Steinmetz and the Concept of Phasor: A Forgotten Story

  • A. E. A. Araújo
  • D. A. V. Tonidandel


The history of the concept of phasor is often neglected when it is introduced in textbooks on circuits. The presentation does not emphasize the historical aspects, which is natural. This paper intends to recover the original and creative way the concept of complex numbers, then almost unknown to engineers, was applied to electric circuits in sinusoidal steady-state. As usual in physics and engineering, the theory of phasor could have been anticipated by earlier researchers, if they had followed their original reasonings. Maxwell and Heaviside had proved the meal, but could not, or were not interested in writing the recipe.


Phasor Steady-state Circuit theory 


  1. Berg, E. J. (1929). Heaviside’s operational calculus: as applied to engineering and physics. New York: McGraw-Hill.Google Scholar
  2. Blanchard, J. (1941). The history of electrical resonance. Bell System Technical Journal, 20(4), 415–433.Google Scholar
  3. Carson, J. (1926). Electric circuit theory and operational calculus. New York: McGraw-Hill.Google Scholar
  4. Cauchy, A. (1825). Mémoire sur les intégrales définies prises entre des limites imaginaires. Oeuvres.Google Scholar
  5. Cauchy, A. (1827). Mémoire sur les intégrales définies. Académie Royale des Sciences, 1(1), 601–799.Google Scholar
  6. Churchill, R. V. (1963). Fourier series and boundary value problems (2nd ed.). New York: McGraw-hill.zbMATHGoogle Scholar
  7. Crowe, M. J. (1985). A history of vector analysis. New York: Dover Publications.Google Scholar
  8. Feynman, R. P., Leighton, R., & Sands, M. (1963). Lectures on physics, (Vol. II), Reading: Addison-Wesley Publishing Company.Google Scholar
  9. Fourier, J. (1878). The analytical theory of heat. Cambridge: Cambridge University Press.Google Scholar
  10. Gibbs, J. (1881). Elements of vector analysis : arranged for the use of students in physics. New Heaven: Tuttle, Morehouse & Taylor.Google Scholar
  11. Grove, W. (1868). An experiment in magneto-electric induction. Philosophical Magazine, 35, 184–185. Google Scholar
  12. Heaviside, O. (1893a). Electromagnetic theory (Vol. 1). Limited. London: The Electrician Printing and Publishing Co.Google Scholar
  13. Heaviside, O. (1893b). On operators in physical mathematics. Proceedings of the Royal Society of London, 52, 504–529.Google Scholar
  14. Johnson, D. E., Hilburn, J. L., & Johnson, J. R. (1990). Fundamentos de Análise de Circuitos Elétricos (4th ed.). Rio de Janeiro: Prentice Hall do Brasil.Google Scholar
  15. Kerchner, R., & Corcoran, G. (1977). Circuitos de corrente alternada (3rd ed.). Editora Globo.Google Scholar
  16. Maxwell J. C. (1868). On Mr. Grove’s “Experiment in magneto-electric induction”, Philosophical Magazine S, The London, Edinburg and Dublin philosophical society, pp. 389–392. In a letter to Mr. Grove, F. R. S.Google Scholar
  17. Nahin, P. J. (1998). An imaginary tale: the story of \(\sqrt{-1}\) (1st ed.). Princeton: Princeton University Press.Google Scholar
  18. Nilsson, J. W., & Riedel, S. A. (2009). Circuitos Elétricos (8th ed.). Upper Saddle River: Pearson Prentice-Hall.Google Scholar
  19. Remscheid, E. J. (1977). Recollections of Steinmetz: a visit to the workshops of Dr. Charles Proteus Steinmetz. Schenectady: General Electric Co., Research and Development Center.Google Scholar
  20. Sadiku, M. N. O. (2008). Fundamentos de circuitos elétricos (3rd ed.). New York: McGraw-Hill.Google Scholar
  21. Smithies, F. (1997). Cauchy and the creation of complex function theory. Cambridge: Cambridge University Press.Google Scholar
  22. Steinmetz, C. P. (1893). Complex quantities and their use in electrical engineering. Proceedings of the International Electrical Congress, Conference of the AIEE : American Institute of Electrical Engineers Proceedings, Chicago (pp. 33–74), Chicago: AIEE.Google Scholar
  23. Steinmetz, C. P. (1897). Theory and calculation of alternating current phenomena. New York: McGraw-Hill.Google Scholar
  24. Steinmetz, C. P. (1908). Theory and calculation of transient electric phenomena and oscilations. New York: McGraw-Hill.Google Scholar
  25. Steinmetz, C. P. (1917). Engineering nathematics: A series of lectures delivered at Union College. New York: McGraw-Hill.Google Scholar
  26. Thomson, W. (1853). On transient electric currents. Philosophical Magazine, 5, 540–553.Google Scholar
  27. Thomson, W. (1854). On the theory of electric telegraph. Proceedings of Royal Society, 7, 382–399.CrossRefGoogle Scholar
  28. Tonidandel, D. A. V., & Araújo, A. E. A. (2012a). Conectando transformadas: Fourier e Laplace, Congresso Brasileiro de Automática, Campina Grande, PB. 1: 32–36. ISBN 978-85-8001-069-5.Google Scholar
  29. Tonidandel, D. A. V., & e Araújo A. E. A. , (2012b). Transformada de Laplace: uma obra de engenharia. Revista Brasileira de Ensino de Física, 34(2), 2601.Google Scholar
  30. Valkenburg, M. V. (1974). Network analysis (3rd ed.). Upper Saddle River: Prentice-Hall International.Google Scholar
  31. WikimediaCommons. (2012). Category: induction coils. Acessed July 11, 2012 from
  32. Wikipedia. (2012). Induction coil. Acessed July 11, 2012 from

Copyright information

© Brazilian Society for Automatics--SBA 2013

Authors and Affiliations

  1. 1.Universidade Federal de Minas Gerais-UFMGBelo HorizonteBrazil
  2. 2.Universidade Federal de Ouro Preto-UFOPOuro PretoBrazil

Personalised recommendations