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On the other hand, the voltage law relies on the fact that the actions of time-varying magnetic fields are confined to individual components, such as inductors. In reality, the induced electric field produced by an inductor is not confined, but the leaked fields are often negligible.
938:
lumped elements. For example, a wire is not an ideal conductor. Unlike an ideal conductor, wires can inductively and capacitively couple to each other (and to themselves), and have a finite propagation delay. Real conductors can be modeled in terms of lumped elements by considering
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is the product of current and the time the current has been flowing. If the net charge in a region is constant, the current law will hold on the boundaries of the region. This means that the current law relies on the fact that the net charge in the wires and components is constant.
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In the low-frequency limit, the voltage drop around any loop is zero. This includes imaginary loops arranged arbitrarily in space – not limited to the loops delineated by the circuit elements and conductors. In the low-frequency limit, this is a corollary of
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Consider some arbitrary circuit. Approximate the circuit with lumped elements, so that time-varying magnetic fields are contained to each component and the field in the region exterior to the circuit is negligible. Based on this assumption, the
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The lumped element approximation for a circuit is accurate at low frequencies. At higher frequencies, leaked fluxes and varying charge densities in conductors become significant. To an extent, it is possible to still model such circuits using
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The current law is dependent on the assumption that the net charge in any wire, junction or lumped component is constant. Whenever the electric field between parts of the circuit is non-negligible, such as when two wires are
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1949:{\displaystyle {\begin{aligned}R_{1}&=100\Omega ,&R_{2}&=200\Omega ,&R_{3}&=300\Omega ,\\{\mathcal {E}}_{1}&=3{\text{V}},&{\mathcal {E}}_{2}&=4{\text{V}}\end{aligned}}}
1775:{\displaystyle {\begin{cases}i_{1}+(-i_{2})+(-i_{3})&=0\\R_{1}i_{1}+R_{2}i_{2}+0i_{3}&={\mathcal {E}}_{1}\\0i_{1}+R_{2}i_{2}-R_{3}i_{3}&={\mathcal {E}}_{1}+{\mathcal {E}}_{2}\end{cases}}}
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in the low-frequency limit. They are accurate for DC circuits, and for AC circuits at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits.
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482:
1500:{\displaystyle {\begin{cases}i_{1}-i_{2}-i_{3}&=0\\-R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}&=0\\-R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}&=0\end{cases}}}
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The current law is applicable to any lumped network irrespective of the nature of the network; whether unilateral or bilateral, active or passive, linear or non-linear.
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Recalling that current is a signed (positive or negative) quantity reflecting direction towards or away from a node, this principle can be succinctly stated as:
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666:{\displaystyle \sum _{i}V_{i}=-\sum _{i}\int _{{\mathcal {P}}_{i}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =\oint \mathbf {E} \cdot \mathrm {d} \mathbf {l} =0}
898:
916:
1036:
890:, this may not be the case. This occurs in high-frequency AC circuits, where the lumped element model is no longer applicable. For example, in a
2076:{\displaystyle {\begin{cases}i_{1}={\frac {1}{1100}}{\text{A}}\\i_{2}={\frac {4}{275}}{\text{A}}\\i_{3}=-{\frac {3}{220}}{\text{A}}\end{cases}}}
147:
901:
In a transmission line, the net charge in different parts of the conductor changes with time. In the direct physical sense, this violates KCL.
291:
Kirchhoff's circuit laws were originally obtained from experimental results. However, the current law can be viewed as an extension of the
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and both depend on the model being applicable to the circuit in question. When the model is not applicable, the laws do not apply.
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The doctrine of description: Gustav
Kirchhoff, classical physics, and the "purpose of all science" in 19th-century Germany
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To model circuits so that both laws can still be used, it is important to understand the distinction between
2323:
Physics for
Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.)
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828:{\displaystyle V_{a\to b}=-\int _{{\mathcal {P}}_{a\to b}}\mathbf {E} \cdot \mathrm {d} \mathbf {l} }
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536:{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}=\mathbf {0} }
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in the exterior region. If each of the components has a finite volume, then the exterior region is
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flowing into that node is equal to the sum of currents flowing out of that node; or equivalently:
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is the total number of branches with currents flowing towards or away from the node.
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The algebraic sum of currents in a network of conductors meeting at a point is zero.
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Assume an electric network consisting of two voltage sources and three resistors.
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Note that this derivation uses the following definition for the voltage rise from
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The current entering any junction is equal to the current leaving that junction.
146:. These laws can be applied in time and frequency domains and form the basis for
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1224:{\displaystyle -R_{3}i_{3}-{\mathcal {E}}_{2}-{\mathcal {E}}_{1}+R_{2}i_{2}=0}
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The second law, again combined with Ohm's law, applied to the closed circuit
127:
2189:(Ph. D.). University of California, Berkeley. p. 52. Docket 3331743.
400:
Similarly to
Kirchhoff's current law, the voltage law can be stated as:
2112:
is actually flowing in the direction opposite to the red arrow labeled
2344:(10. printing. ed.). Upper Saddle River, NJ: Prentice Hall PTR.
551:
in that region. Therefore, for any loop in the circuit, we find that
943:
distributed between the conductors to model capacitive coupling, or
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to model inductive coupling. Wires also have some self-inductance.
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The
Feynman Lectures on Physics, Volume II, Chapter 22: AC Circuits
17:
894:, the charge density in the conductor may be constantly changing.
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27:
Two equalities that deal with the current and potential difference
841:(and thus voltage) can be defined in other ways, such as via the
36:
2233:"The Feynman Lectures on Physics Vol. II Ch. 22: AC Circuits"
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Both of
Kirchhoff's laws can be understood as corollaries of
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The sum of all the voltages around a loop is equal to zero.
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1112:{\displaystyle -R_{2}i_{2}+{\mathcal {E}}_{1}-R_{1}i_{1}=0}
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of each of the components, from one terminal to another.
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High-speed signal propagation : advanced black magic
2094:
has a negative sign which means the assumed direction of
866:
This has practical application in situations involving "
312:
version of
Kirchhoff's current law is the basis of most
122:. They were first described in 1845 by German physicist
1033:, and substituting for voltage using Ohm's law gives:
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Grounding and
Shielding Techniques in Instrumentation
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2202:"Kirchoff's current law and Kirchoff's voltage law"
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30:For other laws named after Gustav Kirchhoff, see
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2302:Physics for Scientists and Engineers (6th ed.)
1024:Applying the second law to the closed circuit
210:, states that, for any node (junction) in an
8:
2364:: CS1 maint: multiple names: authors list (
2298:Serway, Raymond A.; Jewett, John W. (2004).
910:Modelling real circuits with lumped elements
460:is the total number of voltages measured.
395:(voltages) around any closed loop is zero.
2281:Fundamentals of Electric Circuit Analysis
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50:This article includes a list of general
2340:Graham, Howard Johnson, Martin (2002).
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2357:
2254:
2252:
447:{\displaystyle \sum _{i=1}^{n}V_{i}=0}
275:{\displaystyle \sum _{i=1}^{n}I_{i}=0}
2388:Lessons In Electric Circuits Vol 1 DC
2382:Divider Circuits and Kirchhoff's Laws
467:A similar derivation can be found in
464:Derivation of Kirchhoff's voltage law
7:
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1017:{\displaystyle i_{1}-i_{2}-i_{3}=0}
114:(commonly known as voltage) in the
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2185:Oldham, Kalil T. Swain (2008).
697:{\textstyle {\mathcal {P}}_{i}}
320:. The current law is used with
126:. This generalized the work of
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945:parasitic (mutual) inductances
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2156:Duality (electrical circuits)
2394:Lessons In Electric Circuits
965:According to the first law:
2237:feynmanlectures.caltech.edu
334:
314:circuit simulation software
160:
2457:
2426:Linear electronic circuits
2262:Wiley-Interscience (1986)
2161:Faraday's law of induction
2130:flows from left to right.
1235:system of linear equations
857:Faraday's law of induction
29:
2421:Eponymous laws of physics
2283:. John Wiley & Sons.
2279:Paul, Clayton R. (2001).
934:circuit elements and the
208:Kirchhoff's junction rule
130:and preceded the work of
2209:Johns Hopkins University
2166:Lumped matter discipline
921:finite element modelling
477:Maxwell–Faraday equation
391:The directed sum of the
386:, states the following:
100:Kirchhoff's circuit laws
1507:which is equivalent to
843:Helmholtz decomposition
335:Kirchhoff's voltage law
161:Kirchhoff's current law
138:, they are also called
71:more precise citations.
2416:Conservation equations
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378:This law, also called
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2321:Tipler, Paul (2004).
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2148:Electronics portal
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134:. Widely used in
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1804:
1800:
1796:
1792:
1791:
1769:
1762:
1756:
1750:
1745:
1739:
1733:
1730:
1726:
1722:
1716:
1712:
1708:
1703:
1699:
1693:
1689:
1685:
1680:
1676:
1672:
1669:
1668:
1663:
1657:
1651:
1648:
1644:
1640:
1636:
1633:
1628:
1624:
1618:
1614:
1610:
1605:
1601:
1595:
1591:
1587:
1586:
1583:
1580:
1577:
1575:
1570:
1566:
1562:
1559:
1556:
1553:
1548:
1544:
1540:
1537:
1534:
1529:
1525:
1521:
1520:
1518:
1494:
1489:
1486:
1483:
1479:
1475:
1469:
1465:
1461:
1456:
1450:
1444:
1439:
1433:
1427:
1422:
1418:
1412:
1408:
1404:
1401:
1400:
1397:
1394:
1391:
1387:
1383:
1377:
1373:
1369:
1364:
1358:
1352:
1347:
1343:
1337:
1333:
1329:
1326:
1325:
1322:
1319:
1316:
1312:
1308:
1304:
1299:
1295:
1291:
1286:
1282:
1278:
1277:
1275:
1260:
1251:
1242:
1233:This yields a
1220:
1217:
1212:
1208:
1202:
1198:
1194:
1189:
1183:
1177:
1172:
1166:
1160:
1155:
1151:
1145:
1141:
1137:
1124:
1108:
1105:
1100:
1096:
1090:
1086:
1082:
1077:
1071:
1065:
1060:
1056:
1050:
1046:
1042:
1029:
1013:
1010:
1005:
1001:
997:
992:
988:
984:
979:
975:
952:
949:
911:
908:
875:
872:
851:
850:Generalization
848:
823:
818:
814:
810:
802:
799:
796:
790:
783:
779:
776:
771:
768:
765:
761:
740:
720:
691:
685:
662:
659:
655:
650:
646:
642:
638:
635:
631:
626:
622:
618:
610:
604:
597:
591:
587:
583:
580:
575:
571:
565:
561:
531:
527:
521:
518:
512:
508:
502:
499:
495:
491:
488:
462:
443:
440:
435:
431:
425:
420:
417:
414:
410:
370:
363:
356:
349:
336:
333:
326:nodal analysis
305:
302:
271:
268:
263:
259:
253:
248:
245:
242:
238:
195:
188:
181:
174:
162:
159:
95:
94:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2453:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2408:
2406:
2396:
2395:
2390:
2389:
2385:chapter from
2384:
2383:
2379:
2378:
2374:
2367:
2361:
2353:
2351:0-13-084408-X
2347:
2343:
2338:
2334:
2332:0-7167-0810-8
2328:
2324:
2319:
2315:
2313:0-534-40842-7
2309:
2304:
2303:
2296:
2292:
2290:0-471-37195-5
2286:
2282:
2277:
2276:
2269:
2265:
2261:
2255:
2253:
2249:
2238:
2234:
2228:
2226:
2222:
2210:
2203:
2196:
2193:
2188:
2181:
2178:
2171:
2167:
2164:
2162:
2159:
2157:
2154:
2153:
2149:
2143:
2138:
2133:
2131:
2125:
2116:
2107:
2098:
2089:
2083:
2056:
2053:
2048:
2045:
2040:
2036:
2021:
2018:
2013:
2008:
2004:
1989:
1986:
1981:
1976:
1972:
1965:
1934:
1931:
1929:
1922:
1908:
1900:
1897:
1895:
1888:
1872:
1866:
1863:
1861:
1854:
1850:
1844:
1838:
1835:
1833:
1826:
1822:
1816:
1810:
1807:
1805:
1798:
1794:
1760:
1748:
1743:
1731:
1724:
1720:
1714:
1710:
1706:
1701:
1697:
1691:
1687:
1683:
1678:
1674:
1670:
1661:
1649:
1642:
1638:
1634:
1631:
1626:
1622:
1616:
1612:
1608:
1603:
1599:
1593:
1589:
1581:
1578:
1568:
1564:
1560:
1554:
1546:
1542:
1538:
1532:
1527:
1523:
1516:
1487:
1484:
1477:
1473:
1467:
1463:
1459:
1454:
1442:
1437:
1425:
1420:
1416:
1410:
1406:
1402:
1395:
1392:
1385:
1381:
1375:
1371:
1367:
1362:
1350:
1345:
1341:
1335:
1331:
1327:
1320:
1317:
1310:
1306:
1302:
1297:
1293:
1289:
1284:
1280:
1273:
1259:
1250:
1241:
1236:
1231:
1218:
1215:
1210:
1206:
1200:
1196:
1192:
1187:
1175:
1170:
1158:
1153:
1149:
1143:
1139:
1135:
1123:
1106:
1103:
1098:
1094:
1088:
1084:
1080:
1075:
1063:
1058:
1054:
1048:
1044:
1040:
1028:
1011:
1008:
1003:
999:
995:
990:
986:
982:
977:
973:
963:
957:
950:
948:
946:
942:
937:
933:
928:
926:
922:
918:
909:
907:
899:
895:
893:
889:
883:
881:
873:
871:
869:
864:
862:
858:
849:
846:
844:
840:
837:However, the
835:
812:
800:
794:
781:
777:
774:
769:
763:
759:
738:
718:
709:
707:
689:
660:
657:
644:
636:
633:
620:
608:
595:
589:
585:
581:
578:
573:
569:
563:
559:
550:
546:
525:
519:
500:
497:
489:
479:reveals that
478:
472:
470:
461:
454:
441:
438:
433:
429:
423:
418:
415:
412:
408:
397:
396:
394:
387:
385:
381:
369:
362:
355:
348:
341:
332:
329:
327:
323:
319:
315:
311:
303:
301:
298:
294:
289:
286:
269:
266:
261:
257:
251:
246:
243:
240:
236:
225:
224:
219:
217:
214:, the sum of
213:
209:
205:
194:
187:
180:
173:
167:
158:
156:
151:
149:
145:
141:
137:
133:
129:
125:
121:
117:
113:
109:
105:
101:
91:
88:
80:
77:November 2017
70:
66:
60:
59:
53:
48:
39:
38:
33:
19:
2393:
2387:
2381:
2341:
2322:
2301:
2280:
2259:
2240:. Retrieved
2236:
2212:. Retrieved
2208:
2195:
2186:
2180:
2123:
2114:
2105:
2096:
2087:
2085:The current
2084:
1257:
1248:
1239:
1232:
1121:
1026:
964:
961:
935:
931:
929:
913:
904:
884:
877:
865:
853:
836:
710:
705:
549:conservative
473:
468:
466:
455:
399:
390:
389:
383:
379:
377:
367:
360:
353:
346:
330:
307:
290:
284:
227:
222:
221:
207:
203:
201:
192:
185:
178:
171:
152:
143:
139:
99:
98:
83:
74:
55:
874:Limitations
324:to perform
69:introducing
2405:Categories
2268:0471838055
2242:2018-12-06
2214:6 December
2172:References
316:, such as
142:or simply
104:equalities
52:references
2360:cite book
2049:−
1870:Ω
1842:Ω
1814:Ω
1782:Assuming
1707:−
1561:−
1539:−
1443:−
1426:−
1403:−
1368:−
1328:−
1303:−
1290:−
1176:−
1159:−
1136:−
1081:−
1041:−
996:−
983:−
813:⋅
798:→
782:∫
778:−
767:→
645:⋅
637:∮
621:⋅
596:∫
586:∑
582:−
560:∑
517:∂
507:∂
501:−
490:×
487:∇
409:∑
322:Ohm's law
295:, since
237:∑
128:Georg Ohm
2134:See also
932:physical
706:exterior
216:currents
102:are two
2431:Voltage
1128:gives:
951:Example
108:current
65:improve
2397:series
2348:
2329:
2310:
2287:
2266:
673:where
456:Here,
310:matrix
297:charge
282:where
54:, but
2205:(PDF)
936:ideal
382:, or
318:SPICE
206:, or
2366:link
2346:ISBN
2327:ISBN
2308:ISBN
2285:ISBN
2264:ISBN
2216:2018
1990:1100
304:Uses
110:and
2057:220
2022:275
1867:300
1839:200
1811:100
1237:in
923:or
870:".
863:).
731:to
373:= 0
118:of
18:KVL
2407::
2362:}}
2358:{{
2251:^
2235:.
2224:^
2207:.
1264::
1255:,
1246:,
927:.
845:.
751::
471:.
366:+
359:+
352:+
328:.
308:A
191:+
184:=
177:+
150:.
2368:)
2354:.
2335:.
2316:.
2293:.
2245:.
2218:.
2127:3
2124:R
2118:3
2115:i
2109:3
2106:i
2100:3
2097:i
2091:3
2088:i
2063:A
2054:3
2046:=
2041:3
2037:i
2028:A
2019:4
2014:=
2009:2
2005:i
1996:A
1987:1
1982:=
1977:1
1973:i
1966:{
1939:V
1935:4
1932:=
1923:2
1917:E
1909:,
1905:V
1901:3
1898:=
1889:1
1883:E
1873:,
1864:=
1855:3
1851:R
1845:,
1836:=
1827:2
1823:R
1817:,
1808:=
1799:1
1795:R
1761:2
1755:E
1749:+
1744:1
1738:E
1732:=
1725:3
1721:i
1715:3
1711:R
1702:2
1698:i
1692:2
1688:R
1684:+
1679:1
1675:i
1671:0
1662:1
1656:E
1650:=
1643:3
1639:i
1635:0
1632:+
1627:2
1623:i
1617:2
1613:R
1609:+
1604:1
1600:i
1594:1
1590:R
1582:0
1579:=
1574:)
1569:3
1565:i
1558:(
1555:+
1552:)
1547:2
1543:i
1536:(
1533:+
1528:1
1524:i
1517:{
1488:0
1485:=
1478:2
1474:i
1468:2
1464:R
1460:+
1455:1
1449:E
1438:2
1432:E
1421:3
1417:i
1411:3
1407:R
1396:0
1393:=
1386:1
1382:i
1376:1
1372:R
1363:1
1357:E
1351:+
1346:2
1342:i
1336:2
1332:R
1321:0
1318:=
1311:3
1307:i
1298:2
1294:i
1285:1
1281:i
1274:{
1261:3
1258:i
1252:2
1249:i
1243:1
1240:i
1219:0
1216:=
1211:2
1207:i
1201:2
1197:R
1193:+
1188:1
1182:E
1171:2
1165:E
1154:3
1150:i
1144:3
1140:R
1125:2
1122:s
1107:0
1104:=
1099:1
1095:i
1089:1
1085:R
1076:1
1070:E
1064:+
1059:2
1055:i
1049:2
1045:R
1030:1
1027:s
1012:0
1009:=
1004:3
1000:i
991:2
987:i
978:1
974:i
822:l
817:d
809:E
801:b
795:a
789:P
775:=
770:b
764:a
760:V
739:b
719:a
690:i
684:P
661:0
658:=
654:l
649:d
641:E
634:=
630:l
625:d
617:E
609:i
603:P
590:i
579:=
574:i
570:V
564:i
530:0
526:=
520:t
511:B
498:=
494:E
458:n
442:0
439:=
434:i
430:V
424:n
419:1
416:=
413:i
371:4
368:v
364:3
361:v
357:2
354:v
350:1
347:v
285:n
270:0
267:=
262:i
258:I
252:n
247:1
244:=
241:i
196:4
193:i
189:1
186:i
182:3
179:i
175:2
172:i
90:)
84:(
79:)
75:(
61:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
