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 2-D and 3-D magnetic field analysis employing the magnetic vector potential A. For 2-D planar analyses, the flux value is per unit depth.

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

(17–50)

where:

Equation 17–50 can be rewritten in terms of the definition of the vector potential as:

(17–51)

where:

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

(17–52)

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 17–52. 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).

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:

(17–53)

where:

The macro interpolates values of magnetic field intensity, H, to the path and integrates to obtain the I_mmf as in Equation 17–53. In a static analysis or transverse electromagnetic (TEM) and transverse electric (TE) wave guide mode computation, I_mmf can be interpreted as a current passing the surface bounded by the closed contour.

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):

(17–54)

where:

The macro evaluates Equation 17–54 by integrating over the selected element set according to:

(17–55)

where:

n = number of elements

Re{ } = real component of a complex quantity

[ρi] = resistivity tensor (matrix)

= total eddy current density vector for element i

voli = element volume

* = complex conjugate operator

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

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

(17–56)

where:

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

(17–57)

where:

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

(17–58)

where:

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,

(17–59)

where:

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

(17–60)

where:

The terminal voltage is therefore:

(17–61)

where:

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

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:

(17–62)

where:

The magnetic co-energy is calculated as:

(17–63)

where:

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

(17–64)

where:

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

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

(17–65)

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

(17–66)

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.

(17–67)

where:

Emax = maximum element nodal electric field (magnitude)

(17–68)

where:

Dmax = maximum element nodal electric flux density (magnitude)

17.5.9.2. Electromagnetics

17.5.9.2.1. Magnetic Field Intensity

(17–69)

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

(17–70)

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.

(17–71)

where:

Hmax = maximum element nodal magnetic field intensity (magnitude)

(17–72)

where:

Bmax = maximum nodal averaged magnetic flux density (magnitude)

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

(17–73)

where:

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, V_emf can be interpreted as a voltage drop.

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:

(17–74)

(17–75)

where:

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

The voltage term “A” in Equation 17–74 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,

(17–76)

Rearranging we have,

(17–77)

where:

The voltage standing-wave ratio is calculated as:

(17–78)

where:

For a matched load ( = Z_o) 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:

(17–79)

where:

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