Hfss Basics

边界与端口设置

电子科技大学

贾宝富 博士

HFSS Boundary List

„ Perfect E and Perfect H/Natural

„ Ideal Electrically or Magnetically Conducting Boundaries „ ‘Natural’ denotes Perfect E ‘cancellation’ behavior

„ Finite Conductivity

„ Lossy Electrically Conducting Boundary, with user-provided conductivity

and permeability

„ Impedance

„ Used for simulating ‘thin film resistor’ materials, with user-provided

resistance and reactance in Ω/Square

„ Radiation

„ An ‘absorbing boundary condition,’ used at the periphery of a project in

which radiation is expected such as an antenna structure

„ Symmetry

„ A boundary which enables modeling of only a sub-section of a structure

in which field symmetry behavior is assured.

„ “Perfect E” and “Perfect H” subcategories „ Master and Slave

„ ‘Linked’ boundary conditions for unit-cell studies of infinitely replicating

HFSS Boundary Descriptions: Perfect E and Perfect

H/Natural

Ø

Parameters: None

„

Perfect E

is a perfect electrical conductor*

„ Forces E-field perpendicular to the surface „ Represent metal surfaces, ground planes,

ideal cavity walls, etc.

„

Perfect H

is a perfect magnetic conductor

„ Forces H-field perpendicular to surface,

E-field tangential

„ Does not exist in the real world, but

represents useful boundary constraint for modeling

„

Natural

denotes effect of

Perfect H applied

on top of some other (e.g. Perfect E)

boundary

„ ‘Deletes’ the Perfect E condition,

permitting but not requiring tangential electrical fields.

„ Opens a ‘hole’ in the Perfect E plane

Perfect E Boundary* Perfect H Boundary ‘Natural’ Boundary

lar

perpendicu

E

r

continuous

E

r

parallel

E

r

*NOTE: When you define a solid object as a ‘perf_conductor’ in the Material Setup, a Perfect E boundary condition is applied to its exterior surfaces!!

Perfect H for 2D Aperture (I)

Ø

Monopole Over a Ground

plane

Perfect H

Perfect H Surface Interior to the

Problem Space Behaves Like an

Infinitely Thin 2D Aperture

Perfect H for 2D Aperture (II)

Ø

Small Hole Can be “Cut” in infinitely Thin Septum

Between the Upper and Lower Guide Using a Perfect

H Surface at the Hole

HFSS Boundary Descriptions: Finite Conductivity

Ø

Parameters: Conductivity and

Permeability

„

Finite Conductivity is a lossy

electrical conductor

„ E-field forced perpendicular, as with

Perfect E

„ However, surface impedance takes

into account resistive and reactive surface losses

„

User inputs conductivity (in

siemens/meter) and relative

permeability (unitless)

„

Used for non-ideal conductor

analysis*

Finite Conductivity Boundary

g

attenuatin

lar

perpendicu

E

r

,

*NOTE: When you define a solid object as a non-ideal metal (e.g. copper, aluminum) in the Material Setup module, and it is set to ‘Solve Surface’, a Finite Conductivity

boundary is automatically applied to its exterior faces!!

HFSS Boundary Descriptions: Impedance

Ø

Parameters: Resistance and

Reactance, ohms/square (

Ω/χ)

„

Impedance

boundary is a direct,

user-defined surface impedance

„ Use to represent thin film resistors „ Use to represent reactive loads

„ Reactance will NOT vary with

frequency, so does not represent a lumped ‘capacitor’ or ‘inductor’ over a frequency band.

„

Calculate required impedance from

desired lumped value, width, and length

„ Length (in direction of current flow) ÷

Width = number of ‘squares’

„ Impedance per square = Desired

Lumped Impedance ÷ number of squares

EXAMPLE: Resistor in Wilkenson Power Divider

Resistor is 3.5 mils long 4 m

(in direction of flow) and ils wide. Desired lumped value is 35 ohms.

square N R R N lumped sheet 40 / 875 . 35 875 . 0 4 5 . 3 Ω = = = = =

HFSS Boundary Descriptions: Radiation

Ø

Parameters: None

„ A Radiation boundary is an absorbing

boundary condition, used to mimic

continued propagation beyond the boundary plane

„ Absorption is achieved via a

second-order impedance calculation

„ Boundary should be constructed correctly

for proper absorption

„ Distance: For strong radiators (e.g.

antennas) no closer than λ/4 to any structure. For weak radiators (e.g. a bent circuit trace) no closer than λ/10 to any structure

„ Orientation: The radiation boundary

absorbs best when incident energy flow is normal to its surface

„ Shape: The boundary must be

concave to all incident fields from

within the modeled space

Note boundary does not follow ‘break’ at tail end of horn. Doing so would result in a convex surface to interior

radiation.

Boundary is λ/4 away from horn aperture in all directions.

HFSS Boundary Descriptions: Radiation, cont.

„

Radiation boundary absorption profile

vs. incidence angle is shown at left

„ Note that absorption falls off

significantly as incidence exceeds 40 degrees from normal

„ Any incident energy not absorbed is

reflected back into the model, altering the resulting field solution!

„

Implication:

For steered-beam arrays,

the standard radiation boundary may

be insufficient for proper analysis.

„

Solution:

Use a

Perfectly Matched

Layer (PML)

construction instead.

„ Incorporation of PMLs is covered in

the Advanced HFSS training course. Details available upon request.

-100 -80 -60 -40 -20 0 20 Reflection Coefficient (dB) 0 10 20 30 40 50 60 theta (deg) Reflection Coefficient (dB) 70 80 90

Reflection of Radiation Boundary in dB, vs. Angle of Incidence relative to boundary normal (i.e. for normal incidence, θ = 0)

E_TM

HFSS Boundary Descriptions: Symmetry

Ø

Parameters: Type (Perfect E or Perfect H)

„

Symmetry

boundaries permit modeling of

only a fraction of the entire structure under

analysis

„

Two Symmetry Options:

„ Perfect E : E-fields are perpendicular to the

symmetry surface

„ Perfect H : E-fields are tangential to the

symmetry surface

„

Symmetry boundaries also have further

implications to the Boundary Manager and

Fields Post Processing

„ Existence of a Symmetry Boundary will

prompt ‘Port Impedance Multiplier’ verification

„ Existence of a symmetry boundary allows for

near- and far-field calculation of the ‘entire’ structure

Conductive edges, 4 sides

This rectangular waveguide contains a symmetric propagating mode, which could be modeled using half the volume

vertically....

Perfect E Symmetry (top)

...or horizontally.

Perfect H Symmetry (left side)

HFSS Boundary Descriptions: Symmetry, cont.

„

Geometric symmetry does not

necessarily imply field symmetry

for higher-order modes

„

Symmetry boundaries can act as

mode filters

„ As shown at left, the next higher

propagating waveguide mode is

not symmetric about the vertical center plane of the waveguide

„ Therefore one symmetry case is

valid, while the other is not!

„

Implication:

Use caution when

using symmetry to assure that real

behavior in the device is not filtered

out by your boundary conditions!!

Perfect E Symmetry (top)

Perfect H Symmetry (right side) TE20 Mode in WR90

Properly represented with Perfect E Symmetry

Mode can not occur properly with Perfect H Symmetry

HFSS Boundary Descriptions: Master/Slave Boundaries

Ø

Parameters: Coordinate system,

master/slave pairing, and phasing

„

Master

and

Slave

boundaries are used

to model a unit cell of a repeating

structure

„ Also referred to as linked boundaries „ Master and Slave boundaries are

always paired: one master to one slave

„ The fields on the slave surface are

constrained to be identical to those on the master surface, with a phase shift.

„

Constraints:

„ The master and slave surfaces must be

of identical shapes and sizes

„ A coordinate system must be identified

on the master and slave boundary to identify point-to-point correspondence

Unit Cell Model of End-Fire Waveguide Array

WG Port

(bottom) Ground Plane Perfectly Matched Layer

(top) Slave Boundary Master Boundary Origin V-axis U-axis

HFSS Source List

„

Port

„ Most Commonly Used Source. Its use results in S-parameter output

from HFSS.

„ Two Subcategories: ‘Standard’ Ports and ‘Gap Source’ Ports „ Apply to Surface(s) of solids or to sheet objects

„

Incident Wave

„ Used for RCS or Propagation Studies (e.g. Frequency-Selective

Surfaces)

„ Results must be post-processed in Fields Module; no S-parameters

can be provided

„ Applies to entire volume of modeled space

„

Voltage Drop or Current Source

„ ‘Ideal’ voltage or current excitations

„ Apply to Surface(s) of solids or to sheet objects

„

Magnetic Bias

„ Internal H Field Bias for nonreciprocal (ferrite) material problems „ Applies to entire solid object representing ferrite material

HFSS Source Descriptions: Port (II)

Ø

Parameters: Mode Count, Calibration,

Impedance, Polarization, Imp. Multiplier

„

A

port

is an aperture through which

guided electromagnetic field energy is

injected into a 3D HFSS model. There

are two types:

„ Standard Ports: The aperture is solved

using a 2D eigensolution which locates all requested propagating modes

„ Characteristic impedance is

calculated from the 2D solution

„ Impedance and Calibration Lines

provide further control

„ Gap Source Ports: Approximated field

excitation is placed on the gap source port surface

„ Characteristic impedance is

provided by the user during setup

EXAMPLE STANDARD PORTS

Impedance, Calibration and Polarization Lines

Ø

Impedance line, calibration line and polarization line are optional in port

setup.

Ø

They are located in the port and have a starting point and an end point.

I,C, and/or P Line

Port = cross section

of waveguide

Impedance Line

Ø

Without impedance line, HFSS computes port impedance from power

and current: Z

Ø

With impedance line, a voltage can be defined: ∫ E

dl .

Two more port impedances result: Z

and Z

.

Calibration Line

Ø

Calibration lines remove 180

phase ambiguity.

This helps to obtain correct phase in S

and S

.

Polarization Line

Ø

Imposes polarization in case of ambiguity,

e.g. in square or circular guides with degenerate modes.

Port = cross section

of square waveguide

HFSS Source Descriptions: Incident Wave (II)

Parameters: Poynting Vector,

E-field Magnitude and Vector

„

Used for radar cross section (RCS)

scattering problems.

„

Defined by Poynting Vector

(direction of propagation) and

E-field magnitude and orientation

„ Poynting and E-field vectors must

be orthogonal.

„ Multiple plane waves can be

created for the same project.

„

If no ‘ports’ are present in the

model, S-parameter output is not

provided

„ Analysis data obtained by

post-processing on the Fields using the Field Calculator, or by generating RCS Patterns

In the above example, a plane incident wave is directed at a solid made from dielectrics, to view the resultant scattering fields.

HFSS Source Descriptions: Voltage Drop and Current Source (I)

Voltage Drop

HFSS Source Descriptions: Voltage Drop and Current

Source (II)

Example Voltage Drop (between trace and ground) Example Current

Source (along trace or across gap)

Ø

Parameters: Direction and Magnitude

„

A

voltage drop

would be used to

excite a voltage between two metal

structures (e.g. a trace and a ground)

„

A

current source

would be used to

excite a current along a trace, or

across a gap (e.g. across a slot

antenna)

„

Both are ‘ideal’ source excitations,

without impedance definitions

„ No S-Parameter Output

„

User applies condition to a 2D or 3D

object created in the geometry

„ Vector identifying the direction of the

voltage drop or the direction of the current flow is also required

Sources/Boundaries and Eigenmode Solutions

Ø

An

Eigenmode

solution is a direct solution of the resonant

modes of a closed structure

Ø

As a result, some of the sources and boundaries discussed so

far are

not

available for an Eigenmode project. These are:

„

All Excitation Sources:

„ Ports

„ Voltage Drop and Current Sources „ Magnetic Bias

„ Incident Waves

„

The only unavailable boundary type is:

„ Radiation Boundary

„ A Perfectly Matched Layer construction is possible as a

HFSS Source Descriptions: Magnetic Bias

Parameters: Magnitude and

Direction

or

Externally Provided

„

The

magnetic bias

source is used

only to provide internal biasing

H-field values for models containing

nonreciprocal (ferrite) materials.

„ Bias may be uniform field (enter

parameters directly in HFSS)...

„ Parameters are direction and

magnitude of the field

„ ...or bias may be non-uniform

(imported from external

Magnetostatic solution package)

„ Ansoft’s 3D EM Field

Simulator provides this

analysis and output

„

Apply source to selected 3D solid

HFSS Ports: A Detailed Look

Ø

The Port Solution provides the excitation for the 3D FEM

Analysis. Therefore, knowing how to properly define and

create a port is

paramount

to obtaining an accurate analysis.

Ø

Incorrect Port Assignments can cause errors due to...

„

...Excitation of the wrong mode structure

...Bisection by conductive boundary

„

...Unconsidered additional propagating modes

...Improper Port Impedance

„

...Improper Propagation Constants

„

...Differing phase references at multiple ports

„

...Insufficient spacing for attenuation of modes in cutoff

„

...Inability to converge scattering behavior because too many

modes are requested

Ø

Since Port Assignment is so important, the following slides will

HFSS Port Selection: Standard or Gap Source?

„

When would you choose to

use a

Gap Source

Port over a

Standard Port?

„ When the model has

tightly-spaced lines

„ When a port reference

location is difficult to

determine using a Standard port

„ When you’d like to use a

voltage gap, but want S-parameter output

HFSS Ports: Sizing

Ø

A port is an

aperture through which a

guided-wave

mode of some kind

propagates

„

For transmission line structures entirely

enclosed in metal, port size is merely the

waveguide interior carrying the guided

fields

„ Rectangular, Circular, Elliptical, Ridged,

Double-Ridged Waveguide

„ Coaxial cable, coaxial waveguide,

square-ax, Enclosed microstrip or suspended stripline

„

For unbalanced or non-enclosed lines,

however, field propagation in the air

around the structure must also be included

„ Parallel Wires or Strips

„ Stripline, Microstrip, Suspended Stripline „ Slotline, Coplanar Waveguide, etc.

A Coaxial Port Assignment

A Microstrip Port Assignment (includes air above substrate)

HFSS Ports: Sizing, cont.

Ø

The port solver only understands

conductive

boundaries on its borders

„

Electric conductors may be

finite

or

perfect

(including Perfect E symmetry)

„

Perfect H symmetry also understood

Radiation boundaries around the

periphery of the port do

not alter the port

edge termination!!

Ø

Result: Moving the port edges too close

to the circuitry for open waveguide

structures (microstrip, stripline, CPW,

etc.) will allow coupling from the trace

circuitry to the port walls!

„

This causes an incorrect modal solution,

which will suffer an immediate

discontinuity as the energy is injected past

the port into the model volume

Port too narrow (fields couple to side walls)

Port too Short (fields couple to top wall)

HFSS Ports: Sizing Handbook I

Microstrip Port Sizing Guidelines

„

Assume width of microstrip trace is

w

Assume height of substrate dielectric

is

h

Ø

Port Height Guidelines

„

Between 6

h and 10h

„ Tend towards upper limit as dielectric

constant drops and more fields exist in air rather than substrate

„ Bottom edge of port coplanar with the

upper face of ground plane

„ (If real structure is enclosed lower

than this guideline, model the real structure!)

Ø

Port Width Guidelines

„

10

w, for microstrip profiles with w

≥

h

5

w, or on the order of 3h to 4h, for

microstrip profiles with

w < h

w h 6h to 10h 10w, w ≥h or 5w (3h to 4h), w < h

Note: Port sizing guidelines are not inviolable rules true in all cases. For example, if meeting the height and width requirements outlined result in a rectangular aperture bigger than λ/2 on one dimension, the substrate and trace may be ignored in favor of a waveguide mode. When in doubt, build a simple ports-only model and test.

HFSS Ports: Sizing Handbook II

Ø

Stripline Port Sizing Guidelines

„

Assume width of stripline trace is

w

„

Assume height of substrate dielectric is

h

Ø

Port Height Guidelines

„

Extend from upper to lower groundplane,

h

Ø

Port Width Guidelines

„

8

w, for microstrip profiles with w

≥

h

5

w, or on the order of 3h to 4h, for

microstrip profiles with

w < h

Ø

Boundary Note: Can also make side

walls of port

Perfect H boundaries

w h

8w, w ≥h or

HFSS Ports: Sizing Handbook III

Ø

Slotline Port Guidelines

„

Assume slot width is

g

„

Assume dielectric height is

h

Ø

Port Height:

„

Should be at least 4

h, or 4g (larger)

Remember to include air below the

substrate as well as above!

„ If ground plane is present, port should

terminate at ground plane

Ø

Port Width:

„

Should contain at least 3

g to either side

of slot, or 7

g total minimum

„

Port boundary

must intersect both side

ground planes, or they will ‘float’ and

become signal conductors relative to

outline ‘ground’

g

Approx 7g minimum

h

HFSS Ports: Sizing Handbook IV

CPW Port Guidelines

„

Assume slot width is

g

„

Assume dielectric height is

h

Assume center strip width is

s

Ø

Port Height:

„

Should be at least 4

h, or 4g (larger)

„

Remember to include air below the substrate

as well as above!

„ If ground plane is present, port should

terminate at ground plane

Ø

Port Width:

„

Should contain 3-5

g

or

3-5

s of the side

grounds, whichever is larger

„ Total about 10g or 10s

„

Port outline

must

intersect side grounds, or

they will ‘float’ and become additional signal

Larger of approx. 10g or 10s

s h

Larger of 4h or 4g

CPW Wave Ports: Starting Recommendations

Wave Port Size

The standard recommendation for most CPW wave ports is a rectangular aperture Port widthshould be no less than 3 x the overall CPW width, or 3 x (2g+ w)

Port heightshould be no less than 4 x the dielectric height, or 4h

Wave Port Location

The wave port should be centered horizontally on the CPW trace

If the port is on GCPW, the port bottom edge should lie on the substrate bottom ground plane

If the port is on ungrounded CPW, the port height should be roughly centered on the CPW metal layer

Wave Port Restrictions

As with all wave ports, there must be only onesurface normal exposed to the field volume Port should be on exterior model face, or cappedby a perfect conductor block if internal The wave port outline must contact the side grounds (all CPWs) and bottom ground (GCPW)

The wave port size should notexceed lambda/2 in any dimension, to avoid permitting a rectangular waveguide modal excitation 3 (2g + w) w h 4h minimum g Grounded CPW 3 (2g + w) w h 4h minimum g Ungrounded CPW

HFSS Ports: Sizing Handbook V; Gap Source

Ports

Ø

Gap Source

ports behave differently from

Standard Ports

„

Any port edge not in contact with metal structure

or

another port assumed to be a Perfect H

conductor

Ø

Gap Source Port Sizing (microstrip example):

„

“Strip-like”:

[RECOMMENDED] No larger than

necessary to connect the trace width to the

ground

„

“Wave-like”: No larger than 4 times the strip

width and 3 times the substrate height

„ The Perfect H walls allow size to be smaller than

a standard port would be

„ However, in most cases the strip-like application

should be as or more accurate

Ø

Further details regarding Gap Source Port

sizing available as a separate presentation

Perfect H Perfect H Perfect E Perfect E Perfect H Perfect H Perfect E Perfect H

HFSS Port Selection Example: Parallel Traces

Spaced by 8 or more times Trace Width

Inputs sufficiently isolated that no coupling behavior should occur Sufficient room for Wave port apertures around each trace

Use Wave Ports as shown

Spaced by 4 – 8 times Trace Width

Inputs still fairly isolated, little to no coupling behavior should occur

Insufficient room for Wave port apertures around each trace without clipping fringing fields

Use Lumped Ports as shown

Spaced by less than 4 times Trace Width

Traces close enough to exhibit coupling

Evenand Odd modes possible; N modes total for N

conductors and one ground reference [odd mode shown at right]

Lumped Ports from trace to ground neglect coupling behavior and are no longer appropriate

Use multi-mode Wave Port

Terminalline assignments can permit extraction of S-parameters referenced to each ‘trace’

HFSS Ports: Spacing from Discontinuities

„

Structure interior to the modeled volume may

create and reflect

non-propagating modes

„ These modes attenuate rapidly as they travel

along the transmission line

„

If the port is spaced too close to a discontinuity

causing this effect, the improper solution will be

obtained

„ A port is a ‘matched load’ as seen from the

model, but only for the modes it has been designed to handle

„ Therefore, unsolved modes incident upon it are

reflected back into the model, altering the field solution

„

Remedy: Space your port far enough from

discontinuities to prevent non-propagating mode

incidence

„ Spacing should be on order of port size, not

wavelength dependent

Port Extension

HFSS Ports: Single-Direction Propagation

„

Standard ports must be

defined so that only

one

face

can radiate energy into

the model

„ Gap Source Ports have no

such restriction

„

Position Standard Ports on

the

exterior

of the geometry

(one face on

background

) or

provide a port

cap

.

„ Cap should be the same

dimensions as the port aperture, be a 3D solid object, and be defined as a perfect conductor in the Material Setup module

Port on Exterior Face of Model

Port Inside Modeled Air Volume; Back side covered with Solid Cap

HFSS Ports: Mode Count

„

Ports

should

solve for all propagating modes

„ Ignoring a mode which does propagate will result

in incorrect S-parameters, by neglecting mode-to-mode conversion which could occur at

discontinuities

„

However, requesting

too many

modes in the full

solution also negatively impacts analysis

„ Modes in cutoff are more difficult to calculate;

S-parameters for interactions between propagating and non-propagating modes may not converge well

„

What if I don’t know how many modes exist?

„ Build a simple model of a transmission line only,

or run your model in “Ports Only” mode, and check!

„ You can alter the mode count before running the

full solution.

„

Degenerate

mode ordering is controlled with

calibration lines (see next slide)

Circular waveguide, showing two orthogonal TE11 modes and TM01 mode (radial with Z-component). Neglecting the TM01 mode from your solution would cause incorrect results.

HFSS Ports: Degenerate Modes

„

Degenerate

modes have identical impedance,

propagation constants

„

Port solver will arbitrarily pick one of them to

be ‘

mode(n)’ and the other to be ‘mode(n+1)’

„ Thus, mode-to-mode S-parameters may be

referenced incorrectly

„

To enforce numbering, use a

calibration line

and

polarize

the first mode to the line

„

OR

, introduce a dielectric change to slightly

perturb the mode solution and separate the

degenerate modes

„ Example: A dielectric bar only slightly higher in

permittivity than the surrounding medium will concentrate the E-fields between parallel wires, forcing the differential mode to be dominant

„ If dielectric change is very small (approx. 0.001

or less), impedance impact of perturbation is negligible

For parallel lines, a virtual object between them aids mode ordering. Note virtual object need not extend entire length of line to help at port. In circular or square waveguide, use the calibration line to force (polarize) the mode numbering of the two degenerate TE11 modes. This is also useful because without a polarization orientation, the two modes may be rotated to an arbitrary angle inside circular WG.

HFSS Ports: Impedance Definitions

„

HFSS provides port characteristic

impedances calculated using the

power-current

definition (Z

)

„ Incident power is known excitation quantity „ Port solver integrates H-field around port

boundary to calculate current flow

„

For many transmission line types, the

power-voltage

or

voltage-current

definition is

preferred

„ Slot line, CPW: Z_pv preferred „ TEM lines: Z_vi preferred

„

HFSS can provide these characteristic

impedance values, as long as an

impedance

line

is identified

„ The impedance line defines the line along

which the E-field is integrated to obtain a voltage

„ Often it can be identical to the calibration line

For a Coax, the impedance line extends radially from the center to outer conductor (or vice versa). Integrating the E-field along the radius of the coaxial dielectric provides the voltage difference.

In many instances, the impedance and

HFSS Ports: Phase Calibration

„

A second purpose of the

calibration line

is to

control the port

phase references

„ The 2D port eigensolver finds propagating

modes on each port independently

„ The zero degree phase reference is chosen at

a point of maximum E-field intensity on the port face.

„ This occurs twice, with 180 degrees

separation, for each 360 degree cycle

„ Therefore the possibility exists for the software

to select inconsistent phase references from port to port, resulting in S-parameter errors

„ All port-to-port S-parameter phases, e.g.

S21, will be off by 180 degrees

„

Solution: The

calibration line

defines the

preferred direction for the zero degree

reference on each port.

Which of the above field orientations is the zero degree phase reference? Calibration Line defines...

HFSS Ports: Impedance Multiplier

„

When

symmetry is used in a model, the

automatic Z

and impedance

line-dependant Z

and Z

calculations will

be incorrect, since the entire port

aperture is not represented.

„ Split the model with a Perfect E

symmetry case, and the impedance is halved.

„ Split the model with a Perfect H

symmetry case, and the impedance is doubled.

„

The port

impedance multiplier is just a

renormalizing factor, used to obtain the

correct impedance results regardless of

the symmetry case used.

„

The

impedance multiplier is applied to

all ports, and is set during the

assignment of any port in the model.

Whole Rectangular WG (No Symmetry) Impedance Mult = 1.0 Half Rectangular WG (Perfect E Symmetry) Impedance Mult = 2.0 Half Rectangular WG (Perfect H Symmetry) Impedance Mult = 0.5

...and for Quarter Rectangular WG? (Both Perfect E and H Symmetry)