For the AC magnetic field analysis, the most interesting integral values are: the total, eddy and external current, the mechanical force and torque, the magnetic flux and flux linkage, the magnetomotive force, the field energy.
The following notations are used in formulas:
B – complex vector magnetic flux density;
H – complex vector magnetic field intensity (field strength);
jtotal, jeddy, jext – complex values of total, eddy and external current density;
A – z-component of complex magnetic vector potential.
Since AC magnetic problems' formulations use complex values that represent the real world quantities sinusoidally changing with time, the integral values might appear in the following different ways:
As a complex value, with amplitude and phase (e.g. current, flux linkage, magnetomotive force).
As a complex vector, with the endpoint sweeping an ellipse in any complete time period (e.g. induction, magnetic field strength). The characteristics of complex vectors are: the amplitude (per coordinate), the phase, and the polarization coefficient.
As a oscillated value (e.g. ohmic loss power, field energy, etc.) pulsing around its mean value with double frequency. The characteristics of oscillated values are: the mean value, the phase, and the pulsation amplitude.
As a oscillated vector (e.g. mechanical forces) with magnitude and direction varying around its mean value with double frequency. The characteristics of oscillated vectors are: the mean value (length, slant and coordinates), and the variation amplitude (used, for example, to estimate the mechanical force limit for a period).
Name, |
Formula and Description |
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qfInt_Conductance |
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Total current qfInt_Jtotal |
I = sc∫ jtotal·ds Complex value. Electric current through a particular surface. |
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External current qfInt_Jextern |
Iext = sc∫ jext·ds Complex value. External current through a particular surface. |
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Eddy current qfInt_Jeddies |
Ieddy = sc∫ jeddy·ds Complex value. Eddy current through a particular surface. |
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Joule heat qfInt_Power |
P = v∫ j2/g·dv Oscillated value. Joule heat power in a particular volume. g – electric conductivity of the media. |
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Core loss qfInt_Steinmetz |
P = v∫ ( khf B2 + kcf2 B2 + ke(f B)1.5 )dv Average power of core loss, which is the sum of the hysteresis losses, eddy current losses and excess magnetic losses (not covered by first two loss types). |
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Power flow qfInt_EnergyFlow |
PS = s∫(S·n)·ds Oscillated value. Power flow through the given surface (Poynting vector flow) Here S is a Pointing vector S = [E×H]. |
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Maxwell force qfInt_MaxwellForce |
F = 1/2·s∮(H·(n·B) + B·(n·H) - n·(H·B))ds Oscillated vector. Maxwell force acting on bodies contained in a particular volume. The integral is evaluated over the boundary of the volume, and n denotes the vector of the outward unit normal. |
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Maxwell torque qfInt_MaxwellTorque |
T = 1/2·s∮([r×H]·(n·B) + [r×B]·(n·H) - [r×n]·(H·B))ds Oscillated value. Maxwell force torque acting on bodies contained in a particular volume, where r is a radius vector of the point of integration.
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Lorentz force qfInt_LorentzForce |
F = v∫[j×B]dv Oscillated vector. The Lorentz force acting on conductors contained in a particular volume. |
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Lorentz torque qfInt_LorentzTorque |
T = v∫[r×[j×B]]dv Oscillated value. The Lorentz force torque acting on bodies contained in a particular volume. The torque is considered relative to the origin of the coordinate system. |
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Magnetic field energy qfInt_MagneticEnergy |
W = 1/2·v∫H·Bdv Oscillated value. This formula is used for both linear and nonlinear cases. |
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Flux linkage per one turn qfInt_FluxLinkage |
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Magnetomotive force qfInt_KGrad_t_dl |
F = L∫ (H·t)dl Complex value. The integral has to be evaluated over a cross section of the coil, and SC is the area of the cross section. |
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Magnetic flux qfInt_Grad_n_ds |
Φ = s∫ (B·n)ds Complex value. Magnetic flux through a particular surface. |
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Surface energy qfInt_GradKGrad_n_ds |
WS = 1/2·s∫ (B·H)ds Oscillated value. The integral is evaluated over the surface swept by the movement of the contour. |
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Average surface potential qfInt_Potential_ds |
AS = 1/S·s∫ A·ds Complex value. |
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Average volume potential qfInt_Potential_dv |
AV = 1/V·v∫ A·dv Complex value. |
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Average volume flux density qfInt_Grad_dv |
Ba = 1/V·v∫ B·dv Complex vector. |
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Average volume strength qfInt_KGrad_dv |
Ha = 1/V·v∫ H·dv Complex vector. |
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Mean square flux density qfInt_Grad2_dv |
Ba2 = 1/V·v∫ B2·dv Oscillated value. |
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Mean square strenght qfInt_KGrad2_dv |
Ha2 = 1/V·v∫ H2·dv Oscillated value. |
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Line integral of flux density qfInt_Grad_t_dl |
x = L∫ (B·t)dl Complex value. The line integral over the contour of a magnetic flux density. |
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Surface integral of strength qfInt_Grad_n_ds |
x = s∫ (H·n)ds Complex value. |
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Note. The Maxwell force incorporates both the force acting on ferromagnetic bodies and Lorentz force, which acts only on conductors. If the first component is negligible or is not considered, we recommend calculating the electromagnetic force as Lorentz force. Its precision is less sensitive to the contour path, and you can simply select conductors via block selection to calculate the force. With Maxwell force, this method leads to very rough results, and you are recommended to avoid coinciding of your contour parts and material boundaries as described in Calculating integrals.