single-element radiator analysis...
TRANSCRIPT
Single‐Element Radiator Analysis Validation
162
CHAPTER 5
SINGLE-ELEMENT RADIATOR ANALYSIS VALIDATION
The problem formulation and analysis of the proposed radiator based on the Method-
of-Moments developed in the preceding chapters has been utilized to arrive at a
prototype single-element radiator design for operating at C-Band. In the last chapter,
the computer program based on this formulation has been utilized for a series of
parametric studies to assess the effect of varying the different design parameters
involved. Some of the parameter values obtained ab initio using previously published
guidelines or empirical data have been refined using the present analysis algorithms to
yield a more optimum radiator design. The radiator input characteristics as well as the
radiation patterns have been computed for the selected parameter set of the prototype.
In this chapter, we have attempted to validate the developed analysis using an
alternate, commercially-proven e.m. analysis tool (Ansoft® HFSS®.) In this process,
the relative merits / demerits of the present analysis are brought out vis-à-vis this
commercial tool. Also, further analysis has been carried out on the prototype radiator
to arrive at a design that may be fabricated for experimental purposes subsequently.
This chapter is organized as follows. For the sake of completeness, a brief
introduction is first given to FEM which is its underlying analytical method and to
Ansoft® HFSS®, the proven commercial e.m.-analysis tool used for the analysis
validation. Next we consider an FEM analysis of the proposed waveguide shunt-slot
fed microstrip patch antenna radiator using this simulation tool. A description is given
of the HFSS®-simulation model used for the analysis of the C-Band prototype,
highlighting some features that are especially incorporated to ensure an accurate
Single‐Element Radiator Analysis Validation
163
prediction by HFSS®. This is followed by simulated results obtained from the
analysis. These predictions are compared to the MOM-simulations of Chapter 4. In
addition to the baseline simulation model (Case-1), two other cases that were
analyzed are also described. Case-2 uses an extended ground plane to assess the
impact of its size on the radiator performance. Finally, Case-3 includes the effects of
radiation from fold-over currents towards the rear of the ground plane, particularly on
the backlobe. Finally, the requirements of the HFSS®-analysis w.r.t. memory and
execution time are compared to the developed M-o-M-based theoretical analysis and
computer program for the present geometry.
5.1 A Brief Introduction to FEM and Ansoft® HFSS®.
The Finite Element Method (FEM) is one of the two key frequency-domain
computational electromagnetic techniques, the other being the Method of Moments
(M-o-M.) FEM is one of the most robust formulations suited for the analysis of
arbitrarily-shaped electromagnetic structures both in enclosed or open (radiating)
configurations. This method solves Maxwell’s equations by solving the vector wave
equation. The containing structures for the field may be perfectly conducting or lossy
and the intervening medium anisotropic and/or inhomogeneous with regard to both
permittivity and permeability. The vector wave equation is a second-order partial
differential equation for which the solution yields the field inside the problem space.
It is customary to specify a weak form of this equation and to either minimize a
functional or to apply the Rayleigh-Ritz method [50]. In the second case, a vector
function is used as a testing function (usually local.) The scalar expression obtained is
integrated over the testing function domain. Boundary conditions may be specified for
the problem; the source is also treated as a particular case of this.
Single‐Element Radiator Analysis Validation
164
A basic feature of FEM is the segmentation of the problem space into small elements
– hence the term finite element. For a one-dimensional problem like a wire or a
stratified medium, these elements are linear e.g. straight-line segments between node
points. For two-dimensional problems, these would be rectangles, parallelograms or
triangles depending on the geometry to be meshed. Finally, for a three-dimensional
problem; the brick (cuboid), the right prism or the tetrahedron are most popular. Jin
[51] has addressed these three types of FEM formulations in detail, particularly
addressing the optimum aspect ratios of these geometries. For the interested reader, a
large number of references may be found in [50, 51] or on-line. Of the three solid
elements mentioned above, the tetrahedron is the most popular meshing approach
because of its flexibility and ability to approximate arbitrarily-shaped geometries.
The simplest and most widely-employed expansion functions are the lowest mixed-
order edge elements. Higher-order expansion functions and hierarchical functions
have also been used. Apart from the choice of an expansion function, the versatility of
the FEM depends strongly on a robust mesh generator. It is seen that a great amount
of effort has been put by researchers into the development of an efficient meshing
algorithm. The efficacy of the mesh determines the accuracy of the final solution and
the number and type of mesh refinements needed to achieve convergence.
The termination condition of the mesh is another significant aspect, particularly for
radiation-type of problems. Since the entire problem space is meshed in this
formulation, the termination condition limits the problem- (and thus the matrix-) size.
Three types mesh termination conditions are popular: a) Absorbing Boundary
Conditions (ABCs); b) Perfectly Matched Layer (PML); and c) Boundary Integral
(BI). ABCs are the most convenient boundary termination owing to their simplicity.
Single‐Element Radiator Analysis Validation
165
However, for radiation problems, the boundary location must be defined at a certain
(relatively large) distance from the radiating structures. This ensures that the presence
of the ABC does not perturb the near-field and coupled field regions of the actual
problem and also depends on the angle of incidence. The PML requires much lesser
stand-off distance from the actual structure being analyzed. It is a local truncation
technique but needs user-defined parameters to be specified. The BI method is a
global truncation by contrast. It needs no stand-off distance but the obtained matrix is
partially full leading to larger memory requirements and execution time.
The strength of the FEM lies in the problem matrix being sparse. Due to this, the
solution is obtained relatively fast even though the mesh discretization is fine. In
commercial programs, as in the Ansoft® HFSS®, a feature of adaptive meshing is
implemented that makes the method highly versatile. In regions of fast field variation,
the analyzer iteratively introduces finer meshing to better approximate the field
behaviour. Thus, at the expense of minimum incremental matrix elements, a more
accurate solution may be obtained.
With this, the key aspects of the FEM method have been introduced. Features specific
to HFSS® will be briefly addressed in the following section.
5.1.1 The Ansoft® HFSS® Electromagnetic Analysis Software
The HFSS® module from Ansoft® is an FEM-based full-wave electromagnetic
simulator for analyzing passive devices of arbitrary three-dimensional geometry [52].
The basic mesh element used by HFSS® is the tetrahedron. The type of elements used
is termed tangential vector finite elements. Adaptive meshing, as mentioned earlier,
Single‐Element Radiator Analysis Validation
166
allows this simulator to obtain fast, accurate simulations for the specified geometry.
The program also allows mesh seeding. This term means that on selected geometries,
a user-specified density of meshing may be applied even before the first iterative
solution is available. Such a requirement is determined by the user based on an a
priori knowledge of fast field variation across certain types of regions e.g. dielectrics
or thin slots. This expedient can cut down a number of iterations that may otherwise
be needed to reach the correct level of mesh refinement solely by adaptive meshing.
After evaluating fields at the tetrahedral nodes, HFSS® computes the field variation at
all points within the problem geometry. The program uses this information to
compute the device network parameters. The solver integrates the far-field
contributions of the field computed across the outer surface of the ABCs (volume
enclosing the problem) to obtain this. The computed fields can be post-processed to
obtain parameters like directivity, gain, etc. also.
Other features of HFSS® relevant for the problem under analysis will be addressed in
the subsequent sections of this chapter (for a single-element radiator.) For more
details of HFSS®, the User’s Guide [52] may be referred.
5.2 Analysis of the Proposed Single-Element Waveguide Shunt-Slot Fed Microstrip Patch Radiator using Ansoft® HFSS® (Case-1)
The C-Band prototype microstrip radiator described in the previous chapter is
analyzed using the FEM-based solver, Ansoft® HFSS®. This is expected to serve as an
alternate analysis for the validation of the results obtained from the M-o-M
formulation developed in this thesis. In this section we will address the baseline case
(Case-1) as per the final optimized parameter set given in Section 4.9.3 (simulated
input parameters as in Fig. 4.12.) and which uses a reduced ground plane of size
Single‐Element Radiator Analysis Validation
167
100mm X 100mm. By contrast, the M-o-M formulation inherently assumes an infinite
ground in the basic Green’s function derivation (see Chapter 2.) The other features of
the HFSS®-simulation model will be described below, highlighting the aspects
necessary to obtain a convergent solution with reasonable memory and execution-time
requirements. This will be followed by sample simulated results.
5.2.1 The HFSS®-Simulation Model for Baseline (Case-1) Antenna Element
The simulation model of the prototype single-element microstrip patch radiator for the
baseline case is illustrated in Fig. 5.1. Both the feeding waveguide and the dielectric
substrate of the patch radiator are modelled as hollow Box entities. The coupling slot
and patch are modelled as Sheet objects. The slot length is parallel to the waveguide
axis and it is imparted a transverse offset as per the design. The coordinate system is
centred at the slot and the patch is placed above it to the other side of the substrate.
The boundary condition for radiation pattern computation is chosen as ABC; also
modelled as a Box entity.
The material definition for the waveguide and ABC is specified as vacuum from the
HFSS® system library. This has εr = μr = 1 but no other loss factors and is equivalent
to free space for computational purposes. For the substrate, the particular material
chosen (see Section 4.1.3) is available in the system library as Rogers RO3003 (tm)
with the correct parameters. The patch is assigned Perfect E and the slot Perfect H.
All five sides of the ABC (apart from the base under substrate) are defined as a
Radiation Boundary. The four long faces of the waveguide are assigned PerfE. Since
the substrate is completely enclosed in the ABC box volume, no specific boundary
Single‐Element Radiator Analysis Validation
168
Fig. 5.1: Ansoft® HFSS® Simulation Model of Proposed Single-Element Radiator (Case-1)
ABC
Patch
Slot
Waveguide
Substrate
Fig. 5.2: Zoomed View near Coupling Region showing Shunt Slot under Patch Radiator
Single‐Element Radiator Analysis Validation
169
assignment is needed. It is important to observe the order of boundary definition.
Previously defined boundaries are overwritten by the later ones e.g. the slot definition
must be the last one made to ensure the waveguide wall does not “seal off” the
coupling region. A boundary display feature allows verification of the boundaries
before proceeding further. The two ends of the waveguide are defined as Excitations
and the type is WavePort.
An initial execution of the defined geometry led to a problem. The solution
converged within four passes with a very good return loss. However, the field plots
indicated a very low coupling. Upon examining the solution in detail, we found that
the slot had only two triangle elements at convergence; the patch only a few. Since the
absence of a coupling aperture in the top-wall turns the problem into a small length of
waveguide, the solver found it convenient to short out the slot and achieve
convergence quickly.
Hence, it became necessary to apply Mesh Operations which allows selected entities
of the problem definition to be “seeded” before the solver is invoked. A tetrahedron
length of 5mm was specified on the substrate. The patch and slot being planar entities,
triangle size was specified – as 1.5mm and 0.25mm respectively. Fig. 5.3 shows the
mesh plots for these three entities upon convergence of the problem. We observe that
the solver adapts to a finer mesh in the part of the substrate underlying the patch
radiator. The mesh on the slot is very dense, justifying the fine discretization specified
while seeding.
An adaptive solution was specified at 5.8 GHz which is the resonant frequency
observed in Section 4.6.2 for the identical parameter set.
Single‐Element Radiator Analysis Validation
170
(a)
Fig. 5.3: Mesh Plots on Selected Entities at Convergence a) Substrate; b) Patch; and c) Slot
(b)
(c)
Single‐Element Radiator Analysis Validation
171
A frequency sweep of the type Discrete is specified over the range 5.0 to 6.0 GHz
with a LinearStep of 0.01 GHz. This sweep type does not save the fields at each
frequency point thus reducing the memory storage requirements. A second sweep of
the type SinglePoints that saves the fields is specified for 5.59 GHz which is the
resonance observed in the HFSS® analysis – details of this are discussed in the next
subsection. Other options for sweep are also available that may be more suitable for
other types of problems [52].
5.2.2 The HFSS®-Simulated Results for Baseline (Case-1) Antenna Element
The simulator proceeds with an initial mesh to arrive at a solution for all the node
points. The mesh is adaptively refined at the adapt frequency till the network
parameters show that the specified convergence criterion is met. After this the swept
frequency response of the antenna is computed for the relevant sweep(s). For the
present problem, as there are two ports; the parameters s11 and s21 are of interest.
Since the presence of the slot only slightly disturbs the current in the waveguide top-
wall, neither of these two parameters is expected to be particularly sensitive. For this
reason, a derived parameter 1 is used to represent
the power coupling out from the waveguide (see Eqn. 2.104.) This is defined in
HFSS® using an auxillary window called Output Variables where a user-defined
variable derived from the basic computed network parameters may be specified.
Upon executing the proposed antenna geometry, it was found that a very low value of
Pout is predicted. Also, the response increases monotonically with frequency, without
exhibiting any resonant behaviour. The simulation was repeated by tweaking the slot
length to either side of the obtained prototype value of 4.0mm followed by
Single‐Element Radiator Analysis Validation
172
intermediate lengths. Finally, it was found that a slot size of 3.75mm X 0.5mm
exhibits the optimum coupling response. Figs. 5.4 & 5.5 show the computed swept
frequency response for the VSWR and the coupled power.
5.00 5.20 5.40 5.60 5.80 6.00Frequency [GHz]
1.00026
1.00028
1.00030
1.00032
1.00034
1.00036
1.00038
VO
LT
AG
E S
TA
ND
ING
WA
VE
RA
TIO
Ansoft Corporation HFSSDesign1VSWR Quick Report
Curve Info
VSWR(WavePort1)Setup1 : Sw eep1
VSWR(WavePort2)Setup1 : Sw eep1
VSWR(WavePort1)_1Setup1 : Sw eep2
VSWR(WavePort2)_1Setup1 : Sw eep2
Fig. 5.4: HFSS®-Computed VSWR of Proposed Single-Element Radiator (Case-1)
5.00 5.20 5.40 5.60 5.80 6.00Frequency [GHz]
6.00E-007
8.00E-007
1.00E-006
1.20E-006
1.40E-006
1.60E-006
1.80E-006
2.00E-006
Po
ut
Ansoft Corporation HFSSDesign1XY Plot 3
m1
Curve Info
PoutSetup1 : Sw eep1Name X Y
m1 5.5900E+000 1.7050E-006
Fig. 5.5: HFSS®-Computed Coupled-Power, Pout for Proposed Single-Element Radiator (Case-1)
Single‐Element Radiator Analysis Validation
173
The frequency at which Pout exhibits a peak is inferred as the resonant frequency. This
peak value is 1.7050 X 10-6 and occurs at 5.59 GHz (see Fig. 5.5.) As a result, the
fields are recalculated at this frequency in a separate SinglePoints sweep as mentioned
earlier, for field visualization and pattern calculation.
The problem execution was carried out using Ansoft® HFSS® 11.1.1 on a DELL
E8400 with a 2.99GHz-CPU having a 2 GB RAM and an Intel Core-2 Duo Processor.
The convergence behaviour of the input VSWR with number of iterative passes is
illustrated in Fig. 5.6 below. The total CPU-time for the solution was 2h 47m with the
RAM usage of 1.67GB. A total of seven passes were necessary for convergence with
77045 tetrahedra in the final mesh.
The powerful visualization feature of HFSS® allows one to observe the field plots
inside selected geometries. Fig. 5.7 shows E-field contours in two orthogonal
sections of the problem through the slot centre. Several other plots are possible [52].
1 2 3 4 5 6 7Iterative Pass Number
1.000
1.002
1.004
1.006
1.008
1.010
1.012
1.014
1.016
1.018
VS
WR
Co
nv
erg
en
ce
Ansoft Corporation HFSSDesign1VSWR Quick Report1
Curve Info
VSWR(WavePort1)Setup1 : AdaptivePassFreq='5.6GHz'
Fig. 5.6: VSWR Convergence Behaviour for Proposed Single-Element Radiator (Case-1)
Single‐Element Radiator Analysis Validation
174
(a)
(b)
Fig. 5.7: Electric Field Contour Plots through the HFSS®-Solved Problem Geometry (Case-1) a) parallel to waveguide longitudinal section; b) parallel to waveguide cross-section
Single‐Element Radiator Analysis Validation
175
The contour plots indicate a definite coupling mechanism from the waveguide
through the coupling slot to the patch radiator. The fields are confined under the patch
metallization and guided to the ends of the patch. Radiation is seen to occur as the
field couples to the surrounding space. Some energy is seen confined near the
substrate (outside patch margin) which is associated with the surface wave along the
air-dielectric surface.
Pattern cuts are computed in HFSS® by integrating the far-field contributions of the
fields across the specified radiation boundaries. The pattern cuts obtained in the two
principal planes are close to those expected for the dominant TM01 mode on the patch
geometry (see Fig. 5.8.) The 3-dB beamwidths in the principal planes are obtained as
141 X 84. This compares favourably with the predicted pattern beamwidths using
the developed M-o-M code – these are 118 X 84 at 5.8 GHz (see Figs. 4.39, 5.9 &
10.) The change in E-plane beamwidth may be partly on account of frequency
difference, but there is another factor evident in the pattern plot in Fig. 5.8. It is
possible to discern undulations in the E-plane plot that are ascribed to the presence of
a finite ground plane (M-o-M assumes an infinite ground plane by contrast.) Kraus
[53] has treated this aspect in detail and radiation from the ends of the finite ground
plane causes periodic undulations in the E-plane pattern of a slot. The angular
frequency of these undulations is related to the ground plane size. In addition to the
beam broadening, this effect, incidentally, introduces a boresight dip with the peaks in
E-plane occurring at approximately + 45 instead. The null observed in H-plane by
the MOM-simulation is also missing in the FEM prediction (Fig. 5.9); the finite
ground plane is felt to be the reason. The peak directivity obtained is 5.313 dBi which
is expected for such a geometry. A backlobe of -14.55 dB is observed – this is the far-
Single‐Element Radiator Analysis Validation
176
field contribution of the sidewalls of the radiation surface. Significantly, field
predictions in the rear half-space are available (Figs. 5.9 & 10 – note that boresight is
denoted as 90 in these figures) that were not possible with MOM due to the
assumption of an infinite ground plane inherent in the Green’s functions used.
Overall, there is excellent agreement of the pattern predictions by the developed
formulation with the HFSS® results (in the forward half-space).
Fig. 5.8: HFSS®-Computed Radiation Pattern Cuts for Proposed Single-Element Radiator (Case-1)
red H-Plane; and brown E-Plane
-30.00
-20.00
-10.00
90
60
30
0
-30
-60
-90
-120
-150
-180
150
120
Polar Plot
Curve Info
dB10normalize(DirL3Y)Setup1 : Sw eep2Freq='5.59GHz' Phi='0deg'
dB10normalize(DirL3Y)Setup1 : Sw eep2Freq='5.59GHz' Phi='90deg'
Single‐Element Radiator Analysis Validation
177
Fig. 5.9: Comparison of Computed Radiation Patterns for Prototype Antenna Element: HFSS® vs. MOM (H-Plane)
-60
-50
-40
-30
-20
-10
00
30
60
90
120
150
180
-150
-120
-90
-60
-30
MOM
HFSS
Relative Power, dB
Off-Axis Angle, deg
Single‐Element Radiator Analysis Validation
178
Fig. 5.10: Comparison of Computed Radiation Patterns for Prototype Antenna Element: HFSS® vs. MOM (E-Plane)
-60
-50
-40
-30
-20
-10
00
30
60
90
120
150
180
-150
-120
-90
-60
-30
MOM
HFSS
Relative Power, dB
Off-Axis Angle, deg
Single‐Element Radiator Analysis Validation
179
5.2.3 Observations for Baseline (Case-1) Antenna Element Analysis
In this subsection, an HFSS®-analysis of a single-element radiator based on the
proposed geometry has been completed. We close this section with the following
observations.
a) The results of the HFSS®-analysis broadly validate the analysis based on M-o-
M developed in Chapters 2 to 4 of this thesis with the C-band prototype design
parameters as an example.
b) The dimensions frozen with the help of the M-o-M analysis could be used
directly except for the slot length. This needs some tweaking to arrive at the
dimension where resonance may be observed in the HFSS® simulation.
c) The radiation patterns obtained from HFSS® are in close agreement to those
obtained from the developed code except for the effect of the finite ground
plane.
d) Radiation from the ends of the finite ground plane is found to add undulations
to the E-plane pattern of the radiator and to cause beam broadening in both
planes.
e) The directivity and beamwidths obtained are close to the expected values for a
rectangular microstrip antenna excited in its dominant TM01 mode.
As mentioned at the beginning of this chapter, we shall investigate two other cases for
this prototype radiator the first by increasing the ground plane dimensions and the
second by including the effect of currents induced to the rear of the ground plane. The
former is taken up in the next section.
Single‐Element Radiator Analysis Validation
180
5.3 Ansoft® HFSS® Analysis of the Proposed Single-Element Radiator using Extended Ground Plane (Case-2)
It is conventional to use a large ground plane relative to patch size behind the
microstrip patch radiator. This has the advantage of allowing the surface waves to
attenuate before being scattered from the edges of the substrate. Also fringing fields
from the radiating element are shielded from the rear half of the ground plane. This
property may be useful, for instance, when active elements are placed to the rear;
hence the arrangement eliminates the possibility of spurious feedback and oscillation.
Further, the single-radiator is expected to be used as a building block for an array of
patches. Hence, it would be useful to examine its radiation characteristics with a
larger ground plane. As a final consideration, an experimental model has been
implemented for the proposed geometry for which the actual dimensions chosen for
the extended ground are used in this section for analysis.
The proposed radiator geometry analyzed in the previous section is analyzed with a
ground plane of size 180 X 180 and is designated as Case-2. The geometry is
discussed in the following.
5.3.1 The HFSS®-Simulation for Antenna Element with Extended Ground-Plane
(Case-2)
The dimensions and parameters for the Case-2 simulation model are identical to the
previous case except for the ground plane size of 180 X 180 (see Fig. 5.11.) The ABC
is extended 5mm beyond this but the height is retained as previously. This implies a
change in ABC volume by a ratio of 3.24 : 1. Meshing operations similar to the
previous case were applied to the substrate, patch and slot. Boundary conditions and
port excitations are maintained identical.
Single‐Element Radiator Analysis Validation
181
A total CPU-time of 10h 40m was needed for obtaining convergence by the solver for
this version of the problem. The final number of tetrahedra in the adaptive mesh was
132,072. The simulated results obtained for this problem are presented.
5.3.2 The HFSS®-Simulation Results for Single-Element Radiator with Extended
Ground-Plane (Case-2)
The HFSS®-computed VSWR characteristics of the radiating element (Case-2) are
illustrated in Fig. 5.12. These are seen to be in close agreement with those of the
previous case with the smaller ground plane (see Fig. 5.4.) This is expected because
the exciting waveguide and slot region are unchanged and only the ground size and
ABC have been altered.
Fig. 5.11: Ansoft® HFSS® Simulation Model of Proposed Single-Element Radiator with Extended Ground Plane (Case-2)
Single‐Element Radiator Analysis Validation
182
5.00 5.20 5.40 5.60 5.80 6.00FREQUENCY [GHz]
6.00E-007
8.00E-007
1.00E-006
1.20E-006
1.40E-006
1.60E-006
1.80E-006
2.00E-006
Po
ut
Ansoft Corporation HFSSDesign1Coupled Power Plot
m1
Curve Info
PoutSetup1 : Sw eep1Name X Y
m1 5.6300E+000 1.8209E-006
Fig. 5.13: HFSS®-Computed Coupled-Power, Pout for Single-Element Radiator (Case-2)
5.00 5.20 5.40 5.60 5.80 6.00FREQUENCY [GHz]
1.00026
1.00028
1.00030
1.00032
1.00034
1.00036
1.00038
1.00040
VS
WR
Ansoft Corporation HFSSDesign1VSWR Plot
Curve Info
VSWR(WavePort1)Setup1 : Sw eep1
VSWR(WavePort2)Setup1 : Sw eep1
VSWR(WavePort1)_1Setup1 : Sw eep2
VSWR(WavePort2)_1Setup1 : Sw eep2
Fig. 5.12: HFSS®-Computed VSWR of Single-Element Radiator with Extended Ground Plane (Case-2)
Single‐Element Radiator Analysis Validation
183
However, we observe a small shift in the resonance indicated by Pout (Fig. 5.13.) The
maximum power coupling is 1.8209 X 10-6 and takes place at 5.63 GHz now. This is
a relatively small shift and is felt to be as (i) Pout is a difference of two quantities very
close in magnitude, hence minor changes in predicted values will cause greater
impact; and (ii) change in exact meshing in vicinity of the slot / patch entities due to
modified problem definition. The contour plots and radiation patterns presented next
are computed at this new frequency (although the difference is minor.)
The contour plots at the resonant frequency show a clear coupling mechanism from
the waveguide and a field detachment resulting in radiation (see Fig. 5.14.) The H-
plane contours show that the surface wave sustains to a considerable distance from the
patch. This justifies retaining the extended ground plane.
The computed radiation patterns show a scalloped behaviour in both planes (see Fig.
5.15.) The peak directivity is predicted as 6.618 dBi and the half-power beamwidths
in the principal planes are 138 X 104. The E-plane beamwidth is close to the
previous case but H-plane beamwidth is broader. These changes may be ascribed to
the undulations introduced due to the modified ground plane dimensions. It is
interesting to observe that this time the undulations are seen in the H-plane pattern
also.
The predicted backlobe is -14.57 dB which is very close to the number obtained in
Case-1. Thus, even though the pattern variation in the forward half-plane is affected
significantly, the backlobe is affected little. It may be remarked again that the
backlobe is computed from the field across the sidewalls of the radiation boundary.
Single‐Element Radiator Analysis Validation
184
(a)
(b)
Fig. 5.14: Electric Field Contour Plots through the HFSS®-Solved Problem Geometry (Case-2) a) parallel to waveguide longitudinal section; b) parallel to waveguide cross-section
Single‐Element Radiator Analysis Validation
185
Fig. 5.15: HFSS®-Computed Radiation Pattern Cuts for Single-Element Radiator with Extended Ground Plane (Case-2)
red H-Plane; and brown E-Plane
-30.00
-20.00
-10.00
90
60
30
0
-30
-60
-90
-120
-150
-180
150
120
Polar Plot
Curve Info
dB10normalize(DirL3Y)Setup1 : Sw eep2Freq='5.63GHz' Phi='0deg'
dB10normalize(DirL3Y)Setup1 : Sw eep2Freq='5.63GHz' Phi='90deg'
Single‐Element Radiator Analysis Validation
186
5.4 Ansoft® HFSS® Analysis of Proposed Single-Element Radiator including Currents on Rear of Ground Plane (Case-3)
In both the previous analysis cases, the backlobe estimation was based on the far-field
contribution of the fields at the side-walls of the radiation box enclosing the problem
geometry. A more rigorous treatment of this aspect would be to include the fold-over
currents from the edges of the ground plane towards its rear surface in the analysis
and add their far-field contribution also to arrive at the backlobe value. An analysis of
the problem geometry has been carried out with a modified HFSS®-model that allows
the computation of the fold-over currents over the rear surface of the ground plane of
the microstrip radiator. This is designated as Case-3 and the analysis considerations
and computed results are discussed in the following.
5.4.1 The HFSS®-Simulation for Antenna Element including Fold-over Currents
to the Rear of the Ground-Plane (Case-3)
The substrate parameters and radiator dimensions for Case-3 simulation model are
identical to the preceding two cases. A moderate ground plane size of 120 X 120 is
selected to reduce the computational burden. The ABC is extended 5mm outside the
substrate dimensions and the height towards above is still retained identical. However,
the ABC is increased to the lower side (see model in Fig. 5.16.) To allow the
waveguide to pass through either end of the ABC, a clone subtraction procedure was
used. The substrate, slot and patch were seeded prior to invoking the solver as
described previously while other boundary conditions are identical.
The problem was solved in a total CPU-time of 8h 55m for obtaining convergence. At
that instant, the final number of tetrahedra in the adaptive mesh was 110,403 with a
peak memory requirement of 1.77GB. Simulated results are discussed next.
Single‐Element Radiator Analysis Validation
187
5.4.2 The HFSS®-Simulation Results for Antenna Element including Fold-over
Currents to the Rear of the Ground-Plane (Case-3)
The VSWR response of the radiating element is nearly unchanged from the previous
two cases (see Fig. 5.17.) The value of Pout (Fig. 5.18) is different from the previous
case and is 1.7644 X 10-6 in this case and the resonant frequency is 5.59GHz.
The E-field contours through the principal sections of the problem region show
similarity to the previous two cases. Concentration of the field in the substrate is
observed as are the (relatively small) fringing fields in the plane of the substrate
outside its edge (as evident in Fig. 5.19). These fields indicate the presence of fold-
over currents induced at the rear of the ground plane along with scattering from the
edges of the ground plane.
Fig. 5.16: Ansoft® HFSS® Simulation Model of Single-Radiator including Rear Side of Ground Plane (Case-3)
Single‐Element Radiator Analysis Validation
188
5.00 5.20 5.40 5.60 5.80 6.00FREQUENCY [GHz]
1.00026
1.00028
1.00030
1.00032
1.00034
1.00036
1.00038V
SW
RAnsoft Corporation HFSSDesign1VSWR Plot
Curve Info
VSWR(WavePort1)Setup1 : Sw eep1
VSWR(WavePort2)Setup1 : Sw eep1
VSWR(WavePort1)_1Setup1 : Sw eep2
VSWR(WavePort2)_1Setup1 : Sw eep2
Fig. 5.17: HFSS®-Computed VSWR of Single-Radiator including Rear Side of Ground Plane (Case-3)
5.00 5.20 5.40 5.60 5.80 6.00FREQUENCY [GHz]
6.00E-007
8.00E-007
1.00E-006
1.20E-006
1.40E-006
1.60E-006
1.80E-006
2.00E-006
2.20E-006
Po
ut
Ansoft Corporation HFSSDesign1Power Couped Out
m1
Curve Info
PoutSetup1 : Sw eep1Name X Y
m1 5.5900E+000 1.7644E-006
Fig. 5.18: HFSS®-Computed Coupled-Power, Pout of Single-Radiator including Rear Side of Ground Plane (Case-3)
Single‐Element Radiator Analysis Validation
189
(a)
Fig. 5.19: Electric Field Contour Plots through the HFSS®-Solved Problem Geometry (Case-3) a) parallel to waveguide longitudinal section; b) parallel to waveguide cross-section
(b)
Single‐Element Radiator Analysis Validation
190
The computed radiation patterns show lesser undulations compared to the last case on
account of the smaller ground plane size. The H-plane pattern is relatively unaffected
by the scattering from the edges of the ground plane (see Fig. 5.20.) The predicted
peak directivity is 7.096 dBi and the half-power beamwidths are 131 X 61 in the
principal planes. The changes in the beamwidths compared to the previous cases are
small and associated with the size of the ground plane that introduces undulations in
the pattern.
As a notable difference, the predicted backlobe is -18.41 dB which differs from the
result obtained in both the previous cases. This is clearly due to the inclusion of the
fold-over currents in the simulation. For the far-field computation, the fields at the
lower face of the ABC actually are used. This is close to the real condition of the
radiator. It is interesting to observe that even though a relatively small fringing field is
seen at the ground plane edges, the backlobe is significant.
Single‐Element Radiator Analysis Validation
191
Fig. 5.20: HFSS®-Computed Radiation Pattern Cuts for Single-Radiator including Rear Side of Ground Plane (Case-3)
red H-Plane; and brown E-Plane
-40.00
-30.00
-20.00
-10.00
90
60
30
0
-30
-60
-90
-120
-150
-180
150
120
Radiation PatternCurve Info
dB10normalize(DirL3Y)Setup1 : Sw eep2Freq='5.59GHz' Phi='0deg'
dB10normalize(DirL3Y)Setup1 : Sw eep2Freq='5.59GHz' Phi='90deg'
Single‐Element Radiator Analysis Validation
192
5.5 Comparison of the Developed M-o-M Analysis with the Ansoft® HFSS® Analysis
As mentioned earlier, the analysis using Ansoft® HFSS® presented in the preceding
sections was intended to validate the analysis developed using the M-o-M formulation
in the foregoing chapters. Additionally, we have simulated the effect of ground plane
size and its rear aspect that was not possible in the M-o-M analysis. In this section we
will compare the features of the two methods of analysis with regard to 1) the
computational resources needed by the two methods; and 2) the physical details of the
radiating element that may be included in the analysis.
Table 5.1 presents a typical comparison of the computer resources used during the
two analyses. Since the Case-1 described in Section 5.2 is the nominal configuration
in Ansoft® HFSS®; this is selected for comparison. The developed M-o-M based
Table 5.1: Comparison of Computing Resources in Typical Executions of the Developed M-o-M Analysis and the Ansoft® HFSS® Simulation
ANALYSIS COMPUTER
CONFIGURATION
CPU-TIME
(per freq point)
Memory
Requirement
M-o-M (based on
present formulation)
ZENITH PC
1.60GHz, 256MB
RAM,
Intel Pentium-IV
~ 21 s ~ 13MB
Ansoft® HFSS®
(Case-1)
DELL E8400 PC
2.99GHz, 2GB
RAM, Intel®
Core®-2 Duo
~ 159 s ~ 1.7GB
Single‐Element Radiator Analysis Validation
193
program was executed on a moderately sized PC with a small RAM. All the HFSS®-
executions were done on a relatively faster machine with a much larger memory. It is
seen that the developed M-o-M program executes nearly eight times faster than
HFSS®. Even the storage requirements for the FORTRAN program are relatively
small. The HFSS® storage is predominantly required during matrix retrieval. It is
further notable that the other two configurations, Case-2 & -3 need still higher
execution time as well as memory compared to the nominal case shown in Table 5.1.
One may argue that often, more than the time saved, it is also the accuracy of the
solution as well as the ability to solve a variety of problems (such as different slot and
patch shapes) which are more important. It is true that, as compared to HFSS®, the
developed program executes faster only for a very specific case of rectangular
waveguide feeding a rectangular patch through a rectangular aperture. The
formulation would need modification if other waveguide cross-sections / patch or slot
shapes need to be analyzed. Thus, this is not a limitation of the method-of-moments as
such but of the present formulation since a specific geometry was selected. Also, the
present work may be extended to different slot and patch shapes by appropriately
modifying the basis functions. Hence, given these limitations, the speed of execution
of the developed program is faster as indicated.
Keeping in view the above observations and reasoning, we may enumerate some
points in regard to the two analysis methods.
1) The FORTRAN program based on the developed M-o-M formulation is found
to execute much faster even on a relatively moderate PC than an equivalent
HFSS® analysis (for the specific problem geometry selected).
Single‐Element Radiator Analysis Validation
194
2) The M-o-M formulation inherently assumes an infinite ground plane. This is
implicit in the Green’s functions used for deriving the various moment matrix
terms which are for an infinite grounded dielectric slab. However, the
truncation of the matrix term summations does imply a kind of finite ground
size. A large ground plane would be computationally prohibitive on HFSS®.
3) It is not simple to analyze a finite ground plane using the M-o-M formulation.
It will require a modification of the formulation to include currents across the
ground plane also. Whereas in HFSS®, it is relatively simple to analyze with
differing ground plane sizes (as shown by the three cases analyzed.)
4) The M-o-M formulation, however, is derived for this specific geometry and
will not allow minor modifications in geometry like the small taper in patch or
slot shape while this is comparatively simple in HFSS®. Such variations in
shape may represent fabrication imperfections, for instance, that a designer
may need to estimate the impact of.
5) There is a close agreement in the network parameters as well as radiation
behaviour of the antenna element obtained from the two alternate analysis
methods. Also the dimensions of the patch and slot required to obtain
resonance were found to be in close agreement with only the length of the
latter needing minor tweaking.
6) In this regard, we may consider that the developed M-o-M analysis stands
validated through a theoretical comparison with a proven e.m. analysis tool.
7) In conclusion, the computer program based on the developed formulation may
be used for a quick optimization of the radiating element design
parameters in the initial design phase. This is especially useful if the element
is to be used as a part of a larger array. This may be followed by a more
Single‐Element Radiator Analysis Validation
195
rigorous analysis using Ansoft® HFSS® or a suitable tool to tune the slot
dimensions and to assess the impact of other features like a finite ground plane
& fold-over currents.
5.6 Summary
In this chapter, the results of an analysis of the proposed waveguide shunt-slot fed
microstrip patch antenna using the FEM-based, commercially-available, proven e.m.
analysis tool, the Ansoft® HFSS® have been presented. A brief review of the FEM
technique has been provided, discussing the second-order p.d.e. upon which it is
based, the meshing or segmentation of the problem into ‘finite’ elements, boundary
conditions and adaptive meshing that makes FEM highly flexible. Next, the baseline
case, comprising of the WGMPA with a specified ground-plane size of 100mm X
100mm is first described. Details of basic HFSS® library elements invoked to build up
the problem geometry are described. We observe that default meshing of the problem
space turns out to be inadequate especially in the substrate and on the planar entities –
the slot and patch. This is on account of fast field variations locally on these entities.
How this is resolved by mesh seeding has also been discussed at length.
Subsequently, the simulated results for the baseline case have been presented showing
the input impedance characteristics and the power coupling. A slight tweaking of the
slot dimensions from the M-o-M value became necessary to obtain the expected
resonance. Radiation patterns obtained from the HFSS® analysis are compared to and
found to closely resemble the M-o-M predictions and thus validate the developed
formulation and analysis. The merits and limitations of the developed M-o-M based
formulation for the proposed geometry are also discussed. Two additional analyses
that respectively simulate the presence of a larger ground plane and the effect of fold-
Single‐Element Radiator Analysis Validation
196
over currents on its rear are also presented subsequently. The larger ground-plane
slightly changes the resonant frequency but notably introduces undulations to the
pattern. The inclusion of the fold-over currents is seen to significantly alter the
backlobe prediction. This underscores the importance of considering these currents if
the element is to operate in an array environment sensitive to spurious oscillations e.g.
when integrated with active elements. Finally, a comparison is drawn between the
computing resources required in the developed M-o-M program and HFSS® for the
present WGMPA geometry. The relative advantages in speed / memory storage
requirements of the former are highlighted at the expense of assuming an infinite
ground-plane. The utility of the developed formulation for the design and analysis of
such a radiator are also summarized.