HOME

TheInfoList



OR:

Steinmetz's equation, sometimes called the power equation, 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 In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the we ...
. The equation is named after Charles Steinmetz, a German-American electrical engineer, who proposed a similar equation without the frequency dependency in 1890. The equation is: :P_v = k \cdot f^a \cdot B^b where P_v is the time average power loss per unit volume in mW per cubic
centimeter upright=1.35, Different lengths as in respect to the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is in-between 1 meter to 1 millimeter. A centimetre (International spelling) or centimeter (American ...
, f is frequency in
kilohertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), often described as being equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose formal expression in terms of SI base ...
, and B is the peak magnetic
flux density Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phenom ...
; k, a, and b, called the Steinmetz coefficients, are material parameters generally found empirically from the material's ''B-H'' hysteresis curve by curve fitting. In typical magnetic materials, the Steinmetz coefficients all vary with temperature. The energy loss, called core loss, is due mainly to two effects:
magnetic hysteresis Magnetic hysteresis occurs when an external magnetic field is applied to a ferromagnet such as iron and the atomic dipoles align themselves with it. Even when the field is removed, part of the alignment will be retained: the material has become ' ...
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 core A magnetic core is a piece of magnetism, magnetic material with a high magnetic permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, ele ...
s used in
electric motor An electric motor is a machine that converts electrical energy into mechanical energy. Most electric motors operate through the interaction between the motor's magnetic field and electric current in a electromagnetic coil, wire winding to gene ...
s, generators,
transformer In electrical engineering, a transformer is a passive component that transfers electrical energy from one electrical circuit to another circuit, or multiple Electrical network, circuits. A varying current in any coil of the transformer produces ...
s and
inductor An inductor, also called a coil, choke, or reactor, is a Passivity (engineering), passive two-terminal electronic component, electrical component that stores energy in a magnetic field when an electric current flows through it. An inductor typic ...
s excited by sinusoidal current. Core losses are an economically important source of inefficiency in
alternating current Alternating current (AC) is an electric current that periodically reverses direction and changes its magnitude continuously with time, in contrast to direct current (DC), which flows only in one direction. Alternating current is the form in w ...
(AC)
electric power grid An electrical grid (or electricity network) is an interconnected network for electricity delivery from producers to consumers. Electrical grids consist of power stations, electrical substations to step voltage up or down, electric power tran ...
s and appliances. If only hysteresis is taken into account (à la Steinmetz), the coefficient a will be close to 1 and b will be 2 for nearly all modern magnetic materials. However, due to other nonlinearities, a is usually between 1 and 2, and b is between 2 and 3. The equation is a simplified form that only applies when the magnetic field B 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 :P = \frac \int_0^T k_i ^a (\Delta B^) dt where \Delta B is the
flux density Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phenom ...
from peak to peak and k_i is defined by :k_i = \frac 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. 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. 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. The Steinmetz coefficients for magnetic materials may be available from the manufacturers. However, manufacturers of magnetic materials intended for high-power applications usually provide graphs that plot specific core loss (watts per volume or watts per weight) at a given temperature against peak flux density B_, with frequency as a parameter. Families of curves for different temperatures may also be given. These graphs apply to the case where the flux density excursion is ±B_. In cases where the magnetizing field has a DC offset or is unidirectional (i.e. ranges between zero and a peak value), core losses can be much lower but are rarely covered by published data.


See also

*
Core losses A magnetic core is a piece of magnetic material with a high magnetic permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, generators, ...
* Eddy currents * Legg's equation *
Magnetic hysteresis Magnetic hysteresis occurs when an external magnetic field is applied to a ferromagnet such as iron and the atomic dipoles align themselves with it. Even when the field is removed, part of the alignment will be retained: the material has become ' ...


References

{{reflist


External links


Steinmetz's Equation
at
ScienceWorld Wolfram Research, Inc. ( ) is an American multinational company that creates computational technology. Wolfram's flagship product is the technical computing program Wolfram Mathematica, first released on June 23, 1988. Other products include ...
Magnetostatics Electrostatics