3935:
3934:
3919:
3915:
3911:
3903:
3890:
3889:
3871:
3867:
3863:
3855:
3842:
3841:
3826:
3825:
3803:
3801:
3800:
3795:
3793:
3792:
3773:
3771:
3770:
3765:
3763:
3762:
3743:
3741:
3740:
3735:
3733:
3732:
3685:
3683:
3682:
3677:
3675:
3662:
3658:
3638:
3615:
3611:
3607:
3602:
3594:
3582:
3559:
3536:
3532:
3528:
3527:
3522:
3514:
3492:
3488:
3487:
3482:
3474:
3452:
3411:
3409:
3408:
3403:
3401:
3399:
3398:
3389:
3385:
3384:
3374:
3356:
3354:
3353:
3348:
3346:
3342:
3341:
3326:
3325:
3310:
3309:
3290:
3289:
3273:
3268:
3267:
3249:
3244:
3243:
3227:
3222:
3221:
3203:
3198:
3197:
3181:
3176:
3175:
3157:
3148:
3147:
3108:
3106:
3105:
3100:
3089:
3084:
3083:
3082:
3069:
3064:
3063:
3038:
3036:
3035:
3030:
3028:
3027:
3005:
3003:
3002:
2997:
2995:
2989:
2988:
2987:
2978:
2957:
2955:
2954:
2949:
2938:
2936:
2935:
2927:
2918:
2916:
2911:
2898:
2893:
2892:
2864:
2862:
2861:
2856:
2845:
2840:
2839:
2838:
2829:
2828:
2815:
2810:
2809:
2781:
2779:
2778:
2773:
2771:
2767:
2766:
2762:
2761:
2757:
2747:
2739:
2714:
2710:
2700:
2692:
2667:
2663:
2614:
2609:
2608:
2607:
2598:
2597:
2587:
2582:
2581:
2553:
2551:
2550:
2545:
2543:
2539:
2538:
2534:
2524:
2516:
2491:
2469:
2464:
2463:
2462:
2453:
2452:
2442:
2437:
2436:
2414:
2412:
2411:
2406:
2404:
2400:
2399:
2395:
2385:
2377:
2352:
2330:
2325:
2324:
2323:
2314:
2313:
2303:
2298:
2297:
2275:
2273:
2272:
2267:
2265:
2261:
2260:
2256:
2229:
2207:
2202:
2201:
2200:
2191:
2190:
2180:
2175:
2174:
2145:
2143:
2142:
2137:
2135:
2131:
2121:
2113:
2094:
2090:
2086:
2078:
2059:
2058:
2049:
2048:
2036:
2035:
2023:
2022:
2007:
2006:
1984:
1982:
1981:
1976:
1974:
1970:
1960:
1952:
1933:
1929:
1925:
1917:
1898:
1897:
1888:
1887:
1875:
1874:
1862:
1861:
1846:
1845:
1823:
1821:
1820:
1815:
1813:
1809:
1788:
1771:
1770:
1761:
1760:
1748:
1747:
1735:
1734:
1719:
1718:
1693:
1691:
1690:
1685:
1683:
1679:
1669:
1661:
1642:
1641:
1629:
1628:
1606:
1604:
1603:
1598:
1596:
1592:
1582:
1574:
1555:
1554:
1542:
1541:
1519:
1517:
1516:
1511:
1509:
1505:
1484:
1483:
1471:
1470:
1445:
1443:
1442:
1437:
1435:
1433:
1432:
1424:
1418:
1417:
1408:
1403:
1402:
1380:
1378:
1377:
1372:
1370:
1369:
1357:
1349:
1324:
1322:
1321:
1316:
1314:
1309:
1308:
1307:
1294:
1289:
1288:
1263:
1261:
1260:
1255:
1253:
1252:
1230:
1228:
1227:
1222:
1220:
1214:
1213:
1212:
1203:
1185:
1183:
1182:
1177:
1162:
1160:
1159:
1154:
1149:
1147:
1139:
1137:
1132:
1119:
1114:
1113:
1085:
1083:
1082:
1077:
1075:
1067:
1062:
1058:
1054:
1046:
1030:
1029:
1017:
1013:
1009:
1001:
985:
984:
966:
965:
941:
939:
938:
933:
927:
926:
911:
908:
903:
891:
866:
864:
863:
858:
856:
852:
851:
838:
822:
821:
802:
796:
791:
779:
770:
769:
743:
741:
740:
735:
732:
727:
726:
717:
667:
655:
643:
615:
613:
612:
607:
593:
591:
590:
585:
571:
569:
568:
563:
532:
530:
529:
524:
522:
521:
499:
497:
496:
491:
489:
485:
481:
473:
454:
453:
441:
437:
433:
425:
406:
405:
393:
392:
364:
362:
361:
356:
354:
350:
346:
338:
319:
318:
306:
302:
298:
290:
271:
270:
258:
257:
229:
227:
226:
221:
217:
200:
199:
187:
186:
114:
107:
103:
100:
94:
92:
51:
27:
19:
4797:
4796:
4792:
4791:
4790:
4788:
4787:
4786:
4762:
4761:
4760:
4755:
4702:
4699:
4664:
4651:
4648:
4640:
4633:
4629:
4628:
4624:
4616:
4612:
4611:
4607:
4597:
4595:
4591:
4584:
4579:
4578:
4574:
4564:
4562:
4552:
4551:
4547:
4540:
4527:
4526:
4522:
4512:
4510:
4506:
4499:
4495:
4494:
4490:
4486:
4481:
4437:
4425:
4417:motor–generator
4413:phase converter
4379:
4357:
4348:
4317:
4304:
4288:
4275:
4259:
4246:
4176:
4175:
4140:
4124:
4123:
4119:
4078:
4052:
4036:
4031:
4030:
3994:
3990:
3971:
3946:
3942:
3923:
3901:
3897:
3878:
3853:
3849:
3830:
3814:
3809:
3808:
3781:
3776:
3775:
3751:
3746:
3745:
3721:
3716:
3715:
3712:complex numbers
3696:
3673:
3672:
3660:
3659:
3648:
3628:
3613:
3612:
3595:
3589:
3572:
3549:
3534:
3533:
3515:
3506:
3502:
3475:
3466:
3462:
3442:
3429:
3417:
3416:
3390:
3376:
3375:
3362:
3361:
3344:
3343:
3330:
3314:
3298:
3291:
3281:
3275:
3274:
3250:
3232:
3204:
3186:
3158:
3149:
3136:
3127:
3126:
3116:
3074:
3070:
3049:
3044:
3043:
3013:
3008:
3007:
2979:
2966:
2965:
2919:
2899:
2878:
2873:
2872:
2830:
2820:
2816:
2795:
2790:
2789:
2728:
2724:
2681:
2677:
2650:
2646:
2639:
2635:
2619:
2615:
2599:
2589:
2588:
2567:
2562:
2561:
2505:
2501:
2481:
2474:
2470:
2454:
2444:
2443:
2425:
2420:
2419:
2366:
2362:
2342:
2335:
2331:
2315:
2305:
2304:
2286:
2281:
2280:
2243:
2239:
2219:
2212:
2208:
2192:
2182:
2181:
2163:
2158:
2157:
2105:
2101:
2070:
2066:
2050:
2040:
2024:
2011:
1995:
1990:
1989:
1944:
1940:
1909:
1905:
1889:
1879:
1863:
1850:
1834:
1829:
1828:
1799:
1795:
1778:
1762:
1752:
1736:
1723:
1707:
1702:
1701:
1653:
1649:
1633:
1617:
1612:
1611:
1566:
1562:
1546:
1530:
1525:
1524:
1495:
1491:
1475:
1459:
1454:
1453:
1419:
1409:
1394:
1389:
1388:
1358:
1334:
1333:
1299:
1295:
1274:
1269:
1268:
1238:
1233:
1232:
1204:
1191:
1190:
1168:
1167:
1140:
1120:
1099:
1094:
1093:
1038:
1034:
1021:
993:
989:
976:
957:
946:
945:
912:
876:
875:
854:
853:
840:
823:
807:
804:
803:
771:
758:
749:
748:
718:
691:
690:
687:
678:
671:
668:
659:
656:
647:
644:
628:
598:
597:
576:
575:
537:
536:
513:
508:
507:
465:
461:
445:
417:
413:
397:
375:
370:
369:
330:
326:
310:
282:
278:
262:
240:
235:
234:
207:
191:
169:
164:
163:
157:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
4795:
4793:
4785:
4784:
4779:
4774:
4772:Electric power
4764:
4763:
4757:
4756:
4754:
4753:
4748:
4743:
4738:
4728:
4723:
4718:
4713:
4707:
4704:
4703:
4700:
4698:
4697:
4690:
4683:
4675:
4669:
4668:
4662:
4647:
4646:
4643:on 2014-02-21.
4622:
4605:
4572:
4545:
4539:978-1435400290
4538:
4520:
4487:
4485:
4482:
4480:
4479:
4474:
4469:
4464:
4459:
4454:
4452:John Hopkinson
4449:
4444:
4438:
4436:
4433:
4424:
4421:
4378:
4375:
4356:
4353:
4347:
4344:
4343:
4342:
4327:
4324:
4320:
4314:
4311:
4307:
4303:
4298:
4295:
4291:
4285:
4282:
4278:
4274:
4269:
4266:
4262:
4256:
4253:
4249:
4245:
4240:
4235:
4232:
4228:
4224:
4219:
4214:
4211:
4207:
4203:
4198:
4193:
4190:
4186:
4169:
4168:
4156:
4150:
4147:
4143:
4139:
4134:
4131:
4127:
4122:
4116:
4112:
4106:
4103:
4098:
4095:
4088:
4085:
4081:
4077:
4072:
4069:
4062:
4059:
4055:
4051:
4046:
4043:
4039:
4024:
4023:
4011:
4007:
4002:
3999:
3993:
3989:
3986:
3981:
3978:
3974:
3970:
3967:
3963:
3959:
3954:
3951:
3945:
3941:
3938:
3933:
3930:
3926:
3922:
3918:
3914:
3909:
3906:
3900:
3896:
3893:
3888:
3885:
3881:
3877:
3874:
3870:
3866:
3861:
3858:
3852:
3848:
3845:
3840:
3837:
3833:
3829:
3824:
3821:
3817:
3791:
3788:
3784:
3761:
3758:
3754:
3731:
3728:
3724:
3695:
3692:
3687:
3686:
3671:
3668:
3665:
3663:
3661:
3657:
3654:
3651:
3647:
3644:
3641:
3637:
3634:
3631:
3627:
3624:
3621:
3618:
3616:
3614:
3610:
3605:
3601:
3598:
3592:
3588:
3585:
3581:
3578:
3575:
3571:
3568:
3565:
3562:
3558:
3555:
3552:
3548:
3545:
3542:
3539:
3537:
3535:
3531:
3525:
3521:
3518:
3512:
3509:
3505:
3501:
3498:
3495:
3491:
3485:
3481:
3478:
3472:
3469:
3465:
3461:
3458:
3455:
3451:
3448:
3445:
3441:
3438:
3435:
3432:
3430:
3428:
3425:
3424:
3397:
3393:
3388:
3383:
3379:
3372:
3369:
3358:
3357:
3340:
3337:
3333:
3329:
3324:
3321:
3317:
3313:
3308:
3305:
3301:
3297:
3294:
3292:
3288:
3284:
3280:
3277:
3276:
3271:
3266:
3263:
3260:
3257:
3253:
3247:
3242:
3239:
3235:
3230:
3225:
3220:
3217:
3214:
3211:
3207:
3201:
3196:
3193:
3189:
3184:
3179:
3174:
3171:
3168:
3165:
3161:
3155:
3152:
3150:
3146:
3143:
3139:
3135:
3134:
3115:
3112:
3111:
3110:
3098:
3095:
3092:
3087:
3081:
3077:
3073:
3067:
3062:
3059:
3056:
3052:
3026:
3023:
3020:
3016:
2992:
2986:
2982:
2976:
2973:
2959:
2958:
2947:
2944:
2941:
2934:
2930:
2926:
2922:
2915:
2910:
2906:
2902:
2896:
2891:
2888:
2885:
2881:
2866:
2865:
2854:
2851:
2848:
2843:
2837:
2833:
2827:
2823:
2819:
2813:
2808:
2805:
2802:
2798:
2783:
2782:
2770:
2765:
2760:
2756:
2753:
2750:
2745:
2742:
2737:
2734:
2731:
2727:
2723:
2720:
2717:
2713:
2709:
2706:
2703:
2698:
2695:
2690:
2687:
2684:
2680:
2676:
2673:
2670:
2666:
2662:
2659:
2656:
2653:
2649:
2645:
2642:
2638:
2634:
2631:
2628:
2625:
2622:
2618:
2612:
2606:
2602:
2596:
2592:
2585:
2580:
2577:
2574:
2570:
2555:
2554:
2542:
2537:
2533:
2530:
2527:
2522:
2519:
2514:
2511:
2508:
2504:
2500:
2497:
2494:
2490:
2487:
2484:
2480:
2477:
2473:
2467:
2461:
2457:
2451:
2447:
2440:
2435:
2432:
2428:
2416:
2415:
2403:
2398:
2394:
2391:
2388:
2383:
2380:
2375:
2372:
2369:
2365:
2361:
2358:
2355:
2351:
2348:
2345:
2341:
2338:
2334:
2328:
2322:
2318:
2312:
2308:
2301:
2296:
2293:
2289:
2277:
2276:
2264:
2259:
2255:
2252:
2249:
2246:
2242:
2238:
2235:
2232:
2228:
2225:
2222:
2218:
2215:
2211:
2205:
2199:
2195:
2189:
2185:
2178:
2173:
2170:
2166:
2147:
2146:
2134:
2130:
2127:
2124:
2119:
2116:
2111:
2108:
2104:
2100:
2097:
2093:
2089:
2084:
2081:
2076:
2073:
2069:
2065:
2062:
2057:
2053:
2047:
2043:
2039:
2034:
2031:
2027:
2021:
2018:
2014:
2010:
2005:
2002:
1998:
1986:
1985:
1973:
1969:
1966:
1963:
1958:
1955:
1950:
1947:
1943:
1939:
1936:
1932:
1928:
1923:
1920:
1915:
1912:
1908:
1904:
1901:
1896:
1892:
1886:
1882:
1878:
1873:
1870:
1866:
1860:
1857:
1853:
1849:
1844:
1841:
1837:
1825:
1824:
1812:
1808:
1805:
1802:
1798:
1794:
1791:
1787:
1784:
1781:
1777:
1774:
1769:
1765:
1759:
1755:
1751:
1746:
1743:
1739:
1733:
1730:
1726:
1722:
1717:
1714:
1710:
1695:
1694:
1682:
1678:
1675:
1672:
1667:
1664:
1659:
1656:
1652:
1648:
1645:
1640:
1636:
1632:
1627:
1624:
1620:
1608:
1607:
1595:
1591:
1588:
1585:
1580:
1577:
1572:
1569:
1565:
1561:
1558:
1553:
1549:
1545:
1540:
1537:
1533:
1521:
1520:
1508:
1504:
1501:
1498:
1494:
1490:
1487:
1482:
1478:
1474:
1469:
1466:
1462:
1447:
1446:
1431:
1427:
1423:
1416:
1412:
1406:
1401:
1397:
1382:
1381:
1368:
1365:
1361:
1356:
1352:
1348:
1344:
1341:
1327:
1326:
1312:
1306:
1302:
1298:
1292:
1287:
1284:
1281:
1277:
1251:
1248:
1245:
1241:
1217:
1211:
1207:
1201:
1198:
1175:
1164:
1163:
1152:
1146:
1143:
1136:
1131:
1127:
1123:
1117:
1112:
1109:
1106:
1102:
1087:
1086:
1073:
1070:
1065:
1061:
1057:
1052:
1049:
1044:
1041:
1037:
1033:
1028:
1024:
1020:
1016:
1012:
1007:
1004:
999:
996:
992:
988:
983:
979:
975:
972:
969:
964:
960:
956:
953:
930:
925:
922:
919:
915:
907:
902:
898:
894:
888:
884:
868:
867:
850:
847:
843:
837:
833:
829:
826:
824:
820:
817:
814:
810:
806:
805:
800:
795:
790:
787:
783:
777:
774:
772:
768:
765:
761:
757:
756:
730:
725:
721:
714:
710:
707:
703:
699:
686:
683:
677:
676:Balanced loads
674:
673:
672:
669:
662:
660:
657:
650:
648:
645:
638:
627:
624:
623:
622:
619:
618:
617:
605:
595:
583:
559:
556:
553:
550:
547:
544:
534:
520:
516:
501:
500:
488:
484:
479:
476:
471:
468:
464:
460:
457:
452:
448:
444:
440:
436:
431:
428:
423:
420:
416:
412:
409:
404:
400:
396:
391:
388:
385:
382:
378:
366:
365:
353:
349:
344:
341:
336:
333:
329:
325:
322:
317:
313:
309:
305:
301:
296:
293:
288:
285:
281:
277:
274:
269:
265:
261:
256:
253:
250:
247:
243:
231:
230:
216:
213:
210:
206:
203:
198:
194:
190:
185:
182:
179:
176:
172:
156:
153:
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
4794:
4783:
4780:
4778:
4775:
4773:
4770:
4769:
4767:
4752:
4749:
4747:
4744:
4742:
4741:paper folding
4739:
4736:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4711:apportionment
4709:
4708:
4705:
4696:
4691:
4689:
4684:
4682:
4677:
4676:
4673:
4665:
4663:0-07-061285-4
4659:
4655:
4650:
4649:
4639:
4632:
4626:
4623:
4615:
4609:
4606:
4590:
4583:
4576:
4573:
4560:
4556:
4549:
4546:
4541:
4535:
4531:
4524:
4521:
4518:public domain
4505:
4498:
4492:
4489:
4483:
4478:
4477:Y-Δ transform
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4439:
4434:
4432:
4430:
4422:
4420:
4418:
4414:
4409:
4407:
4403:
4399:
4395:
4393:
4388:
4384:
4376:
4374:
4370:
4368:
4363:
4354:
4352:
4345:
4325:
4322:
4318:
4312:
4309:
4305:
4301:
4296:
4293:
4289:
4283:
4280:
4276:
4272:
4267:
4264:
4260:
4254:
4251:
4247:
4243:
4238:
4233:
4230:
4226:
4222:
4217:
4212:
4209:
4205:
4201:
4196:
4191:
4188:
4184:
4174:
4173:
4172:
4154:
4148:
4145:
4141:
4137:
4132:
4129:
4125:
4120:
4114:
4110:
4104:
4101:
4096:
4093:
4086:
4083:
4079:
4075:
4070:
4067:
4060:
4057:
4053:
4049:
4044:
4041:
4037:
4029:
4028:
4027:
4009:
4005:
4000:
3997:
3991:
3987:
3984:
3979:
3976:
3972:
3968:
3965:
3961:
3957:
3952:
3949:
3943:
3939:
3936:
3931:
3928:
3924:
3920:
3916:
3912:
3907:
3904:
3898:
3894:
3891:
3886:
3883:
3879:
3875:
3872:
3868:
3864:
3859:
3856:
3850:
3846:
3843:
3838:
3835:
3831:
3827:
3822:
3819:
3815:
3807:
3806:
3805:
3789:
3786:
3782:
3759:
3756:
3752:
3729:
3726:
3722:
3713:
3707:
3705:
3704:zero sequence
3701:
3693:
3691:
3669:
3666:
3664:
3655:
3652:
3649:
3645:
3642:
3639:
3635:
3632:
3629:
3625:
3622:
3619:
3617:
3608:
3603:
3599:
3596:
3590:
3586:
3583:
3579:
3576:
3573:
3569:
3566:
3563:
3560:
3556:
3553:
3550:
3546:
3543:
3540:
3538:
3529:
3523:
3519:
3516:
3510:
3507:
3503:
3499:
3496:
3493:
3489:
3483:
3479:
3476:
3470:
3467:
3463:
3459:
3456:
3453:
3449:
3446:
3443:
3439:
3436:
3433:
3431:
3426:
3415:
3414:
3413:
3395:
3391:
3386:
3381:
3377:
3370:
3367:
3338:
3335:
3331:
3327:
3322:
3319:
3315:
3311:
3306:
3303:
3299:
3295:
3293:
3286:
3282:
3278:
3269:
3264:
3261:
3258:
3255:
3251:
3245:
3240:
3237:
3233:
3228:
3223:
3218:
3215:
3212:
3209:
3205:
3199:
3194:
3191:
3187:
3182:
3177:
3172:
3169:
3166:
3163:
3159:
3153:
3151:
3144:
3141:
3137:
3125:
3124:
3123:
3121:
3113:
3096:
3093:
3090:
3085:
3079:
3075:
3071:
3065:
3060:
3057:
3054:
3050:
3042:
3041:
3040:
3024:
3021:
3018:
3014:
2990:
2984:
2980:
2974:
2971:
2962:
2945:
2942:
2939:
2928:
2920:
2913:
2908:
2904:
2900:
2894:
2889:
2886:
2883:
2879:
2871:
2870:
2869:
2852:
2849:
2846:
2841:
2835:
2831:
2825:
2821:
2817:
2811:
2806:
2803:
2800:
2796:
2788:
2787:
2786:
2768:
2763:
2758:
2754:
2751:
2748:
2743:
2740:
2735:
2732:
2729:
2725:
2721:
2718:
2715:
2711:
2707:
2704:
2701:
2696:
2693:
2688:
2685:
2682:
2678:
2674:
2671:
2668:
2664:
2660:
2657:
2654:
2651:
2647:
2643:
2640:
2636:
2632:
2629:
2626:
2623:
2620:
2616:
2610:
2604:
2600:
2594:
2590:
2583:
2578:
2575:
2572:
2568:
2560:
2559:
2558:
2540:
2535:
2531:
2528:
2525:
2520:
2517:
2512:
2509:
2506:
2502:
2498:
2495:
2492:
2488:
2485:
2482:
2478:
2475:
2471:
2465:
2459:
2455:
2449:
2445:
2438:
2433:
2430:
2426:
2418:
2417:
2401:
2396:
2392:
2389:
2386:
2381:
2378:
2373:
2370:
2367:
2363:
2359:
2356:
2353:
2349:
2346:
2343:
2339:
2336:
2332:
2326:
2320:
2316:
2310:
2306:
2299:
2294:
2291:
2287:
2279:
2278:
2262:
2257:
2253:
2250:
2247:
2244:
2240:
2236:
2233:
2230:
2226:
2223:
2220:
2216:
2213:
2209:
2203:
2197:
2193:
2187:
2183:
2176:
2171:
2168:
2164:
2156:
2155:
2154:
2152:
2132:
2128:
2125:
2122:
2117:
2114:
2109:
2106:
2102:
2098:
2095:
2091:
2087:
2082:
2079:
2074:
2071:
2067:
2063:
2060:
2055:
2051:
2045:
2041:
2037:
2032:
2029:
2025:
2019:
2016:
2012:
2008:
2003:
2000:
1996:
1988:
1987:
1971:
1967:
1964:
1961:
1956:
1953:
1948:
1945:
1941:
1937:
1934:
1930:
1926:
1921:
1918:
1913:
1910:
1906:
1902:
1899:
1894:
1890:
1884:
1880:
1876:
1871:
1868:
1864:
1858:
1855:
1851:
1847:
1842:
1839:
1835:
1827:
1826:
1810:
1806:
1803:
1800:
1796:
1792:
1789:
1785:
1782:
1779:
1775:
1772:
1767:
1763:
1757:
1753:
1749:
1744:
1741:
1737:
1731:
1728:
1724:
1720:
1715:
1712:
1708:
1700:
1699:
1698:
1680:
1676:
1673:
1670:
1665:
1662:
1657:
1654:
1650:
1646:
1643:
1638:
1634:
1630:
1625:
1622:
1618:
1610:
1609:
1593:
1589:
1586:
1583:
1578:
1575:
1570:
1567:
1563:
1559:
1556:
1551:
1547:
1543:
1538:
1535:
1531:
1523:
1522:
1506:
1502:
1499:
1496:
1492:
1488:
1485:
1480:
1476:
1472:
1467:
1464:
1460:
1452:
1451:
1450:
1425:
1414:
1410:
1404:
1399:
1395:
1387:
1386:
1385:
1366:
1363:
1359:
1350:
1342:
1339:
1332:
1331:
1330:
1310:
1304:
1300:
1296:
1290:
1285:
1282:
1279:
1275:
1267:
1266:
1265:
1249:
1246:
1243:
1239:
1215:
1209:
1205:
1199:
1196:
1187:
1173:
1150:
1144:
1141:
1134:
1129:
1125:
1121:
1115:
1110:
1107:
1104:
1100:
1092:
1091:
1090:
1071:
1068:
1063:
1059:
1055:
1050:
1047:
1042:
1039:
1035:
1031:
1026:
1022:
1018:
1014:
1010:
1005:
1002:
997:
994:
990:
986:
981:
977:
973:
970:
967:
962:
958:
954:
951:
944:
943:
942:
928:
923:
920:
917:
913:
905:
900:
896:
892:
886:
882:
873:
848:
845:
841:
835:
831:
827:
825:
818:
815:
812:
808:
798:
793:
788:
785:
781:
775:
773:
766:
763:
759:
747:
746:
745:
728:
723:
719:
712:
708:
705:
701:
697:
684:
682:
675:
666:
661:
654:
649:
642:
637:
635:
633:
625:
620:
603:
596:
581:
574:
573:
557:
554:
551:
548:
545:
542:
535:
518:
514:
506:
505:
504:
486:
482:
477:
474:
469:
466:
462:
458:
455:
450:
446:
442:
438:
434:
429:
426:
421:
418:
414:
410:
407:
402:
398:
394:
389:
386:
383:
380:
376:
368:
367:
351:
347:
342:
339:
334:
331:
327:
323:
320:
315:
311:
307:
303:
299:
294:
291:
286:
283:
279:
275:
272:
267:
263:
259:
254:
251:
248:
245:
241:
233:
232:
214:
211:
208:
204:
201:
196:
192:
188:
183:
180:
177:
174:
170:
162:
161:
160:
154:
152:
150:
145:
141:
137:
129:
123:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
4777:Nikola Tesla
4750:
4735:introduction
4653:
4638:the original
4625:
4608:
4596:. Retrieved
4575:
4563:. Retrieved
4559:the original
4548:
4529:
4523:
4511:. Retrieved
4491:
4462:Nikola Tesla
4426:
4410:
4391:
4387:transformers
4380:
4371:
4358:
4349:
4170:
4025:
3708:
3697:
3688:
3359:
3117:
2963:
2960:
2867:
2784:
2556:
2148:
1696:
1448:
1383:
1328:
1188:
1165:
1088:
869:
688:
679:
629:
502:
158:
133:
105:
99:October 2022
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
4598:24 November
4565:24 November
155:Definitions
4766:Categories
4721:bookmaking
4513:2012-11-21
4484:References
4394:connection
632:alternator
69:newspapers
4398:rectifier
4383:polyphase
4302:−
4273:−
4244:−
4138:−
4076:−
4050:−
4006:π
3988:
3958:π
3940:
3913:π
3895:
3865:π
3847:
3653:θ
3646:
3640:−
3633:θ
3626:
3600:π
3587:
3577:θ
3570:
3554:θ
3547:
3520:π
3508:θ
3500:
3480:π
3471:−
3468:θ
3460:
3447:θ
3440:
3279:−
3262:−
3216:−
3170:−
3097:φ
3094:
2946:φ
2943:
2853:φ
2850:
2755:φ
2752:−
2749:π
2736:−
2733:θ
2722:
2708:φ
2705:−
2702:π
2689:−
2686:θ
2675:
2661:φ
2658:−
2655:θ
2644:
2633:−
2630:φ
2627:
2532:φ
2529:−
2526:π
2513:−
2510:θ
2499:
2493:−
2486:φ
2479:
2393:φ
2390:−
2387:π
2374:−
2371:θ
2360:
2354:−
2347:φ
2340:
2254:φ
2251:−
2248:θ
2237:
2231:−
2224:φ
2217:
2129:φ
2126:−
2123:π
2110:−
2107:θ
2099:
2088:π
2075:−
2072:θ
2064:
1968:φ
1965:−
1962:π
1949:−
1946:θ
1938:
1927:π
1914:−
1911:θ
1903:
1807:φ
1804:−
1801:θ
1793:
1783:θ
1776:
1677:φ
1674:−
1671:π
1658:−
1655:θ
1647:
1590:φ
1587:−
1584:π
1571:−
1568:θ
1560:
1503:φ
1500:−
1497:θ
1489:
1367:φ
1174:θ
1056:π
1043:−
1040:θ
1032:
1011:π
998:−
995:θ
987:
971:θ
968:
832:∑
552:π
543:θ
483:π
467:θ
459:
435:π
422:−
419:θ
411:
387:−
348:π
332:θ
324:
300:π
287:−
284:θ
276:
252:−
212:θ
205:
181:−
128:frequency
4589:Archived
4504:Archived
4435:See also
4406:harmonic
626:Diagrams
4392:Scott T
503:where:
83:scholar
4746:Sudoku
4660:
4536:
4367:torque
3774:, and
2149:Using
149:phases
85:
78:
71:
64:
56:
4641:(PDF)
4634:(PDF)
4617:(PDF)
4592:(PDF)
4585:(PDF)
4507:(PDF)
4500:(PDF)
90:JSTOR
76:books
4658:ISBN
4600:2012
4567:2012
4534:ISBN
62:news
3985:sin
3937:cos
3892:sin
3844:cos
3643:sin
3623:sin
3584:cos
3567:sin
3544:sin
3497:sin
3457:sin
3437:sin
3091:cos
2940:cos
2868:or
2847:cos
2719:cos
2672:cos
2641:cos
2624:cos
2496:cos
2476:cos
2357:cos
2337:cos
2234:cos
2214:cos
2096:sin
2061:sin
1935:sin
1900:sin
1790:sin
1773:sin
1644:sin
1557:sin
1486:sin
1023:sin
978:sin
959:sin
456:sin
408:sin
321:sin
273:sin
202:sin
134:In
45:by
4768::
4502:.
4402:DC
3744:,
3412::
3122:.
3006:,
2153::
138:,
4737:)
4733:(
4694:e
4687:t
4680:v
4666:.
4619:.
4602:.
4569:.
4542:.
4516:.
4326:3
4323:L
4319:I
4313:2
4310:L
4306:I
4297:3
4294:L
4290:I
4284:1
4281:L
4277:I
4268:2
4265:L
4261:I
4255:1
4252:L
4248:I
4239:2
4234:3
4231:L
4227:I
4223:+
4218:2
4213:2
4210:L
4206:I
4202:+
4197:2
4192:1
4189:L
4185:I
4155:)
4149:3
4146:L
4142:I
4133:2
4130:L
4126:I
4121:(
4115:2
4111:3
4105:j
4102:+
4097:2
4094:1
4087:3
4084:L
4080:I
4071:2
4068:1
4061:2
4058:L
4054:I
4045:1
4042:L
4038:I
4010:)
4001:3
3998:4
3992:(
3980:3
3977:L
3973:I
3969:j
3966:+
3962:)
3953:3
3950:4
3944:(
3932:3
3929:L
3925:I
3921:+
3917:)
3908:3
3905:2
3899:(
3887:2
3884:L
3880:I
3876:j
3873:+
3869:)
3860:3
3857:2
3851:(
3839:2
3836:L
3832:I
3828:+
3823:1
3820:L
3816:I
3790:3
3787:L
3783:I
3760:2
3757:L
3753:I
3730:1
3727:L
3723:I
3670:0
3667:=
3656:)
3650:(
3636:)
3630:(
3620:=
3609:)
3604:3
3597:2
3591:(
3580:)
3574:(
3564:2
3561:+
3557:)
3551:(
3541:=
3530:)
3524:3
3517:2
3511:+
3504:(
3494:+
3490:)
3484:3
3477:2
3464:(
3454:+
3450:)
3444:(
3434:=
3427:i
3396:P
3392:V
3387:R
3382:N
3378:I
3371:=
3368:i
3339:3
3336:L
3332:I
3328:+
3323:2
3320:L
3316:I
3312:+
3307:1
3304:L
3300:I
3296:=
3287:N
3283:I
3270:R
3265:N
3259:3
3256:L
3252:V
3246:=
3241:3
3238:L
3234:I
3229:,
3224:R
3219:N
3213:2
3210:L
3206:V
3200:=
3195:2
3192:L
3188:I
3183:,
3178:R
3173:N
3167:1
3164:L
3160:V
3154:=
3145:1
3142:L
3138:I
3109:.
3086:Z
3080:2
3076:V
3072:3
3066:=
3061:T
3058:O
3055:T
3051:P
3025:T
3022:O
3019:T
3015:P
2991:2
2985:p
2981:V
2975:=
2972:V
2933:|
2929:Z
2925:|
2921:2
2914:2
2909:P
2905:V
2901:3
2895:=
2890:T
2887:O
2884:T
2880:P
2842:2
2836:P
2832:I
2826:P
2822:V
2818:3
2812:=
2807:T
2804:O
2801:T
2797:P
2769:}
2764:]
2759:)
2744:3
2741:8
2730:2
2726:(
2716:+
2712:)
2697:3
2694:4
2683:2
2679:(
2669:+
2665:)
2652:2
2648:(
2637:[
2621:3
2617:{
2611:2
2605:P
2601:I
2595:P
2591:V
2584:=
2579:T
2576:O
2573:T
2569:P
2541:]
2536:)
2521:3
2518:8
2507:2
2503:(
2489:)
2483:(
2472:[
2466:2
2460:P
2456:I
2450:P
2446:V
2439:=
2434:3
2431:L
2427:P
2402:]
2397:)
2382:3
2379:4
2368:2
2364:(
2350:)
2344:(
2333:[
2327:2
2321:P
2317:I
2311:P
2307:V
2300:=
2295:2
2292:L
2288:P
2263:]
2258:)
2245:2
2241:(
2227:)
2221:(
2210:[
2204:2
2198:P
2194:I
2188:P
2184:V
2177:=
2172:1
2169:L
2165:P
2133:)
2118:3
2115:4
2103:(
2092:)
2083:3
2080:4
2068:(
2056:P
2052:I
2046:P
2042:V
2038:=
2033:3
2030:L
2026:I
2020:3
2017:L
2013:V
2009:=
2004:3
2001:L
1997:P
1972:)
1957:3
1954:2
1942:(
1931:)
1922:3
1919:2
1907:(
1895:P
1891:I
1885:P
1881:V
1877:=
1872:2
1869:L
1865:I
1859:2
1856:L
1852:V
1848:=
1843:2
1840:L
1836:P
1811:)
1797:(
1786:)
1780:(
1768:P
1764:I
1758:P
1754:V
1750:=
1745:1
1742:L
1738:I
1732:1
1729:L
1725:V
1721:=
1716:1
1713:L
1709:P
1681:)
1666:3
1663:4
1651:(
1639:P
1635:I
1631:=
1626:3
1623:L
1619:I
1594:)
1579:3
1576:2
1564:(
1552:P
1548:I
1544:=
1539:2
1536:L
1532:I
1507:)
1493:(
1481:P
1477:I
1473:=
1468:1
1465:L
1461:I
1430:|
1426:Z
1422:|
1415:P
1411:V
1405:=
1400:P
1396:I
1364:j
1360:e
1355:|
1351:Z
1347:|
1343:=
1340:Z
1325:.
1311:R
1305:2
1301:V
1297:3
1291:=
1286:T
1283:O
1280:T
1276:P
1250:T
1247:O
1244:T
1240:P
1216:2
1210:p
1206:V
1200:=
1197:V
1151:.
1145:R
1142:2
1135:2
1130:P
1126:V
1122:3
1116:=
1111:T
1108:O
1105:T
1101:P
1072:2
1069:3
1064:=
1060:)
1051:3
1048:4
1036:(
1027:2
1019:+
1015:)
1006:3
1003:2
991:(
982:2
974:+
963:2
955:=
952:p
929:R
924:T
921:O
918:T
914:P
906:2
901:P
897:V
893:1
887:=
883:p
849:i
846:L
842:P
836:i
828:=
819:T
816:O
813:T
809:P
799:R
794:2
789:i
786:L
782:V
776:=
767:i
764:L
760:P
729:R
724:2
720:V
713:=
709:I
706:V
702:=
698:P
604:f
582:t
558:t
555:f
549:2
546:=
519:P
515:V
487:)
478:3
475:2
470:+
463:(
451:P
447:V
443:=
439:)
430:3
427:4
415:(
403:P
399:V
395:=
390:N
384:3
381:L
377:V
352:)
343:3
340:4
335:+
328:(
316:P
312:V
308:=
304:)
295:3
292:2
280:(
268:P
264:V
260:=
255:N
249:2
246:L
242:V
215:)
209:(
197:P
193:V
189:=
184:N
178:1
175:L
171:V
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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