HFSS Theory[1]

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magnetic field inside a strucmentation is largely transking the most effective use
d and its implementation incomputed from the simurted to “nodal” or “voltage”
s Conversion
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Technical NotesThe simulation technique used to calculate the full 3D electroture is based on the finite element method. Although its impleparent, a general understanding of the method is useful in maof Ansoft HFSS.
This section provides an overview of the finite element methoAnsoft HFSS. It also describes how modal Sparameters arelated electric and magnetic fields and how they can be convebased pseudoSparameters used in circuit theory.
Information is included on the following:
• The Finite Element Method • SParameters
• Software Modules • Radiated Fields
• Derivation of Wave Equation • Modes to Node
• Implementation
hnical NotesFinite Element Method
lementationtware Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
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HFSS employs the finite elee full problem space intosubregion (element) with a
into a large number of tetraramid. This collection of tet
eld) at points inside each tett each vertex, Ansoft HFSSree edges of the tetraheector field at the midpoint ofdge (as shown below). Thevalues.
nsform Maxwell’s equationsal methods.
ield that areof an element
the vertices.
ld that isf an elementis explicitly
of selected
ield at anlated from
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hnical NotesFinite Element Methodepresentation of aield Quantity
asis Functionsize of Mesh Versusccuracy
ield Solutionsigenmodeseak vs. RMS phasorslementationtware Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
The Finite Element MethodIn order to generate an electromagnetic field solution, Ansoftment method. In general, the finite element method divides ththousands of smaller regions and represents the field in eachlocal function.
In Ansoft HFSS, the geometric model is automatically dividedhedra, where a single tetrahedron is basically a foursided pyrahedra is referred to as the finite element mesh.
Representation of a Field Quantity
The value of a vector field quantity (such as the Hfield or Efirahedron is interpolated from the vertices of the tetrahedron. Astores the components of the field that are tangential to the thdron. In addition, the system can store the component of the vselected edges that is tangential to a face and normal to the efield inside each tetrahedron is interpolated from these nodal
By representing field quantities in this way, the system can trainto matrix equations that are solved using traditional numeric
The components of a ftangential to the edgesare explicitly stored at
The component of a fietangential to the face oand normal to an edgestored at the midpointedges.The value of a vector finterior point is interpothe nodal values.
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to interpolate field values
field values from both nodal
hedron.rtices only — and therefore
ron.hedron.
level of accuracy, and the
e individual elements (tetraing thousands of elementssing relatively few elements.ment must occupy a regiond from the nodal values.
ix with approximately asith a large number of eleputing power and memory.n accurate field solution butand processing power.
rocess in which the mesh isolution based on a coarseensity and generates a newired limit, the system breaks
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hnical NotesFinite Element Methodepresentation of a Fielduantity
asis Functionsize of Mesh Versusccuracy
ield Solutionsigenmodeseak vs. RMS phasorslementationtware Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Basis Functions
Various interpolation schemes (basis functions) can be usedfrom nodal values.
• A 1st order tangential element basis function interpolatesvalues at vertices and on edges.1st order tangential elements have 20 unknowns per tetra
• A 0th order basis function makes use of nodal values at veassumes that the field varies linearly inside each tetrahed0th order tangential elements have six unknowns per tetra
Size of Mesh Versus Accuracy
There is a tradeoff between the size of the mesh, the desiredamount of available computing resources.
The accuracy of the solution depends on how small each of thhedra) is. Generally speaking, solutions based on meshes usare more accurate than solutions based on coarse meshes uTo generate a precise description of a field quantity, each elethat is small enough for the field to be adequately interpolate
However, generating a field solution involves inverting a matrmany elements as there are tetrahedra nodes. For meshes wments, such an inversion requires a significant amount of comTherefore, it is desirable to use a mesh fine enough to obtain anot so fine that it overwhelms the available computer memory
To produce the optimal mesh, Ansoft HFSS uses an iterative pautomatically refined in critical regions. First, it generates a sinitial mesh. Then, it refines the mesh in areas of high error dsolution. When selected parameters converge to within a desout of the loop.
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lly stabilize before the fullthe field solution associ
criteria that are tighter than
agnetic field (Hfield) is lessse the Hfield is computed
ode solver finds the resoant frequencies.
ix equation (for a lossless
mesh.
eigenmode solver sets b to
lectric field solution, and kohe wave number ko isllowing:
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nical NotesFinite Element Methodepresentation of a Fielduantity
asis Functionsize of Mesh Versus Accuacyeld Solutionsigenmodeseak vs. RMS phasorslementationware Modulesrameters
vation of Wave Equationiated Fieldses to Nodes Conversion
Field Solutions
During the iterative solution process, the Sparameters typicafield solution. Therefore, when you are interested in analyzingated with a structure, it may be desirable to use convergenceusual.
In addition, for any given number of adaptive iterations, the maccurate than the solution for the electric field (Efield) becaufrom the Efield using the relationship:
Eigenmodes
Eigenmodes are the resonances of the structure. The eigenmnant frequencies of the structure and the fields at those reson
For a Driven Solution, Ansoft HFSS solves the following matrcase):
where:
• Sand T are matrices that depend on the geometry and the• x is the electric field solution.• ko is the freespace wave number.• b is the value of the source defined for the problem.
However, in order to find the resonances of the structure, thezero, and solves the equation:
for sets of (ko,x), one ko for every x. The variable x is still the eis the free space wave number corresponding to that mode. Trelated to the frequency of the resonant modes through the fo
H E∇×jωµ–
=
Sx ko2Tx+ b=
Sx ko2Tx+ 0=Sx ko
2Tx+ 0=
fkoc=
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of lossy as well as losslessthe quality factor, and is aQ is the energy lost due to
ed for eigenmode problems,s.
er can calculate the loaded
ithin Ansoft HFSS. Someknow port Sparameters or
bsolute field values, forypes of field representation,
epresentation of the steadythe instantaneous (timefrom the phasor representa
ting a timeharmonic electricd Ex(t), is computed from:
nverge long before the
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nical NotesFinite Element Methodepresentation of a Fielduantity
asis Functionsize of Mesh Versus Accuacyield Solutionsigenmodeseak vs. RMS phasorslementationware Modulesrameters
vation of Wave Equationiated Fieldses to Nodes Conversion
where c is the speed of light.
The Ansoft HFSS eigenmode solver can find the eigenmodesstructures, and can calculate the unloaded Q of a cavity. Q ismeasure of how much energy is lost in the system. Unloadedlossy materials. Because ports and other sources are restrictthe Q calculated does not include losses due to those source
In addition to calculating the unloaded Q, the eigenmode solvQ of structures only for power lost through PML structures.
Peak vs. RMS phasors
This section concerns how field quantities are represented wusers will not need this information, such as those who wish torelative amplitudes of field solutions. Those that wish to find aexample, will need to review the difference between the two tpeak and RMS.
HFSS solves in the frequency domain and obtains a phasor rstate finite element field solution. Physical quantities such asdomain) electric field are then obtained as derived quantitiestion.
If Ex is the xcomponent of a “peak” phasor quantity represenfield, the physical electric field xcomponent at time t, denote
where:
• ℜ is the real part of a complex number or function.• ω is angular frequency, 2πf.• j is the imaginary unit, .• t is the time.
Note: Like the driven solution, the eigenmodes tend to cofield solutions.
Ex t( ) ℜ Exejωt[ ]=Ex t( ) ℜ Exejωt[ ]=
1–
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tor of is required:
, max (Ex(t)) observed overfor RMS
averaged power flowhe complex Poynting vectorx Poynting vector S depends
t of timeaveraged power., if a voltage gap sourceehaves as v(t) = 5cosωt.
s, if the plane wavegnitude is
is a timeaveraged quantity
ynting vector button in theak phasors, S = 1/2E x H*.
s time domain quantities
2
ax Ex t( )( ) 2 Ex=
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hnical NotesFinite Element Methodepresentation of a Fielduantity
asis Functionsize of Mesh Versusccuracy
ield Solutionsigenmodeseak vs. RMS phasorslementationtware Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
On the other hand, if Ex is an “RMS” phasor, an additional fac
As a consequence of these equations, the peak physical fielda full time cycle is for peak phasors andphasors.
Additionally, given field phasors E and H, to compute the timethrough a surface, the normal component of the real part of tis integrated over the surface. The correct form of the compleon which phasor representation is used. For peak phasors:
For RMS phasors:
While using Ansoft HFSS, the conventions are as follows:
• Each propagating mode incident on a port contains 1 wat• Circuit gap sources are specified in a peak sense. That is
magnitude is 5 volts, then the time domain circuit source bLikewise for a current gap source.
• Planewave sources are specified in a peak sense. That imagnitude is 5 V/m, then the plane wave incident field ma
.• Radiated power, as computed in the 3D Post Processor,
computed using the complex Poynting vector.• Phasors in the Data Calculator are peak phasors. The Po
calculator therefore implements the Poynting vector for peCalculations that compute either average or instantaneoumust adhere to the peak phasor conventions.
Ex t( ) ℜ 2Exejωt[ ]=
max Ex t( )( ) Ex= m
S 12E H∗×=S 12E H∗×=
S E H∗×=S E H∗×=
E t( ) 5 k r⋅ ωt+( )cos=
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tem does the following:
e supported by art.tructure, assuming that one
flection and transmission
nsmitted and reflected sigreducing the full 3D electro
rcuit parameters.
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hnical NotesFinite Element Method
lementationolution ProcessSingle FrequencySolution
Fast Frequency SweepDiscrete FrequencySweep
InterpolatingFrequency Sweep
tware Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
ImplementationTo calculate the Smatrix associated with a structure, the sys
• Divides the structure into a finite element mesh.• Computes the modes on each port of the structure that ar
transmission line having the same crosssection as the po• Computes the full electromagnetic field pattern inside the s
mode is excited at a time.• Computes the generalized Smatrix from the amount of re
that occurs.
The final result is an Smatrix that allows the magnitude of tranals to be computed directly from a given set of input signals,magnetic behavior of a structure to a set of high frequency ci
Solution Process
There are four variations to the solution process:
• Single frequency solution• Fast frequency sweep• Discrete frequency sweep• Interpolating frequency sweep
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ptive solution at a single freeep. An adaptive solution islly refined in the areas of
ive solutions. The procedure
No
Refine the meshinside the structure.
Yes
acceptable?∆S
rameters and field
Yes
ailable.
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hnical NotesFinite Element Method
lementationolution ProcessSingle FrequencySolution
Fast Frequency SweepDiscrete FrequencySweep
InterpolatingFrequency Sweep
tware Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Single Frequency Solution
A single frequency solution generates an adaptive or nonadaquency and is often the first step in performing a frequency swone in which a finite element mesh is created and automaticahighest error — increasing the accuracy of succeeding adaptfor performing a single frequency solution is shown below:
Select a starting mesh.
Compute excitation currentpattern for each port atω = ωtest.
Refine the meshat the ports.
Test accuracy of excitationsignal at each port by
∇xE to H.comparing ∇xH to E and
No YesAcceptable?
Compute field solution
ω = ωtest.
excitation at ports forinside structure due to
Make circuit pa
Adaptivesolution?
No
distributions av
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ution over a range of frep (ALPS)based solver topecify the starting and endnerated. The same mesh is
utions can be computed andis the most accurate. The
nly sweep at frequencies
meters available forlizing, and de
ble field post processing
ystem across the
within the sweep’s.
.
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hnical NotesFinite Element Method
lementationolution ProcessSingle FrequencySolution
Fast FrequencySweep
Discrete FrequencySweep
InterpolatingFrequency Sweep
tware Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Fast Frequency Sweep
This type of solution uses an existing mesh to generate a solquencies. The system uses an Adaptive LanczosPadé Sweeextrapolate an entire bandwidth of solution information. You sing frequencies and the interval at which new solutions are geused for each solution, regardless of the frequency. While solviewed at any frequency, the solution at the center frequencyprocedure for a fast frequency sweep is shown below:
Select a starting mesh.
Compute the ALPS matrices.
Compute eigenvectors for ALPS
Perform portso
Make circuit paraviewing, renormaembedding. Ena
“fsweeppart1”
Invert the lowest order ALPS matrix.“Solver”
system.“mr”
Recover Sparameters“fsweeppart2”
chosen by the s
If a single frequency solution wasselected, perform a single frequencysolution at the single frequency or
at any frequencyfrequency range
at the center frequency.
frequency range
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xisting mesh to generate aand ending frequencies andmesh is used for each solu
frequency sweep is showne sweep:
circuit parameters availfor viewing, renormalizing,eembedding. Enable
post processing at thefrequency or at the
r frequency. (Field postssing is only availablee single frequency or the
er frequency.)
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hnical NotesFinite Element Method
lementationolution ProcessSingle FrequencySolution
Fast Frequency SweepDiscrete FrequencySweep
InterpolatingFrequency Sweep
tware Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Discrete Frequency Sweep
To perform a discrete frequency sweep, the system uses an esolution over a range of frequencies. You specify the startingthe interval at which new solutions are generated. The sametion, regardless of the frequency. The procedure for a discretebelow, where n equally spaced frequencies are included in th
n = 1
Compute the Smatrix forω = ωn.
Compute the field solution
frequencyMore
points?
n = n + 1
No
Yes
Compute the excitationmode pattern at each portfor ω = ωn.
inside the structure due toexcitation at ports forω = ωn.
Select a starting mesh.
If a single frequency solution wasselected, perform a single frequencysolution at the center frequency.
Makeableand dfieldsinglecenteprocefor thcent
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s an existing mesh to interstarting and ending frequenby the system. The same
uniform error toleranceerpolating frequency sweepm are included in the
e circuit parameters availfor viewing, renormalizing,deembedding. Enablepost processing at the
le frequency or at theer frequency. (Field postessing is only availablehe single frequency or theter frequency.)
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hnical NotesFinite Element Method
lementationolution ProcessSingle FrequencySolution
Fast Frequency SweepDiscrete FrequencySweep
InterpolatingFrequency Sweep
tware Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Interpolating Frequency Sweep
To perform an interpolating frequency sweep, the system usepolate solutions over a range of frequencies. You specify thecies and solutions are generated at frequency points chosenmesh is used for each solution, regardless of the frequency. Aexists throughout the entire solution. The procedure for an intis shown below, where n frequencies determined by the systesweep:
n = 1
Compute the Smatrix forω = ωn.
Compute the field solution
n = n + 1
Yes
NoCompute the excitationmode pattern at each portfor ω = ωn.
inside the structure due toexcitation at ports forω = ωn.
All parameters(S, Zpi, etc.)
converge?
Obtain the next interpolationfrequency ωn.
Select a starting mesh.
If a single frequency solution wasselected, perform a single frequencysolution at the center frequency.
Makableandfieldsingcentprocfor tcen
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rious software modules ton the following table:
eometric models.
materials to objects.
boundaries and
es existing meshes.ally refining the mesh.
rn at each port.
the structure.
e finite element mesh.
des.
e data calculated durt Processors are avail
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Software ModulesIn the process of generating a solution, Ansoft HFSS calls vaexecute specialized tasks. The role of each module is listed i
Module Function
Executive Performs all executive functions.
3D Modeler Provides the interface for drawing g
Material Manager Provides the interface for assigning
3D Boundary Manager Provides the interface for assigningsources.
Mesh3d Generates the initial mesh and refinAlso provides the interface for manu
wave Determines the excitation field patte
abc3d Computes the full 3D solution inside
solver Inverts the matrix associated with th
eigen Calculates real or complex eigenmo
Post Processors Provide the interface for analyzing thing the solution process. Three Posable.
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ess the Project Manager.oft High Frequency Strucware module.
ks. It controls the Executivee Material Manager, the 3D
s. It also coordinates thenecessary.
irectory containing the cur
ent points.
trices.
formation.
.) Stores the eigences.
es the magnetostatic
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
The Executive Module
Choose the Maxwell Control Panel Projects command to accFrom there, you must create a project, classify its type as Ansture Simulator 8 , and then open it to start the Executive soft
The Executive module is responsible for all executive level tasCommands menu, and provides access to the 3D Modeler, thBoundary Manager, the Meshmaker, and the Post Processorsolution process by calling the modules wave and abc3d as
Files
The following files created by Ansoft HFSS are stored in the drent project:
File Name Extension Contents
projectname .adp Stores the adaptive refinem
projectname .szg Stores the contents of Sma
projectname .tst Stores the emissions test in
projectname .kvc (Fast frequency sweeps onlyvectors of the reduced matri
projectname .hsf.msf
(Ferrite problems only.) Storfield information.
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e geometry for a structure,eler screen appears.
aterials to objects or defineanager. The Material Man
e final geometric
ts.
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
The 3D Modeler Module
When you choose the Draw command to create or modify thAnsoft HFSS passes control to the 3D Modeler. The 3D Mod
Files
Geometric objects are written to the following file:
The Material Manager Module
When you choose the Setup Materials command to assign mnew materials, Ansoft HFSS passes control to the Material Mager window appears.
Files
Material information is written to the following file:
File Name Extension Contents
projectname .sm3 Stores objects for use in thmodel.
File Name Extension Contents
projectname .mat Stores material definitions.
projectname .asn Stores material assignmen
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d to define boundary condiundary Manager. The 3D
e:
rts and boundaries.
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
The 3D Boundary Manager Module
When you choose the Setup Boundaries/Sources commantions and sources, Ansoft HFSS passes control to the 3D BoBoundary Manager window appears.
Files
Boundary and source information is written to the following fil
File Name Extension Contents
projectname .bnd Stores information about po
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ing mesh, Ansoft HFSSalso access the Meshmaker
eometry into a number of. The Meshmaker uses theer points are added to servest mesh.
igh error density have beenthose areas. After refiningmesh files.
etrahedra.
etrahedra faces.
etrahedra edges.
etrahedra vertices.
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
The Meshmaker Module
To create the initial finite element mesh and to refine an existpasses control to the Meshmaker software module. You mayinterface to manually refine the mesh.
Initial Mesh
Initially, the Meshmaker automatically divides the structure’s grelatively coarse tetrahedra, each having four triangular facesvertices of objects as the initial set of tetrahedra vertices. Othas the vertices of tetrahedra only as needed to create a robu
Automatic Mesh Refinement
After the initial field solution has been created and areas of hlocated, the Meshmaker is called again to refine the mesh inthe mesh in the selected region, the Meshmaker updates the
Files
The finite element mesh is written out to the following files:
File Name Extension Contents
projectname .hyd Specifies the location of t
projectname .fac Specifies the location of t
projectname .lin Specifies the location of t
projectname .pnt Specifies the location of t
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ThTTM
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be calculated, it is necescalculate the excitation pat. This module calculates the
ission structure with thes serve as boundary condi
aveguide that has the samefield associated with travel
t is connected:
that determines, at a given
sssection of the port; the z
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nical NotesFinite Element Method
lementationware Modulese Executive Module
he 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Module
e 2D Wave Modulee 3D Field Solver Mod
leost Processor Modules
rametersvation of Wave Equationiated Fieldses to Nodes Conversion
The 2D Wave Module
Before the full 3D electromagnetic field inside a structure cansary to determine the excitation field pattern at each port. Totern at each port, the system calls the wave software modulenatural field patterns (or modes) that can exist inside a transmsame crosssection as the port. The resulting 2D field patterntions for the full 3D problem.
Excitation Fields
The assumption is that each port is connected to a uniform wcrosssection as the port. Therefore, the excitation field is theing waves propagating along the waveguide to which the por
where:
• ℜ is the real part of a complex number or function.• E(x,y) is a phasor field quantity.• γ=α + jβ is the complex propagation constant, where:
• α is the attenuation constant of the wave.• β is the propagation constant associated with the wave
time t, how the phase angle varies with z.• ω is angular frequency, 2πf.• j is the imaginary unit, .
In this context, the x and yaxes are assumed to lie in the croaxis lies along the direction of propagation.
E x y z t, , ,( ) ℜ E x y,( )ejωt γz–[ ]=
1–
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e determined by solvingodule solves is derived
.
citation field pattern in thefor H(x,y) using the corre
ependent of z and t; only.
is valid only at a given frech frequency point of inter
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Wave Equation
The field pattern of a traveling wave inside a waveguide can bMaxwell’s equations. The following equation that the wave mdirectly from Maxwell’s equation:
where:
• E(x,y) is a phasor representing an oscillating electric field• k0 is the free space wave number, .• ω is the angular frequency, 2πf.• is the complex relative permeability.• is the complex relative permittivity.
When the wave module solves this equation, it obtains an exform of a phasor solution, E(x,y). It also solves independentlysponding wave equation in H. These phasor solutions are indafter being multiplied by eγz do they become traveling waves
Also note that the excitation field pattern computed by wavequency. A different excitation field pattern is computed for eaest.
1µr E x y,( )∇×
∇× k02εrE x y,( )– 0=
ω µ0ε0
µr x y,( )εr x y,( )
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pattern as a 2D finite elethe 2D mesh of trianglesface. The wave module per
the Meshmaker.
of the initial mesh, waveelectric field, E.
e following equations:
the appropriate waveults to the solved E. It
ved H.rance, the solution is
ned and wave performs
re read out to the existingsh the next time the Mesh
odule implements, refer to
ve Analysis of DielectricEE Transactions on1991.
on, there is a series of basicecific frequency. Any linearefault, the system instructs
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
2D Mesh Refinement
The wave module treats its computation of the excitation fieldment problem. The mesh associated with each port is simplycorresponding to the face of tetrahedra that lie on the port surforms an iterative refinement of this 2D mesh without calling
The refinement procedure is as follows:
1. Using the triangular mesh formed by the tetrahedra facescalculates solutions for both the magnetic field, H, and the
2. To determine if the 2D solution is accurate, wave uses th
where H(x,y) and E(x,y) are phasors.3. The module first calculates E and H independently using
equations. Next, it computes and compares the resthen computes and compares the results to the sol
4. If the reciprocal comparison falls within an acceptable toleaccepted. Otherwise, the 2D mesh on the port face is refianother iteration.
Any mesh points that have been added to the face of a port amesh files. These points are incorporated into the full 3D memaker is called.
For a detailed understanding of the theory which the wave mthe following:
JinFa Lee, DinKow Sun, and Zoltan J. Cendes, “FullWaWaveguides Using Tangential Vector Finite Elements,” IEMicrowave Theory and Techniques, vol. 39, No 8, August
Modes
For a waveguide or transmission line with a given crosssectifield patterns (modes) that satisfy Maxwell’s equations at a spcombination of these modes can exist in the waveguide. By dwave to compute only the dominant mode field pattern.
H∇× σE jωεE+=E∇× jωµH–=
∇ H×∇ E×
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order modes because the1 (dominant) field at one2 field pattern at another,
2 field.
tation signal of one specificdue to discontinuities in thee excitation port or transmitodes should be calculated.
ither because of attenuationode — there is no need toavoid the need for comput
length of waveguide in thein. For example, if the mode0.5 mm, then the “constante port should be at least 0.5Sparameters must betion segment that has to beenuation constant, α.
ith frequency. However, theuency. When performing free likelihood of higher order
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Mode Conversion
In some cases it is necessary to include the effects of higherstructure acts as a mode converter. For example, if the modeport is converted (as it passes through a structure) to a modethen it is necessary to obtain the Sparameters for the mode
Modes, Reflections, and Propagation
It is also possible for a 3D field solution generated by an excimode to contain reflections of higher order modes which arisestructure. If these higher order modes are reflected back to thted to another port, the Sparameters associated with these m
If the higher order mode decays before reaching any port — efrom losses or because it is a nonpropagating evanescent mobtain the Sparameters for that mode. Therefore, one way toing the Sparameters for a higher order mode is to include amodel that is long enough for the higher order mode to decay2 wave associated with a certain port decays to near zero incrosssection” portion of the geometric model leading up to thmm long. Otherwise, for accurate Sparameters, the mode 2included in the Smatrix. The length of the constant crosssecincluded in the model depends on the value of the mode’s att
Frequency
The field patterns associated with each mode generally vary wpropagation constants and impedances always vary with freqquency sweeps, be aware that as the frequency increases, thmodes being propagating modes also increases.
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ting strips side by side asrate, the system simulates
ransmission structures. It iser, in actuality, there will be
single port with multiplethe port crosssection, atxample, if the port consistsnclosure, N = 3; therefore atual number of terminals as
the 3D model, additionale terminals on a multiconnetic (TEM) modes wills. Therefore, rather thanend the port outward untiltted from consideration.
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Multiple Ports on the Same Face
Visualize a port face on a microstrip that contains two conductwo separate ports. If the two ports are defined as being sepathe case in which the two ports are connected to uncoupled tas if a conductive wall separates the excitation waves. Howevelectromagnetic coupling between the two strips.
To model this coupling accurately, analyze the two ports as amodes. In general, if there are N disconnected conductors inleast N  1 modes are required for an accurate solution. For eof two adjacent microstrip lines surrounded by a conducting eleast two modes should be defined on the port. Assign an eqmodes. Refer to Defining Terminals for more information.
If the multiconductor port plane is near discontinuities withinmodes beyond N  1 may be necessary. However, if you definductor port, the presence of nonquasi transverse electromagadversely affect the entries of any computed terminal matriceincrease the number of modes beyond the required N  1, extany higherorder modes have sufficient attenuation to be omi
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xample, the files associatedes is produced for each port.
on numbers to frequency.
finement process.
sh at port 1.
edance line for port 1. Apedance line has been
tructure.
the port.
f the 2D object.
he 2D field pattern.
D triangular elements lie.
the 2D mesh for port 1.
aces in the 2D mesh.
ectric field at port 1.
agnetic field at port 1.
for port 1.
r of modes for port 1.
l the port.
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Files
The wave module writes out a set of files for each port. For ewith port 1 are shown in the following table. A similar set of fil
File Name Extension Contents
port .frq Contains the mapping of port soluti
port1 .ref Stores the status of the adaptive re
port1 .apt Contains the point list for the 2D me
port1 .i1 Contains the coordinates of the impset of .in files is created only if an imdefined for at least one port in the s
port1 .nvc Specifies vectors that are normal to
port1 .sat Specifies the material parameters o
port1 .sln Contains boundary information for t
port1 .sob Specifies the objects in which the 2
port1 .spt Specifies the location of vertices in
port1 .sti Specifies the location of triangular f
port1_0 .e1 Contains the field solution for the el
port1_0 .h1 Contains the field solution for the m
port1_0 .imp Contains characteristic impedances
port1_0 .spc Specifies the frequency and numbe
port1_1 .tvc Specifies vectors that are tangentia
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reated. For example, if two
t1_2.tvc ._0.e2 ._0.h2 .
created. For example, a disdominant mode only yields
_1.e1 ._1.h1 .rt1_1.imp .rt1_1.spc .
iles are created. For examdominant mode only yields
1.sln .1_1.sob ..sti .
n specifies the mode.e 1 is port1_1.tvc while
, the _n indicates the fre0. That is, the file for thethe second frequency
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver Modleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Multiple Modes
If multiples modes have been requested, additional files are cmodes have been requested, the following files are created:
• The file port1_1.tvc is replaced with port1_1.tvc and por• The file port1_0.e1 is replaced with port1_0.e1 and port1• The file port1_0.h1 is replaced with port1_0.h1 and port1• The file port1.i1 is replaced with port1.i1 and port1.i2 .
Frequency Sweeps
If a frequency sweep has been performed, additional files arecrete frequency sweep containing two frequency points for thethe following:
• The file port1_0.e1 is replaced with port1_0.e1 and port1• The file port1_0.h1 is replaced with port1_0.h1 and port1• The file port1_0.imp is replaced with port1_0.imp and po• The file port1_0.spc is replaced with port1_0.spc and po
PortsOnly Solutions
If a portsonly solution has been requested, other additional fple, a frequency sweep containing two frequency points for thethe following:
• The file port1.sln is replaced with port1_0.sln and port1_• The file port1.sob is replaced with port1_0.sob and port• The file port1.sti is replaced with port1_0.sti and port1_1
Note: The .tvc files are the only wave files for which the _Mode numbering starts at 1. That is, the file for modthe file for mode 2 is port1_2.tvc . For all other filesquency point. Frequencypoint numbering starts atfirst frequency point is port1_0.e1, while the file forpoint is port1_1.e1 .
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bc3d module. This module
ating electric field.
calculating the 2D field patme that the electric field is aat the phasor E is a functionpart of the product of the
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver
oduleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
The 3D Field Solver Module
To calculate the full 3D field solution, Ansoft HFSS calls the asolves the following wave equation:
where:
• E(x,y,z) is a complex vector phasor representing an oscill• k0 is the free space phase constant, .• ω is the angular frequency, 2πf.• is the complex relative permeability.• is the complex relative permittivity.
This is the same equation that the wave module solves for intern at each port. The difference is that abc3d does not assutraveling wave propagating in a single direction. It assumes thof x, y, and z. The physical electric field, E(x,y,z,t), is the realphasor, E(x,y,z), and ejωt:
1µr E x y z, ,( )∇×
∇× k02εrE x y z, ,( )– 0=
ω µ0ε0
µr x y,( )εr x y,( )
E x y z t, , ,( ) ℜ E x y z, ,( )ejωt[ ]=
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posed to the edge of theand all surfaces exposed toare recognized by abc3d :
signal of one specific mode
quest that the Sparameters
rts — either because ofodes — there is no need to
nnected to each port mustcted to a coaxial cable, thencrosssection of the cable.
of the attenuation constant,e mode 2 wave associated
n the constant crosssectionbe 0.5 millimeters long. OthSmatrix.
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver
oduleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Boundary Conditions
Ansoft HFSS imposes boundary conditions at all surfaces exmeshed problem region — which includes all outer surfacesvoids within the structure. The following boundary conditions
Evanescent Modes and Attenuation
It is possible for a 3D field solution generated by an excitationto contain reflections consisting of higherorder modes.
• If these higherorder reflections propagate to the ports, reassociated with these modes be calculated.
• If these higherorder modes decay before reaching the polosses or because they are nonpropagating evanescent mobtain the Sparameters for that mode.
A small length of the waveguide or transmission line that is cobe included in the structure. For example, if one port is connethe portion of the model leading up to the port must have the
The required length of the port section depends on the valueα, associated with each higherorder mode. For example, if thwith a certain port decays to near zero in 0.5 millimeters, theportion of the geometric model leading up to the port shoulderwise you should include the effect of a mode 2 wave in the
• Port• Incident wave• Voltage drop• Current Source• Perfect E• Perfect H/Natural• Finite conductivity
• Impedance• Lumped RLC• Radiation• Symmetry• Master• Slave
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n to the following set of files:
ost Process menu, Ansoftlowing Post Processors are
ts
ich explicit values are
of E at specific nodes.
ta for the field solution.
ows you to display contour,mated plots of fields. Alsos and access the field calcu
which allows you to viewpedances, and propagation
hich allows you to plot ancy.
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hnical NotesFinite Element Method
lementationtware Moduleshe Executive Modulehe 3D Modeler Modulehe Material Manager
odulehe 3D Boundary Manger Modulehe Meshmaker Modulehe 2D Wave Modulehe 3D Field Solver
oduleost Processor Modulesarametersivation of Wave Equationiated Fieldses to Nodes Conversion
Files
After a solution has been calculated, abc3d writes the solutio
Post Processor Modules
When you select one of the three Post Processors from the PHFSS passes control to the selected Post Processor. The folavailable from the Post Process menu:
The selected Post Processor screen appears.
File Name Extension Conten
projectname .bsp Specifies the nodes at whstored.
projectname .btn Specifies the field solution
projectname .bst Specifies interpolation da
Fields Accesses the 3D Post Processor, which allshaded, vector, value vs. distance, and aniallows you to import and export solution filelator.
Matrix Data Accesses the Matrix Data Post Processor,matrices computed for the Sparameters, imconstants during each adaptive solution.
Matrix Plot Accesses the Matrix Plot Post Processor, wvariety of parameters versus time or freque
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ciated with a given field
ing one watt of power.
d field at port i.ation field and all reflected/
t and reflected/transmitted
n port i at t = 0. (By default,modes. For lossless cutoff
smitted field with respect to
on field at port j is reflectedompute the amount ofport 3. The phase of S31m port 1 to port 3.
n port, it has no informae, if the port mode has
ay be 180 degrees out of
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hnical NotesFinite Element Method
lementationtware Modulesarametersigher Order Modesrequency Pointsenormalized SMatriceseembeddingvanescent Modes and Sarameters
ivation of Wave Equationiated Fieldses to Nodes Conversion
SParametersA generalized Smatrix describes what fraction of power assoexcitation is transmitted or reflected at each port.
The Smatrix for a threeport structure is as follows:
where:
• All quantities are complex numbers.• The magnitudes of a and b are normalized to a field carry
• ai2 represents the excitation power at port i.
• bi2 represents the power of the transmitted or reflecte
• The full field pattern at a port is the sum of the port’s excittransmitted fields.
• The phase of ai and bi represent the phase of the incidenfield at t=0.• represents the phase angle of the excitation field o
it is zero for lossy port modes and lossless propagatingmodes, it is 90.)
• represents the phase angle of the reflected or tranthe excitation field.
• Sij is the Sparameter describing how much of the excitatiback or transmitted to port i. For example, S31 is used to cpower from the port 1 excitation field that is transmitted tospecifies the phase shift that occurs as the field travels fro
Note: When wave computes the excitation field for a givetion indicating which way is “up” or “down.” Therefornot been calibrated, the calculated Sparameters mphase with the expected solution.
b1
b2
b3
S11 S12 S13
S21 S22 S23
S31 S32 S33
a1
a2
a3
=
ai∠
bi∠
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sociated with the dominanter order modes if it is specifrepresents another entry inhe two most dominantscribe the behavior of the
arry one watt of power.ld at port i.power of the mode m field
itation fields at that mode
itted and reflected fields.ing mode m field at port i.propagating modes. For
nsmitted field of mode m at
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nical NotesFinite Element Method
lementationware Modulesrameters
igher Order Modesequency Pointsenormalized SMatriceseembeddingvanescent Modes and Sarametersvation of Wave Equationiated Fieldses to Nodes Conversion
Higher Order Modes
By default, Ansoft HFSS only computes the Sparameters asmode at each port. However, the system can account for highically requested to do so. Each additional mode at each portthe Smatrix. For example, in a threeport structure in which tmodes are analyzed, a 6x6 Smatrix would be required to destructure. The full Smatrix required in such a case is:
where:
• All quantities are complex numbers.• The magnitudes of a and b are normalized to fields that c
• represents the power of the mode m excitation fie• represents the transmitted/reflected component’s
at port i.
The full field pattern of a mode at a port is the sum of the excand all reflected/transmitted fields of the same mode.
• The phases of a and b are the phase angles of the transm• represents the phase angle (at t=0) of the incom
(By default, it is zero for lossy port modes and losslesslossless cutoff modes, it is 90.)
• represents the phase angle of the reflected or traport i with respect to the excitation fields.
b11
b21
b12
b22
b13
b23
S1111
S1112
S1211
S1212
S1311
S1312
S1121
S1122
S1221
S1222
S1321
S1322
S2111
S2112
S2211
S2212
S2311
S2312
S2121
S2122
S2221
S2222
S2321
S2322
S3111
S3112
S3211
S3212
S3311
S3312
S3121
S3122
S3221
S3222
S3321
S3322
a11
a21
a12
a22
a13
a23
=
ami
2
bmi
2
a∠ im
bim∠
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de n excitation field at port j
atrix involving higher order
s relating a “port 1, mode 2"matrix is displayed:
f frequency. Therefore, sepequency point of interest.
dé Sweep (ALPS)basednter frequency, solutions forl for analyzing the behaviormuch faster than solving the
ed for the structure at eachupport any single frequency
rpolated for the structurency sweeps are useful for
or structures with ferritearies, or cutoff frequencies.ecause an accurate solutionmal number of frequency
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lementationtware Modulesarametersigher Order Modesrequency Pointsenormalized SMatriceseembeddingvanescent Modes and Sarameters
ivation of Wave Equationiated Fieldses to Nodes Conversion
• is the Sparameter that describes how much of a mois transmitted (or reflected) as a mode m field at port i.
To simplify how you view the data, the system displays an Smmodes as a series of smaller modetomode matrices.
For example, when you request a display of the Sparameterfield to a “port 2, mode 1" field, the following modetomode S
Frequency Points
The Sparameters associated with a structure are a function oarate field solutions and Smatrices are generated for each frAnsoft HFSS supports three types of frequency sweeps:
• Fast frequency sweeps, in which an Adaptive LanczosPasolver is used to extrapolate, from a single solution at a cea range of frequencies. Fast frequency sweeps are usefuof high Q structures. For large frequency ranges, they areproblem at individual frequencies.
• Discrete frequency sweeps, in which a solution is generatfrequency point you specify. Discrete frequency sweeps ssolution.
• Interpolating frequency sweeps, in which a solution is intebased upon solved frequency points. Interpolating frequeanalyzing the behavior of scatterer or wideband structuresmaterials, highly lossy materials, master and slave boundThis type of sweep is much faster than a discrete sweep bcan be obtained by calculating a field solution from a minipoints.
Sijmn
b21
b12
S1122
S1221
S2112
S2211
a21
a12
=
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h frequency circuit simulatorbe normalized to the approbeen normalized to 50
directly from signals that are
e, the system first calculatesefined as follows:
e (Z0) of each port as a
impedance matrix using this
nce and admittance asalized to 50 ohms, then ZΩ
en renormalized to the charl matrix containing the charove equation, the result
se see the section in thesenalysis.
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hnical NotesFinite Element Method
lementationtware Modulesarametersigher Order Modesrequency PointsenormalizedMatricesZMatrixYMatrixCharacteristic Impedances
eembeddingvanescent Modes and Sarameters
ivation of Wave Equationiated Fieldses to Nodes Conversion
Renormalized SMatrices
Before a structure’s generalized Smatrix can be used in a higto compute the reflection and transmission of signals, it mustpriate impedance. For example, if a generalized Smatrix hasohms, it can be used to compute reflection and transmissionnormalized to 50 ohms.
To renormalize a generalized Smatrix to a specific impedanca unique impedance matrix Z, associated with the structure d
where:
• S is the nxn generalized Smatrix.• I is an nxn identity matrix.• Z0 is a diagonal matrix having the characteristic impedanc
diagonal value.
The renormalized Smatrix is then calculated from the uniquerelationship:
where:
• Z is the structure’s unique impedance matrix.• ZΩ and YΩ are diagonal matrices with the desired impeda
diagonal values. For example, if the matrix is being renormwould have diagonal values of 50.
Visualize the generalized Smatrix as an Smatrix that has beacteristic impedances of the structure. Therefore, if a diagonaacteristic impedances of the structure is used as ZΩ in the abwould be the generalized Smatrix again.
For information about renormalized terminal Smatrices, pleaTechnical Notes entitled TerminalBased Models for Circuit A
Z Z0 I S–( ) 1–I S+( ) Z0=
SΩ YΩ Z ZΩ–( ) Z ZΩ+( ) 1–ZΩ=
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follows:
e (Z0) of each port as a
nce matrix, Z.
and Y) associated with a
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lementationtware Modulesarametersigher Order Modesrequency PointsenormalizedMatricesZMatrixYMatrixCharacteristic Impedances
eembeddingvanescent Modes and Sarameters
ivation of Wave Equationiated Fieldses to Nodes Conversion
ZMatrix
The impedance matrix, Z, is calculated from the Smatrix as
where:
• S is the nxn generalized Smatrix.• I is an nxn identity matrix.• Z0 is a diagonal matrix having the characteristic impedanc
diagonal value.YMatrix
The admittance matrix, Y, is simply the inverse of the impeda
To display the unique impedance and admittance matrices (Zstructure, use the Matrix Data Post Processor.
Z Z0 I S–( ) 1–I S+( ) Z0=
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of each port in order tocharacteristic impedance of
sed in the renormalization
of power (P) and current (I):
ted fields. The power pass
around the port:
the current of interest is thate, I+. In integrating around
to each and uses the aver
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hnical NotesFinite Element Method
lementationtware Modulesarametersigher Order Modesrequency PointsenormalizedMatricesZMatrixYMatrixCharacteristic Impedances
eembeddingvanescent Modes and Sarameters
ivation of Wave Equationiated Fieldses to Nodes Conversion
Characteristic Impedances
Ansoft HFSS needs to calculate the characteristic impedancecompute a renormalized Smatrix. The system computes theeach port in three ways — as Zpi, Zpv, and Zvi impedances.
You have the option of specifying which impedance is to be ucalculations.
PI Impedance
The Zpi impedance is the impedance calculated from values
The power and current are computed directly from the simulaing 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
While the net current computed in this way will be near zero,flowing into the structure, I, or that flowing out of the structurthe port, the system keeps a running total of the contributionsage of the two in the computation of impedances.
ZpiP
I I•=
P E Hds×s°=
I H dl•l°=
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of power (P) and voltage (V):
simulated fields. The powerge is computed as follows:
impedance line — which isine for a port, select the twoThe system has no way ofs you define an impedance
lower boundaries to a port’spproaches a port’s actual
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hnical NotesFinite Element Method
lementationtware Modulesarametersigher Order Modesrequency PointsenormalizedMatricesZMatrixYMatrixCharacteristic Impedances
eembeddingvanescent Modes and Sarameters
ivation of Wave Equationiated Fieldses to Nodes Conversion
PV Impedance
The Zpv impedance is the impedance calculated from values
where the power and voltage are computed directly from theis computed in the same way as the Zpi impedance. The volta
The path over which the system integrates is referred to as thedefined when the ports are set up. To define the impedance lpoints across which the maximum voltage difference occurs.knowing where the maximum voltage difference will be unlesline.
VI Impedance
The Zvi impedance is given by:
For TEM waves, the Zpi and Zpv impedances form upper andactual characteristic impedance. Therefore, the value of Zvi aimpedance for TEM waves.
ZpvV V•
P=
V E dl•l°=
Zvi ZpiZpv=
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a structure to be cut in half),g an impedance multiplier.ow the use of symmetry
e in two, only onehalf of theputed by the system. There
nce multiplier of 2 must be
e in two, only onehalf of theial is present. Therefore,computed impedances thatof 0.5 must be specified in
tructure is modeled, you
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hnical NotesFinite Element Method
lementationtware Modulesarametersigher Order Modesrequency PointsenormalizedMatricesZMatrixYMatrixCharacteristic Impedances
eembeddingvanescent Modes and Sarameters
ivation of Wave Equationiated Fieldses to Nodes Conversion
Impedance Multiplier
If a symmetry plane has been defined (allowing the model ofthe impedance computations have to be adjusted by specifyinThe need for this multiplier can be understood by looking at haffects the computation of Zpv.
In cases where a perfect E plane of symmetry splits a structurvoltage differential and onehalf of the power flow can be comfore, since the Zpv impedance is given by:
the computed value is onehalf the desired value. An impedaspecified in such cases.
In cases where a perfect H plane of symmetry splits a structurpower flow is seen by the system but the full voltage differentstructures split in half with perfect H symmetry planes result inare twice those for the full structure. An impedance multipliersuch cases.
If multiple symmetry planes are used or if only a wedge of a smust adjust the impedance multiplier accordingly.
ZpvV V•
P=
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modal Smatrix, you mustZvi.
t’s actual impedance and
iate to use the Zpi
uides), Zpv impedance is the
r to the following:
della, “Someharacteristic ImpedanceTheory and Techniques,
“FrequencyDependentational URSI Symposiumpp. 335 B/13.
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hnical NotesFinite Element Method
lementationtware Modulesarametersigher Order Modesrequency PointsenormalizedMatricesZMatrixYMatrixCharacteristic Impedances
eembeddingvanescent Modes and Sarameters
ivation of Wave Equationiated Fieldses to Nodes Conversion
Choice of Impedance
When the system is instructed to renormalize the generalizedspecify which value to use in the computations — Zpi, Zpv, or
• For TEM waves, the Zvi impedance converges on the porshould be used.
• When modeling microstrips, it is sometimes more approprimpedance.
• For slottype structures (such as finline or coplanar wavegmost appropriate.
For more information on the computation of impedances, refe
Bruno Bianco, Luigi Panini, Mauro Parodi, and Sandro RiConsiderations about the Frequency Dependence of the Cof Uniform Microstrips,” IEEE Transactions on Microwavevol. MTT26 No. 3, March 1978.
Edward F. Kuester, David C. Chang, and Leonard Lewin,Definitions of Microstrip Characteristic Impedance,” Internon Electromagnetic Waves, Munich, 2629 August 1980,
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d from) a port, the Smatrixg relationship:
ion line at port i.en added to or removeda length of transmission line
wave module.
n of power implies that:
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hnical NotesFinite Element Method
lementationtware Modulesarametersigher Order Modesrequency Pointsenormalized SMatriceseembeddingvanescent Modes andParameters
ivation of Wave Equationiated Fieldses to Nodes Conversion
Deembedding
If a uniform length of transmission line is added to (or removeof the modified structure can be calculated using the followin
where:
• is a diagonal matrix with the following entries:
• γ=α + jβ is the complex propagation constant, where:• α is the attenuation constant of the wave.• β is the propagation constant of the uniform transmiss
• li is the length of the uniform transmission line that has befrom the structure at port i. A positive value indicates thathas been removed from the structure.
The value of γ for each port is automatically calculated by the
Evanescent Modes and SParameters
In structures containing only propagating modes, conservatio
S' eγl S e
γl=
eγl
eγ1l1 0 0
0 eγ2l2 0
0 0 eγ3l3
Sij2
j 1=
n
1≤
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However, in structures containing evanescent (nonpropagating) modes, this relationship
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modes is zero, which introlculation of Sparameters —for evanescent modes
ed with these modes on the
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lementationware Modulesrameters
vation of Waveuationaxwell’s Equationshasor NotationssumptionsConductivityDielectric Loss TangentMagnetic Loss TangentDefinition of FreespaceWave Number
iated Fieldses to Nodes Conversion
is not necessarily true. The propagation constant, β, of theseduces a term for the reactive power in the structure into the caincreasing the structure’s apparent power. The Sparametersshould be considered to represent the field strength associatport(s).
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equation:
ld.e angular frequency, 2πf.
es that the electric field is a
nd z:
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lementationware Modulesrameters
vation of Waveuationaxwell’s Equationshasor NotationssumptionsConductivityDielectric Loss TangentMagnetic Loss TangentDefinition of FreespaceWave Number
iated Fieldses to Nodes Conversion
Derivation of Wave EquationThe wave and abc3d modules both solve the following wave
where:
• E(x,y,z) is a phasor representing an oscillating electric fie• k0 is the free space wave number, , where ω is th• is the complex relative permeability.• is the complex relative permittivity.
The difference between wave and abc3d is that wave assumtraveling wave with this form:
while abc3d assumes that the phasor E is a function of x, y, a
1µr E∇×
∇× k02εrE– 0=
ω µ0ε0µrεr
E x y z t, , ,( ) ℜ E x y,( )ejωt γz–[ ]=
E x y z t, , ,( ) ℜ E x y z, ,( )ejωt[ ]=
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Maxwell’s equations, which
permittivity.
ex permeability.
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hnical NotesFinite Element Method
lementationtware Modulesarametersivation of Waveuationaxwell’s Equationshasor NotationssumptionsConductivityDielectric LossTangent
Magnetic LossTangent
Definition of FreespaceWave Number
iated Fieldses to Nodes Conversion
Maxwell’s Equations
The wave equation solved by wave and abc3d is derived fromin their timedomain form are:
where:
• E(t) is the electric field intensity.• D(t) is the electric flux density, εE(t), and ε is the complex• H(t) is the magnetic field intensity.• B(t) is the magnetic flux density, µH(t), and µ is the compl• J(t) is the current density, σE(t).• ρ is the charge density.
H t( )∇× J t( )t∂
∂D t( )+=
∇ B t( )• 0=
E t( )∇×t∂
∂ Bt( )–=
∇ D t( )• ρ=
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ating at the same frequency,r by e(jωt—γz) (in wave ).
e:
ionships:
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vation of Waveuationaxwell’s Equationshasor NotationssumptionsConductivityDielectric LossTangent
Magnetic LossTangent
Definition of FreespaceWave Number
iated Fieldses to Nodes Conversion
Phasor Notation
Because all timevarying electromagnetic quantities are oscillthey can be treated as phasors multiplied by ejωt (in abc3d ) o
In the general case which abc3d solves, the equations becom
By factoring out the quantity ejωt and using the following relat
Maxwell’s equations in phasor form reduce to:
where B, H, E, and D are phasors in the frequency domain.
Hejωt∇× Je
jωt
t∂∂ De
jωt+=
Eejωt∇×
t∂∂
Bejωt
–=
Dejωt∇• ρe
jωt=
Bejωt∇• 0=
t∂∂
Eejωt
jωEejωt
=
t∂∂
Hejωt
jωHejωt
=
H∇× J jωD+=
E∇× jωB–=
D∇• ρ=
B∇• 0=
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nd , Maxwell’s
complex permeability, and ε
magnitude and phase or as
J σE=
ating atfrequency
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lementationtware Modulesarametersivation of Waveuationaxwell’s Equationshasor NotationssumptionsConductivityDielectric Loss TangentMagnetic Loss TangentDefinition of FreespaceWave Number
iated Fieldses to Nodes Conversion
Now, using the constitutive relationships , , aequations in phasor form become:
where H and E are phasors in the frequency domain, µ is theis the complex permittivity.
H and E are stored as phasors, which can be visualized as aa complex quantity.
B µH= D εE=
H∇× jω ε σjω+
E jωεcE= =
∇ H× jωεE=
E∇× jωµH–=
εE∇• ρ=
µH∇• 0=
Phasor rotan angularof ω
Fm cos(ωt+θ)
ωt
θ
Real
Imaginary
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tion in terms of E to obtain:
produce:
l either as a bulk materialce boundary condition
ary condition is problemver the conductor is muchse, the unknowns within thea smaller matrix and a
to the skin depth, the bulklution. With this assumption,
oundary condition isThe boundary conditionboundary manager.
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lementationware Modulesrameters
vation of Waveuationaxwell’s Equationshasor NotationssumptionsConductivityDielectric Loss Tangent
Magnetic Loss Tangent
Definition of FreespaceWave Number
iated Fieldses to Nodes Conversion
Assumptions
To generate the final field equation, place H in the equa
Then, substitute this expression for H in the equation to
Conductivity
Ansoft HFSS is capable of including conductivity in the modeloss factor, similar to dielectric loss tangent, or as an impedanapplied to the outer surfaces of the object.
The choice between bulk material loss and the surface bounddependent. The boundary condition should be applied whenethicker than the skin depth at the solution frequency. In this caconductor are not included in the unknown vector, resulting infaster solution. However, if the conductor is not thick relativematerial conductivity must be used to arrive at an accurate sothe wave equation reduces to:
Note: The choice between the bulk material loss and the bmade using Solve Inside in the Material Manager.can also be applied when setting boundaries in the
E∇×
H1
jωµ E∇×–=
H∇×
1jωµ E∇×–
∇× jωεE=
1–jωµ E∇×
∇× jωε σ+( )E=
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e permittivity, , is complex:
, the complex
hat µr is complex.
, the complex relative
εr
δen εr'' εr'⁄=
r'' µr'⁄ )
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hnical NotesFinite Element Method
lementationtware Modulesarametersivation of Waveuationaxwell’s Equationshasor NotationssumptionsConductivityDielectric LossTangent
Magnetic LossTangent
Definition of Freespace Wave Number
iated Fieldses to Nodes Conversion
Dielectric Loss Tangent
Dielectric losses can be modeled by assuming that the relativ
Expressed in terms of the dielectric (electric) loss tangent,relative permittivity, becomes:
Magnetic Loss Tangent
Losses in magnetic materials can be modeled by assuming t
Expressed in terms of the magnetic loss tangent,permeability becomes:
ε ε' jε''–=
taεr
εr εr' jεr' δetan–=
µr µr' jµr''–=
δmtan µ(=
µr µr' 1jµr''
µr'–
µr' 1 j δmtan–( )= =
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uation being solved can be
fined as , the
e.
ve equation takes the form:
ed using finite elements,
t Method for Modelingand Techniques, vol.
k02 ω2µ0ε0=
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vation of Waveuationiated Fieldsntenna PropertiesPolarizationAxial RatioPolarization RatioBeam AreaDirectivityGainRadiated PowerAccepted PowerRadiation EfficiencyMax Ues to Nodes Conversion
Definition of Freespace Wave Number
Using the relationships and , the wave eqplaced in this form:
Now, if the freespace phase constant (or wave number) is deabove reduces to:
which is the equation that the modules abc3d and wave solv
When materials with finite conductivities are included, the wa
where:
• Z0 is the characteristic impedance of free space:
For more information on the wave equation and how it is solvrefer to the following:
Zoltan J. Cendes and JinFa Lee, “The Transfinite ElemenMMIC Devices,” IEEE Transactions on Microwave Theory36, pp. 16391649, Dec. 1988.
ε ε0εr= µ µ0µr=
1µr E∇×
∇× ω2µ0ε0εrE– 0=
1µr E∇×
∇× k02εrE– 0=
1µr E∇×
∇× k02εr jk0Z0σ–( )E– 0=
Z0
µ0
ε0=
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the radiation surface aree. This space is typicallyegion. The nearfield regiond E(x,y,z) external to the
(1)
tial to the surface.l to the surface.l to the surface.
points on the surface.
nction can be approximated
normal∇G ) sd
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lementationtware Modulesarametersivation of Waveuationiated Fieldsntenna PropertiesPolarizationAxial RatioPolarization RatioBeam AreaDirectivityGainRadiated PowerAccepted PowerRadiation EfficiencyMax Ues to Nodes Conversion
Radiated FieldsWhen calculating radiation fields, the values of the fields overused to compute the fields in the space surrounding the devicsplit into two regions — the nearfield region and the farfield ris the region closest to the source. In general, the electric fielregion bounded by a closed surface may be written as:
where:
• s represents the radiation boundary surfaces.• j is the imaginary unit, .• ω is the angular frequency, 2πf.• µ0 is the relative permeability of the free space.• Htan is the component of the magnetic field that is tangen• Enormal is the component of the electric field that is norma• Etan is the component of the electric field that is tangentia• G is the free space Green’s function, given by:
where:• k0 is the free space wave number, .• r and represent, respectively, field points and source
In the far field where r>>r' (and usually r>>λ0), the Green’s fuby:
E x y z, ,( ) jωµ0H tan G Etan ∇G× E+ +(s=
1–
Ge
jk0 r r '––
r r '–=
ω µ0ε0r'
Ge
jk0r–ejk0r r'⋅
r≈
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elds that result have an r
is a key feature of far fields.ion:
FSS uses the generalecify the radial coordinate r.
s from the radiating struc
usly discussed farfieldpoints in the farfield region.
rested in the antenna propwing antenna properties:
calculate the fields in ahich the fields are calcu
ution region, then thettered fields dependingthe radius is outside thee scattered fields.
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When this form of G is used in the farfield calculations, the fidependence in the form of:
This r dependence is characteristic of a spherical wave, whichThe far field is a spherical TEM wave with the following equat
where η0 is the intrinsic impedance of free space.
When you choose Radiation/Compute/Near Field , Ansoft Hexpressions given in (eq. 1). For this command, you must spBecause it can be used to compute fields at an arbitrary radiuture, this command can be useful in EMC applications.
When you choose Radiation/Compute/Far Field , the previoapproximations are used, and the result is valid only for field
Antenna Properties
Generally, when dealing with radiated fields, you are also inteerties of the radiated bodies. Ansoft HFSS calculates the follo
Note: If you are using Radiation/Compute/Near Field toproblem containing an incident wave, the radius at wlated is very important. If the radius is within the solfields calculated are either the total fields or the scaupon which is selected using Data/Edit Sources . Ifsolution region, then the fields calculated are only th
beam area accepted powerdirectivity radiation efficiencygain maximum intensityradiated power polarizationaxial ratio polarization ratio
ejkr–
r
E η0H r×=
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electric and magnetic fieldtric field vector traces out aion state of the field.
n ellipse. The wave is saidld traces out an ellipse. As a
case the wave is circularlyt states, left and right, distinme of these figures, oration is out of the screen.
generates to a straight line.
ld, two independent compocriptions:
ircular polarization
elliptical polarization
ω
ω
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hnical NotesFinite Element Method
lementationtware Modulesarametersivation of Waveuationiated Fieldsntenna PropertiesPolarizationAxial RatioPolarization RatioBeam AreaDirectivityGainRadiated PowerAccepted PowerRadiation EfficiencyMax Ues to Nodes Conversion
Polarization
At each aspect angle in the far field of a radiating source, thevectors lie in a fixed plane. Over time, the instantaneous elecfigure or shape in this plane. This figure defines the polarizat
In general, this figure is an ellipse and is called the polarizatioto be elliptically polarized when the instantaneous electric fiespecial case, the polarization ellipse may be a circle, in whichpolarized. Elliptical and circular polarization have two differenguished by the sense of rotation of the electric field vector. Sostates, are shown below. In each case the direction of propag
A second special case occurs when the polarization ellipse deIn this case the wave is linearly polarized.
To completely describe the polarization state of a radiated fienents are required. Ansoft HFSS supports three types of des
• Spherical polar• Ludwig3• Circular
Circular
Elliptical
Righthand circular polarization Lefthand c
Righthand elliptical polarization Lefthand
ω
ω
ω is the rotation radian frequency.
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the electric field phasorference coordinate system.
ss polarization. In particular,mponents that are typicallyn, the field may be written
l coordinate system.
zed fields, a meaningfulthis description, the field
s of the polarization ellipse.
ularly polarized and an axialzed.
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lementationtware Modulesarametersivation of Waveuationiated Fieldsntenna PropertiesPolarizationAxial RatioPolarization RatioBeam AreaDirectivityGainRadiated PowerAccepted PowerRadiation EfficiencyMax Ues to Nodes Conversion
Spherical Polar
The most fundamental description is spherical polar, which isresolved in the directions of unit theta and phi vectors of the reIn this description, the field may be written as E = (Eθ, Eφ).
Ludwig3
Arthur C. Ludwig wrote a classic paper on the definition of crohis third definition is often used since it describes the field comeasured on a farfield antenna test range. Using his definitioas E = (Ex, Ey) where:
and phi is the usual azimuthal angle in the reference spherica
Circular
For antennas designed to receive or transmit circularly polaridescription is in terms of pure left and right circular states. Inmay be written as E = (ER, EL) where:
Axial Ratio
Axial ratio is defined as the ratio of the major to the minor axiBy definition, the axial ratio has the following range:
Here an axial ratio of one indicates that the electric field is circratio of infinity indicates that the electric field is linearly polari
Ex Eθ φcos Eφ φsin–=
Ey Eθ φsin Eφ φcos+=
ER1
2 Eθ jEφ–( )=
EL1
2 Eθ jEφ+( )=
1 AR ∞≤ ≤
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iven field vector at a point inof two specified orthogo
as been resolved.” [Ref.]each selected aspect angle:
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lementationtware Modulesarametersivation of Waveuationiated Fieldsntenna PropertiesPolarizationAxial RatioPolarization RatioBeam AreaDirectivityGainRadiated PowerAccepted PowerRadiation EfficiencyMax Ues to Nodes Conversion
Polarization Ratio
The IEEE defines the (complex) polarization ratio as, “For a gspace, the (magnitude of the) ratio of the complex amplitudesnally polarized field vectors into which the given field vector hAnsoft HFSS computes the following six polarization ratios at
Circular LHCP⁄ EL
ER=
Circular RHCP⁄ ER
EL=
Spherical Phi⁄ EφEθ=
Spherical Theta⁄ EθEφ=
Ludwig 3 X⁄ Ex
Ey=
Ludwig 3 Y⁄ Ey
Ex=
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m the antenna would flow ifr the beam area. It is mea
dian.
of the radiation intensity inirections. It is a dimension
ection specified.
. However, if the antennae radiation efficiency of the
nel is the directivity in
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lementationtware Modulesarametersivation of Waveuationiated Fieldsntenna PropertiesPolarizationAxial RatioPolarization RatioBeam AreaDirectivityGainRadiated PowerAccepted PowerRadiation EfficiencyMax Ues to Nodes Conversion
Beam Area
This is the solid angle through which all power emanating frothe maximum radiation intensity is constant for all angles ovesured in steradians and is represented by:
where:
• ΩA is the solid angle in steradians.• Prad is the radiated power in watts.• Umax is the maximum radiation intensity in watts per stera
Directivity
The directivity of an antenna is commonly defined as the ratioa given direction to the radiation intensity averaged over all dless quantity represented by:
where:
• U is the radiation intensity in watts per steradian in the dir• Prad is the radiated power in watts.
For a lossless antenna, the directivity will be equal to the gainhas inherent losses, the directivity is related to the gain by thantenna.
Note: The directivity given in the Far Field Parameters pathe direction of maximum radiation intensity, Umax.
ΩA
Prad
Umax=
directivity4πUPrad=
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a given direction to the net. In Ansoft HFSS the conrt.
SS:
ection specified.
nt for lossless antennas.ntenna. If the radiation effi
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Gain
Gain is 4π times the ratio of an antenna’s radiation intensity inpower accepted by the antenna from a connected transmitternected transmitter is a surface that has been defined as a po
The following equation is used to calculate gain in Ansoft HF
where:
• U is the radiation intensity in watts per steradian in the dir• Pacc is the accepted power in watts entering the antenna.
Gain can be confused with directivity, since they are equivaleGain is related to directivity by the radiation efficiency of the aciency is 100%, they are equal.
gain 4π UPacc=
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tts) exiting a radiating
omputed as:
of the 3D model.
nds on the accuracy of Eed radiated power may
utput signal at the port, ad.
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vation of Waveuationiated Fieldsntenna PropertiesPolarizationAxial RatioPolarization RatioBeam AreaDirectivityGainRadiated PowerAccepted PowerRadiation EfficiencyMax Ues to Nodes Conversion
Radiated Power
Radiated power is the amount of timeaveraged power (in waantenna structure through a radiation boundary.
For a general radiating structure in HFSS, radiated power is c
where:
• Prad is the radiated power in watts.• ℜ is the real part of a complex number.• s represents the radiation boundary surfaces.• E is the radiated electric field.• H* is the conjugate of H.• ds is the local radiationboundary unit normal directed out
Note: The accuracy of the computed radiated power depeand H. In some cases it is possible that the computdeviate slightly from the actual radiated power.
Because the radiated power is calculated from the oport must be defined for this quantity to be displaye
Prad ℜ E H∗× ds⋅s=
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n watts) entering a radiatingith a single port, acceptedatch loss at the port plane.
computed as:
3D HFSS model.
taining a single propagating
/Edit Sources command.ut renormalization)
input signal at the port, ad.
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Accepted Power
The accepted power is the amount of timeaveraged power (iantenna structure through one or more ports. For antennas wpower is a measure of the incident power reduced by the mism
For a general radiating structure in HFSS, accepted power is
where:
• Pacc is the accepted power in watts.• ℜ is the real part of a complex number.• A is the union of all port boundaries in the model.• E is the radiated electric field.• H* is the conjugate of H.• ds is the local portboundary unit normal directed into the
For the simple case of an antenna with one lossless port conmode, the above expression reduces to:
where:
• a is the complex modal excitation specified using the Data• s11 is the singleentry generalized scattering matrix (witho
computed by HFSS.
Note: Because the accepted power is calculated from theport must be defined for this quantity to be displaye
Pacc ℜ E H∗ ds⋅×A=
Pacc a2
1 s112
–( )=
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BNr
Tf
TI
EMS
e accepted power given by:
power, a port is required for
na per unit solid angle.in which it has the maxi
red in watts per steradian
.
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lementationtware Modulesarametersivation of Waveuationiated Fieldses to Nodesnversionackgroundodal Voltages and CurentserminalBased Modelsor Circuit Analysiserminal Characteristicmpedance Matrixxporting Terminal Satrices for FullWavepice Analysis
Radiation Efficiency
The radiation efficiency is the ratio of the radiated power to th
where:
• Prad is the radiated power in watts.• Pacc is the accepted power in watts.
Since the radiation efficiency is calculated from the acceptedthe radiation efficiency.
Max U
The radiation intensity, U, is the power radiated from an antenAnsoft HFSS calculates the radiation intensity in the directionmum value. The maximum intensity of the radiation is measuand is calculated by:
where:
• U(θ,φ) is the radiation intensity in watts per steradian.• E is the magnitude of the Efield.• η0 is the intrinsic impedance of free space — 376.7 ohms• r is the distance from the antenna, in meters.
ePrad
Pacc=
U θ φ( , )E 2
η0r2=
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RadModCo
BNr
TeC
TeIm
EMS
, computed using AnsoftudoSparameters used in
of the incident and reflectedlly and physically rigorous,
odes can propagate simultaission lines on printed circuit
lectrical connectors. Foruasitransverse electromagmeasure a single mode. In
several modes
simulation of terminal curermine the needed linear
the transverse componentsrm crosssection can be
kward traveling modaltic field patterns. Note thathere, while and have
ussion’s purpose, assumexponential terms in the
x y,( )eγz
x y,( )eγz
en hn
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vation of Waveuationiated Fieldses to Nodesnversionackgroundodal Voltages and CurentsrminalBased Models forircuit Analysisrminal Characteristicpedance Matrix
xporting Terminal Satrices for FullWavepice Analysis
Modes to Nodes ConversionThis section describes the conversion of modal SparametersHFSS’s fullwave field solver, into nodal or voltagebased psecircuit theory.
The Smatrix solutions in HFSS express their results in termspowers of waveguide modes. This description is mathematicabut does not lend itself to problems where several different mneously. Examples of such situations include coupled transmboards, multiconductor cables, and many common types of ethese types of structures, which generally support multiple, qnetic (TEM) modes of propagation, it is difficult to excite andany practical, or laboratorylike, measurement situation:
• measurements (e.g. of voltage) contain contributions from• the applied stimuli excite several modes simultaneously
Nodal or “terminal” support was added to HFSS to enable therents and voltages directly, without requiring engineers to detcombinations of waveguide modes.
Background
Recall from electromagnetic field theory that the solutions forof the electric and magnetic fields in a waveguide with a unifowritten as:
Here the and are the intensities of the forward and bacwaves, and and are the transverse electric and magnethe and are considered to be dimensionless quantitiesthe normal units for electric and magnetic fields. For this discthat each port is defined at z=0; this permits the removal of eabove equations.
Et anen x y,( )eγz–
n bnen
n+=
Ht anhn x y,( )eγz–
n bnhn
n–=
an bnen hn
an bn
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ing a number of differentEM modes of propagation,
, ..., , and define N diflas:
, when you define the termi
t of closed contours of inte
ration contours separately.e chosen voltage contours
odal intensities and the voltanded as a linear combina
2 CN
l
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Nodal Voltages and Currents
It is possible to define a set of voltages for a port by establishintegration paths across the port. If a port supports N quasiTone can set up N different open contours of integration, ,ferent voltages, , , ..., according to the integral formu
These integration paths are set in the 3D Boundary Managernal voltage lines on a port.
Alternatively, one can define a set of currents using a segration :
Ansoft HFSS does not require you to set up the current integThe software automatically infers the current contours from thusing a power conservation relationship.
Each definition produces a set of relationships between the mages or currents. Since the total transverse fields can be exption of modes, we can rewrite the previous equations as:
C1 Cv1 v2 vN
vk Et ld⋅Ck
–=
i k Dk
ik Ht ld⋅Dk
°=
vk an en x y,( ) ld⋅Ck
n– bn en x y,( ) d⋅
Ck
n–=
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M modes, there are NxN
l
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nical NotesFinite Element Method
lementationware Modulesrameters
vation of Waveuationiated Fieldses to Nodesnversionackgroundodal Voltages and CurentsrminalBased Modelsr Circuit Analysisrminal Characteristicpedance Matrix
xporting Terminal Satrices for FullWavepice Analysis
To simplify, introduce the terms:
Now vk and ik become:
Since there are N possible voltage definitions and N quasiTE
ik an hn x y,( ) ld⋅Dk
n bn hn x y,( ) d⋅
Dk
n–=
tkn en x y,( ) ld⋅Ck
–=
ukn hn x y,( ) ld⋅Dk
=
vk antkn
n bntkn
n+=
i k anukn
n bnukn
n–=
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dal intensities:
sent these transformations.peres.
ifferent modal field patternsach terminal contour line.S’s modestoterminals con
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lementationtware Modulesarametersivation of Waveuationiated Fieldses to Nodesnversionackgroundodal Voltages and CurentserminalBased Modelsor Circuit Analysiserminal Characteristicmpedance Matrixxporting Terminal Satrices for FullWavepice Analysis
matrices that relate the nodal voltages and currents to the mo
Now introduce the matrices and to repreNote that the units of are volts, while the units of are am
In summary, the 3D Post Processor could be used to load dand manually perform the integrals defined previously over eThis would be a tedious and errorprone process, which HFSversion feature alleviates.
vk[ ] tkn[ ] an[ ] bn[ ]+( )=
ik[ ] ukn[ ] an[ ] bn[ ]–( )=
T tkn[ ]= U ukn[ ]=T U
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ompute terminalbasedirst consider the admittancehe vector of terminal cur
. It is known that:
are related by ,:
n then be solved for in terms
d from the above expres
d impedance matrix:
nd impedance matrices intoe impedance must be
d to convert the terminal
v vk[ ]=
b Sa=
Zk
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TerminalBased Models for Circuit Analysis
It is possible to use the transformations developed above to cadmittance (Y), impedance (Z) and pseudoSmatrices ( ). Fcase. A relationship of the form is needed that gives trents as a function of the vector of terminal voltages
and:
It is also known that the incident and scattered modal powerswhere is the modal Smatrix computed by HFSS. Therefore
Here denotes an identity matrix of the same size as . caof by eliminating the incident wave variable :
The terminalbased (or nodal) admittance matrix is identifiesion as
A similar relationship can be developed for the terminalbase
It is also possible to convert the terminalbased admittance aa terminalbased “pseudoSmatrix” . To do this, a referencdefined for each terminal . Then standard formulas are useimpedance matrix into the terminal Smatrix:
Spi Yv=
i i k[ ]=
v T a b+( )=
i U a b–( )=
S
v T a Sa+( ) T I S+( )a= =
i U a Sa–( ) U I S–( )a= =
I S iv a
a I S+( ) 1–T
1–v=
i U I S–( ) I S+( ) 1–T
1–v=
Y
Y U I S–( ) I S+( ) 1–T
1–=
Z T I S+( ) I S–( ) 1–U
1–=
Spk
Z
Sp Zref1 2⁄
Z Z+ ref( ) 1–Z Zref–( )Zref
1 2⁄–=
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e reference impedances .
t and reflected pseudo
the reflected pseudowavetts1/2. The terminal voltagess:
ave no associated propagans of several modes, whichpable of performing deplished by first deembedon back to a terminalbased
Zk
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Here is a diagonal matrix whose entries are th
The terminal Smatrix relates the intensities of the incidenwaves at the terminals:
These pseudowaves are defined by:
Here is the incident pseudowave at terminal , and isat the same terminal. Note that the units of and are waand currents can also be written in terms of the pseudowave
Unlike true waveguide modes, the pseudowaves and htion constant. The pseudowaves represent linear combinatiomay all have differing propagation constants. HFSS is still caembedding on the terminalbased Smatrix, but this is accomding the modal Smatrix and then performing the transformatiSmatrix.
Zref diag Zk( )=
Sp
βk[ ] Sp αk[ ]=
αk12Zk
1 2⁄–vk Zkik+( )=
βk12Zk
1 2⁄–vk Zkik–( )=
αk k βkαk βk
vk Zk1 2⁄ αk βk+( )=
i k Zk1 2⁄– αk βk–( )=
αk βk
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ne multimode waveguidee contains internal sources,de port and strike the imped
in some sense, a reflectionhere it is interpreted as an
ad returning as an incident(the minus sign is due to
rents with their modal expan
ishes. Corresponding to this
ltimode waveguide port.mbination of modal wavessigners wishing to controluide, it can be shown thatthen easy to synthesize a
i
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Terminal Characteristic Impedance Matrix
Consider the situation illustrated below: A 3D structure with oport is loaded by an Nport matrix impedance Z. The structurwhich generate outgoing waves that exit through the waveguiance . If does not match the impedance of the waveguidewill occur from this load and will return to the 3D structure, wincident wave.
An optimal choice of Z will prevent any reflections from the lowave a. Note that the circuit equations at the load arethe sense of the current .) By replacing the voltages and cursions, the voltage becomes:
Rearranging this to isolate and , it is determined that:
Now notice that if we select ; the incident wave vancondition is an “optimal” choice for the impedance :
is the terminal characteristic impedance matrix for the muThis value of impedance will completely absorb any linear coleaving the port. As such, it should be of interest to circuit dereflections. In the important, special case of a lossless waveg
is a realvalued, symmetric impedance matrix. It isnetwork of resistors with the specified matrix impedance.
Z Z
Z
+
_
v
i
a
b3D structure
v Z–=i
v T a b+( ) ZU a b–( )–= =
a b
T ZU+( )a T ZU–( )b+ 0=
T ZU= aZ0 Z
Z0 TU1–
=
Z0
Z0 TTT
=
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hing frequencies above oneparameter data calculated
l Spice software, transformaon the Sparameter data.
ysis.
Spice or HSPICE, HFSSinput and output terminals,riginal device analyzed inansferred to circuit simulafrequency responses ishen terminal Sparameterre, HFSS creates a newthat includes the same fre
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Exporting Terminal SMatrices for FullWave Spice Ana
Significant coupling that occurs in closely located circuits reacGHz can now be included in circuit analysis. When terminal Sby Ansoft HFSS is exported to PSpice, HSPICE, and Maxweltion data between voltages and currents is constructed basedTherefore fullwave effects can be modeled in the circuit anal
When terminal Sparameter data is exported from HFSS to Pcreates an equivalent circuit, represented by a black box withthat includes the same frequency response behavior as the oHFSS. Frequency domain information from Ansoft HFSS is trtors that operate in the time domain. Broadband matching ofensured while the fullwave solution accuracy is preserved. Wdata is exported from HFSS to Ansoft’s Maxwell Spice softwa“fullwave Nport subcircuit”, also represented by a black boxquency response behavior as the original device.
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