17.5. POST1 - Electromagnetic Macros

Electromagnetic macros are macro files created to perform specific postprocessing operations for electromagnetic field analysis. Macros performing computational analysis are detailed in this section.

17.5.1. Flux Passing Thru a Closed Contour

The flux passing through a surface defined by a closed line contour (PPATH command) is computed (using the FLUXV command macro). The macro is applicable to 2D and 3D magnetic field analysis employing the magnetic vector potential A. For 2D planar analyses, the flux value is per unit depth.

The flux passing through a surface S can be calculated as:

(17–48)

where:

φ = flux enclosed by the bounding surface S
{B} = flux density vector
{n} = unit normal vector
area = area of the bounding surface S

Equation 17–48 can be rewritten in terms of the definition of the vector potential as:

(17–49)

where:

{A} = magnetic vector potential

By applying Stokes theorem, the surface integral reduces to a line integral of A around a closed contour;

(17–50)

where:

= length of the bounding contour line

The macro interpolates values of the vector potential, A, to the closed contour path (defined by the PPATH command) and integrates to obtain the flux using Equation 17–50. In the axisymmetric case, the vector potential is multiplied by 2πr to obtain the total flux for a full circumferential surface (where "r" is the x-coordinate location of the interpolation point).

17.5.2. Magnetomotive Forces

The magnetomotive force (current) along a contour or path (defined by the PPATH command) is calculated (using the MMF command macro) according to Amperes' theorem:

(17–51)

where:

Immf = magnetomotive force
{H} = magnetic field intensity vector

The macro interpolates values of magnetic field intensity, H, to the path and integrates to obtain the Immf as in Equation 17–51. In a static analysis or transverse electromagnetic (TEM) and transverse electric (TE) wave guide mode computation, Immf can be interpreted as a current passing the surface bounded by the closed contour.

17.5.3. Power Loss

The power dissipated in a conducting solid body under the influence of a time-harmonic electromagnetic field is computed (using the POWERH command macro). The r.m.s. power loss is calculated from the equation (see Harmonic Analysis Using Complex Formalism for further details):

(17–52)

where:

Prms = rms power loss
r = material resistivity
Jt = total current density
~ = complex quantity

The macro evaluates Equation 17–52 by integrating over the selected element set according to:

(17–53)

where:

n = number of elements
Re{ } = real component of a complex quantity
i] = resistivity tensor (matrix)
= total eddy current density vector for element i
voli = element volume
* = complex conjugate operator

For 2D planar analyses, the resulting power loss is per unit depth.

17.5.4. Energy Supplied

The energy supplied to the coil for a linear system is calculated as:

(17–54)

where:

W = energy input to coil
{A} = nodal vector potential
{Js} = d.c. source current density
vol = volume of the coil

17.5.5. Terminal Inductance

The inductance as seen by the terminal leads of the coil is calculated as:

(17–55)

where:

L = terminal inductance
i = coil current (per turn)

17.5.6. Flux Linkage

The total flux linkage of a coil can be calculated from the terminal inductance and coil current,

(17–56)

where:

λ = flux linkage

17.5.7. Terminal Voltage

For a coil operating with an a.c. current at frequency ω (Hz), a voltage will appear at the terminal leads. Neglecting skin effects and saturation, a static analysis gives the correct field distribution. For the assumed operating frequency, the terminal voltage can be found. From Faraday's law,

(17–57)

where:

u = terminal voltage

Under a sinusoidal current at an operating frequency ω, the flux linkage will vary sinusoidally

(17–58)

where:

λm = zero-to-peak magnitude of the flux linkage

The terminal voltage is therefore:

(17–59)

where:

U = ωλm = zero-to-peak magnitude of the terminal voltage (parameter VLTG returned by the macro)

For 2D planar analyses, the results are per unit depth.

17.5.8. Energy in a Magnetic Field

The stored energy and co-energy in a magnetic field are calculated (by the SENERGY command macro). For the static or transient analysis, the stored magnetic energy is calculated as:

(17–60)

where:

Ws = stored magnetic energy

The magnetic co-energy is calculated as:

(17–61)

where:

Wc = stored magnetic co-energy
Hc = coercive force

For time-harmonic analysis, the r.m.s. stored magnetic energy is calculated as:

(17–62)

where:

Wrms = r.m.s. stored energy

For 2D planar analyses, the results are per unit depth.

17.5.9. Relative Error in Electrostatic or Electromagnetic Field Analysis

The relative error in an electrostatic or electromagnetic field analysis is computed (by the EMAGERR command macro). The relative error measure is based on the difference in calculated fields between a nodal-averaged continuous field representation and a discontinuous field represented by each individual element's-nodal field values. An average error for each element is calculated. Within a material, the relative error is calculated as:

17.5.9.1. Electrostatics

17.5.9.1.1. Electric Field

(17–63)

where:

Eei = relative error for the electric field (magnitude) for element i
Ej = nodal averaged electric field (magnitude)
Eij = electric field (magnitude) of element i at node j
n = number of vertex nodes in element i

17.5.9.1.2. Electric Flux Density

(17–64)

where:

Dei = relative error for the electric flux density (magnitude) for element i
Dj = nodal averaged electric flux density (magnitude)
Dij = electric flux density (magnitude) of element i at node j

A normalized relative error norm measure is also calculated based on the maximum element nodal calculated field value in the currently selected element set.

(17–65)

where:

Emax = maximum element nodal electric field (magnitude)

(17–66)

where:

Dmax = maximum element nodal electric flux density (magnitude)

17.5.9.2. Electromagnetics

17.5.9.2.1. Magnetic Field Intensity

(17–67)

where:

Hei = relative error for the magnetic field intensity (magnitude) for element i
Hj = nodal averaged magnetic field intensity (magnitude)
Hij = magnetic field intensity (magnitude) of element i at node j

17.5.9.2.2. Magnetic Flux Density

(17–68)

where:

Bei = relative error for the magnetic flux density (magnitude) for element i
Bj = nodal averaged magnetic flux density (magnitude)
Bij = magnetic flue density (magnitude) of element i at node j

A normalized relative error measure is also calculated based on the maximum element nodal calculated field value in the currently selected element set.

(17–69)

where:

Hmax = maximum element nodal magnetic field intensity (magnitude)

(17–70)

where:

Bmax = maximum nodal averaged magnetic flux density (magnitude)

17.5.10. Electromotive Force

The electromotive force (voltage drop) between two conductors defined along a path contour (PATH command) is computed (using the EMF command macro):

(17–71)

where:

Vemf = electromotive force (voltage drop)
{E} = electric field vector

The macro interpolates values of the electric field, E, to the path (defined by the PATH command) and integrates to obtain the electromotive force (voltage drop). The path may span multiple materials of differing permittivity. At least one path point should reside in each material transversed by the path. In static analysis or transverse electromagnetic (TEM) and transverse magnetic (TM) wave guide mode computation, Vemf can be interpreted as a voltage drop.

17.5.11. Computation of Equivalent Transmission-line Parameters

The equivalent transmission-line parameters for a guiding wave structure are calculated. For a lossless guiding structure, the total mode voltage, V(Z), and mode current, I(Z), associated with a +Z propagating field take on the form:

(17–72)

(17–73)

where:

Zo = characteristic impedance for any mode
A = amplitude of the incident voltage wave (see below)
B = amplitude of the backscattered voltage wave (see below)

We can consider the propagating waves in terms on an equivalent two-wire transmission line terminated at Z = by a load impedance .

Figure 17.10: Equivalent Two-Wire Transmission Line

Equivalent Two-Wire Transmission Line

The voltage term "A" in Equation 17–72 can be considered as the amplitude of the incident wave, and voltage term "B" as the amplitude of the mode voltage wave backscattered off the load impedance .

Thus,

(17–74)

Rearranging we have,

(17–75)

where:

Γ = voltage reflection coefficient

The voltage standing-wave ratio is calculated as:

(17–76)

where:

S = voltage standing-wave ratio (output as VSWR)

For a matched load ( = Zo) there is no reflection (Γ = 0) and the S = 1. If is a short circuit, B = -A, Γ = -1, and the S is infinite. If is an open circuit, B = A, Γ = +1, and the S once again is infinite.

The reflection coefficient is frequently expressed in dB form by introducing the concept of return loss defined by:

(17–77)

where:

LR = return loss in dB (output as RL)

The macro calculates the above transmission line parameters in terms of the incident, reference and total voltage.