Steinmetz's equation

Steinmetz's equation, sometimes called the power equation,[1] is an empirical equation used to calculate the total power loss (core losses) per unit volume in magnetic materials when subjected to external sinusoidally varying magnetic flux.[2][3] The equation is named after Charles Steinmetz, a German-American electrical engineer, who proposed a similar equation without the frequency dependency in 1890.[4][5] The equation is:[2][3]

where Pv is the time average power loss per unit volume in mW per cubic centimeter, is frequency in kilohertz, and is the peak magnetic flux density; , , and , called the Steinmetz coefficients, are material parameters generally found empirically from the material's B-H hysteresis curve by curve fitting. The Steinmetz coefficients for most magnetic materials are available from the manufacturers. This energy loss, called core loss is due mainly to two effects, magnetic hysteresis and, in conductive materials eddy currents, which consume energy from the source of the magnetic field, dissipating it as waste heat in the magnetic material. The equation is used mainly to calculate core losses in ferromagnetic magnetic cores used in electric motors, generators, transformers and inductors excited by sinusoidal current. Core losses are an economically important source of inefficiency in alternating current (AC) electric power grids and appliances.

If only hysteresis is taken into account (a la Steinmetz), the coefficient will be close to 1 and will be 2 for nearly all modern magnetic materials. However, due to other nonlinearities, is usually between 1 and 2, and is between 2 and 3. The equation is a simplified form that only applies when the magnetic field has a sinusoidal waveform and does not take into account factors such as DC offset. However, because most electronics expose materials to non-sinusoidal flux waveforms, various improvements to the equation have been made. An improved generalized Steinmetz equation, often referred to as iGSE, can be expressed as[2][3]

where is the flux density from peak to peak and is defined by

where a, b, and k are the same parameters used in the original equation. This equation can calculate losses with any flux waveform using only the parameters needed for the original equation, but it ignores the fact that the parameters, and therefore the losses, can vary under DC bias conditions.[4] DC bias cannot be neglected without severely affecting results, but there is still not a practical physically-based model that takes both dynamic and nonlinear effects into account.[6] However, this equation is still widely used because most other models require parameters that are not usually given by manufacturers and that engineers are not likely to take the time and resources to measure.[1]

See also

References

  1. 1 2 Venkatachalam; et al. (2012). "Accurate Prediction of Ferrite Core Loss with Nonsinusoidal Waveforms Using Only Steinmetz Parameters" (PDF). Dartmouth College. Retrieved 2013-07-31.
  2. 1 2 3 Sudhoff, Scott D. (2014). Power Magnetic Devices: A Multi-Objective Design Approach. John Wiley and Sons. pp. 168–169. ISBN 978-1118824634.
  3. 1 2 3 Rashid, Muhammad H. (2017). Power Electronics Handbook, 4th Ed. Butterworth-Heinemann. p. 573. ISBN 978-0128114087.
  4. 1 2 Mühlethaler et al. (February 2012). "Core Losses Under the DC Bias Condition Based on Steinmetz Parameters" (PDF). IEEE Transactions on Power Electronics. 27 (2): 953.
  5. ↑ Steinmetz, Charles P. (1892). "On the law of hysteresis". Trans. AIEE. 9 (2): 3–62. doi:10.1109/PROC.1984.12842. Retrieved November 28, 2017.
  6. ↑ Reinert, J.; Brockmeyer, A.; De Doncker, R.W. (1999). "Calculation of losses in ferro- and ferrimagnetic materials based on the modified Steinmetz equation". Proceedings of 34th Annual Meeting of the IEEE Industry Applications Society. 3: 2087–92.
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