Editing Electrical impedance (section)

==History==
Perhaps the earliest use of complex numbers in circuit analysis was by Johann Victor Wietlisbach in 1879 in analysing the [[Maxwell bridge]].  Wietlisbach avoided using differential equations by expressing AC currents and voltages as [[exponential function]]s with [[imaginary number|imaginary]] exponents (see {{section link|#Validity of complex representation}}).  Wietlisbach found the required voltage was given by multiplying the current by a complex number (impedance), although he did not identify this as a general parameter in its own right.<ref>Kline, Ronald R., ''Steinmetz: Engineer and Socialist'', Johns Hopkins University Press, 1992 {{isbn|9780801842986}}, p. 78.</ref>

The term ''impedance'' was coined by [[Oliver Heaviside]] in July 1886.<ref>''Science'', p.&nbsp;18, 1888{{Full citation needed|date=March 2023}}{{failed verification|date = January 2024}}</ref><ref>Oliver Heaviside, ''The Electrician'', p.&nbsp;212, 23 July 1886, reprinted as ''Electrical Papers, Volume II'', p 64, AMS Bookstore, {{ISBN |0-8218-3465-7}}</ref>  Heaviside recognised that the "resistance operator" (impedance) in his [[operational calculus]] was a complex number.  In 1887 he showed that there was an AC equivalent to [[Ohm's law]].<ref>Kline, p. 79.</ref>

[[Arthur Kennelly]] published an influential paper on impedance in 1893. Kennelly arrived at a complex number representation in a rather more direct way than using imaginary exponential functions.  Kennelly followed the graphical representation of impedance (showing resistance, reactance, and impedance as the lengths of the sides of a right angle triangle) developed by [[John Ambrose Fleming]] in 1889.  Impedances could thus be added [[Vector (mathematics and physics)|vectorially]].  Kennelly realised that this graphical representation of impedance was directly analogous to graphical representation of complex numbers ([[Argand diagram]]).  Problems in impedance calculation could thus be approached algebraically with a complex number representation.<ref>Kline, p. 81-2.</ref><ref>Kennelly, Arthur,[https://archive.org/details/transactionsame31engigoog/page/175/mode/1up "Impedance"], ''Transactions of the American Institute of Electrical Engineers'', vol. 10, pp. 175–232, 18 April 1893.</ref>  Later that same year, Kennelly's work was generalised to all AC circuits by [[Charles Proteus Steinmetz]].  Steinmetz not only represented impedances by complex numbers but also voltages and currents.  Unlike Kennelly, Steinmetz was thus able to express AC equivalents of DC laws such as Ohm's and Kirchhoff's laws.<ref>Kline, p. 85.</ref>  Steinmetz's work was highly influential in spreading the technique amongst engineers.<ref>Kline, p. 90-1.</ref>