smr.razavizadeh@ieee.org
[Dispersion diagram: CST Microwave Studio]
One of the most important reports in metamaterial topics is “Dispersion curve”, which
determines the band-gap behavior of the band-gap structures. In this article two examples that
could clearly show the right way of getting this report including two extra Matlab code scripts, are
presented.
I hope this article could be helpful for the students who are love Electromagnetic!
“Thanks to Dr Miguel Navarro-Cía Imperial College London for his valuable comment”.
Dispersion Diagram Example 1: Slow Wave Helix
General Description
This example demonstrates an eigenmode calculation using periodic boundaries in z-direction
and a very meaningful example for showing how we can get the dispersion diagram in CST MWS
for periodic structures. This example already is available in example folder of CST MWS in path
of:
“C:\Program Files (x86)\CST STUDIO SUITE 2013\Examples\MWS\Eigenmode\TET\Slow
Wave\helix.cst”
Structure description:
Inner conductor (Fig.1(a)) is a helical metal ribbon which has been hold by three dielectric
bricks (d) inside a PEC cylindrical hole (b) as outer conductor.
Fig.1 (a) inner conductor
(b) outer conductor
(c) vacuum
Frequency Range
The frequency range starts at DC and ends at 20 GHz.
Background Material:
Fig.2 Background material setting.
1
(d) dielectric holder
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[Dispersion diagram: CST Microwave Studio]
Mesh setting:
Click on the objects in navigation tree and R. Click on the object; then choose the “Local mesh
properties” and set it as the following suggestions:
Vacuum: Meshgroup1
The automesh option is disabled for the two
inlets which are not parallel to the coordinate
axes.
Other elements: Meshgroup2
The automesh option is enabled.
(Inner&outer conductor, holders)
Fig.3 mesh setting of the model components.
Boundary Conditions
The boundary conditions for x and y planes are set to “electric” or Et=0, which is equivalent
to PEC , while for the two boundary conditions in z-direction have been defined as “periodic” in
order to model an infinite extent of the proposed helical inner conductor coax unit ell.
A parameter “phase” is assigned to the periodic boundary, so that the phase shift can be used in
a parameter sweep, for measuring the frequency dispersive behavior of the present structure.
Fig.4 Boundary conditions and phase shift setting of the one dimensional periodic structure.
*A userdefined watch is already available and may serve as a template for building custom
watches. The userdefined parameter sweep watch adds the group velocity, phase velocity and
dispersion plots of the first mode to the navigation tree in the folder '1D Results'. Also, the power
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[Dispersion diagram: CST Microwave Studio]
flow and Pierce Impedance can be seen in the folder '1D Results'. In parameter sweep window,
select "Edit" to view or modify the source code.
>
Fig.5 the user-defined VBA codes of the important reports of the simulation as dispersion
curve, phase & group velocity and Pierce impedance.
But I preferred that the following method for setting the dispersion diagram.
Post Processing:
The dispersion diagram, which is a graph based on “frequency” vs. spatial phase variations of
the predefined parameter of “phase”, could obtained by the following step by step process:
1) Recall the TBP and choose the “2D and 3D Field Results>3D Eigenmode Result”
Fig.6 the outline of defining the postprocessing setup of dispersion plot in CST
MWS.
2) For obtaining the dispersion data of the modes start one by one from the first mode, then
after finishing simulation operation export the data in “txt” format(available in “Table”
folder in Navigation tree) for our future postprocessing in Matlab environment.
3) Evaluate
Solver Setup
Whenever the “Eigenmode Solver” is started, a specific number of the lowest resonance
frequencies of the structure are calculated. Since only the fundamental mode is of interest, the
number of modes is reduced to “1”, and the JDM eigenmode solver is chosen, which is faster for
the given example.
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Fig.7 Eigenmode Solver setting for calculating the dispersion characteristics of the proposed
periodic structure.
After the parameter sweep has been selected from the Eigenmode Solver's dialog, a new
sequence is added, and the parameter "phase" is chosen to be swept from 0 to 180 degrees in 19
steps for an equal steps.
Fig.8 Parametric sweep setup for phase shift parameter of “phase” in the space periodicity
direction.
Start:
If you set lower sweep limit at zero, the machine temporary would interrupt the process and
alert we the parameter is zero, don’t care about it and click Ok to continue the operation.
>>
Fig.10
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Results
During the run process the real time dispersion results is accessible from the Table folder, as a
plot based on frequency vs. phase parameter, as following:
(a)
(b)
Fig. 11 The dispersion diagram based on (a) TBP result template, and (b) VBA
userdefined.
As shown in Fig. 11, all the data are the same but in Fig.11(a) the horizontal axis is the phase.
Export the data of the Mode “1”
>>
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[Dispersion diagram: CST Microwave Studio]
>>
Fig. 12 extraction of table dispersion data
Open via excel
Open a new Excel file and open from it your txt file of “Mode1.txt”:
>
>
>
>
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[Dispersion diagram: CST Microwave Studio]
Fig. 13 The preparation of the dispersion text data file for producing the
compatible *.csv file
We should keep just the phase and value column and eliminate others, then edit the title of the
value column as “Frequency(Mode1)”, finally save as a CSV file format, as shown in Fig. 13. We
should repeat the simulation for remained modes frequencies as presented in Fig. 14, and extract
the favorite data for updating the previous “mode1.csv” data file.
Fig. 15 How we can update the calculation for the higher mode?
Fig. 16 all the dispersion data gathered in one csv file for plotting the final
dispersion diagram for first forth modes of the proposed 1D-periodic structure.
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[Dispersion diagram: CST Microwave Studio]
Mtalab Code for plotting the dispersion diagram
For publishing all of your plot result, especially in IEEE publications, we should import our
data into Matlab environment, as a perfect engineering tool to sketch various plots in a unique plot
in a high resolution view. Here we present a brief quick guide about it.
1)Run Matlab
2) import the dispersion CSV file.
>
4) disable the header and text box.
>>
5) Run the plot m.file(see the appendix)
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[Dispersion diagram: CST Microwave Studio]
Final result:
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Mode1
Mode2
Mode3
Mode4
18
Frequency(GHz)
16
14
12
10
BAND GAP
8
6
4
2
0
0
20
40
60
80
100
Phase(Deg)
120
140
160
180
Fig. the final sketch of the dispersion diagram obtained by a MATLAB plotting
program (Appendix).
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Example2. 2D-periodic structure dispersion diagram
In this section, dispersion characterization of surface waves propagating on a mushroom-like
periodic structure is investigated using CST MWS.
Mushroom-like EBG structures
Wavenumber k is an important parameter to describe the propagation property of
electromagnetic waves. In a lossless case, the phase constant is β = k.
Usually, β is a function of frequency ω. Once the phase constant is obtained, the phase velocity
(vp) and group velocity (vg) can be derived:
=
=
(2-1)
Furthermore, the field distribution can also be determined, such as the field variation in a
transverse direction. For a plane wave in free space, the relation between β and ω is a linear
function:
= = (2-2)
For surface waves propagating in an EBG structure, it is usually difficult to give an explicit
expression for the wavenumber k. One has to either solve an eigen-value equation or perform a full
wave simulation to determine the wavenumber. It is important to point out that the solution of an
eigen-value equation may not be unique.
In another words, there may exist several different propagation constants at the same
frequency. Each one is known as a specific mode with its own phase velocity, group velocity, and
field distribution. The relation between β and ω is often plotted out and referred to as the
dispersion diagram.
For a periodic structure such as the EBG, the field distribution of a surface wave is also
periodic with a proper phase delay determined by the wavenumber β and periodicity p.
Thus, each surface wave mode can be decomposed into an infinite series of space harmonic waves.
, , = ∑&'
!' , !"β#$ %
, β% = β% + (2-3)
Here, we assume the periodic and propagation direction is the x direction. Although these space
harmonics have different phase velocities, they share the same group velocity. Furthermore, these
space harmonics cannot exist individually because each single harmonic does not satisfy the
boundary conditions of the periodic structure. Only their summation satisfies the boundary
conditions. Thus, they are considered to be the same mode.
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Another important observation from (2-3) is that the dispersion curve β% is periodic along the
β- axis with a periodicity of 2π/p. Therefore, we only need to plot the dispersion relation within
one single period, namely, 0 ≤ β% ≤ 2,/.% , which is known as the Brillouin zone. This
concept can be easily extended to two-dimensional periodic structures, where the Brillouin zone
becomes a two-dimensional square area:
0 ≤ β% ≤ 2,/.% , 0 ≤ β/ ≤ 2,/./
Figure 2-1 shows the dispersion diagram of the mushroom-like EBG structure.
The vertical axis shows the frequency and the horizontal axis represents the
values of the transverse wavenumbers (βx, βy). Three specific points are: Γ, X
and M.
Let us consider a plane wave impinging on a periodic surface. For surface waves, the angle of
incidence is equal to 90° (parallel with the surface). For this value, the wave propagation on a
periodic structure cannot be investigated using the plane wave response. The exact position of pass
bands and band gaps in the frequency spectrum can be obtained only by the dispersion relation of
surface waves (i.e. calculating the resonant frequencies of eigenmodes) along the contour of the
irreducible Brillouin zone.
Brillouin Zone in CST Microwave Studio
In dispersion diagram of a 2D-periodic structure, we are faced with an special definition of the
frequency dispersion effects of the surface wave phase variations in three main directions usually
labeled as Γ to X, X to M and M to Γ, which we know as “Brillouin Zone”, as shown in Fig..
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[Dispersion diagram: CST Microwave Studio]
Fig.2-2. Brillouin zone definition of a 2D-periodic structure (in a full period of
px=py=d, we have 2π radian phase change).
Now we should describe three directions phase constant variations with a 2D-plot. In this plot
the independent variable of β-axis is a set of ordered pairs of:
0
.ℎ2 _4
.ℎ2 _
Then, the first set has been called as “Γ to X”:
0
.ℎ2 _: 06180 9.4
.ℎ2 _ = 0
And the second set, “X to M”:
0
.ℎ2 _ = 180 9.4
.ℎ2 _:06180 9
Finally, the last set is “M to Γ”:
0
.ℎ2 _: 06180 9.4
; "=>266ℎ ?>@.ℎ2 A B 62 CC .6 "
.ℎ2 _: 06180 9
Fig.2-3. example of “dispersion diagram” of a 2D-periodic structure, based on
Brillouin Zone definition (in this example the step of sweep is 30deg).
This simulator uses finite integration technique (FIT). The unit cell of the structure under
investigation has to be drawn with periodic boundary conditions applied in the appropriate
directions. The phase shift is changed along the boundary of the irreducible Brillouin zone and
frequencies of eigenmodes are obtained in each step. Because of slow-wave behavior of surface
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[Dispersion diagram: CST Microwave Studio]
waves, dispersion curves are calculated only in the region under the light line, as shown in Fig.
2.1. Band gaps occur in frequency intervals, where no dispersion curves in the slow-wave region
are present.
Model and Dimension
The most popular mushroom-like EBG structure which is used here as an example of 2Dperiodic structures, is composed of a grounded FR4 substrate with periodic square patch above,
which the middle of the patch is connected with a via to the ground plane.
Parameter List of our example:
Substrate: (xs,ys) 15×15mm2 , FR4(εr=4.5, loss tan. 0.025,
hs=1.6mm)
Patch size: (xs-gap, ys-gap) 14×14mm2 gap=1mm,
Via: radius Rp=0.5mm
Conductor: PEC, thickness(tm):0.035mm
Fig.2-4.
Fig. 2.5 shows the unit cell setup of a periodic structure with the proposed mushroom-like
square lattice, for dispersion analysis in CST MWS. The main difference between the
computational models in CST MWS and HFSS consists in the following fact. HFSS uses a
perfectly matched layer (PML) to represent an infinite air layer above the unit cell, while in CST
MWS, open boundaries are not allowed in combination with periodic walls, but only perfect
electric conductor (PEC) or perfect magnetic conductor (PMC) boundary conditions can be
applied.
(a)
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(b)
Fig.2-5. The setup of (a) Background material and (b) six boundaries of the unit cell.
The Eigenmode Solver of CST Microwave Studio does not support open boundaries. In this
case, the boundary condition of side walls of the unit cell can be considered periodic.
After many computer simulations with this program and comparing the obtained results with
analytical and experimental considerations, a practical rule for the correct surface wave dispersion
diagram computation was stated for background material and boundary setting.
Therefore, the key parameter in the computer model is then the height of the air space above of
the dielectric substrate to emulate the free space over the structure (in “Background Setting” this
space is “Upper Z-distance”), an airbox with the height of about many times of the dielectric slab
thickness has to be placed over the unit cell. Based on numerous computer simulations and by
comparing the obtained results with analytical and experimental considerations, the correct choice
of approximately ten times the substrate thickness was established, “Upper Z-distance ≅10·hs”[1].
The boundaries at the top and the bottom of the model should be defined as electric conductor
or PEC (Et=0).
If the EBG surfaces are used as ground plane of antenna, like monopole this Figure 3(b) shows
the frequency response of transmission coefficient S21, both TM and TE surface waves measured
by using a pair of small monopole antennas oriented normally (TM mode) and parallel (TE mode)
to the EBG surface.
Frequency Setting: 0-8GHz
Mesh setting:
For not complicated structures like this poject mesh setting is not required but if you want more
and more precise results you can pay time for more dense mesh size!
Phase Shift Setting
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[Dispersion diagram: CST Microwave Studio]
“Phase_x” and “phase_y” are the phase variation of the slow surface wave on the periodic
structure, in x- and y-directions, respectively. We know those parameters have a obvious relation
with correspondent wave numbers as the following:
.ℎ2 _ ≡ % ∙ 2.C . @@C@6A96ℎ x-axis
.ℎ2 _ ≡ % ∙ 2
.ℎ2 _ ≡ / ∙ 2.C . @@C@6A96ℎ x-axis
.ℎ2 _ ≡ / ∙ 2
For symmetry with respect to the x- and y-axis’s, we define both of xs and ys with a new
parameter of “d”.
Γ to X plot:
Sweep the “phase_x (βxd)” from 0 t0 180, with sample: 19 and “phase_y (βyd)=0”
X to M plot:
Sweep the “phase_y (βyd)” from 0 t0 180, with sample: 19 and “phase_x (βxd)=180”
Eigenmode Solver Setting
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[Dispersion diagram: CST Microwave Studio]
Start:
Don’t care about alert zero value for phase_x parameter.
Γ to X plot:
Sweep the “phase_x (βxd)” from 0 t0 180, with sample: 19 and “phase_y (βyd)=0” , the
simulation takes not much more time, about 25 minutes by a typical laptop.
The figure shows phase variation linearity below 2.2GHz for Mode#1 and for surface wave
propagation phase constant with respect to x-direction.
Export the data and save as “Mode1_Gamma2X.txt”:
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[Dispersion diagram: CST Microwave Studio]
The last value of phase_x is 180, that’s perfect! So go to Eigenmode>para sweep and delete
phase_x and define phase_y from 0 to 180, then open “Result Template…” and do “Evaluate” for
reset for new plot, “X to M”, then click the “Start”:
During the new parameter simulation, still the previous plot would be shown, don’t care about
it! The real time new results are available in table format:
Updating the phase_y sweep data:
Recall the table plot, set the phase_y as, “abscissa” and phase_x at 180 as fixed x direction
phase which is required for X to M plot,
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It’s obvious that the upper limit of Frequency Band Gap region is about 2.13GHz.
Export:
Nav. Tree>R.Click on Table and choose the table. Then change phase_x t0 180 and apply,
finally export:
>>
M to Γ Plot:
For convenience to extract this step data we first clean all the stored data of previous sweep by
drawing for example a Brick then apply it to message of deletion would be shown, click Ok. In the
next step delete that object and now the project is free of any data.
The last plot of Brillouin zone is based on diagonal propagation phase shift which is equivalent
to kx=ky or phase_x=phase_y. So we can satisfy this by setting value of phase_y with phase_x, in
parameters list window.
>>
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Now recall Eigenmode solver and sweep par. And delete the phase_y and define phae_x: 180
to 0deg, with 19 samples, this constraint the calculations performed with descent variation, which
is required for M to Γ plot.
Start the Sweep!
Results:
Export the results of the last parametric calculations as “M2Gamma_Mode1.txt”.
Preparing the final Dispersion diagram CSV file
In this section we should create the CSV file format of three correspondent text files of Γto X,
X to M and M to Γ, respectively.
Γto X
X to M
M to Γ
In this step we should prepared a new excel book which contains in a serial format all the three
calculated data set, compatible with the Brillouin zone definition, as following:
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[Dispersion diagram: CST Microwave Studio]
(a)
(b)
For plotting the desirable Brillouin zone diagram with MATLAB, We insert a new sequential
numbering column, as shown in Fig. (b). It is important to determine boarder row numbers of the
CSV data file, this issue has shown in the following table.
Plot Name
Γ to X
X to M
M to Γ
Row range in main CSV data file
1-19
19-38
38-57
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Matlab Plotting:
Now we can import the data files in Matlab command and use the m.file script as mentioned in
Appendix (b), the resulted plot after a few editing illustrated below:
(a)
(b)
Fig. (a) the sketch of dispersion curve based on just the first mode calculation using Matlab, (b) example of “dispersion
diagram” of a 2D-periodic structure, based on Brillouin Zone definition (in this example the step of sweep is 30deg)..
Since the exact Band Gap region should be determined after obtaining some higher modes and
plotting the entire dispersion plots with repeating the previous procedure for next mode numbers
greater than dominate mode, e.g. 2 and 3, we have done it for mode number “2” and all two modes
data were sketched using Matlab code for 2D-periodic structures, presented in appendix (b), as
following:
5.5
5
Frequency(GHz)
4.5
4
3.5
3
2.5
Band Gap
Band Gap
2
1.5
1
0.5
0
1
56
Beta*d
Fig. Matlab figure obtained using two mode Eigenmode solver calculations.
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[Dispersion diagram: CST Microwave Studio]
Appendix:
a) Matlab Source code for 1D periodic structures
b) Matlab code for 2D-periodic structures
clc;
x=data(:,1); %phase parameter
y=data(:,2); %Frequency (Mode1)
plot(x,y,'black');
hold on
y=data(:,3); %Frequency (Mode2)
plot(x,y,'Red');
hold on
y=data(:,4); %Frequency (Mode3)
plot(x,y,'Green');
hold on
y=data(:,5); %Frequency (Mode4)
plot(x,y,'blue');
legend('Mode1','Mode2','Mode3','Mode4');
xlabel('Phase(Deg)');
ylabel('Frequency(GHz)')
clc;
i=1:19;
x1=data(i,1); %phase parameter
y1=data(i,3); %Frequency (Mode1)
plot(x1,y1,'blue');
hold on
i=19:38;
x1=data(i,1); %phase parameter
y1=data(i,3); %Frequency (Mode1)
plot(x1,y1,'red');
hold on
i=38:57;
x1=data(i,1); %phase parameter
y1=data(i,3); %Frequency (Mode1)
plot(x1,y1,'green');
hold on
legend('2D Dispersion Diagram');
xlabel('Beta*d');
ylabel('Frequency(GHz)')
Reference:
[1] F. YANG, and Y. Rahmat-Samii, Electromagnetic Band Gap Structures in Antenna Engineering, 2008.
[2] P. KOVÁCS, Electromagnetic Band Gap Structures Practical Tips and Advice for Antenna Engineers.
(www.radioeng.cz/fulltexts/2012/12_01_0414_0421.pdf)
[3] Baharak Mohajer Iravani, Electromagnetic Interference Reduction Using Electromagnetic Band gap structures in
Packages, Enclosures, Cavities and Antenna, PhD Thesis, University of Maryland, 2007.
Good Luck!
Me, my little son and Prof. Samii at APS/URSI
2011,Spokane, Uinted States
An Old Fashion Iranian Home!
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