Theory of Operation for EMDS for ADS

The simulation technique used to calculate the full three-dimensional electromagnetic field inside a structure is based on the finite element method. Although its implementation is largely transparent, a general understanding of the method is useful in making the most effective use of Electromagnetic Design System (EMDS) for Advanced Design System (ADS).

This section provides an overview of the finite element method, its implementation in EMDS for ADS, and a description of how S-parameters are computed from the simulated electric and magnetic fields.

The Finite Element Method

To generate an electromagnetic field solution from which S-parameters can be computed, EMDS for ADS employs the finite element method. In general, the finite element method divides the full problem space into thousands of smaller regions and represents the field in each sub-region (element) with a local function.

In EMDS for ADS, the geometric model is automatically divided into a large number of tetrahedra, where a single tetrahedron is formed by four equilateral triangles.

Representation of a Field Quantity

The value of a vector field quantity (such as the H-field or the E-field) at points inside each tetrahedron is interpolated from the vertices of the tetrahedron. At each vertex, EMDS for ADS stores the components of the field that are tangential to the three edges of the tetrahedron. In addition, the component of the vector field at the midpoint of selected edges that is tangential to a face and normal to the edge can also be stored. (See Field Quantities are Interpolated from Nodal Values). The field inside each tetrahedron is interpolated from these nodal values.

Field Quantities are Interpolated from Nodal Values

The components of a field that are tangential to the edges of an element are explicitly stored at the vertices.
The component of a field that is tangential to the face of an element and normal to an edge is explicitly stored at the midpoint of the selected edges.

The value of a vector field at an interior point is interpolated from the nodal values.
By representing field quantities in this way, Maxwell's equations can be transformed into matrix equations that are solved using traditional numerical methods.

Basis Functions

A first-order tangential element basis function interpolates field values from both nodal values at vertices and on edges. First-order tangential elements, like those shown in Field Quantities are Interpolated from Nodal Values, have 20 unknowns per tetrahedra. (For simplicity, all 20 elements are not shown in Field Quantities are Interpolated from Nodal Values.)

Size of Mesh Versus Accuracy

There is a trade-off between the size of the mesh, the desired level of accuracy, and the amount of available computing resources.

On one hand, the accuracy of the solution depends on how small each of the individual elements (tetrahedra) are. Solutions based on meshes that use a large number of elements are more accurate than solutions based on coarse meshes using relatively few elements. To generate a precise description of a field quantity, each tetrahedron must occupy a region that is small enough for the field to be adequately interpolated from the nodal values.

On the other hand, generating a field solution for meshes with a large number of elements requires a significant amount of computing power and memory. Therefore, it is desirable to use a mesh that is fine enough to obtain an accurate field solution but not so fine that it overwhelms the available computer memory and processing power.

To produce the optimal mesh, EMDS for ADS uses an iterative process in which the mesh is automatically refined in critical regions. First, it generates a solution based on a coarse initial mesh. Then, it refines the mesh based on suitable error criteria and generates a new solution. When selected S-parameters converge to within a desired limit, the iteration process ends.

Field Solutions

During the iterative solution process, the S-parameters typically stabilize before the full field solution. Therefore, when you are interested in analyzing the field solution associated with a structure, it may be desirable to use convergence criteria that is tighter than usual.

In addition, for any given number of adaptive iterations, the magnetic field (H-field) is less accurate than the solution for the electric field (E-field) because the H-field is computed from the E-field using the following relationship:

thus making the polynomial interpolation function an order lower than those used for the electric field.

Implementation Overview

To calculate the S-matrix associated with a structure, the following steps are performed:

  1. The structure is divided into a finite element mesh.
  2. The waves on each port of the structure that are supported by a transmission line having the same cross section as the port are computed.
  3. The full electromagnetic field pattern inside the structure is computed, assuming that each of the ports is excited by one of the waves.
  4. The generalized S-matrix is computed from the amount of reflection and transmission that occurs.

The final result is an S-matrix that allows the magnitude of transmitted and reflected signals to be computed directly from a given set of input signals, reducing the full three-dimensional electromagnetic behavior of a structure to a set of high frequency circuit values.

The Solution Process

There are three variations to the solution process:

Adaptive Solution

An adaptive solution is one in which a finite element mesh is created and automatically refined to increase the accuracy of succeeding adaptive solutions. The adaptive solution is performed at a single frequency. (Often, this is the first step in generating a non-adaptive frequency sweep or a fast frequency sweep.).

Non-adaptive Discrete Frequency Sweep

To perform this type of solution, an existing mesh is used to generate a solution over a range of frequencies. You specify the starting and ending frequency, and the interval at which new solutions are generated. The same mesh is used for each solution, regardless of the frequency.

Non-adaptive Fast Frequency Sweep

This type of solution is similar to a discrete frequency sweep, except that a single field solution is performed at a specified center frequency. From this initial solution, the system employes asymptotic waveform evaluation (AWE) to extrapolate an entire bandwidth of solution information. While solutions can be computed and viewed at any frequency, the solution at the center frequency is the most accurate.

The Mesher

A mesh is the basis from which a simulation begins. Initially, the structure's geometry is divided into a number of relatively coarse tetrahedra, with each tetrahedron having four triangular faces. The mesher uses the vertices of objects as the initial set of tetrahedra vertices. Other points are added to serve as the vertices of tetrahedra only as needed to create a robust mesh. Adding points is referred to as seeding the mesh.

After the initial field solution has been created, if adaptive refinement is enabled, the mesh is refined further.

2D Mesh Refinement

For 2D objects or ports, the mesher treats its computation of the excitation field pattern as a two-dimensional finite element problem. The mesh associated with each port is simply the 2D mesh of triangles corresponding to the face of tetrahedra that lie on the port surface.

The mesher performs an iterative refinement of this 2D mesh as follows:

  1. Using the triangular mesh formed by the tetrahedra faces of the initial mesh, solutions for the electric field, E, are calculated.
  2. The 2D solution is verified for accuracy.
  3. If the computed error falls within a pre-specified tolerance, the solution is accepted. Otherwise, the 2D mesh on the port face is refined and another iteration is performed.
    Any mesh points that have been added to the face of a port are incorporated into the full 3D mesh.

The 2D Solver

Before the full three-dimensional electromagnetic field inside a structure can be calculated, it is necessary to determine the excitation field pattern at each port. The 2D solver calculates the natural field patterns (or modes) that can exist inside a transmission structure with the same cross section as the port. The resulting 2D field patterns serve as boundary conditions for the full three-dimensional problem.

Excitation Fields

The assumption is that each port is connected to a uniform waveguide that has the same cross section as the port. The port interface is assumed to lie on the z=0 plane. Therefore, the excitation field is the field associated with traveling waves propagating along the waveguide to which the port is connected:

where:

Re is the real part of a complex number or function.

E(x,y) is a phasor field quantity.

j is the imaginary unit,

ω is angular frequency, 2πf.

γ=α + j β is the complex propagation constant, α is the attenuation constant of the wave.

β is the propagation constant associated with the wave that determines, at a given time t , how the phase angle varies with z.

In this context, the x and y axes are assumed to lie in the cross section of the port; the z axis lies along the direction of propagation.

Wave Equation

The field pattern of a traveling wave inside a waveguide can be determined by solving Maxwell's equations. The following equation that is solved by the 2D solver is derived directly from Maxwell's equation.

where:

E(x,y) is a phasor representing an oscillating electric field.

k 0 is the free space wave number, ,

ω is the angular frequency, 2πf.

µr( x, y ) is the complex relative permeability.

εr( x, y ) is the complex relative permittivity.

To solve this equation, the 2D solver obtains an excitation field pattern in the form of a phasor solution, E(x,y) . These phasor solutions are independent of z and t ; only after being multiplied by e-γ z do they become traveling waves.

Also note that the excitation field pattern computed is valid only at a single frequency. A different excitation field pattern is computed for each frequency point of interest.

Modes

For a waveguide or transmission line with a given cross section, there is a series of basic field patterns (modes) that satisfy Maxwell's equations at a specific frequency. Any linear combination of these modes can exist in the waveguide.

Modes, Reflections, and Propagation

It is also possible for a 3D field solution generated by an excitation signal of one specific mode to contain reflections of higher-order modes which arise due to discontinuities in a high frequency structure. If these higher-order modes are reflected back to the excitation port or transmitted onto another port, the S-parameters associated with these modes should be calculated.

If the higher-order mode decays before reaching any port-either because of attenuation due to losses or because it is a non-propagating evanescent mode-there is no need to obtain the S-parameters for that mode. Therefore, one way to avoid the need for computing the S-parameters for a higher-order mode is to include a length of waveguide in the geometric model that is long enough for the higher-order mode to decay.

For example, if the mode 2 wave associated with a certain port decays to near zero in 0.5 mm, then the "constant cross section" portion of the geometric model leading up to the port should be at least 0.5 mm long. Otherwise, for accurate S-parameters, the mode 2 S-parameters must be included in the S-matrix.

The length of the constant cross section segment to be included in the model depends on the value of the mode's attenuation constant, α.

Modes and Frequency

The field patterns associated with each mode generally vary with frequency. However, the propagation constants and impedances always vary with frequency. Therefore, when a frequency sweep has been requested, a solution is calculated for each frequency point of interest.

When performing frequency sweeps, be aware that as the frequency increases, the likelihood of higher-order modes propagating also increases.

Modes and Multiple Ports on a Face

Visualize a port face on a microstrip that contains two conducting strips side by side as two separate ports. If the two ports are defined as being separate, they are treated as two ports are connected to uncoupled transmission structures. It is as if a conductive wall separates the excitation waves.

However, in actuality, there will be electromagnetic coupling between the two strips. The accurate way to model this coupling is to analyze the two ports as a single port with multiple modes.

The 3D Solver

To calculate the full 3D field solution, the following wave equation is solved:

where:

E(x,y,z) is a complex vector representing an oscillating electric field.

mr( x, y ) is the complex relative permeability.

k 0 is the free space phase constant, ,

ω is the angular frequency, 2πf.

εr( x, y ) is the complex relative permittivity.

This is the same equation that the 2D solver solves for in calculating the 2D field pattern at each port. The difference is that the 3D solver does not assume that the electric field is a traveling wave propagating in a single direction. It assumes that the vector E is a function of x, y, and z. The physical electric field, E(x,y,z,t) , is the real part of the product of the phasor, E(x,y,z) , and ε j ω t :

Boundary Conditions

EMDS for ADS imposes boundary conditions at all surfaces exposed to the edge of the meshed problem region. This includes all outer surfaces and all surfaces exposed to voids and surface discontinuities within the structure.
The following types of boundary conditions are recognized by the 3D solver:

Port Boundaries

The 2D field solutions generated by the 2D solver for each port serve as boundary conditions at those ports. The final field solution that is computed for the structure must match the 2D field pattern at each port.

EMDS for ADS solves several problems in parallel. Consider the case of analyzing modes 1 and 2 in a two-port device. To compute how much of a mode 1 excitation at port 1 is transmitted as a mode 2 wave at port 2, the 3D mesher uses the following as boundary conditions:

To compute the full set of S-parameters, solutions involving other boundary conditions must also be solved. Because the S-matrix is symmetric for reciprocal structures (that is, S 12 is the same as S 21 ), only half of the S-parameters need to be explicitly computed.

Perfect H Boundaries

A Perfect H boundary forces the magnetic field (H-field) to have a normal component only. A symmetric H boundary can be used to model a plane of symmetry for a mode in which the H-field is normal to the symmetry plane.

Perfect E Boundaries

By default, the electric field is assumed to be normal to all surfaces exposed to the background, representing the case in which the entire structure is surrounded by perfectly conducting walls. This is referred to as a Perfect E boundary. The final field solution must match the case in which the tangential component of the electric field goes to zero at Perfect E boundaries.

It is also possible to assign Perfect E boundaries to surfaces within a structure. Using Perfect E boundaries in this way enables users to model perfectly conducting surfaces. The surfaces of all objects that have been defined to be perfectly conducting materials are automatically assigned to be Perfect E boundaries.

Conductor Boundaries

Conductor boundaries can be assigned to surfaces of imperfect conductors or resistive loads such as thick film resistors. At such boundaries, the following condition holds:

where:

is the is the unit vector that is normal to the surface.

E tan is the component of the E-field that is tangential to the surface.

H is the H-field.

Z s is the surface impedance of the boundary. (1+ j )/(δσ).

δ is the skin depth, , of the conductor being modeled.

ω is the frequency of the excitation wave.

σ is the conductivity of the conductor.

The fact that the E-field has a tangential component at the surface of imperfect conductors simulates the case in which the surface is lossy. The amount of loss will be proportional to the component of
that flows into the surface.

The field inside these objects is not computed; the conductor boundary approximates the behavior of the field at the surfaces of the objects.

Resistor Boundaries

Resistor boundaries model surfaces that represent resistive loads such as thin films on conductors. The following condition holds at resistor boundaries:

where:

E tan is the component of the E-field that is tangential to the surface.

H is the H-field.

R is the resistance at the boundary in ohms per square meter.

Radiation Boundaries

Radiation boundaries model surfaces that represent open space. Energy is allowed to radiate from these boundaries instead of being contained within them. At these surfaces, the second order radiation boundary condition is employed:

where:

E tan is the component of the E-field that is tangential to the surface.

is the unit vector normal to the radiation surface.

k 0 is the free space phase constant, .

j is equal to .

To ensure accurate results, radiation boundaries should be applied at least one quarter of a wavelength away from the source of the signal. However, they do not have to be spherical. The only restriction regarding their shape is that they be convex with regard to the radiation source.

Computing Radiated Fields

Electromagnetic Design System maps the E-field computed by the 3D solver on the radiation surfaces to plane registers and then calculates the radiated E-field using the following equation:

where:

s represents the radiation surfaces.

j is the imaginary unit, .

ω is the angular frequency, 2πf.

µ0 is the relative permeability of the free space.

H tan is the component of the magnetic field that is tangential to the surface.

H normal is the component of the magnetic field that is normal to the surface.

E tan is the component of the electric field that is tangential to the surface.

G is the free space Green's Function, given by:

where:

k 0 is the free space wave number, .

r and represent, respectively, points on the radiation surface and points beyond the surface as shown in Implementing Green's Function When Computing Radiated Fields.

Implementing Green's Function When Computing Radiated Fields

Displaying Field Solutions

The 3D solver is also used to manipulate field quantities for display. The system enables you to display or manipulate the field associated with any excitation wave at any port-for example, the field inside the structure due to a discrete mode 2 excitation wave at port 3. Waves excited on different modes can also be superimposed, even if they have different magnitudes and phases-for example, the waves excited on mode 1 at port 1 and mode 2 at port 2. In addition, far-field radiation in structures with radiation boundaries can be displayed.

The available fields depend on the type of solution that was performed:

Ports-only Solutions and Impedance Computations

This section addresses how impedances are computed for multi-conductor transmission line ports. Some examples of such structures are:

For structures with one or two conductors, you will need to define a single line segment, called an "impedance line", for each mode. Some examples of such structures include:

For these structures, the port solver will compute the voltage V along the impedance line which is used to calculate Z pv = V 2 /(2*Power). The power is always normalized to 1 Watt.

If one models N-conductor structures where N>2, then EMDS for ADS uses a different algorithm for computing Z pv and Z pi . The user must define an impedance line for each interior conductor. The impedance lines should go from the center of each interior conductor to the outer conductor.

The port solver computes a voltage along the first N line segments for each of the first N modes when the port solver detects that there are N+1 conductors. This becomes a "voltage vector"
(of length N) for each of the N quasi-TEM modes. Then, when computing Z pv , the square of the scalar voltage is now replaced by the dot product of the voltage vectors, for example:

For a more detailed explanation as to why this is done, refer to reference [1] at the end of this section.
For Z pi , the current is generally computed by adding the currents flowing into and out of the port and taking the average of the two. (If the simulator computed currents to perfect accuracy, the inward and outward currents would be identical.) For all mode numbers >= N, where N = number of conductors, the currents are calculated in this way.
For the first (N-1) quasi-TEM modes, the currents are computed on the N-1 interior conductors producing a current eigenvector. Then the impedance:

The result is that the EMDS for ADS impedance computations for such structures as coupled microstrip lines match the published equations for even- and odd-mode impedances.

As an example, take a CPW modeled as three interior strips surrounded by an enclosure. The ground strips do not touch the enclosure. Such a model is in the examples directory of EMDS for ADS and is called cpwtaper . The port solver shows us that the desired CPW mode is not the dominant mode, but is actually mode 3. To identify the modes, one can use the arrow plots in the Port Calibration menu or the Arrow display of the E-field in the post processor.

Each port consists of a 4 conductor system, the outer (ground) conductor, the inner strip, and the two "ground" strips. This results in 3 quasi-TEM modes. Mode 1 has E-field lines predominately in the substrate, all pointing in the same direction. This is the common mode (+V, +V, +V). Mode 2 also has E-field lines predominately in the substrate, but in opposite directions under the two "ground" strips. This is the slot mode (-V, 0, +V). Mode 3 has nearly zero E-fields everywhere because the fields are predominately between the inner strip and the "ground" strips. This is the CPW mode (0, +V, 0).

For further help in identifying modes in such a structure, one can look at the distributions with the "full" scale. One will notice that modes 1 and 3 obviously have the same "even" symmetry in the E fields, while mode 2 has an odd symmetry. The CPW mode has an "even" symmetry, so it has to be mode 1 or 3.

Modes 1 and 2 have significant E-field strengths in the substrate, especially under the "ground" strips. So, there is a potential difference between the "ground" strips and the outer ground for these two modes. However, for mode 3, the "ground" strips are at the same potential as the outer ground, which is consistent with the CPW mode.

Thus one can identify the modes. For mode 3, the "ground" strips are at 0 volts with respect to the outer ground, and the signal line has +V. The Z pv impedance computed for this mode using a dot product of the voltage vector for mode 3 gives the same Z pv as by computing the simple voltage between center strip and either "ground" strip. That is because the voltage vector for mode 3 along the three impedance lines is =[0,V,0] and

However, the impedances for the other modes now match accepted impedance definitions found in the literature for multi-conductor transmission lines.

  1. G.G. Gentili and M. Salazar-Palma, "The definition and computation of modal characteristic impedance in quasi-TEM coupled transmission lines," IEEE Trans. Microwave Theory Tech., Feb. 1995, pp. 338-343.

Calculating S-Parameters

A generalized S-matrix describes what fraction of power associated with a given field excitation is transmitted or reflected at each port.

The S-matrix for a three port structure is shown below:

where:

All quantities are complex numbers.

The magnitudes of a and b are normalized to a field carrying one watt of power.

|a i | 2 represents the excitation power at port i.

|b i | 2 represents the power of the transmitted or reflected field at port i.

The full field pattern at a port is the sum of the port's excitation field and all reflected/transmitted fields.

The phase of a i and b i represent the phase of the incident and reflected/transmitted field at t=0,

represents the phase angle of the excitation field on port i at t=0. (By default, it is zero.)

represents the phase angle of the reflected or transmitted field with respect to the excitation field.

S ij is the S-parameter describing how much of the excitation field at port j is reflected back or transmitted to port i.

For example, S 31 is used to compute the amount of power from the port 1 excitation field that is transmitted to port 3. The phase of S 31 specifies the phase shift that occurs as the field travels from port 1 to port 3.

Note
When the 2D solver computes the excitation field for a given port, it has no information indicating which way is "up" or "down". Therefore, if ports have not been calibrated, it is possible to obtain solutions in which the S-parameters are out of phase with the expected solution.

Frequency Points

The S-parameters associated with a structure are a function of frequency. Therefore, separate field solutions and S-matrices are generated for each frequency point of interest. EMDS for ADS supports two types of frequency sweeps:

Fast frequency sweeps are useful for analyzing the behavior of high Q structures. For wide bands of information, they are much faster than solving the problem at individual frequencies.

Note
Within a fast frequency solution, there is a bandwidth where the solution results are most accurate. This range is indicated by an error criterion using a matrix residue that measures the accuracy of the solution. For complex frequency spectra that have many peaks and valleys, a fast sweep may not be able to accurately model the entire frequency range. In this case, additional fast sweeps with different expansion frequencies will automatically be computed and combined into a single frequency response.

Renormalized S-Matrices

Before a structure's generalized S-matrix can be used in a high frequency circuit simulator to compute the reflection and transmission of signals, the generalized S-matrix must be normalized to the appropriate impedance. For example, if a generalized S-matrix has been normalized to 50 ohms, it can be used to compute reflection and transmission directly from signals that are normalized to 50 ohms, as in:

where the input signals, Vi i , and output signals, Vo i , are both normalized to 50 ohms.

To renormalize a generalized S-matrix to a specific impedance, the system first calculates a unique impedance matrix associated with the structure. This unique impedance matrix, Z, is defined as follows:

where:

S is the n x n generalized S-matrix.

I is an n x n identity matrix.

Z 0 is a diagonal matrix having the characteristic impedance (Z 0 ) of each port as a diagonal value.

The renormalized S-matrix is then calculated from the unique impedance matrix using this relationship:

where:

Z is the structure's unique impedance matrix.

and are diagonal matrices with the desired impedance and admittance as diagonal values. For example, if the matrix is being renormalized to 50 ohms, then would have diagonal values of 50.

Visualize the generalized S-matrix as an S-matrix that has been renormalized to the characteristic impedances of the structure. Therefore, if a diagonal matrix containing the characteristic impedances of the structure is used as in the above equation, the result would be the generalized S-matrix again.

Z- and Y-Matrices

Calculating and displaying the unique impedance and admittance matrices (Z and Y) associated with a structure is performed in the post processor.

Characteristic Impedances

EMDS for ADS calculates the characteristic impedance of each port in order to compute a renormalized S-matrix, Z-matrix, or Y-matrix. The system computes the characteristic impedance of each port in three ways-as Z pi , Z pv , and Z vi impedances.

You have the option of specifying which impedance is to be used in the renormalization calculations.

PI Impedance

The Z pi impedance is the impedance calculated from values of power (P) and current (I):

The power and current are computed directly from the simulated fields. The power passing through a port is equal to the following:

where the surface integral is over the surface of the port.

The current is computed by applying Ampere's Law to a path around the port:

While the net current computed in this way will be near zero, the current of interest is that flowing into the structure, I - or that flowing out of the structure, I + . In integrating around the port, the system keeps a running total of the contributions to each and uses the average of the two in the computation of impedances.

PV Impedance

The Z pv impedance is the impedance calculated from values of power (P) and voltage (V):

where the power and voltage are computed directly from the simulated fields. The power is computed in the same way as for the Z pi impedance. The voltage is computed as follows:

The path over which the system integrates is referred to as the impedance line, which is defined when setting up the ports. To define the impedance line for a port, select the two points across which the maximum voltage difference occurs. EMDS for ADS cannot determine where the maximum voltage difference will be unless you define an impedance line.

VI Impedance

The Z vi impedance is given by:

For TEM waves, the Z pi and Z pv impedances form upper and lower boundaries to a port's actual characteristic impedance. Therefore, the value of Z vi approaches a port's actual impedance for TEM waves.

Choice of Impedance

When the system is instructed to renormalize the generalized S-matrix or compute a Y- or Z-matrix, you must specify which value to use in the computations, Z pi , Z pv , or Z vi.

For TEM waves, the Z vi impedance converges on the port's actual impedance and should be used.

When modeling microstrips, it is sometimes more appropriate to use the Z pi impedance.

For slot-type structures (such as finline or coplanar waveguides), Z pv impedance is the most appropriate.

De-embedding

If a uniform length of transmission line is added to (or removed from) a port, the S-matrix of the modified structure can be calculated using the following relationship:

where:

is a diagonal matrix with the following entries:

γ = α + jβ is the complex propagation constant, where:

αi is the attenuation constant of the wave of port i.
βi is the propagation constant associated with the uniform transmission line at port i.
l i is the length of the uniform transmission line that has been added to or removed from the structure at port i. A positive value indicates that a length of transmission line has been removed from the structure.

The value of γ for the dominant mode of each port is automatically calculated by the 2D solver.

Equations

The sections below describe some of the equations that are solved in a simulation or used to define elements of a structure.

Derivation of Wave Equation

The solution to the following wave equation is found during a simulation:

where:

E( x,y,z ) is a phasor representing an oscillating electric field
k 0 is the free space wave number, .
ω is the angular frequency, 2πf.
µr( x,y,z ) is the complex relative permeability.
εr( x,y,z ) is the complex relative permittivity.

The difference between the 2D and 3D solvers is that the 2D solver assumes that the electric field is a traveling wave with this form:

while the 3D solver assumes that the phasor E is a function of x, y, and z:

Maxwell's Equations

The field equation solved during a simulation is derived from Maxwell's Equations, which in their time-domain form are:

where:

E (t) is the electric field intensity.

D (t) is the electric flux density, ε E (t), and ε is the complex permittivity.

H (t) is the magnetic field intensity.

B (t) is the magnetic flux density, µ H (t), and µ is the complex permeability.

J (t) is the current density, σ E (t).

ρ is the charge density.

Phasor Notation

Because all time-varying electromagnetic quantities are oscillating at the same frequency, they can be treated as phasors multiplied by e j ω t (in the 3D solver) or by e j ω t -γ z (in the 2D solver).
In the general case with the 3D solver, the equations become:

By factoring out the quantity e j ω t and using the following relationships:

Maxwell's Equations in phasor form reduce to:

where B , H , E , and D are phasors in the frequency domain. Now, using the relationships B = µ H , D = ε E , and J = α E , Maxwell's Equations in phasor form become:


for σ = 0


where H and E are phasors in the frequency domain, µ is the complex permeability, and ε is the complex permittivity.

H and E are stored as phasors, which, as illustrated in Electric and Magnetic Fields Can Be Represented as Phasors, can be visualized as a magnitude and phase or as a complex quantity.

Electric and Magnetic Fields Can Be Represented as Phasors

Assumptions

To generate the final field equation, place H in the equation in terms of E to obtain:

Then, substitute this expression for H in the equation to produce:

Conductivity

Although good conductors can be included in a model, the system does not solve for any fields inside these materials. Because fields penetrate lossy conductors only to one skin depth (which is a very small distance in good conductors), the behavior of a field can be represented with an equivalent impedance boundary.

For perfect conductors, the skin depth is zero and no fields exist inside the conductor. Perfect conductors are assumed to be surrounded with Perfect E boundaries.

Dielectric Loss Tangent

Dielectric losses can be modeled by assuming that the relative permittivity, , is complex:

Expressed in terms of the dielectric (electric) loss tangent, , the complex relative permittivity, becomes:

Magnetic Loss Tangent

Losses in magnetic materials can be modeled by assuming that µr is complex.

Expressed in terms of the magnetic loss tangent, , the complex relative permeability becomes:

Definition of Freespace Phase Constant

Using the relationships and , the wave equation being solved can be placed in this form:

Now, if the freespace phase constant (or wave number) is defined as, , the above reduces to:

which is the equation that the 2D and 3D engines solve.

 

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