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Contents

1. IntroductionIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-

2. TheoryIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-

Edge Elements

COSMOSCAVITY Modules . . . . . . . . . . . . . . . . . . . . . . . . . 2-2

CAV3D: The Three-Dimensional Cavity Field Solver . . . . 2-2

CAVAXI: The Axisymmetric Cavity Field Solver . . . . . . . 2-3

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4

Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5

Conductor Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6

Secondary Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6

The Quality Factor and RLC Equivalent Circuit Calculation 2-6

3. Description of CommandsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-

Detailed Description of Commands . . . . . . . . . . . . . . . . . . . . 3-Common Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2

Material Property Commands . . . . . . . . . . . . . . . . . . . . . . 3-2

Boundary Condition Commands . . . . . . . . . . . . . . . . . . . 3-3

Integration Paths Commands . . . . . . . . . . . . . . . . . . . . . 3-1

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Analysis Options Commands . . . . . . . . . . . . . . . . . . . . . 3-12

Performing the Analysis Commands . . . . . . . . . . . . . . . 3-1

Available Results Commands . . . . . . . . . . . . . . . . . . . . . 3-1

Postprocessing Commands . . . . . . . . . . . . . . . . . . . . . . . 3-14

Graphing Results Commands . . . . . . . . . . . . . . . . . . . . . 3-15

Module-Specific Commands . . . . . . . . . . . . . . . . . . . . . . . 3-17

COSMOSCAVITY Commands . . . . . . . . . . . . . . . . . . . 3-17

4. Detailed ExampleIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-

MICAV1: A Conical Dielectric Resonator Inside

a Cylindrical Cavity (CAVAXI) . . . . . . . . . . . . . . . . . . . . . . . 4-

Creating the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-

Assigning Material Properties . . . . . . . . . . . . . . . . . . . . . . . 4-8

Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-

Refining Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

Applying Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 4-15

Running Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-16

Visualization of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

5. Verification ProblemsIntroduction 25

MICAVV1: Multi-Mode Calculations for

Homogeneously Filled Rectangular Cavities . . . . . . . . . . . . 5-26

MICAVV2: An Inhomogeneously Filled Rectangular

Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-27

MICAVV3: A Rectangular Cavity with a Dielectric

Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2

MICAVV4: Dominant Mode Calculations for

Inhomogeneously Filled Cylindrical Cavities; a

High-Q Dielectric Resonator . . . . . . . . . . . . . . . . . . . . . . . . 5-29

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MICAVV5: Dominant Mode Calculations for

Inhomogeneously Filled Cylindrical Cavities; a

Dielectric Resonator Over a Microstrip Substrate . . . . . . . . 5-30

MICAVV6: Multi-Mode Calculations for

Inhomogeneously Filled Cylindrical Cavities . . . . . . . . . . . . 5-3

MICAVV7: Equivalent Lumped Resonant

Circuits of a Cylindrical Cavity . . . . . . . . . . . . . . . . . . . . . . 5-32

A.Material Constants . . . . . . . . . . . . . . . . . . . . . . . . . .A-1

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I-

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1. Introduction

Introduction

For years the finite element method (FEM) has been the key design and simulation

tool for engineers working in a wide range of disciplines. The principle beneficiarie

of the flexibility and power of this method have traditionally been people working

on mechanical, structural, fluid, and thermal problems. Those working in the area o

high frequency electromagnetics (from radio frequencies, RF, to optics) have, on the

other hand, relied more on analytical approaches, whenever possible, on empirical

and semi-empirical models, or on simple solution techniques with limited accuracyand range of applicability. Several numerical difficulties associated with the nature

of the high frequency electromagnetic fields and their representation in a descritized

space have slowed the introduction of the FEM as a reliable tool in RF, microwave

millimeter-wave, and optical designs. Now, Integrated Microwave Technologies

Inc. and Structural Research and Analysis Corporation, bring the power of the FEM

to you through the COSMOSHFS, a High Frequency Simulation suite that contains

three basic components marked with accuracy, speed, efficiency and ease of use. The

three basic solvers are COSMOSHFS 2D, COSMOSCAVITY, and COSMOSHFS

3D as shown in Figure 1.1.

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Figure 1.1 Components of the COSMOSHFS Suite

This manual describes the COSMOSCAVITY package by presenting the theory

behind it, its implementation, some detailed step-by-step examples, and a number

of verification problems.

COSMOSCAVITYis a general frequency domain program for the analysis of

resonant structures. Its applications include the analysis and design of cavities,

dielectric resonators, frequency meters, connectors, cavity filters, and oscillators.

It solves the vector wave equation for the resonant frequency and the correspond-

ing modal field distributions. Depending on the geometry of the problem being

solved, one of two sub-modules (MICAV-3D and MICAV-AXI) will be invoked

(see Figure 1.2). MICAV-3D is a fully three-dimensional program for arbitrary-

shaped cavities. It uses an edge-based finite elements approach to represent the

electric or magnetic field in tetrahederal elements. MICAV-AXI, on the other handis a program for axially symmetric cavities. It uses a hybrid node/edge approach

to represent the electric or magnetic fields on the edges and nodes of triangular

elements. Both MICAV-3D and MICAV-AXI give spurious modes-free solutions

COSMOSCAVITIES

Axi-symmetric and Arbitrary3D Cavities and Resonant

Structures

The C OSMOSHFS System

COSMOSHFS2D

2D Guiding StructuresHigh sSpeed Digital

Interconnects

COSMOSHFS 3 D

Arbitrary 3D PassiveStructure S-parameter

Simulator

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Figure 1.2 COSMOSCAVITY Modules

COSMOSCAVITY modules handle arbitrary conductors, dielectric and ferrite

shapes as well as dielectric and conductor losses.

This manual is intended to be used in conjunction with the standard COSMOSMdocumentation. In particular, COSMOSM Users Guide and the Command

Reference manuals in addition to the on-line help in GEOSTAR are essential

complement to the this manual. The on-line help should be consulted for detailed

explanation of the commands described in Chapter 3 and the ones used in the

detailed examples of Chapter 4. In addition, Chapters 2, 3 and 5 of the COSMOSM

Users Guide can help you have a global picture of the COSMOSM system and

will give you a clearer understanding of GEOSTAR, the pre- and postprocessing

interface.

2DXTALK

Quasi-static Solution ofArbitrary Ttransmission Lines

with Transient Analysis

COSMOSH FS 2D

2DHFQR

Full-wave Solut ion ofArbitrary 2D Guiding

Structures

XTALK

Transient Analysis ofHigh Speed Digital

Interconnects

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2. Theory

Introduction

In this chapter, a general overview of the theory and implementation of the two

main modules ofCOSMOSCAVITY namely CAVAXI and CAV3D will be given.

Since both modules of the COSMOSCAVITY system rely on the use of vector

basis functions, called edge elements, to represent the electromagnetic fields in the

domain of computation, a brief discussion of these elements is first presented.

Edge Elements [1] [2]

The edge-based finite element method is based on using vector basis functions

designed specifically for the solution of vector field problems and constructed to be

divergence free. For a tetrahedral element, in 3D problems, and triangular element

in 2D and axisymmetric problems, the vector basis function is defined as:

(2-1)

where i and j are the node numbers of the tetrahedral or triangular elements. The sare the regular node-based finite element shape functions. Clearly, the divergence

of such vector basis functions or edge element is zero. Therefore, unlike node-

based finite elements, there is no need to enforce a gauge by a penalty function or in

a least squares-sense. Since the vector field quantities are expanded in terms of

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these basis functions, they will, in turn, be divergence free leading to a complete

elimination of the vector parasites or the spurious modes. In addition, the unknown

coefficients in this approach are the tangential components of the electromagnetic

fields; hence enforcing a Dirichlet boundary condition for the electric field

formulation can be easily achieved. The edge elements also produce less populated

matrices than does the node-based approach. Such elements allow for the direct

discretization of the curl-curl form of the vector Helmholtz equation and yield a

straightforward boundary value problem that does not require any modification or

any special treatment at the boundaries. In addition, as physically required, only the

tangential components of the field are forced to be continuous and the normal

components are allowed to change along material interfaces.

COSMOSCAVITY Modules

This COSMOSCAVITYpackage combines two modules

for the analysis of resonant

cavity structures as illustrated

in Figure 2.1. For arbitrary,

three-dimensional structures, the

CAV3D sub-module is used with

a fully three-dimensional mesh

of four-node/six-edge

tetrahedrons. When the cavity

has an axial symmetry,

considerable savings can be

achieved by using the CAVAXI

sub-module with only a two-dimensional mesh of hybrid node-edge triangular

elements. Both modules implement a full-wave analysis as will be illustrated

below.

CAV3D: The Three-Dimensional Cavity Field Solver

This sub-module analyzes fully three-dimensional cavities without assuming any

symmetry, hence, the structure has to be meshed in its entirety. The boundary value

problem governing these structures is also represented by a vector wave equation

and a set of boundary conditions.

Figure 2.1. A Schematic Representationof a General Resonant CavityStructure

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Upon discretization using the edge elements, we obtain a generalized eigenvalue

problem which is then solved for the eigenvalues (the resonant frequencies) and the

eigenvectors (the modal electric field distributions) for a specified number of

modes.

(2-2)

where:

ko is the free space wavenumber and is equal to ,

r is the complex relative permittivity, and

r is the complex relative permeability.

To complete the specification of the boundary value problem to be solved, the

following boundary conditions are used:

(2-3)

(2-4)

The unknown

electric field is

represented by first

order tetrahedral

elements as shown in

Figure 2.2. Clearly,each tetrahedron

has six unknowns

associated with its

edges.

CAVAXI: The

Axisymmetric

Cavity Field Solver

For axisymmetric geometries, a quasi-two-dimensional analysis is performed by

considering a cross section of the cavity at any arbitrary -plane. The vector waveequation (2-2) with the boundary conditions given by equations (2-3) and (2-4) are

still valid for these geometries. However, given the symmetry of the structure, the

field is assumed to have the following -dependence:

( ) Node1

Node2

Node3 Edge2 E t2

Node4

( )

Edge1

E t1( )

Edge4

E t4

Edge

5

E

t5

(

)

Edge6

( )E t6

Edge3

( )E t3

Figure 2.2. Unknowns on a Tetrahedral Element in MICAV-3D

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(2-5)

where a cylindrical coordinate system is used.

By substituting the electric field

expression of equation (2-5) back

in the vector wave equation (2-2),

we obtain a generalized eigenvalueproblem which is then solved for

the eigenvalues (the resonant

frequencies) and the eigenvectors

(the modal electric field distribu-

tions) for a specified number of

modes. Note that m, the harmonic

number, is a parameter of the

problem that must be specified at

the time of the solution.

Since at a given constant -plane, the azimuthal direction is purely normal to thatplane, a hybrid node/edge approach is used whereby a nodal representation is used

for the azimuthal component of the field and a vector representation is used for the

transverse component as shown in Figure 2.3.

Again, each triangle has three unknowns associated with its nodes and another

three associated with its edges. Finally, note that the axis of the symmetry of the

cavity must coincide with the y-axis of the x-y coordinate system.

Boundary Conditions

In high frequency electromagnetics, there are several possible boundary conditions

COSMOSCAVITY recognizes the following:

Perfect electric conductor (fc/gc):

Surfaces/curves of grounded conductors (gc) and/or floating conductors (fc) could

be assigned this type of boundary condition. As a result, COSMOSCAVITY forces

the tangential component of the electric field on those surfaces/curves to be zero.

( )( )

Node1

E

1

( )Node3E3

( )

Node2

E

2

Edge1

Et1

( )

Edge3

Et3 ( )

Edge2

Et2

Figure 2.3. Unknowns on a TriangularElement in MICAV-AXI

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Perfect magnetic conductor (pmc):

COSMOSCAVITY forces the component of the magnetic field that is tangential to

perfect magnetic conducting surfaces/curves (usually surfaces of symmetry) to be

zero. This boundary condition could be used to terminate the mesh for open outer

boundaries.

A special axial boundary condition for the CAVAXI sub-module is applied

internally to axisymmetric cavities and the user need not specify it explicitly. Axia

element nodes/edges should just be left as free nodes/edges or have oob-type

boundary condition.

Material Properties

COSMOSCAVITY can treat isotropic dielectric and ferrite materials with a

complex relative permittivity ( = r o, with ), a complex relativepermeability ( = ro, with ), and an electrical conductivity ()

In MKS units, the free space permittivity o and permeability o have the values

8.8541853x10-12 F/m and 410-7 H/m, respectively, and is in S/m. For materialshaving non-zero electrical conductivity, the complex permittivity used by

COSMOSCAVITY is the following:

(2-6)

where

is the angular frequency.

For each material used in the model, the user needs to specify the real and

imaginary parts of the relative permittivity and permeability as well as the electrica

conductivity. The default values are those of free space. See the on-line help for the

MPROP and USER_ MAT commands (Propsets > Material Property and UserMaterial Lib) for more details).

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Conductor Properties

In applying the boundary conditions discussed above, all conductors are treated as

perfect conductors (i.e., infinite conductivity and zero penetration depth). However

the finite conductivity and the relative permeability of the metals are taken into

account when calculating such quantities as the attenuation constant of a

waveguide or the quality factor of a resonator due to the conductor loss. The defaul

conductor properties used are those of copper. For other metals, the user shouldspecify the values of the relative permeability and the electrical conductivity of the

metal.

Secondary Calculations

So far, only a description of the fundamental theory and the basic solutionsavailable through the various COSMOSCAVITY modules have been given. In

this section, we present the theoretical formulation used for the various secondary

calculations available within COSMOSCAVITY.

The Quality Factor and RLC Equivalent Circuit Calculation

In the MICAV module, the user has the option of calculating the quality factor and

of determining the RLC equivalent circuit of the resonator analyzed.

The Quality Factor

The quality factor Q of a resonator is defined as [4]:

(2-7)

where r is the angular resonant frequency and

(2-8)

However, the average power dissipated can be due to conductor loss and/or

dielectric loss; we therefore define the quality factor due to conductor loss as:

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(2-9)

where Rm is the skin-effect surface resistance and is given by:

(2-10)

where and are the conductivity and the permeability of the metal. The qualityfactor due to dielectric loss is defined as:

(2-11)

where ffi is the filling factor for the ith

dielectric region which is the ratio of theelectric energy stored in the ith region to the total electric energy stored in the

cavity. The total quality factor of the cavity is then given by:

(2-12)

Note that each mode has a different field distribution, and hence a different quality

factor.

The RLC Equivalent Circuit

The RLC equivalent circuit of a resonator is often informative, especially in giving

an idea about what minor perturbations will do to the properties of the resonator [4]

The elements R, L, and C are responsible for the power dissipated, the stored

magnetic and electric energy, respectively. Since the magnetic and electric energies

are equivalent at resonance, we can write:

(2-13)

and from the average electric energy we can compute C using the following

relationship:

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(2-14)

However, the voltage across the resonator needs to be computed first as

(2-15)

where Path follows the curve of maximum electric field. It should be clear here

that such integration path need not be a straight line and is in general formed by N

straight line segments each of which is defined by two end points. Note that the

determination of the path will depend on the particular mode being considered and

the resulting field distribution. The user may wish to make an initial examination of

the results before defining the integration path for subsequent calculations.

Once V is computed, C is obtained from equation (32). Next, L is computed using

equation (31). Finally, the equivalent resistance R, is related to the quality factor Qand the equivalent inductance L via

(2-16)

It should be clear that each mode will have a different RLC equivalent circuit and

should have an appropriate integration path to define the voltage associated with it

References

[1] Daniel R. Lynch and Keith D. Paulsen, Origin of vector parasites in

numerical Maxwell solutions, IEEE Trans. Microwave Theory Tech. March

1991.

[2] A. Bossavit and I. Mayergoyz, Edge-element for scattering problems, IEEE

Trans. Magn, vol. MAG-25, pp. 2816-2821, 1989.

[3] A. Khebir, A. B. Kouki, and R. Mittra, Asymptotic boundary conditions for

finite element analysis of three-dimensional transmission line

discontinuities, IEEE Trans. Microwave Theory Tech., vol 38, pp 1427-

1432, 1990.

[4] O. P. Gandhi, Microwave Engineering and Applications, Pergamon Press,

1987.

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3. Description of Commands

Introduction

The use ofCOSMOSCAVITY for solving high frequency electromagnetic

problems involves generating a proper finite element mesh, specifying the materia

properties, imposing the boundary conditions, and specifying the appropriate

solution parameters. All of this is done through the GEOSTAR preprocessor.

Similarly, using GEOSTAR postprocessor, the results of the various COSMOS

CAVITY modules can be viewed in graphical and text formats. The general

commands for model creation, mesh generation and postprocessing are documentedin the COSMOSM Users Guide Volume (1) and will not be described here. Only

commands that are specific to COSMOSCAVITY or that have an implementation

related to it will be described in this chapter.

Detailed Description of Commands

This section is divided into two sub-sections. The first describes commands that are

common to all COSMOSCAVITY modules while the second describes the

module-specific commands.

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ommands

Common Commands

These commands cover the definition of material properties, boundary conditions,

integration paths and the context-sensitive postprocessing features. They are

described in the following eight sub-sections.

Material Property Commands

You may use the library or define numerical properties directly.

USER_MAT

(Menu: PROPSET > User Material Library)

The USER_MAT command accesses COSMOSM library for electromagnetic

materials.

Where:

Material set

Material set number between 1 and 90

(default is highest set number defined + 1)

Material name

Name of the material property. Select a material from the drop-down menu.

Unit-label

Units used.

MPROP

(Menu: PROPSET > Material Property)

The MPROP command is a general purpose GEOSTAR command for specifying

the material properties for different model regions. The pertinent material propertie

for HFESAP are given below and can be set as follows:

USER_MAT Material set Material name unit-label

MPROP set name1 value1 name2 value2 ...

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Where:

set

Material set number between 1 and 99

(default is highest set number defined + 1)

name1, name2, ...

Name of the material property.

EQ. permit_r Real part of the relative permittivity.EQ. permit_i Imaginary part of the relative permittivity.

EQ. mperm_r Real part of the relative permeability.

EQ. mperm_i Imaginary part of the relative permeability.

EQ. econ The electric conductivity.

value1, value2, ...

Corresponding real values to the material properties with defaults:

permit_r 1.0.

permit_i 0.0.

mperm_r 1.0.

mperm_i 0.0.

econ 0.0.

At least one property must be defined for each material set.

Example: MPROP, 1, permit_r, 10.0, permit_i, 1.e-03

This command defines the real part of the real permittivity for

material set 1 to be 10 and imaginary part be 0.001. The remainingproperties (mperm_r, mperm_i, econ) assume their default values.

Boundary Condition Commands

The boundary condition commands are the following: CBEL, CBEDEL, CBCR,

CBCDEL, CBSF, CBSDEL, CBRG, CBRDEL, CBPLOT and CBLIST. They are

described next.

CBEL

(Menu: LOADS-BC > E_MAGNETIC > Hi-Freq_B-C > Define by Elements)

The CBEL command specifies a boundary condition on faces of elements in the

specified pattern.

CBEL bel bc cond_num conductivity

permeability face_num eel increment

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Where:

bel

Beginning element in the pattern.

bc

Boundary condition type.

EQ. fc Floating conductor.

EQ. gc Grounded conductor.EQ. pmc Perfect magnetic conductor.

EQ. oob Open outer boundary.

(default is fc)

cond_num

Conductor number associated with the boundary condition.

conductivity

Conductivity of the conductor number cond_num.

permeability

Relative permeability of the conductor number cond_num.

face_num

Face of the elements on which the boundary condition is to be applied.

eel

Ending element in the pattern.

increment

Increment between elements in the pattern.

Example: CBEL, 4, fc, 2,,, 5 ,3, 5,,

This command defines a floating conductor number 2 on face number5 of elements 4 and 5 using the default conductivity and permeability(copper).

CBCR

(Menu: LOADS-BC > E_MAGNETIC > Hi-Freq_B-C > Define by curves)The CBCR command defines a boundary condition on a pattern of curves.

CBCR bcurve bc cond_num conductivity permeability ecurve increment

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Where:

bcurve

Beginning curve in the pattern.

bc





(default is fc)

cond_num


conductivity


permeability


ecurve

Ending curve in the pattern.

increment

Increment in curve numbering.

Example 1: CBCR, 2, fc, 1,,, 2,,This command defines a floating conductor on curve 2 of defaultconductivity and permeability (copper).

Example 2: CBCR, 5, gc, 1, 6.1e7,, 9, 2,

This command defines curves 5, 7 and 9 to be grounded conductorsmade of silver.

CBSF

(Menu: LOADS-BC > E_MAGNETIC > Hi-Freq_B-C > Define by Surfaces)

The CBSF command defines a boundary condition on a pattern of surfaces.

CBSF bsurface bc cond_num conductivity

permeability esurface increment

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Where:

bsurface

Beginning surface in the pattern.

bc





(default is fc)

cond_num


conductivity


permeability


esurface

Ending surface in the pattern.

increment

Increment in surface numbering.

Example: CBSF, 1, gc, 1,,, 6,,This command defines surfaces 1 through 6 to be grounded conductor#1 and to have the default conductivity and permeability (copper).

CBRG

(Menu: LOADS-BC > E_MAGNETIC > Hi-Freq_B-C > Define by Regions)

The CBRG command defines a boundary condition on a pattern of regions.

Where:

bregion

Beginning region in the pattern.

CBRG bregion bc cond_num conductivity

permeability eregion increment

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bc



EQ. gc Grounded conductor.

EQ. pmc Perfect magnetic conductor.


(default is fc)

cond_num


conductivity


permeability


eregion

Ending region in the pattern.

increment

Increment in region numbering.

Example: CBRG, 3, fc, 2,3.43e+07,, 3,,

This command defines a floating conductor (number 2) on region 3 ofconductivity 3.43e+07 mho/m and default permeability (aluminum).

CBEDEL(Menu: LOADS-BC > E_MAGNETIC > Hi-Freq_B-C > Delete by Elements)

The CBEDEL command deletes previously defined High Frequency (HF) boundary

conditions for the specified face for a pattern of elements.

Where:

bel


face

Face number of the elements for which existing HF boundary condition is to be

deleted.

CBEDEL bel face eel inc

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eel


(default is bel)

inc


(default is 1)

Example: CBEDEL, 3, 2, 10, 1

This command deletes the boundary conditions on face 2 of elements3 through 10.

CBCDEL

(Menu: LOADS-BC > E_MAGNETIC > Hi-Freq_B-C > Delete by Curves)

The CBCDEL command deletes previously defined HF boundary conditions for

elements associated with a pattern of curves.

Where:

bcr

Beginning curve in the pattern.

ecr

Ending curve in the pattern.

(default is bcr)

inc

Increment between curves in the pattern.

(default is 1)

Example: CBCDEL, 1, 10, 1

This command deletes HF boundary conditions for elementsassociated with curves 1 through 10.

CBSDEL(Menu: LOADS-BC > E_MAGNETIC > Hi-Freq_B-C > Delete by Surfaces)

The CBSDEL command deletes previously defined HF boundary conditions for

elements associated with a pattern of surfaces.

CBCDEL bcr ecr inc

CBSDEL bsf esf inc

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Where:

bsf

Beginning surface in the pattern.

esf

Ending surface in the pattern.

(default is bsf)

inc

Increment between surfaces in the pattern.

(default is 1)

Example: CBSDEL, 1, 10, 1

This command deletes HF boundary conditions for elementsassociated with surfaces 1 through 10.

CBRDEL

(Menu: LOADS-BC > E_MAGNETIC > Hi-Freq_B-C > Delete by Regions)

The CBRDEL command deletes previously defined HF boundary conditions for

elements associated with a pattern of regions.

Where:

brg

Beginning region in the pattern.

erg

Ending region in the pattern.

(default is brg)

inc

Increment between regions in the pattern.

(default is 1)

Example: CBRDEL, 1, 10, 1

This command deletes HF boundary conditions for elementsassociated with regions 1 through 10.

CBPLOT

(Menu: LOADS-BC > E_MAGNETIC > Hi-Freq_B-C > Plot)

CBRDEL brg erg inc

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The CBPLOT command plots a predefined symbol at elements with prescribed HF

boundary condition for a pattern of elements. The symbol is shown in the STATUS2

table.

Where:

belBeginning element in the pattern.

(default is 1)

eel


(default is elmax)

inc


(default is 1)

Example CBPLOT;

The above command plots a predefined symbol at elements withprescribed HF boundary conditions.

CBLIST

(Menu: LOADS-BC > E_MAGNETIC > Hi-Freq_B-C > List)

The CBLIST command lists element HF boundary conditions for a pattern of

elements.

Where:

bel


(default is 1)

eel


(default is elmax)

inc


(default is 1)

CBPLOT bel eel inc

CBLIST bel eel inc

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Example: CBLIST, , 10, 2,

The above command lists all the specified HF boundary conditionsfor elements 1, 3, 5, 7 and 9.

For all boundary condition commands, it is recommended that the conductor

numbering for conductors (particularly for floating conductors) be sequential

starting from one.

Integration Paths Commands

HF_PATH

(Menu: ANALYSIS > Hi-Freq_Emagnetic > Integration Path > Define)

The HF_PATH command defines one or more integration paths for the 2-dimen-

sional field simulator or for cavity analysis. The integration paths are used in voltage

computation based on electric field line integrals.

Where:

PN

Path number. The maximum number of paths is 2.

Xn,Yn,Zn

X, Y, Z coordinate triplets that define the integration paths straight line

segments. The minimum number of triplets per path is 2 and the maximum is 13The list of triplets is terminated by entering a ; or by repeating the last triplet

The path X, Y, Z triplets can be picked using the mouse on any specified

plane. For this, the grid must be turned on with the GRIDON command.

Example: HF_PATH,1,0.0,0.0,0.0,0.0,1.0,1.0,0.0,1.0,2.0;

This commands defines integration path #1 by 3 points (i.e., 2straight line segments).

HF_PATHDEL

(Menu: ANALYSIS > Hi-Freq_Emagnetic > Integration Path > Delete)

The HF_PATHDEL command deletes an integration path previously defined by the

HF_PATH command.

HF_PATH PN X1 Y1 Z1 X2Y2 Z2 X3 Y3 Z3.....

HF_PATHDEL path_number

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Where:

path_number

Path number. (1 or 2)

Example: HF_PATHDEL, 2

This commands deletes the second integration path defined.

HF_PATHLIST

(Menu: ANALYSIS > Hi-Freq_Emagnetic > Integration Path> List)

The HF_PATHLIST command lists coordinate triplets of an integration path defined

by the HF_PATH.

Where:

path_number

Path number. (1 or 2)

(default is 1)

Example: HF_PATHLIST, 1

This commands lists coordinate triplets making up the firstintegration path.

Analysis Options Commands

A_HFRQEM

(Menu: ANALYSIS > Hi-Freq_Emagnetic > Analysis Options)

The A_HFRQEM command defines the high frequency analysis to be run and sets

the distance units to be used in the analysis.

Where:

option

Analysis option.

EQ. 2dhfrq Run the 2-dimensional full-wave field solver.

HF_PATHLIST path_number

A_HFRQEM option unit

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EQ. 2dxtalk Run the 2-dimensional quasi-static field solver to compute

RLCG matrices then the time-domain cross-talk simulator

to compute cross-talk and distortion.

EQ. xtalk Run the time domain cross-talk simulator with pre-computed

RLCG matrices.

EQ. cavaxi Run the time axisymmetric cavity field solver.

EQ. cav3d Run COSMOSCAVITY (the time 3D cavity field solver).

EQ. sparam Run COSMOSHFS 3D (S-Parameter Simulator).

(default is 2dhfrq)

unit

Unit for distance measurement to be used.

EQ. 0 Dimensions are in mm.

EQ. 1 Dimensions are in cm.

EQ. 2 Dimensions are in m.

EQ. 3 Dimensions are in mils.

EQ. 4 Dimensions are in inches.

EQ. 5 Dimensions are in microns.

(default is 0)

Example: A_HFRQEM, xtalk, 4

This command sets the high-frequency analysis option to run thecross-talk time domain simulator using pre-computed RLCGmatrices with lengths specified in inches.

Performing the Analysis Commands

R_HFRQEM

(Menu: ANALYSIS > Hi-Freq_Emagnetic > Run Analysis)

The R_HFRQEM command runs the electromagnetic analysis specified by the

A_HFRQEM command.

Available Results Commands

HF_RESLIST

(Menu: RESULTS > LIST > HF_RESLIST)

The HF_RESLIST command lists the results of the performed high-frequency

electromagnetic analysis based on the analysis options chosen and the solution

parameters.

R_HFRQEM

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RESULTS?

(Menu: RESULTS > Available_Results)

The RESULTS? command lists the available nodal and/or elemental results for

postprocessing from the performed analysis. For 2DHFRQ, the command lists the

frequency points (freq), mode number (Mode), mode flag (M_Flag) and Frequency

(GHz). This listing is used to establish a correspondence between the frequency

point number and the actual simulation frequency in GHz. The mode flag indicates

whether the computed mode is propagating (M_Flag = 1) or evanescent (M_Flag

= -1). For 2DXTALK the command lists the fundamental modes calculated.

Postprocessing Commands

MAGPLOT

(Menu: RESULTS > Plot > Electromagnetics)

The MAGPLOT command is a postprocessing command that plots the results of the

analysis.

Where:

freqn

Time step number (use RESULTS? for corresponding frequency values).

Prompted only for 2DHFRQ as frequency step number.

(default is 1)

nd/el

Flag to activate results at nodes or centers of elements.

EQ. 1 Nodes.

EQ. 2 Elements.

(default is 1)

comp

Field component. Admissible components depend on the type of analysis

performed as follows:

HF_RESLIST

RESULTS?

ACTMAG freqn moden nd/el comp

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For 2DHFRQ and CAV3D:

EQ. EX Electric field intensity in the X-direction. Real.

EQ. EY Electric field intensity in the Y-direction. Real.

EQ. EZ Electric field intensity in the Z-direction. Real.

EQ. ER Resultant electric field intensity. Real.

EQ. HX Magnetic field intensity in the X-direction. Real.

EQ. HY Magnetic field intensity in the Y-direction. Real.

EQ. HZ Magnetic field intensity in the Z-direction. Real.

EQ. HR Resultant magnetic field intensity. Real.

For 2DXTALK analysis:

EQ. POT Electrostatic potential. Real.

EQ. EX Electric field intensity in the X-direction. Real.

EQ. EY Electric field intensity in the Y-direction. Real.

EQ. ER Resultant electric field intensity. Real.

This command is not needed for XTALK analysis.

MAGLIST

(Menu: RESULTS > List > Electromagnetics)

The MAGLIST command is a postprocessing command that lists results of the

analysis.

MAGMAX

(Menu: RESULTS > Extremes > Electromagnetics)

The MAGMAXcommand is a postprocessing command that lists the extremes of the

results of the analysis.

Graphing Results Commands

ACTXYPOST

(Menu: DISPLAY > XY_Plots > Activate Post-proc)The ACTXYPOST is a postprocessing command that sets the parameters to be used

for viewing X-Y type results using the XYPLOT command.

ACTXYPOST graph-num mode y-axis (line)

graph-color line-style symbol-type graph-id

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Where:

graph-num

Graph number. (1 to 6)

(default is highest defined + 1)

mode

Mode number.

(defaults is1)y-axis

For COSMOSHFS 2D:

The y-axis may be one of the components described below. The x-axis is not

prompted for and is fixed to be frequency in GHz.

ALPHA Real part of propagation constant. Non-zero for decaying

modes only.

BETA Imaginary part of propagation constant in m-1.EPSEFF Effective dielectric constant.

PHASEV Phase velocity in m/s.

ALPHAC Attenuation constant in dB/m due to conductor losses in dB/m

ALPHAD Attenuation constant in dB/m due to dielectric losses in dB/m

(default is EPSEFF)

The following components are computed only when the number of conductors is

non-zero and are based on the power-current definitions.

ZMI Modal impedance ().LMI Modal inductance (nH/m).

CMI Modal capacitance (pF/m).RMI Modal resistance (/m).GMI Modal conductance (S/m).

The following components are computed only when the number of integration

paths is non-zero and are based on the power-voltage definitions.

ZMV Modal impedance ().LMV Modal inductance (nH/m).

CMV Modal capacitance (pF/m).

RMI Modal resistance (/m).GMV Modal conductance (S/m).

For XTALK and 2DXTALK:

The following components are plotted versus mode number for 2DXTALK only

BETA Imaginary part of propagation constant in m-1.EPSEFF Effective dielectric constant.

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PHASEV Phase velocity in m/s.

ALPHAC Attenuation constant in dB/m due to conductor losses.

ALPHAD Attenuation constant in dB/m due to dielectric losses.

ZM Modal impedance ().LM Modal inductance (nH/m).

CM Modal capacitance (pF/m).

RM Modal resistance (/m).GM Modal conductance (S/m).

(default is EPSEFF)

The following components are plotted versus time for both XTALK and

2DXTALK.

VTLSNEAR Near end voltages (V).

VLTSFAR Far end voltages (V).

(line)

Line number (prompted only when the y-axis is VLTSNEAR or VLTSFAR in

XTALK OR 2DXTALK).

graph-color

Color to be used for plotting.

line-style

Line style to plot graph.

symbol-type

Symbol type for plotting at points on the x-y graph.

graph-id

Graph identification. Default depends on the y-axis entry.

Note:

Refer to the COSMOSM Command Reference Manual for more help on graph-

color, line-style, symbol-type and graph-id.

Module-Specific Commands

COSMOSCAVITY Commands

HF_CAVSOLN

(Menu: ANALYSIS > HF_Emag > Cavities > HF_Cav-Soln)

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The HF_CAVSOLN command defines the solution options for the cavity simulator

Where:

model_flag

Flag to specify model type.

EQ. 0 Axisymmetric cavities.EQ. 1 3-dimensional cavities.

(default is 0)

nmodes

Number of desired modes.

(default 1)

fharmonic

First harmonic (for axisymmetric cavities).

(default 0)

lharmonic

Last harmonic (for axisymmetric cavities).

(default 0)

Example: HF_CAVSOLN, 0, 2, 1, 3

This command sets the cavity simulators solution options tosimulate an axisymmetric cavity and compute its 2 most

dominant modes for each of the harmonics: 1, 2, 3.

HF_CAVOUT

(Menu: ANALYSIS > HF_Emag > Cavities > HF_Cav-Out)

The HF_CAVOUT command sets the output options for axisymmetric and 3-D

cavity solvers.

Where:

compQ_flag

Flag for cavity quality factor computation.

EQ. 0 Quality factor is not computed.

EQ. 1 Quality factor is computed.

(default is 0)

HF_CAVSOLN model_flag nmodes fharmonic lharmonic

HF_CAVOUT compQ_flag compRLC_flag output_flag

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compRLC_flag

Flag for equivalent RLC circuit computation.

EQ. 0 Equivalent RLC circuit is not computed.

EQ. 1 Equivalent RLC circuit is computed.

(default is 0)

output_flag

Flag to specify the type of output.EQ. 0 No output.

EQ. 1 Output nodal values only.

EQ. 2 Output element values only.

EQ. 3 Output both nodal and element values.

(default is 0)

Notes:

1. If compQ_flag is set to 0, compRLC_flag is ignored and the RLC equivalen

circuit is not computed.

2. If compQ_flag is set to 1 and compRLC_flag is also set to 1, an integration

path must be specified for RLC computation.

Example: HF_CAVOUT, 1, 1, 1

This command sets the cavity simulators output options to computethe cavitys quality factor and its equivalent RLC circuit and to outputnodal values only of the modal electric and magnetic fields.

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Introduction

This chapter presents step by step procedures for solving a problem with

COSMOSCAVITY.

MICAV1: A Conical Dielectric Resonator Inside

a Cylindrical Cavity (CAVAXI)

The geometry of the conical resonator,

its dielectric ring support and the metal-

lic enclosure cavity are shown in Figure

4.1. The dimensions of the problem are

h = r = 4 mm for the resonator, R = 8

mm and H = 7 mm for the cavity and,

for the ring support with square cross-

section, a = 1 mm with inner and outer

radii of 1 mm and 2 mm, respectively.

The material of the resonator is non-

magnetic and slightly lossy with r =

35.7 - j4.2x10-4 while the ring support

has a dielectric constant of 2.2.h

r

a

r

R

H

y

x

r1

2

Figure 4.1.Geometry of the ConicalResonator with a Dielectric RingSupport Inside a Metallic Cavity

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To start a new problem in GEOSTAR:

1. Launch GEOSTAR. GEOSTAR starts and the Open Problem Files dialog box

opens.

2. Browse to the directory which you want to use for the new problem.

3. In the File name field, enter

micav1, for example, for the

problem name.

4. ClickOpen. GEOSTAR sets the

new problem and creates all

related database files in the

specified folder.

Creating the Model

To set up the proper working plane and the view:

1. From the Geometry menu, select Grid, Plane. The Plane dialog box opens.

2. Click Ok to use the default settings.

3. From the Display menu, select View Parameter, View. The View dialog box

opens.

4. Click OK to use the default Y-view.

The dimensions of the structure are all integer multiples of 1 mm in both x and y

directions. Therefore, defining a grid based on these values will significantlysimplify the task of model construction.

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To setup a drawing grid in the active plane:

1. From the Geometry menu, select Grid, Grid On. The Grid On dialog box opens

2. Enter the following values:

Origin x-Coordinate Value [0] >

(accept default)

Origin y-Coordinate Value [0] >

(accept default)

X- increment [5] > 1Z- increment [5] > 1

No of X- increments [20] > 8

No of Z- increments [20] > 7

Grid line color index [2] >

(accept default)

3. Click OK.

To re-scale the grid and fit in the display window:

1. From the Geo Panel, click the Scale Auto button .

At this stage it is a good idea to give a descriptive title of the problem we are abou

to solve.

To give a title to the current problem:

1. From the Control menu, select Miscellaneous, Write Title. The Title dialog box

opens.2. Type the title in the Message field as follows:

Message > Conical resonator inside a cylindrical cavity

3. Click OK.

Next, we create the seven outer curves. These curves include the cavity walls the

central axis of the cavity. Note that the axis of symmetry must coincide with the x=0

axis. The following procedure creates the necessary curves (the mouse can be used

to pick the points on the grid).

To create the seven outer curves:

1. From the Geometry menu, select Curves, Draw Polyline. The CRPCORD

dialog box opens.

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2. Enter the following coordinates:

Curve [1] > (accept default)

X, Y, Z Coordinates of keypoint 1> 0, 0, 0








3. Click OK.

Next, we create three additional curves which, along with curve 2, will delimit the

dielectric support rings region.

To create the curves delimiting the dielectric support rings region:


dialog box opens.2. Enter the following coordinates:


X, Y, Z Coordinates of keypoint 1 > 2, 0, 0





3. Click OK.

Finally, we create the remaining curves to delimit the region of the cone resonator

To create the curves delimiting the region of the cone resonator:


dialog box opens.

2. Enter the following coordinates:







3. Click OK. This completes the creation of all necessary 13 curves.

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To plot the curve labels:

1. Click the button. The Status 1 setting

box opens.

2. Click the curve label checkbox as shown in the

figure.

3. Click Save.

4. Click the Repaint button to plot the model.

The model at this stage should look as follows:

Figure 4.2. Curves Defining the Geometry of the Model

Next, we build contours using of the curves just created.

1. From the Geometry menu, select Contours, Define. The CT dialog box opens.

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2. Enter the following options:

Contour [1] > (accept default)

Mesh flag 0 = Esize 1= Num. elems [0] > (accept default)

Average element size > 0.5

Number of reference boundary curves [1] > 3

Pick/Input Curve 1 > 1



Use selection set 0 = No 1= Yes [0] > (accept default)Redefinition Criterion 0=Prev 1=Redef 2=Max 3=Min elements [1] > (accept defaul

3. Click OK. Contour 1 is now created and plotted in different color.

To define the second contour:




Mesh flag 0 = Esize 1= Num. elems [0] > (accept default)Average element size > 0.5










Use selection set 0 = No 1= Yes [0] > (accept default)

Redefinition Criterion 0=Prev 1=Redef 2=Max 3=Min elements [1] > (accept defaul

3. Click OK.

To define the third contour:



Contour [3] > (accept default)Mesh flag 0 = Esize 1= Num. elems [0] > (accept default)






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3. Click OK.

To define the last fourth contour:




Mesh flag 0 = Esize 1= Num. elems [0] > (accept default)









3. Click OK.

With the contours complete, the next step is to generate the regions to be meshed.

To create the first region:

1. From the Geometry menu, select Regions, Define. The RG dialog box opens.


Region [1] > (accept default)

Number of contours [1] > (accept

default)

Pick/Input Outer Contour > 1

Underlying surface [0] > (accept default)

3. Click OK.

To define the second region similarly:

1. From the Geometry menu, select Regions,

Define. The RG dialog box opens.



Number of contours [1] > (accept default)



3. Click OK.

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To define the third region:







3. Click OK.

To define the fourth region:







3. Click OK.

Assigning Material Properties

We are now ready to proceed with material definition and mesh generation. Note

that regions 1 and 2 are both air and can, therefore, be meshed together under the

same material property set. We define the first material property set for air.

To define material property for air:

1. From the PropSets menu, select Material Property. The MPROP dialog box

opens.


Material property set [1] >

(accept default)

Material Property Name > permit_r

Property value [0] > 1.0

Material Property Name >

(to end this command)

3. Click OK.

4. Click Cancel button to end the command.

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Meshing

To mesh region 1 and 2:

1. From the Meshing menu, select Auto_Mesh, Regions. The MA_RG dialog box

opens.


Pick/Input Beginning Region > 1Pick/Input Ending Region > 2

Increment [1] > (accept default)

Number of smoothing iterations [0] > (accept default)

Method 0 = Sweeping 1= Hierarchical [0] > (accept default)

Element order 0=Low 1=High [0] > (accept default)

3. Click OK.

Next, we define the second material property set to be that of the dielectric ring

support.

To define material property for the dielectric ring support:


opens.


Material property set [1] > 2



Material Property Name > (to end this command)

3. Click OK.

4. Click Cancel button to end the command.

To mesh the corresponding region 3:


opens.


Pick/Input Beginning Region > 3

Pick/Input Ending Region > 3



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3. Click OK.

Finally, we define the third material property set to be that of the resonator using

permit_rto define the real part of the permittivity and permit_i to define the

imaginary part as follows.

To define the third material property set for the resonator:


opens.

2. Enter the following options:Material property set [1] > 3



Material Property Name > permit_i

Property value [0] > 4.2e-04Material Property Name > (to end this command)

3. Click OK.

4. Click Cancel button to end this command.

To mesh the resonator region 4:


opens.

2. Enter the following options:Pick/Input Beginning Region > 4

Pick/Input Ending Region > 4





3. Click OK.

This completes the initial meshing of the model which should now look as follows

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Figure 4.3 Initial Mesh of the Model

It is a good idea at this stage to turn on element colors based on material prop

erties.

To turn on element colors based on material properties:

1. From the Meshing menu, select Elements,

Activate Element Color. The ACTECLRdialog box opens.

2. From the Color Flag drop-down menu,

select Yes.

3. From the Set Label option, select Material Property.

4. Click OK. You will then be able to easily distinguish the different regions based

on their material properties by repainting the screen.

Refining Mesh

So far, the initial mesh thus obtained is coarse and fairly uniform throughout the

model. However, because the dielectric resonator region has such a high permit-

tivity (i.e., the wavelength inside it is much shorter), we must refine the mesh

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DetailedExample

inside it so as to insure that we keep a ratio of around 10 nodes per wavelength. To

do this, we must first select the elements to be refined. Because of the geometry of

the resonator, it is best to select the elements based on their reference entity. In this

case we would like to select all elements in the resonators region (region number

4).

To select the elements to be refined:

1. From the Control menu, select Select, by Reference. The SELREF dialog boxopens.


Selection Entity [EL] > (accept default)

Reference Entity [SF] > RG

3. Click Continue.


Pick/Input Beginning Region > 4Pick/Input Ending Region > 4


Boundary element flag [0] > 1

5. Click OK

A number of elements (119) are then selected and highlighted with a different.

Note that the boundary flag should be set to 1 to avoid generating hanging nodes

Next, we proceed with refining the selected elements as follows:

To refine the selected elements:

1. From the Meshing menu, select Elements, Refine Mesh. The EREFINE dialog

box opens.

2. Click OK to accept all the default settings.

We need to ensure smooth elements (i.e., good aspect ratios) after the first

refinement.

To smooth the mesh:

1. From the Meshing menu, select Elements, Smoothen Mesh. The ESMOOTH

dialog box opens.


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DetailedExample

Additional refinement passes on the resonator region can be carried out using the

above outlined steps. Here, we chose to refine the areas near the tips of the cone

since they can be considered as a mild singularity points. However, before each

additional refinement pass, we must make sure to unselect the elements from the

previous selection set. This can be done by re-initializing the selection set with the

command.

To initialize the selection set:

1. From the Control menu, select Select,

Initialize. The INITSEL dialog box

opens.

2. Click OK to accept all the default

settings.

Also, click the REPAINT button to view the new mesh. To start the second

refinement pass, we must again select which elements to refine. We will select theelements that lie within a circular region centered at the resonators top right tip.


1. From the Control menu, select Select, by

Windowing. The SELWIN dialog box

opens.

2. Enter the following options:Entity Name [EL] > (accept default)

Window type 0 = Box 1 = Circle 2 = Polygon [0] > 1

Selection set number [1] > (accept default)

3. Click OK.

4. Select center point of circular window.

5. Select point on perimeter. Choose a circle centered at (4, 4) and of radius

roughly equal to 1.5.

The selected elements are once again highlighted and can now be refined.



box opens.


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We need to ensure smooth elements (i.e., good aspect ratios) after the refinement.

To smooth the mesh:


dialog box opens.


The last refinement pass is once again accomplished by selecting the elements that

lie within a circle centered at the origin and having a radius of roughly 0.75 and

refining them.

To initialize the selection set:

1. From the Control menu, select Select, Initialize. The INITSEL dialog box

opens.



1. From the Control menu, select Select, by

Windowing. The SELWIN dialog box

opens.


Entity Name [EL] > (accept default)

Window type 0 = Box 1= Circle 2= Polygon [0] > 1

Selection set number [1] > (accept default)

3. Click OK.

4. Select center point of circular window. Choose the origin.

5. Select point on perimeter. Pick a point at roughly 0.75 mm from the center.



box opens.


We need to ensure smooth elements (i.e., good aspect ratios) after the first

refinement.

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To smooth the mesh:


dialog box opens.


The resulting mesh at this stage is deemed acceptable for this problem. Before

applying the boundary conditions, we must first merge all duplicate nodes resulting

from the meshing of different regions.

To merge nodes:

1. From the Meshing menu, select Nodes, Merge.

2. Click OK to accept all default settings.

Applying Boundary Conditions

Next, we apply the boundary conditions of this problem. The only boundary

condition needed is for the conducting cavity walls which is defined as follows:

To apply the gc boundary condition to curves 1 to 5:

1. From the LoadsBC menu, select

E-Magnetic, Hi-Freq_B-C, Define

by Curves. The CBCR dialog box

opens.2. Enter the following options:

Pick/Input Beginning Curve > 1

Boundary condition type

(fc, gc, pmc, oob) [fc] > gc

3. Click Continue.

Conductor Number > 1

Conductivity value [5.8e+007] >

(accept default)

Relative permeability value [1] > (accept default)

Pick/Input Ending Curve > 5


4. Click OK.

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The boundary condition symbols are then plotted on all edges of the elements that

fall on the above curves. The resulting final mesh with the applied boundary

conditions is shown in Figure 4.4. Note that no particular boundary condition need

be applied to the axis of the structure since the solver does take care of it internally

Figure 4.4 Final Model Mesh with Boundary Conditions

Running Analysis

We are now ready to submit the model for solution. First, we set the analysis optionto CAVAXI using the units of mm.

To set the analysis options:

1. From the Analysis menu, select Hi-Freq_EMagnetic, Analysis Option. The

A_HFRQEM dialog box opens.

2. From the Analysis Option drop-down menu,

select CAVAXI.

3. From the Units option, select mm.

4. Click OK.

Next, we set the solution options, choosing the axisymmetric model and requesting

3 modes each for harmonics 0 and 1.

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To set the solution options:

1. From the Analysis menu, select Hi-Freq_EMagnetic, Cavities, Set Option. The

HF_CAVSOLN dialog box opens.


Model flag 0=Axisymmetric 1=3D [0] >

(accept default)

Number of modes [1] > 3

3. Click Continue.


First harmonic [0] > (accept default)

Last harmonic [0] > 1

Couple to thermal analysis 0=No 1=Yes [0]

> (accept default)

5. Click Continue.

Next, we set the output options to compute the quality factor and give the nodal

fields only.

To specify the output options:

1. From the Analysis menu, select

Hi-Freq_EMagnetic, Cavities,

Output Option. The HF_CAVOUT

dialog box opens.


Compute quality factor 0=No 1=Yes [0] > 1

Compute RLC equivalent circuit 0=No 1=Yes [0] > (accept default)

Output option 0=None 1=Nodal 2=Elem 3=Both [0] > 1

3. Click OK.

We can now run the analysis.

To run the analysis:

1. From the Analysis menu, select Hi-Freq_EMagnetic, Run Analysis. The

analysis starts.

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DetailedExample

Upon completion of the solution (of which a log is kept in the file micav1.zlg),

control is returned to GEOSTAR for postprocessing. As the results of the analysis

comprise both mesh-related electric and magnetic field distributions as well as

modal quantities (resonant frequency, conductor and dielectric quality factors and

in the general case, RLC equivalent circuit parameters) two types of postprocessing

analyses can be performed. First, we examine the resulting field distribution for a

given mode and of a given harmonic.

To list available results:

1. From the Results menu, select Available Results.

This gives you a list of available nodal and element results in the form of harmonic

number, mode number and mode flag. The mode flag indicates whether the

solution for that mode converged (1) or did not converge (-1). We start by

examining the fundamental mode of the structure (i.e., harmonic 0 and mode 1).

Visualization of Results

To plot the fundamental mode of the structure:

1. From the Results menu, select Plot, Electromagnetic. The ACTMAG dialog

box opens.

2. Enter the following options.

Harmonic number [1] > 0

Mode number [1] > (accept default)Entity flag 1=ND 2=EL [1] > (accept default)

Component [Ero] > Er

3. Click the Contour Plot button. The MAGPLOT dialog box opens.


Line Flag 0=FILL 1=LINE 2=VECT [0] > (accept default)

Beginning Element [1] > (accept default)

Ending Element [1410] > (accept default)


5. Click OK.

Make sure the selection list is re-initialized by selecting Select, Initialize

command from the Control menu otherwise only the field at the last selected

elements will be plotted.

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You can/should turn off mesh plotting by selecting Display Option, Set

Bound Plot from the Display menu.

You can plot the points by selecting Plot, Points from the Edit menu to see the

end points of the conductors and the curves by selecting Plot, Curves from the

Edit menu to see the different curves and region boundaries.

The resulting field distribution is shown in Figure 4.5. Note that for this mode Ero

and Ez are both zero (nearly zero numerically). This can be verified by examining

the field plots for the individual field components and noting the relative

corresponding scales.

Figure 4.5 Resultant Electric Field Distribution for Harmonic 0, Mode 1

The other field distributions for other (harmonic, mode number) combinations can

be examined in a similar manner to the (0, 1) combination above. One way to decide

on which combination might be more interesting to examine is to view the modal

data results. The summary of these results can be viewed with the command:

To list the results:

1. From the Results menu, select List, HF Emag Result.

The resulting listing is as follows:

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Note from the above results the relatively low Q of harmonic 0, mode 2. A close

examination of the field distribution associated with this mode reveals the reasons

the field is concentrated mostly near the bottom tip of the cone and close to the

cavity conducting walls as shown in Figure 4.6. It is prudent to make additional

mesh refinement in the high-intensity field region to insure accurate results for this

particular mode.

Harmonic: 0 Mode: 1=======================

Resonant Frequency: 9960.8659 MHzConductor Q Factor: 51413.9778Dielectric Q Factor: 87485.3252Overall Quality Factor: 32382.9456Equivalent Resistance: Not computedEquivalent Inductance: Not computedEquivalent Capacitance: Not computed

Harmonic: 0 Mode: 2

=======================Resonant Frequency: 10895.7230 MHzConductor Q Factor: 7833.8955Dielectric Q Factor: 564302.5347Overall Quality Factor: 7726.6310Equivalent Resistance: Not computedEquivalent Inductance: Not computedEquivalent Capacitance: Not computed

Harmonic: 0 Mode: 3=======================

Resonant Frequency: 15560.5081 MHzConductor Q Factor: 63340.9538Dielectric Q Factor: 95728.6514Overall Quality Factor: 38118.8102Equivalent Resistance: Not computedEquivalent Inductance: Not computed

Equivalent Capacitance: Not computed

Harmonic: 1 Mode: 1=======================


Harmonic: 1 Mode: 2=======================


Harmonic: 1 Mode: 3=======================


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Figure 4.6 Resultant Electric Field Distribution for Harmonic 0, Mode 2

Finally, we can use xy-plots to examine the variation of the various modal

quantities that have been computed versus harmonic number. For example, to

examine the change in resonant frequency versus harmonic number for the three

modes, we use the following command sequence:

To load the resonant frequency versus harmonic data for mode 1:

1. From the Display menu, select XY_Plots, Activate Post-Proc. The

ACTXYPOST dialog box opens.


Graph Number [1] > (accept default)

Mode number [1] > (accept default)

Y_Variable [RFREQ] > (accept default)

Graph Color [12] > (accept default)

Graph line style [1] > (accept default)

Graph symbol style [1] > (accept default)

Graph id [1N] > MODE_1

3. Click OK.




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Graph Number [1] > 2

Mode number [1] > 2

Y_Variable [RFREQ] > (accept default)

Graph Color [12] > 13


Graph symbol style [1] > 2


3. Click OK.





Graph Number [1] > 3

Mode number [1] > 3

Y_Variable [RFREQ] > (accept default)Graph Color [12] > 14


Graph symbol style [1] > 3


3. Click OK.

To plot the data:

1. From the Display menu, select XY_Plots, Plot Curves. The XYPLOT dialog

box opens.

2. Enter the following option:

Plot graph 1 0=No, 1=Yes [1] > (accept default)



3. Click OK.

The resulting plot is presented in Figure 4.7 showing the resonant frequencies inHz versus the harmonic number for the first three modes. Similar plots for the

different cavity Qs can be made by using the above two commands.

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Figure 4.7 Variation of the Resonant Frequency with Harmonic Numberfor the First Three Modes

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5. Verification Problems

Introduction

This chapter contains verification problems to check the accuracy of the variou

solvers. The input files for these verification problems are included in th

vprobls\hfs sub-directory in the COSMOSM installation folder. Each file may b

read to GEOSTAR through the File, Load... command. All files have the .gfm

extension

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Description:

The geometry of the problem is depicted

in Figure 5.1. The closed form solutionfor this structure can be found in [1], for

example. The particular dimensions used

in this example are: a = 0.7m, b = 0.3m,

and c = 1m. The cavity walls are assumed

to be made of copper (conductivity

= 5.8x107mhos/m, skin depth = 66.1 /f Mhz).

Results:

The first four modes for the cavity shown in Figure 5.1 with the above dimensions

and air filling are summarized in Table 5.1 and compared to the closed form

formulas [1].

References:

1. R. E. Collin, Field Theory of Guided Waves, New York: McGraw-Hill, 1960.

Table 5.1 Resonant Frequency and Quality Factor for the First Four

Modes of the Cavity Shown in Figure 5.1

MICAVV1: Multi-Mode Calculations forHomogeneously Filled Rectangular Cavities

ModeCOSMOSCAVITY Computed [1]

f(MHz) Q f(MHz)

TE101 261.27 41756 261.38

TE102 368.08 52083 368.41

TE201 452.54 53513 453.75TE103 497.38 62171 498.07

a

bc

Figure 5.1 Geometry of theHomogeneously Filled

Rectangular Cavity

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Problems

Description:

The geometry of the problem is depicted in

Figure 5.2. This structure was originallyinvestigated in [1] by both an analytical

approach and a perturbational method. The

particular case considered is for b1 = b2 = l

and the normalized resonant frequency kl =

r(l/co) is computed.

Results:

The first two dominant resonant frequenciesand the quality factors of the cavity shown

Figure 5.2 have been computed using

COSMOSCAVITY for four different materials. The results are summarized in

Table 5.2 and are compared to those obtained in [1], which were computed for the

second mode only. The metallic walls are assumed to be made of copper.

References:

1. J. Van Bladel, High-permittivity dielectrics in waveguides and resonators,IEEE Trans. on Microwave Theory and Tech., Vol. 22, pp. 32-37, Jan. 1974.

Table 5.2 Normalized Resonant Frequencies and Quality Factors for

First Two Dominant Modes of the Cavity of Figure 5.2

MICAVV2: An Inhomogeneously FilledRectangular Cavity

r ModeResults from [1] COSMOSCAVITY

Analytical Perturbation r(2fl/co) Q

2.251

2

-

2.5053

-

2.5220

2.2655

2.4995

4.770 x 104

4.348 x 104

4.01

2

-

2.5987

-

2.6048

2.3642

2.5988

2.458 x 104

2.098 x 104

9.01

2

-

2.6617

-

2.6628

2.4293

2.6641

9.132 x 103

7.569 x 103

16.01

2

-

2.6829

-

2.6833

2.4508

2.6892

4.473 x 103

3.682 x 103

a

b1c

b2 r

Figure 5.2 Geometry of the

Inhomogeneously FilledRectangular Cavity

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5erificationProblems

Description:

The geometry of the problem is depicted in

Figure 5.3. This structure was investigated in[1] by an integral equation approach as well

as measurements. The particular structure

considered is with the dielectric block

centered in the bottom plane of the

rectangular cavity and with a = 9m,

b = 4m, c = 5m, w = 4.5m, h = 2.5m and l =

2m. The first three dominant resonant modes

(resonant frequencies and quality factors for

copper walls) are computed for r = 2.05.

Results:

The resonant frequencies of the first three dominant modes and the corresponding

quality factors for cavity of Figure 5.3 have been computed using COSMOS

CAVITY. The results are summarized in Table 5.8 and are compared to those

obtained in [1] where the normalized wavenumber (koa = a/co) has been computedfor the first mode only.

References:

1. M. Albani and P. Bernardi, A numerical method based on the descretization of

Maxwells equations in integral form, IEEE Trans. on Microwave Theory and

Tech., Vol. 22, pp. 446-450, Apr. 1974.

Table 5.3 Normalized Resonant Frequencies and Quality Factors for the

First Three Dominant Modes of the Cavity of Figure 5.3

MICAVV3: A Rectangular Cavitywith a Dielectric Block

ModeResults from [1] COSMOSCAVITY

Computed Measured koa Q

1 5.55 5.22 5.459 235.5

2 - - 6.858 1126.6

3 - - 7.332 1236.0

a

b cw

h

lr

Figure 5.3 Geometry of theRectangular Cavity

with a Dielectric Block

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Problems

Description:

The geometry of the structure analyzed by COSMOSCAVITY in this example is

depicted in Figure 5.4. The dielectrics used are r1=1.031 and r2=24.6-j8.54x10-4

This high-Q dielectric resonator has been investigated by [1]. The cavity walls are

assumed to be made of copper (conductivity = 5.8x107mhos/m, skin depth = 66.1 /f MHz).

Figure 5.4 Geometry of the High-q Dielectric Resonator Investigated in [1]

Results:

References:

1. Y. Kobayashi, Y. Kabe, Y. Kogami and T. Yamagishi, Frequency and low-

temperature characteristics of high-Q dielectric resonators, 1989 IEEE MTT-S

Digest, pp. 1239-1242.

MICAVV4: Dominant Mode Calculations forInhomogeneously Filled Cylindrical Cavities;

a High-Q Dielectric Resonator

Measured [1] COSMOSCAVIYT

f

(GHz)

Q

total

f

(GHz)

Q

dielectric

Q

conductor

Q

total

8.383 19000 8.384 29712 84015 21951

z

3.37mm

5mm

5mm

8.05mm

15.61

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