651:
flow for an out-of-phase system, however real power at certain times will not be transferred, because there will be points during which instantaneous current is positive while instantaneous voltage is negative, or vice versa, implying negative power transfer. Hence, real work is not performed when power transfer is "negative". However, current still flows even when a system is out-of-phase, which causes transmission lines to heat up due to current flow. Consequently, transmission lines can only heat up so much (or else they would physically sag too much, due to the heat expanding the metal transmission lines), so transmission line operators have a "ceiling" on the amount of current that can flow through a given line, and excessive inductive reactance can limit the power capacity of a line. Power providers utilize capacitors to shift the phase and minimize the losses, based on usage patterns.
2628:
2411:
141:
reactive element but is stored instead. Third, reactances can be negative so that they can 'cancel' each other out. Finally, the main circuit elements that have reactance (capacitors and inductors) have a frequency dependent reactance, unlike resistors which have the same resistance for all frequencies, at least in the ideal case.
639:
changing as a result of current that oscillates back and forth. It is this change in magnetic field that induces another electric current to flow in the same wire (counter-EMF), in a direction such as to oppose the flow of the current originally responsible for producing the magnetic field (known as Lenz's Law). Hence,
650:
In electric power systems, inductive reactance (and capacitive reactance, however inductive reactance is more common) can limit the power capacity of an AC transmission line, because power is not completely transferred when voltage and current are out-of-phase (detailed above). That is, current will
140:
There are several important differences between reactance and resistance, though. First, reactance changes the phase so that the current through the element is shifted by a quarter of a cycle relative to the phase of the voltage applied across the element. Second, power is not dissipated in a purely
638:
Inductive reactance is a property exhibited by an inductor, and inductive reactance exists based on the fact that an electric current produces a magnetic field around it. In the context of an AC circuit (although this concept applies any time current is changing), this magnetic field is constantly
128:
Reactance is similar to resistance in that larger reactance leads to smaller currents for the same applied voltage. Further, a circuit made entirely of elements that have only reactance (and no resistance) can be treated the same way as a circuit made entirely of resistances. These same techniques
646:
For an ideal inductor in an AC circuit, the inhibitive effect on change in current flow results in a delay, or a phase shift, of the alternating current with respect to alternating voltage. Specifically, an ideal inductor (with no resistance) will cause the current to lag the voltage by a quarter
623:
Driven by an AC supply (ideal AC current source), a capacitor will only accumulate a limited amount of charge before the potential difference changes polarity and the charge is returned to the source. The higher the frequency, the less charge will accumulate and the smaller the opposition to the
2623:{\displaystyle {\begin{aligned}\mathbf {Z} _{C}&={1 \over \omega C}e^{-\mathbf {j} {\pi \over 2}}=\mathbf {j} \left({-{\frac {1}{\omega C}}}\right)=\mathbf {j} X_{C}\\\mathbf {Z} _{L}&=\omega Le^{\mathbf {j} {\pi \over 2}}=\mathbf {j} \omega L=\mathbf {j} X_{L}\quad \end{aligned}}}
291:
There are two choices in the literature for defining reactance for a capacitor. One is to use a uniform notion of reactance as the imaginary part of impedance, in which case the reactance of a capacitor is the negative number,
2662:
phase difference) with the sinusoidal current through the component. The component alternately absorbs energy from the circuit and then returns energy to the circuit, thus a pure reactance does not dissipate power.
1096:
368:
444:
2277:
1924:
930:
1270:
1356:
2403:
2416:
61:
occurs in reactance; instead, the reactance stores energy until a quarter-cycle later when the energy is returned to the circuit. Greater reactance gives smaller current for the same applied
1675:
1611:
1506:
1145:
782:
2141:
2079:
2660:
2358:
2325:
2110:
1210:
1464:
1183:
494:
1768:
1746:
1724:
1698:
1534:
2016:
571:
263:
2199:
2172:
2039:
1990:
1957:
1826:
1799:
679:
213:
1437:
1412:
523:
1846:
1632:
1564:
1293:
1007:
987:
964:
855:
835:
808:
729:
706:
591:
551:
286:
240:
110:
1774:
When both a capacitor and an inductor are placed in series in a circuit, their contributions to the total circuit impedance are opposite. Capacitive reactance
2360:
radians for an inductive reactance. Without knowledge of both the resistance and reactance the relationship between voltage and current cannot be determined.
2888:
1015:
298:
533:
from flowing through the dielectric). As frequency increases, the magnitude of reactance decreases, allowing more current to flow. As
153:
1378:
has a time-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.
380:
2207:
1854:
2873:
2291:
863:
17:
1218:
1301:
2366:
2725:
1637:
1573:
1472:
1155:
1150:
Any conductor of finite dimensions has inductance; the inductance is made larger by the multiple turns in an
1104:
682:
736:
2893:
2118:
2056:
450:
In this case however one needs to remember to add a negative sign for the impedance of a capacitor, i.e.
2636:
2334:
2301:
31:
2087:
1191:
2760:
1445:
1164:
1701:
453:
1751:
1729:
1707:
1681:
1517:
116:
has zero reactance, whereas ideal reactors have no shunt conductance and no series resistance. As
2201:
are assumed both positive by definition, then the intermediary formula changes to a difference:
80:
alternating current going through a circuit element. Like resistance, reactance is measured in
2827:
216:
2363:
The origin of the different signs for capacitive and inductive reactance is the phase factor
2001:
1567:
1415:
1186:
556:
248:
175:
2177:
2150:
2021:
1968:
1935:
1804:
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1537:
1440:
1387:
1151:
657:
620:
associated with the charge exactly balances the applied voltage, the current goes to zero.
191:
134:
54:
8:
2633:
For a reactive component the sinusoidal voltage across the component is in quadrature (a
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1159:
502:
171:
42:
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1831:
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due to the accumulated charge is the source of the opposition to the current. When the
576:
536:
271:
225:
95:
1748:
is used so as not to confuse the imaginary unit with current, commonly represented by
1101:
making it appear as if the inductive reactance to a square wave was about 19% smaller
129:
can also be used to combine elements with reactance with elements with resistance but
2745:
2721:
609:
605:
243:
526:
530:
188:
is an opposition to the change of voltage across an element. Capacitive reactance
2764:
2717:
2706:
1634:
is the reactance, measured in ohms. It is the imaginary part of the impedance:
1363:
613:
601:
130:
2882:
2713:
1367:
1362:
The counter-emf is the source of the opposition to current flow. A constant
594:
120:
increases, inductive reactance increases and capacitive reactance decreases.
73:
58:
2703:
525:, the magnitude of the capacitor's reactance is infinite, behaving like an
2677:
2290:
The phase of the voltage across a purely reactive device (i.e. with zero
1993:
1371:
936:
266:
165:
57:; however, while both elements involve transfer of electrical energy, no
50:
1960:
944:
811:
788:
709:
633:
374:
Another choice is to define capacitive reactance as a positive number,
179:
46:
967:
815:
686:
617:
220:
117:
77:
69:
1091:{\displaystyle I_{L}={A\pi ^{2} \over 8\omega L}={A\pi \over 16fL}}
940:
113:
363:{\displaystyle X_{C}=-{\frac {1}{\omega C}}=-{\frac {1}{2\pi fC}}}
62:
439:{\displaystyle X_{C}={\frac {1}{\omega C}}={\frac {1}{2\pi fC}}}
2769:
Transactions of the
American Institute of Electrical Engineers
2272:{\displaystyle {X=X_{L}-X_{C}=\omega L-{\frac {1}{\omega C}}}}
1919:{\displaystyle {X=X_{L}+X_{C}=\omega L-{\frac {1}{\omega C}}}}
643:
is an opposition to the change of current through an element.
133:
are typically needed. This is treated below in the section on
2042:
925:{\displaystyle I_{L}={A \over \omega L}={A \over 2\pi fL}.}
2704:
Young, Hugh D.; Roger A. Freedman; A. Lewis Ford (2004) .
1570:, measured in ohms. It is the real part of the impedance:
30:"Reactance (physics)" redirects here. For other uses, see
1541:
148:
was first suggested by French engineer M. Hospitalier in
81:
2750:(1st ed.). New York: Tab Books. pp. 229, 232.
1265:{\displaystyle {\mathcal {E}}=-{{d\Phi _{B}} \over dt}}
731:, which depends on the physical shape of the inductor:
2641:
2339:
2306:
2122:
2091:
2060:
1351:{\displaystyle {\mathcal {E}}=-N{d\Phi _{B} \over dt}}
1195:
1185:(voltage opposing current) due to a rate-of-change of
966:
in series with a square wave AC voltage source of RMS
53:. Along with resistance, it is one of two elements of
2843:
Hughes E., Hiley J., Brown K., Smith I.McK., (2012).
2639:
2414:
2398:{\displaystyle e^{\pm \mathbf {j} {\frac {\pi }{2}}}}
2369:
2337:
2304:
2210:
2180:
2153:
2121:
2090:
2059:
2024:
2004:
1971:
1938:
1857:
1834:
1807:
1780:
1754:
1732:
1710:
1684:
1640:
1620:
1576:
1552:
1520:
1475:
1448:
1423:
1398:
1366:
has a zero rate-of-change, and sees an inductor as a
1304:
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1221:
1194:
1167:
1107:
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2010:
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488:
438:
362:
280:
257:
234:
207:
152:on 10 May 1893. It was officially adopted by the
104:
27:Opposition to current by inductance or capacitance
2787:, page 274. New York: John Wiley & Sons, Inc.
1370:(it is typically made from a material with a low
2880:
2143:, the total reactance is said to be capacitive.
1670:{\displaystyle {X={\text{Im}}{(\mathbf {Z} )}}}
1606:{\displaystyle {R={\text{Re}}{(\mathbf {Z} )}}}
1158:of electromagnetic induction gives the counter-
2771:, vol. 11, pp. 640–648, January–December 1894.
2081:, the total reactance is said to be inductive;
1501:{\displaystyle \mathbf {Z} =R+\mathbf {j} X}
2845:Hughes Electrical and Electronic Technology
2814:Introduction to Circuit Analysis and Design
604:voltage across a capacitor causes positive
123:
1275:For an inductor consisting of a coil with
154:American Institute of Electrical Engineers
2874:National magnet lab, inductive reactance
2808:
2806:
2747:The Benchtop Electronics Reference Manual
2112:, then the impedance is purely resistive;
943:, the average current flowing through an
2860:, 5th ed., Cengage Learning, pp. 554-558
2839:
2837:
1147:than the reactance to the AC sine wave.
92:reactance. It is denoted by the symbol
59:dissipation of electrical energy as heat
2779:
2777:
2708:Sears and Zemansky's University Physics
2327:radians for a capacitive reactance and
1140:{\displaystyle X_{L}={16 \over \pi }fL}
787:The average current flowing through an
608:to accumulate on one side and negative
573:, the capacitor's reactance approaches
159:
14:
2881:
2803:
2737:
939:has multiple amplitudes at sinusoidal
777:{\displaystyle X_{L}=\omega L=2\pi fL}
627:
2889:Electrical resistance and conductance
2858:Circuit Analysis: Theory and Practice
2834:
2743:
2285:
612:to accumulate on the other side; the
2847:, 11th edition, Pearson, pp. 237-241
2774:
2282:but the ultimate value is the same.
2136:{\displaystyle \scriptstyle X<0}
2074:{\displaystyle \scriptstyle X>0}
24:
2796:Hayt, W.H., Kimmerly J.E. (2007).
2785:Basic Engineering Circuit Analysis
2655:{\displaystyle {\tfrac {\pi }{2}}}
2353:{\displaystyle {\tfrac {\pi }{2}}}
2320:{\displaystyle {\tfrac {\pi }{2}}}
1828:contribute to the total reactance
1328:
1307:
1242:
1224:
1170:
560:
88:reactance and negative indicating
84:, with positive values indicating
25:
2905:
2867:
2856:Robbins, A.H., Miller W. (2012).
2601:
2587:
2567:
2538:
2518:
2482:
2462:
2421:
2379:
2105:{\displaystyle \scriptstyle X=0}
1756:
1734:
1712:
1686:
1658:
1594:
1522:
1491:
1477:
1451:
1205:{\displaystyle \scriptstyle {B}}
2615:
41:is the opposition presented to
2850:
2819:
2800:, 7th ed., McGraw-Hill, p. 388
2790:
2754:
1662:
1654:
1598:
1590:
1459:{\displaystyle {\mathbf {Z} }}
1178:{\displaystyle {\mathcal {E}}}
13:
1:
2825:Horowitz P., Hill W. (2015).
2683:
489:{\displaystyle Z_{c}=-jX_{c}}
68:Reactance is used to compute
2798:Engineering Circuit Analysis
2744:Veley, Victor F. C. (1987).
2693:John Wiley & Sons, 2011.
1996:reactance, measured in ohms;
1963:reactance, measured in ohms;
1763:{\displaystyle \mathbf {i} }
1741:{\displaystyle \mathbf {j} }
1726:in non-electrical formulas.
1719:{\displaystyle \mathbf {i} }
1693:{\displaystyle \mathbf {j} }
1529:{\displaystyle \mathbf {Z} }
1381:
170:A capacitor consists of two
7:
2698:Foundations of Electronics,
2666:
10:
2910:
2689:Shamieh C. and McComb G.,
2018:is the angular frequency,
1385:
631:
163:
29:
2761:Charles Proteus Steinmetz
1704:, usually represented by
814:AC voltage source of RMS
685:to the sinusoidal signal
2691:Electronics for Dummies,
1801:and inductive reactance
1702:square root of minus one
1212:through a current loop.
124:Comparison to resistance
37:In electrical circuits,
2700:Cengage Learning, 2002.
2041:times the frequency in
2011:{\displaystyle \omega }
566:{\displaystyle \infty }
258:{\displaystyle \omega }
18:Reactance (electronics)
2828:The Art of Electronics
2812:Glisson, T.H. (2011).
2656:
2624:
2399:
2354:
2321:
2273:
2195:
2168:
2137:
2106:
2075:
2035:
2012:
1986:
1953:
1920:
1842:
1822:
1795:
1764:
1742:
1720:
1694:
1671:
1628:
1607:
1560:
1530:
1502:
1460:
1433:
1408:
1352:
1289:
1266:
1206:
1179:
1141:
1092:
1003:
983:
960:
926:
851:
831:
804:
778:
725:
702:
675:
587:
567:
547:
519:
490:
440:
364:
282:
259:
236:
217:inversely proportional
209:
150:L'Industrie Electrique
106:
2657:
2625:
2400:
2355:
2322:
2274:
2196:
2194:{\displaystyle X_{C}}
2169:
2167:{\displaystyle X_{L}}
2147:Note however that if
2138:
2107:
2076:
2036:
2034:{\displaystyle 2\pi }
2013:
1987:
1985:{\displaystyle X_{C}}
1954:
1952:{\displaystyle X_{L}}
1921:
1843:
1823:
1821:{\displaystyle X_{L}}
1796:
1794:{\displaystyle X_{C}}
1765:
1743:
1721:
1695:
1672:
1629:
1608:
1561:
1531:
1503:
1461:
1434:
1409:
1353:
1290:
1267:
1207:
1187:magnetic flux density
1180:
1142:
1093:
1004:
984:
961:
927:
852:
832:
805:
779:
726:
703:
676:
674:{\displaystyle X_{L}}
600:The application of a
588:
568:
548:
520:
491:
441:
365:
283:
260:
237:
210:
208:{\displaystyle X_{C}}
107:
2763:, Frederick Bedell,
2637:
2412:
2367:
2335:
2302:
2292:parasitic resistance
2208:
2178:
2151:
2119:
2088:
2057:
2022:
2002:
1969:
1936:
1855:
1832:
1805:
1778:
1752:
1730:
1708:
1682:
1638:
1618:
1574:
1550:
1518:
1473:
1446:
1421:
1396:
1388:Electrical impedance
1302:
1279:
1219:
1192:
1165:
1152:electromagnetic coil
1105:
1016:
993:
973:
950:
864:
841:
821:
794:
737:
715:
692:
658:
654:Inductive reactance
577:
557:
537:
503:
454:
381:
299:
272:
249:
226:
192:
186:Capacitive reactance
160:Capacitive reactance
96:
2894:Physical quantities
1432:{\displaystyle {R}}
1407:{\displaystyle {X}}
1376:alternating current
641:inductive reactance
628:Inductive reactance
518:{\displaystyle f=0}
43:alternating current
2816:, Springer, p. 408
2783:Irwin, D. (2002).
2673:Magnetic reactance
2652:
2650:
2620:
2618:
2405:in the impedance.
2395:
2350:
2348:
2317:
2315:
2286:Phase relationship
2269:
2191:
2164:
2133:
2132:
2102:
2101:
2071:
2070:
2031:
2008:
1982:
1949:
1916:
1838:
1818:
1791:
1760:
1738:
1716:
1690:
1667:
1624:
1603:
1556:
1526:
1498:
1456:
1439:are components of
1429:
1404:
1348:
1295:loops this gives:
1285:
1262:
1202:
1201:
1175:
1137:
1088:
999:
979:
956:
922:
847:
827:
800:
774:
721:
698:
671:
593:, behaving like a
583:
563:
543:
515:
486:
436:
360:
278:
255:
232:
205:
178:, also known as a
102:
2649:
2579:
2507:
2474:
2451:
2391:
2347:
2314:
2266:
1913:
1841:{\displaystyle X}
1651:
1627:{\displaystyle X}
1587:
1559:{\displaystyle R}
1346:
1288:{\displaystyle N}
1260:
1129:
1086:
1060:
1002:{\displaystyle f}
982:{\displaystyle A}
959:{\displaystyle L}
917:
893:
850:{\displaystyle f}
830:{\displaystyle A}
810:in series with a
803:{\displaystyle L}
724:{\displaystyle L}
701:{\displaystyle f}
586:{\displaystyle 0}
546:{\displaystyle f}
434:
410:
358:
331:
281:{\displaystyle C}
244:angular frequency
235:{\displaystyle f}
105:{\displaystyle X}
16:(Redirected from
2901:
2861:
2854:
2848:
2841:
2832:
2831:, 3rd ed., p. 42
2823:
2817:
2810:
2801:
2794:
2788:
2781:
2772:
2758:
2752:
2751:
2741:
2731:
2711:
2661:
2659:
2658:
2653:
2651:
2642:
2629:
2627:
2626:
2621:
2619:
2614:
2613:
2604:
2590:
2582:
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2580:
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2547:
2546:
2541:
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2513:
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2404:
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2324:
2323:
2318:
2316:
2307:
2278:
2276:
2275:
2270:
2268:
2267:
2265:
2254:
2240:
2239:
2227:
2226:
2200:
2198:
2197:
2192:
2190:
2189:
2173:
2171:
2170:
2165:
2163:
2162:
2142:
2140:
2139:
2134:
2111:
2109:
2108:
2103:
2080:
2078:
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2072:
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2032:
2017:
2015:
2014:
2009:
1991:
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1355:
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1337:
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1335:
1322:
1311:
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1294:
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1286:
1271:
1269:
1268:
1263:
1261:
1259:
1251:
1250:
1249:
1236:
1228:
1227:
1211:
1209:
1208:
1203:
1200:
1184:
1182:
1181:
1176:
1174:
1173:
1146:
1144:
1143:
1138:
1130:
1122:
1117:
1116:
1097:
1095:
1094:
1089:
1087:
1085:
1074:
1066:
1061:
1059:
1048:
1047:
1046:
1033:
1028:
1027:
1008:
1006:
1005:
1000:
988:
986:
985:
980:
965:
963:
962:
957:
931:
929:
928:
923:
918:
916:
899:
894:
892:
881:
876:
875:
856:
854:
853:
848:
836:
834:
833:
828:
809:
807:
806:
801:
783:
781:
780:
775:
749:
748:
730:
728:
727:
722:
707:
705:
704:
699:
680:
678:
677:
672:
670:
669:
592:
590:
589:
584:
572:
570:
569:
564:
552:
550:
549:
544:
529:(preventing any
524:
522:
521:
516:
495:
493:
492:
487:
485:
484:
466:
465:
445:
443:
442:
437:
435:
433:
416:
411:
409:
398:
393:
392:
369:
367:
366:
361:
359:
357:
340:
332:
330:
319:
311:
310:
287:
285:
284:
279:
264:
262:
261:
256:
241:
239:
238:
233:
214:
212:
211:
206:
204:
203:
174:separated by an
111:
109:
108:
103:
21:
2909:
2908:
2904:
2903:
2902:
2900:
2899:
2898:
2879:
2878:
2870:
2865:
2864:
2855:
2851:
2842:
2835:
2824:
2820:
2811:
2804:
2795:
2791:
2782:
2775:
2759:
2755:
2742:
2738:
2728:
2712:(11 ed.).
2686:
2669:
2640:
2638:
2635:
2634:
2617:
2616:
2609:
2605:
2600:
2586:
2571:
2566:
2565:
2561:
2548:
2542:
2537:
2536:
2533:
2532:
2526:
2522:
2517:
2499:
2494:
2490:
2486:
2481:
2466:
2461:
2457:
2453:
2443:
2438:
2431:
2425:
2420:
2419:
2415:
2413:
2410:
2409:
2383:
2378:
2374:
2370:
2368:
2365:
2364:
2338:
2336:
2333:
2332:
2331:the current by
2305:
2303:
2300:
2299:
2298:the current by
2288:
2258:
2253:
2235:
2231:
2222:
2218:
2211:
2209:
2206:
2205:
2185:
2181:
2179:
2176:
2175:
2158:
2154:
2152:
2149:
2148:
2120:
2117:
2116:
2089:
2086:
2085:
2058:
2055:
2054:
2023:
2020:
2019:
2003:
2000:
1999:
1976:
1972:
1970:
1967:
1966:
1943:
1939:
1937:
1934:
1933:
1905:
1900:
1882:
1878:
1869:
1865:
1858:
1856:
1853:
1852:
1833:
1830:
1829:
1812:
1808:
1806:
1803:
1802:
1785:
1781:
1779:
1776:
1775:
1755:
1753:
1750:
1749:
1733:
1731:
1728:
1727:
1711:
1709:
1706:
1705:
1685:
1683:
1680:
1679:
1657:
1653:
1648:
1641:
1639:
1636:
1635:
1619:
1616:
1615:
1593:
1589:
1584:
1577:
1575:
1572:
1571:
1551:
1548:
1547:
1536:is the complex
1521:
1519:
1516:
1515:
1490:
1476:
1474:
1471:
1470:
1450:
1449:
1447:
1444:
1443:
1424:
1422:
1419:
1418:
1399:
1397:
1394:
1393:
1392:Both reactance
1390:
1384:
1338:
1331:
1327:
1323:
1321:
1306:
1305:
1303:
1300:
1299:
1280:
1277:
1276:
1252:
1245:
1241:
1237:
1235:
1223:
1222:
1220:
1217:
1216:
1196:
1193:
1190:
1189:
1169:
1168:
1166:
1163:
1162:
1121:
1112:
1108:
1106:
1103:
1102:
1075:
1067:
1065:
1049:
1042:
1038:
1034:
1032:
1023:
1019:
1017:
1014:
1013:
994:
991:
990:
974:
971:
970:
951:
948:
947:
903:
898:
885:
880:
871:
867:
865:
862:
861:
842:
839:
838:
822:
819:
818:
795:
792:
791:
744:
740:
738:
735:
734:
716:
713:
712:
693:
690:
689:
665:
661:
659:
656:
655:
647:cycle, or 90°.
636:
630:
578:
575:
574:
558:
555:
554:
538:
535:
534:
504:
501:
500:
480:
476:
461:
457:
455:
452:
451:
420:
415:
402:
397:
388:
384:
382:
379:
378:
344:
339:
323:
318:
306:
302:
300:
297:
296:
273:
270:
269:
250:
247:
246:
227:
224:
223:
199:
195:
193:
190:
189:
168:
162:
131:complex numbers
126:
97:
94:
93:
35:
28:
23:
22:
15:
12:
11:
5:
2907:
2897:
2896:
2891:
2877:
2876:
2869:
2868:External links
2866:
2863:
2862:
2849:
2833:
2818:
2802:
2789:
2773:
2753:
2735:
2734:
2733:
2732:
2726:
2718:Addison Wesley
2701:
2694:
2685:
2682:
2681:
2680:
2675:
2668:
2665:
2648:
2645:
2631:
2630:
2612:
2608:
2603:
2599:
2596:
2593:
2589:
2585:
2578:
2575:
2569:
2564:
2560:
2557:
2554:
2551:
2549:
2545:
2540:
2535:
2534:
2529:
2525:
2520:
2516:
2512:
2505:
2502:
2498:
2493:
2489:
2484:
2480:
2473:
2470:
2464:
2460:
2456:
2449:
2446:
2442:
2437:
2434:
2432:
2428:
2423:
2418:
2417:
2390:
2387:
2381:
2377:
2373:
2346:
2343:
2313:
2310:
2287:
2284:
2280:
2279:
2264:
2261:
2257:
2252:
2249:
2246:
2243:
2238:
2234:
2230:
2225:
2221:
2217:
2214:
2188:
2184:
2161:
2157:
2145:
2144:
2131:
2128:
2125:
2113:
2100:
2097:
2094:
2082:
2069:
2066:
2063:
2047:
2046:
2030:
2027:
2007:
1997:
1979:
1975:
1964:
1946:
1942:
1927:
1926:
1911:
1908:
1904:
1899:
1896:
1893:
1890:
1885:
1881:
1877:
1872:
1868:
1864:
1861:
1837:
1815:
1811:
1788:
1784:
1772:
1771:
1758:
1736:
1714:
1688:
1677:
1664:
1660:
1656:
1647:
1644:
1623:
1613:
1600:
1596:
1592:
1583:
1580:
1555:
1545:
1540:, measured in
1524:
1509:
1508:
1497:
1493:
1489:
1486:
1483:
1479:
1453:
1427:
1402:
1386:Main article:
1383:
1380:
1364:direct current
1360:
1359:
1344:
1341:
1334:
1330:
1326:
1320:
1317:
1314:
1309:
1284:
1273:
1272:
1258:
1255:
1248:
1244:
1240:
1234:
1231:
1226:
1199:
1172:
1136:
1133:
1128:
1125:
1120:
1115:
1111:
1099:
1098:
1084:
1081:
1078:
1073:
1070:
1064:
1058:
1055:
1052:
1045:
1041:
1037:
1031:
1026:
1022:
998:
989:and frequency
978:
955:
933:
932:
921:
915:
912:
909:
906:
902:
897:
891:
888:
884:
879:
874:
870:
846:
837:and frequency
826:
799:
773:
770:
767:
764:
761:
758:
755:
752:
747:
743:
720:
697:
668:
664:
632:Main article:
629:
626:
614:electric field
582:
562:
542:
514:
511:
508:
483:
479:
475:
472:
469:
464:
460:
448:
447:
432:
429:
426:
423:
419:
414:
408:
405:
401:
396:
391:
387:
372:
371:
356:
353:
350:
347:
343:
338:
335:
329:
326:
322:
317:
314:
309:
305:
277:
254:
231:
219:to the signal
202:
198:
164:Main article:
161:
158:
125:
122:
101:
26:
9:
6:
4:
3:
2:
2906:
2895:
2892:
2890:
2887:
2886:
2884:
2875:
2872:
2871:
2859:
2853:
2846:
2840:
2838:
2830:
2829:
2822:
2815:
2809:
2807:
2799:
2793:
2786:
2780:
2778:
2770:
2766:
2762:
2757:
2749:
2748:
2740:
2736:
2729:
2727:0-8053-9179-7
2723:
2719:
2715:
2714:San Francisco
2710:
2709:
2702:
2699:
2695:
2692:
2688:
2687:
2679:
2676:
2674:
2671:
2670:
2664:
2646:
2643:
2610:
2606:
2597:
2594:
2591:
2583:
2576:
2573:
2562:
2558:
2555:
2552:
2550:
2543:
2527:
2523:
2514:
2510:
2503:
2500:
2496:
2491:
2487:
2478:
2471:
2468:
2458:
2454:
2447:
2444:
2440:
2435:
2433:
2426:
2408:
2407:
2406:
2388:
2385:
2375:
2371:
2361:
2344:
2341:
2330:
2311:
2308:
2297:
2293:
2283:
2262:
2259:
2255:
2250:
2247:
2244:
2241:
2236:
2232:
2228:
2223:
2219:
2215:
2212:
2204:
2203:
2202:
2186:
2182:
2159:
2155:
2129:
2126:
2123:
2114:
2098:
2095:
2092:
2083:
2067:
2064:
2061:
2052:
2051:
2050:
2044:
2028:
2025:
2005:
1998:
1995:
1977:
1973:
1965:
1962:
1944:
1940:
1932:
1931:
1930:
1909:
1906:
1902:
1897:
1894:
1891:
1888:
1883:
1879:
1875:
1870:
1866:
1862:
1859:
1851:
1850:
1849:
1835:
1813:
1809:
1786:
1782:
1703:
1678:
1645:
1642:
1621:
1614:
1581:
1578:
1569:
1553:
1546:
1543:
1539:
1514:
1513:
1512:
1495:
1487:
1484:
1481:
1469:
1468:
1467:
1442:
1425:
1417:
1400:
1389:
1379:
1377:
1373:
1369:
1368:short-circuit
1365:
1342:
1339:
1332:
1324:
1318:
1315:
1312:
1298:
1297:
1296:
1282:
1256:
1253:
1246:
1238:
1232:
1229:
1215:
1214:
1213:
1197:
1188:
1161:
1157:
1156:Faraday's law
1153:
1148:
1134:
1131:
1126:
1123:
1118:
1113:
1109:
1082:
1079:
1076:
1071:
1068:
1062:
1056:
1053:
1050:
1043:
1039:
1035:
1029:
1024:
1020:
1012:
1011:
1010:
1009:is equal to:
996:
976:
969:
953:
946:
942:
938:
919:
913:
910:
907:
904:
900:
895:
889:
886:
882:
877:
872:
868:
860:
859:
858:
857:is equal to:
844:
824:
817:
813:
797:
790:
785:
771:
768:
765:
762:
759:
756:
753:
750:
745:
741:
732:
718:
711:
695:
688:
684:
666:
662:
652:
648:
644:
642:
635:
625:
621:
619:
615:
611:
607:
603:
598:
596:
595:short circuit
580:
540:
532:
528:
512:
509:
506:
497:
481:
477:
473:
470:
467:
462:
458:
430:
427:
424:
421:
417:
412:
406:
403:
399:
394:
389:
385:
377:
376:
375:
354:
351:
348:
345:
341:
336:
333:
327:
324:
320:
315:
312:
307:
303:
295:
294:
293:
289:
275:
268:
252:
245:
229:
222:
218:
200:
196:
187:
183:
181:
177:
173:
167:
157:
156:in May 1894.
155:
151:
147:
142:
138:
136:
132:
121:
119:
115:
99:
91:
87:
83:
79:
75:
71:
66:
64:
60:
56:
52:
48:
44:
40:
33:
19:
2857:
2852:
2844:
2826:
2821:
2813:
2797:
2792:
2784:
2768:
2756:
2746:
2739:
2707:
2697:
2690:
2632:
2362:
2328:
2295:
2289:
2281:
2146:
2048:
1928:
1848:as follows:
1773:
1510:
1391:
1361:
1274:
1149:
1100:
934:
786:
733:
683:proportional
653:
649:
645:
640:
637:
622:
599:
527:open circuit
498:
449:
373:
290:
185:
184:
169:
149:
145:
143:
139:
127:
89:
85:
67:
38:
36:
2765:"Reactance"
2678:Susceptance
1372:resistivity
937:square wave
553:approaches
267:capacitance
166:Capacitance
112:. An ideal
76:changes of
51:capacitance
2883:Categories
2696:Meade R.,
2684:References
1994:capacitive
1568:resistance
1416:resistance
945:inductance
935:Because a
812:sinusoidal
789:inductance
710:inductance
634:Inductance
265:) and the
180:dielectric
172:conductors
90:capacitive
78:sinusoidal
47:inductance
2644:π
2592:ω
2574:π
2556:ω
2501:ω
2492:−
2469:π
2459:−
2445:ω
2386:π
2376:±
2342:π
2309:π
2260:ω
2251:−
2245:ω
2229:−
2029:π
2006:ω
1961:inductive
1907:ω
1898:−
1892:ω
1538:impedance
1441:impedance
1382:Impedance
1329:Φ
1316:−
1243:Φ
1233:−
1127:π
1072:π
1054:ω
1040:π
968:amplitude
941:harmonics
908:π
887:ω
816:amplitude
766:π
754:ω
687:frequency
624:current.
618:potential
561:∞
471:−
425:π
404:ω
349:π
337:−
325:ω
316:−
253:ω
221:frequency
176:insulator
146:reactance
144:The term
135:impedance
118:frequency
86:inductive
70:amplitude
55:impedance
39:reactance
32:Reactance
2667:See also
708:and the
114:resistor
2049:Hence:
1992:is the
1959:is the
1929:where:
1700:is the
1566:is the
1511:where:
1374:). An
531:current
63:voltage
2724:
610:charge
606:charge
2329:leads
74:phase
2722:ISBN
2296:lags
2174:and
2127:<
2065:>
1542:ohms
1414:and
242:(or
82:ohms
72:and
49:and
2115:if
2084:if
2053:if
1160:emf
681:is
499:At
215:is
45:by
2885::
2836:^
2805:^
2776:^
2767:,
2720:.
2716::
2294:)
2043:Hz
1650:Im
1586:Re
1466:.
1154:.
1124:16
1077:16
784:.
602:DC
597:.
496:.
288:.
182:.
137:.
65:.
2730:.
2647:2
2611:L
2607:X
2602:j
2598:=
2595:L
2588:j
2584:=
2577:2
2568:j
2563:e
2559:L
2553:=
2544:L
2539:Z
2528:C
2524:X
2519:j
2515:=
2511:)
2504:C
2497:1
2488:(
2483:j
2479:=
2472:2
2463:j
2455:e
2448:C
2441:1
2436:=
2427:C
2422:Z
2389:2
2380:j
2372:e
2345:2
2312:2
2263:C
2256:1
2248:L
2242:=
2237:C
2233:X
2224:L
2220:X
2216:=
2213:X
2187:C
2183:X
2160:L
2156:X
2130:0
2124:X
2099:0
2096:=
2093:X
2068:0
2062:X
2045:.
2026:2
1978:C
1974:X
1945:L
1941:X
1910:C
1903:1
1895:L
1889:=
1884:C
1880:X
1876:+
1871:L
1867:X
1863:=
1860:X
1836:X
1814:L
1810:X
1787:C
1783:X
1770:.
1757:i
1735:j
1713:i
1687:j
1663:)
1659:Z
1655:(
1646:=
1643:X
1622:X
1599:)
1595:Z
1591:(
1582:=
1579:R
1554:R
1544:;
1523:Z
1496:X
1492:j
1488:+
1485:R
1482:=
1478:Z
1452:Z
1426:R
1401:X
1358:.
1343:t
1340:d
1333:B
1325:d
1319:N
1313:=
1308:E
1283:N
1257:t
1254:d
1247:B
1239:d
1230:=
1225:E
1198:B
1171:E
1135:L
1132:f
1119:=
1114:L
1110:X
1083:L
1080:f
1069:A
1063:=
1057:L
1051:8
1044:2
1036:A
1030:=
1025:L
1021:I
997:f
977:A
954:L
920:.
914:L
911:f
905:2
901:A
896:=
890:L
883:A
878:=
873:L
869:I
845:f
825:A
798:L
772:L
769:f
763:2
760:=
757:L
751:=
746:L
742:X
719:L
696:f
667:L
663:X
581:0
541:f
513:0
510:=
507:f
482:c
478:X
474:j
468:=
463:c
459:Z
446:.
431:C
428:f
422:2
418:1
413:=
407:C
400:1
395:=
390:C
386:X
370:.
355:C
352:f
346:2
342:1
334:=
328:C
321:1
313:=
308:C
304:X
276:C
230:f
201:C
197:X
100:X
34:.
20:)
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