Electromagnetic fields are governed by the following Maxwell's equations (Smythe([150])):
(5–1) |
(5–2) |
(5–3) |
(5–4) |
where:
x = curl operator |
= divergence operator |
{H} = magnetic field intensity vector |
{J} = total current density vector |
{J_{s}} = applied source current density vector |
{J_{e}} = induced eddy current density vector |
{J_{vs}} = velocity current density vector |
{D} = electric flux density vector (Maxwell referred to this as the displacement vector, but to avoid misunderstanding with mechanical displacement, the name electric flux density is used here.) |
t = time |
{E} = electric field intensity vector |
{B} = magnetic flux density vector |
ρ = electric charge density |
The continuity equation follows from taking the divergence of both sides of Equation 5–1.
(5–5) |
The continuity equation must be satisfied for the proper setting of Maxwell's equations. Users should prescribe J_{s} taking this into account.
The above field equations are supplemented by the constitutive relation that describes the behavior of electromagnetic materials. For problems considering saturable material without permanent magnets, the constitutive relation for the magnetic fields is:
(5–6) |
where:
μ = magnetic permeability matrix, in general a function of {H} |
The magnetic permeability matrix [μ] may be input either as a function of temperature or field. Specifically, if [μ] is only a function of temperature,
(5–7) |
where:
μ_{o} = permeability of free space (input on EMUNIT command) |
μ_{rx} = relative permeability in the x-direction (input as MURX on MP command) |
If [μ] is only a function of field,
(5–8) |
where:
μ_{h} = permeability derived from the input B versus H curve (input with TB,BH). |
Mixed usage is also permitted, e.g.:
(5–9) |
When permanent magnets are considered, the constitutive relation becomes:
(5–10) |
where:
{M_{o}} = remanent intrinsic magnetization vector |
Rewriting the general constitutive equation in terms of reluctivity it becomes:
(5–11) |
where:
[ν] = reluctivity matrix = [μ]-1 |
The constitutive relations for the related electric fields are:
(5–12) |
(5–13) |
where:
σ_{xx} = conductivity in the x-direction (input as inverse of RSVX on MP command) |
ε_{xx} = permittivity in the x-direction (input as PERX on MP command) |
The solution of magnetic field problems is commonly obtained using potential functions. Two kinds of potential functions, the magnetic vector potential and the magnetic scalar potential, are used depending on the problem to be solved. Factors affecting the choice of potential include: field dynamics, field dimensionality, source current configuration, domain size and discretization.
The applicable regions are shown below. These will be referred to with each solution procedure discussed below.
where:
Ω_{0} = free space region |
Ω_{1} = nonconducting permeable region |
Ω_{2} = conducting region |
μ = permeability of iron |
μ_{o} = permeability of air |
M_{o} = permanent magnets |
S_{1} = boundary of W1 |
σ = conductivity |
Ω = Ω_{1} + Ω_{2} + Ω_{0} |
The scalar potential method as implemented in SOLID5, SOLID96, and SOLID98 for 3-D magnetostatic fields is discussed in this section. Magnetostatics means that time varying effects are ignored. This reduces Maxwell's equations for magnetic fields to:
(5–14) |
(5–15) |
In the domain Ω_{0} and Ω_{1} of a magnetostatic field problem (Ω_{2} is not considered for magnetostatics) a solution is sought which satisfies the relevant Maxwell's Equation 5–14 and Equation 5–15 and the constitutive relation Equation 5–10 in the following form (Gyimesi([141]) and Gyimesi([149])):
(5–16) |
(5–17) |
where:
{H_{g}} = preliminary or “guess” magnetic field |
φ_{g} = generalized potential |
The development of {H_{g}} varies depending on the problem and the formulation. Basically, {H_{g}} must satisfy Ampere's law (Equation 5–14) so that the remaining part of the field can be derived as the gradient of the generalized scalar potential φ_{g}. This ensures that φ_{g} is singly valued. Additionally, the absolute value of {H_{g}} must be greater than that of Δφ_{g}. In other words, {Hg} should be a good approximation of the total field. This avoids difficulties with cancellation errors (Gyimesi([149])).
This framework allows for a variety of scalar potential formulation to be used. The appropriate formulation depends on the characteristics of the problem to be solved. The process of obtaining a final solution may involve several steps (controlled by the MAGOPT solution option).
As mentioned above, the selection of {H_{g}} is essential to the development of any of the following scalar potential strategies. The development of {H_{g}} always involves the Biot-Savart field {H_{s}} which satisfies Ampere's law and is a function of source current {J_{s}}. {H_{s}} is obtained by evaluating the integral:
(5–18) |
where:
{J_{s}} = current source density vector at d(volc) |
{r} = position vector from current source to node point |
volc = volume of current source |
The above volume integral can be reduced to the following surface integral (Gyimesi et al.([174]))
(5–19) |
where:
surfc = surface of the current source |
Evaluation of this integral is automatically performed upon initial solution execution or explicitly (controlled by the BIOT command). The values of {J_{s}} are obtained either directly as input by:
SOURC36 - Current Source |
or indirectly calculated by electric field calculation using:
SOLID5 - 3-D Coupled-Field Solid |
LINK68 - Coupled Thermal-Electric Line |
SOLID98 - Tetrahedral Coupled-Field Solid |
PLANE230 - 2-D Electric Solid |
SOLID231 or SOLID232 - 3-D Electric Solids |
Depending upon the current configuration, the integral given in Equation 5–19 is evaluated in a closed form and/or a numerical fashion (Smythe([150])).
Three different solution strategies emerge from the general framework discussed above:
Reduced Scalar Potential (RSP) Strategy |
Difference Scalar Potential (DSP) Strategy |
General Scalar Potential (GSP) Strategy |
Applicability
If there are no current sources ({J_{s}} = 0), the RSP strategy is applicable. The RSP strategy is also applicable, in general, if there are current sources but there is no iron ([μ] = [μ_{o}]) within the problem domain. This formulation is developed by Zienkiewicz([75]).
Procedure
The RSP strategy uses a one-step procedure (MAGOPT,0). Equation 5–16 and Equation 5–17 are solved making the following substitution:
(5–20) |
Saturation is considered if the magnetic material is nonlinear. Permanent magnets are also considered.
Applicability
The DSP strategy is applicable when current sources and singly connected iron regions exist within the problem domain ({J_{s}} ≠ {0}) and ([μ] ≠ [μ_{o}]). A singly connected iron region does not enclose a current. In other words, a contour integral of {H} through the iron must approach zero as u → .
(5–21) |
This formulation is developed by Mayergoyz([119]).
Procedure
The DSP strategy uses a two-step solution procedure. The first step (MAGOPT,2) makes the following substitution into Equation 5–16 and Equation 5–17:
(5–22) |
subject to:
(5–23) |
This boundary condition is satisfied by using a very large value of permeability in the iron (internally set by the program). Saturation and permanent magnets are not considered. This step produces a near zero field in the iron region which is subsequently taken to be zero according to:
(5–24) |
and in the air region:
(5–25) |
The second step (MAGOPT,3) uses the fields calculated on the first step as the preliminary field for Equation 5–16 and Equation 5–17:
(5–26) |
(5–27) |
Here saturation and permanent magnets are considered. This step produces the following fields:
(5–28) |
and
(5–29) |
which are the final results to the applicable problems.
Applicability
The GSP strategy is applicable when current sources ({J_{s} ≠ {0}) exist within the problem domain in conjunction with a multiply connected iron ([μ] ≠ [μ_{o}]) region. A multiply connected iron region encloses some current source. This means that a contour integral of {H} through the iron region is not zero:
(5–30) |
where:
= refers to the dot product |
This formulation is developed by Gyimesi([141], [149], [202]).
Procedure
The GSP strategy uses a three-step solution procedure. The first step (MAGOPT,1) performs a solution only in the iron with the following substitution into Equation 5–16 and Equation 5–17:
(5–31) |
subject to:
(5–32) |
Here S_{1} is the surface of the iron air interface. Saturation can optimally be considered for an improved approximation of the generalized field but permanent magnets are not. The resulting field is:
(5–33) |
The second step (MAGOPT,2) performs a solution only in the air with the following substitution into Equation 5–16 and Equation 5–17:
(5–34) |
subject to:
(5–35) |
This boundary condition is satisfied by automatically constraining the potential solution φ_{g} at the surface of the iron to be what it was on the first step (MAGOPT,1). This step produces the following field:
(5–36) |
Saturation or permanent magnets are of no consequence since this step obtains a solution only in air.
The third step (MAGOPT,3) uses the fields calculated on the first two steps as the preliminary field for Equation 5–16 and Equation 5–17:
(5–37) |
(5–38) |
Here saturation and permanent magnets are considered. The final step allows for the total field to be computed throughout the domain as:
(5–39) |
The vector potential method, implemented in PLANE13 and PLANE233 for 2-D electromagnetic fields, is discussed in this section. Considering static and dynamic fields and neglecting displacement currents (quasi-stationary limit), the following subset of Maxwell's equations apply:
(5–40) |
(5–41) |
(5–42) |
The usual constitutive equations for magnetic and electric fields apply as described by Equation 5–11 and Equation 5–12. Although some restrictions on anisotropy and nonlinearity do occur in the formulations mentioned below.
In the entire domain, Ω, of an electromagnetic field problem a solution is sought which satisfies the relevant Maxwell's Equation 5–40 thru Equation 5–41. See Figure 5.1: Electromagnetic Field Regions for a representation of the problem domain Ω.
A solution can be obtained by introducing potentials which allow the magnetic field {B} and the electric field {E} to be expressed as (Biro([120])):
(5–43) |
(5–44) |
where:
{A} = magnetic vector potential |
V = electric scalar potential |
These specifications ensure the satisfaction of two of Maxwell's equations, Equation 5–41 and Equation 5–42. What remains to be solved is Ampere's law, Equation 5–40 in conjunction with the constitutive relations, Equation 5–11, and the divergence free property of current density. The resulting differential equations are:
(5–45) |
(5–46) |
(5–47) |
These equations are subject to the appropriate boundary conditions.
This system of simplified Maxwell's equations with the introduction of potential functions is used for the solutions of 2-D static and dynamic fields. Silvester([72]) presents a 2-D static formulation. Chari([69]), Brauer([70]) and Tandon([71]) discuss the 2-D eddy current problem and Weiss([94]) and Garg([95]) discuss 2-D eddy current problems which allow for skin effects (eddy currents present in the source conductor).
An edge-based method, using edge-based elements with a discontinuous normal component of magnetic vector potential, is implemented in the 3-D electromagnetic SOLID236 and SOLID237 elements.
The differential equations governing SOLID236 and SOLID237 elements are the following:
(5–48) |
(5–49) |
(5–50) |
These equations are subject to the appropriate magnetic and electrical boundary conditions.
The uniqueness of edge-based magnetic vector potential is ensured by the tree gauging procedure (GAUGE command) that sets the edge-flux degrees of freedom corresponding to the spanning tree of the finite element mesh to zero.
In a general dynamic problem, any field quantity, q(r,t) depends on the space, r, and time, t, variables. In a harmonic analysis, the time dependence can be described by periodic functions:
(5–51) |
or
(5–52) |
where:
r = location vector in space |
t = time |
ω = angular frequency of time change. |
a(r) = amplitude (peak) |
φ(r) = phase angle |
c(r) = measurable field at ωt = 0 degrees |
s(r) = measurable field at ωt = -90 degrees |
In an electromagnetic analysis, q(r,t) can be the flux density, {B}, the magnetic field, {H}, the electric field, {E}, the current density, J, the vector potential, {A}, or the scalar potential, V. Note, however, that q(r,t) can not be the Joule heat, Q^{j}, the magnetic energy, W, or the force, F^{jb}, because they include a time-constant term.
The quantities in Equation 5–51 and Equation 5–52 are related by
(5–53) |
(5–54) |
(5–55) |
(5–56) |
In Equation 5–51) a(r), φ(r), c(r) and s(r) depend on space coordinates but not on time. This separation of space and time is taken advantage of to minimize the computational cost. The originally 4 (3 space + 1 time) dimensional real problem can be reduced to a 3 (space) dimensional complex problem. This can be achieved by the complex formalism.
The measurable quantity, q(r,t), is described as the real part of a complex function:
(5–57) |
Q(r) is defined as:
(5–58) |
where:
j = imaginary unit |
Re { } = denotes real part of a complex quantity |
Q_{r}(r) and Q_{i}(r) = real and imaginary parts of Q(r). Note that Q depends only on the space coordinates. |
The complex exponential in Equation 5–57 can be expressed by sine and cosine as
(5–59) |
Substituting Equation 5–59 into Equation 5–57 provides Equation 5–58
(5–60) |
Comparing Equation 5–51 with Equation 5–60 reveals:
(5–61) |
(5–62) |
In words, the complex real, Q_{r}(r), and imaginary, Q_{i}(r), parts are the same as the measurable cosine, c(r), and sine, s(r), amplitudes.
A harmonic analysis provides two solution sets: the real and imaginary components of a complex solution. According to Equation 5–51, and Equation 5–61 the magnitude of the real and imaginary sets describe the measurable field at t = 0 and at ωt = -90 degrees, respectively. Comparing Equation 5–52 and Equation 5–61 provides:
(5–63) |
(5–64) |
Equation 5–63 expresses the amplitude (peak) and phase angle of the measurable harmonic field quantities by the complex real and imaginary parts.
The time average of harmonic fields such as A, E, B, H, J, or V is zero at point r. This is not the case for P, W, or F because they are quadratic functions of B, H, or J. To derive the time dependence of a quadratic function - for the sake of simplicity - we deal only with a Lorentz force, F, which is product of J and B. (This is a cross product but, for simplicity, the components are not shown. The space dependence is also omitted.)
(5–65) |
where:
F^{jb} = Lorentz Force density (output as FMAG on PRESOL command) |
The time average of cos^{2} and sin^{2} terms is 1/2 whereas that of the sin cos term is zero. Therefore, the time average force is:
(5–66) |
Thus, the force can be obtained as the sum of “real” and “imaginary” forces. In a similar manner the time averaged Joule power density, Q^{j}, and magnetic energy density, W, can be obtained as:
(5–67) |
(5–68) |
where:
W = magnetic energy density (output as SENE on PRESOL command) |
Q^{j} = Joule Power density heating per unit volume (output as JHEAT on PRESOL command) |
The time average values of these quadratic quantities can be obtained as the sum of real and imaginary set solutions.
The element returns the integrated value of F^{jb} is output as FJB and W is output as SENE. Q^{j} is the average element Joule heating and is output as JHEAT. For F and Q^{j} the 1/2 time averaging factor is taken into account at printout. For W the 1/2 time factor is ignored to preserve the printout of the real and imaginary energy values as the instantaneous stored magnetic energy at t = 0 and at ωt = -90 degrees, respectively. The element force, F, is distributed among nodes to prepare a magneto-structural coupling. The average Joule heat can be directly applied to thermoelectric coupling.
Neglecting the time-derivative of magnetic flux density (the quasistatic approximation), the system of Maxwell's equations (Equation 5–1 through Equation 5–4) reduces to:
(5–69) |
(5–70) |
(5–71) |
(5–72) |
As follows from Equation 5–70, the electric field {E} is irrotational, and can be derived from:
(5–73) |
where:
V = electric scalar potential |
In the time-varying electromagnetic field governed by Equation 5–69 through Equation 5–72, the electric and magnetic fields are uncoupled. If only the electric solution is of interest, replacing Equation 5–69 by the continuity Equation 5–5 and eliminating Equation 5–71 produces the system of differential equations governing the quasistatic electric field.
Repeating Equation 5–12 and Equation 5–13 without velocity effects, the constitutive equations for the electric fields become:
(5–74) |
(5–75) |
where:
ρ_{xx} = resistivity in the x-direction (input as RSVX on MP command) |
ε_{xx} = permittivity in the x-direction (input as PERX on MP command) |
The conditions for {E}, {J}, and {D} on an electric material interface are:
(5–76) |
(5–77) |
(5–78) |
where:
E_{t1}, E_{t2} = tangential components of {E} on both sides of the interface |
J_{n1}, J_{n2} = normal components of {J} on both sides of the interface |
D_{n1}, D_{n2} = normal components of {D} on both sides of the interface |
ρ_{s} = surface charge density |
Two cases of the electric scalar potential approximation are considered below.
In this analysis, the relevant governing equations are Equation 5–73 and this continuity equation:
(5–79) |
Substituting the constitutive Equation 5–74 and Equation 5–75 into Equation 5–79, and taking into account Equation 5–73, one obtains the differential equation for electric scalar potential:
(5–80) |
Equation 5–80 is used to approximate a time-varying electric field in elements PLANE230, SOLID231, and SOLID232. It takes into account both the conductive and dielectric effects in electric materials. Neglecting time-variation of the electric potential, Equation 5–80 reduces to the governing equation for steady-state electric conduction:
(5–81) |
In the case of a time-harmonic electric field analysis, the complex formalism allows Equation 5–80 to be re-written as:
(5–82) |
where:
ω = angular frequency |
Equation 5–82 is the governing equation for a time-harmonic electric analysis using elements PLANE121, SOLID122, and SOLID123.
In a time-harmonic analysis, the loss tangent tan δ can be used instead of or in addition to the electrical conductivity [σ] to characterize losses in dielectric materials. In this case, the conductivity matrix [σ] is replaced by the effective conductivity [σ^{eff}] defined as:
(5–83) |
where:
tan δ = loss tangent (input as LSST on MP command) |
Electric scalar potential equation for electrostatic analysis is derived from governing Equation 5–72 and Equation 5–73, and constitutive Equation 5–75:
(5–84) |
Equation 5–84, subject to appropriate boundary conditions, is solved in an electrostatic field analysis of dielectrics using elements PLANE121, SOLID122, and SOLID123.