QuickField Help

Main >> Support >> QuickField Help

Integral quantities in DC and Transient magnetics

Generally the integral quantities of interest in magnetic analysis are: mechanical force and torque, magnetic flux and flux linkage, magnetomotive force (MMF), magnetic field energy.
The following notations are used in formulas:

A - z-component of magnetic vector potential;
B - magnetic flux density vector;
H - vector of magnetic field strength;
B(H) - magnetization curve of a ferromagnetic, that assumed to be isotropic.

Name, ActiveField constant	Formula and Description
Magnetomotive force qfInt_KGrad_t_dl	\[ F = \int_L (\mathbf H · \mathbf t) dl, \] Magnetomotive force is a line integral around the contour of magnetic field strength. According Ampere's law the magnetomotive force around a closed line is equal to total current through the contour.
Flux linkage per one turn qfInt_FluxLinkage	For planar case \[ Ψ = \frac{1}{S_c} \oint_{S_c} Ads \] For axisymmetric case \[ Ψ = \frac{1}{S_c} 2π\oint_{S_c} (rA)ds \] The integral has to be evaluated over a cross section of the coil, and `S` is the area of the cross section.
Mechanical force qfInt_MaxwellForce	\[ \mathbf f = \frac{1}{2}\oint\left( \mathbf H(\mathbf n · \mathbf B) + \mathbf B(\mathbf n · \mathbf H) - \mathbf n(\mathbf H · \mathbf B) \right) ds \] Total magnetic force acting on bodies contained in a particular volume, where integral is evaluated over the boundary of the volume.
Mechanical torque qfInt_MaxwellTorque	\[ \mathbf T = \frac{1}{2}\oint\left( (\mathbf r × \mathbf H)(\mathbf n · \mathbf B) + (\mathbf r × \mathbf B)(\mathbf n · \mathbf H) - (\mathbf r × \mathbf n)(\mathbf H · \mathbf B) \right) ds \] Total torque of magnetic forces acting on bodies contained in a particular volume. where r is a radius vector of the point of integration. The torque vector is parallel to z-axis in the planar case, and is identically equal to zero in the axisymmetric one. The torque is considered relative to the origin of the coordinate system. The torque relative to any other arbitrary point can be obtained by adding extra term of [F×r₀], where F is the total force and r₀ is the radius vector of the point.
Magnetic field energy qfInt_MagneticEnergy	Linear case \[ W = \frac{1}{2} \int (\mathbf H·\mathbf B)dV \] Nonlinear case \[ W = \int \left ( \int_0^B H(B')dB' \right)dV \]
Magnetic field co-energy qfInt_MagneticCoenergy	For linear material the co-energy is equal to the magnetic energy. Nonlinear case \[ W_{co} = \int \left ( \int_0^H B(H')dH' \right)dV \]
Linearized field energy qfInt_ElectrostaticEnergy	For linear case the linearized energy is equal to the ordinary magnetic energy. Nonlinear case \[ W_l = \frac{1}{2} \int (\mathbf H·\mathbf B)dV \]

Name, ActiveField constant	Formula and Description
Average surface potential qfInt_Potential_ds	\[ A = \frac{1}{S}\int A ds \]
Average volume potential qfInt_Potential_dv	\[ A = \frac{1}{V}\int A dv \]
Root mean square value of the surface flux density	\[ B = \sqrt {\frac{1}{S}\int \|\mathbf B\|^2 ds } \]
Root mean square value of the volume flux density	\[ B = \sqrt {\frac{1}{V}\int \|\mathbf B\|^2 dv } \]
Root mean square value of the surface strength	\[ H = \sqrt {\frac{1}{S}\int \|\mathbf H\|^2 ds } \]
Root mean square value of the volume strength	\[ H = \sqrt {\frac{1}{V}\int \|\mathbf H\|^2 dv } \]

With transient problems only:
Name, ActiveField constant	Formula and Description
Total current qfInt_Jtotal	\[ I = \int_{S_c} J_{total} ds, \] Electric current through a particular surface. The integral is evaluated over a cross section of the coil, and `S_C` is the area of the cross section. `J`_total - is a total current density in z-direction.
External current qfInt_Jextern	\[ I_{external} = \int_{S_c} J_{external} ds, \] External current through a particular surface. `J`_external - density of external current.
Eddy current qfInt_Jeddies	\[ I_{eddy} = \int_{S_c} J_{eddy} ds, \] Eddy current through a particular surface. `J`_eddy - density of eddy current.
Joule heat qfInt_Power	\[ P = \int \frac{J_{total}^2}{σ} dv, \] Joule heat power in a volume. `g` - electric conductivity, `J`_total - total current density.
Power flow qfInt_EnergyFlow	\[ P_S = \int \left (\mathbf S · \mathbf n \right )ds \] Power flow through the given surface (Poynting vector flow). Here S = [E×H] is a Pointing vector.
Lorentz force qfInt_LorentzForce	\[ \mathbf f = \int [\mathbf J_{total} × \mathbf B] dv\] Oscillated vector. The Lorentz force acting on conductors contained in a particular volume.
Lorentz torque qfInt_LorentzTorque	\[ \mathbf T = \int [\mathbf r × [\mathbf J_{total} × \mathbf B]] dv\] 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.