# Accurate Analysis of Practical 3D Periodic  Leaky-Wave Patch Antennas

Authors: Mohammad R. Zunoubi; Hassan A. Kalhor
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

This paper presents a new analysis method based on the finite integral technique (FIT) of practical three-dimensional leaky-wave antennas comprised of a finite number of patches arranged in arrays. Radiation patterns and scattering parameters are calculated and compared with corresponding measured results to demonstrate the adequacy and the accuracy of the proposed method. The technique is then used as an antenna design tool to obtain the desired radiation patterns and beam scanning by selective choice of array elements dimensions, spacing, and substrate parameters.


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## Body

## 1. Introduction

The foundation for surface-wave and leaky-wave antennas was
laid by Sommerfeld in 1899 [1]. After this pioneering work, leaky-wave antennas
(LWAs) have been the subject of many investigations [2, 3]. The planar
printed-circuit leaky-wave antenna arrays with discrete periodic energy leakage
have, however, been the focus of research in recent years [4, 5]. The interest
in these newer antenna types stems from their ability to produce both broadside
pencil beams and scanned conical beams, and their structural simplicity such as
low-profile construction and ease of integration with other planar structures.
The beam scanning ability of early leaky-wave antennas was achieved by using
substrates with dielectric materials of permittivity as high as 100 which were
clearly not realistic [6, 7].
Alternatively, the idea of using multilayered dielectric substrates was introduced
which used lower permittivity but added significant complexity to the
theoretical analysis and antenna fabrication [8]. Later, Zhao et al. reported
a new planar structure that consisted of infinite periodic arrays of metallic
patches separated from a ground plane by substrates that had lower dielectric
permittivity, which permitted beam scanning from broadside to close to end-fire
by adjusting other antenna parameters [9]. Their analysis was based on a method
of moments approach along with the application of reciprocity theorem to obtain
the far-field patterns of a short electric dipole positioned in the middle of
the dielectric substrate. Instead of the actual radiation problem, they
calculated the scattering of the entire structure under plane wave illumination.
Although their contribution offers major insights into the operation of
periodic planar leaky-wave antennas, their approach is limited to 2-dimensional
infinite structures. The analysis of practical antennas with finite number of
elements is substantially more difficult.The purpose of
this research is to present a complete analysis and design of 3-dimensional
finite periodic planar leaky-wave patch antennas which have practical
significance. Such structure is seen in Figure1.Figure 1
The geometry of the practical printed-circuit leaky-wave periodic patch antenna.The patch antenna consists of a
simple finite periodic planar array of metal strips that are printed on a
grounded dielectric substrate as depicted above. A small horizontal dipole
located in the middle of the substrate with a length ofld is
used as the source in modeling because it is easy to analyze and launche the
desired leaky waves on the antenna. In experimental verifications, more
convenient methods of excitation may be used without changing the antenna
pattern appreciably. In our measurements, we employed a coaxial line that was
insulated from and passed through the ground plane and fed the horizontal
dipole antenna. The inner conductor was connected to one arm, and the outer
conductor was shorted to the second arm of the dipole. Also as shown in Figure 1, w and l are the width and length of array patches, a and b are their periodicities in x and y directions,
respectively, and h is the substrate thickness. The finite integration technique
(FIT) [10] and perfectly matched layer (PML) absorbing boundary condition [11]
are combined in a numerical technique introduced earlier by the authors to
analyze general diffraction gratings [12] and periodic slot antennas [13]. This
method is now extended, specialized, and used to solve the present finite
periodic patch antenna to obtain convergent and accurate results. The main
contribution of this work is that it analyzes a practical finite size antenna
which is analytically more demanding than an idealized infinite structure in
which infinite periodicity can be used to advantage. Different parameters
including substrate height and permittivity, the patch width w and
length l, the periodicities a and b, and the frequency are
varied to investigate the effects on the antenna scanning properties and to
obtain patterns that may be either a pencil beam at broadside, or a conical
beam, depending on the desired scan angle. The results are cast into a few
empirical design equations that can serve as convenient design tools.


## 2. Formulation

One of the direct
methods of solving Maxwell’s equations is the finite integration technique
(FIT) that is based on the integral form of these equations which are(1)∮cE→(r→,t)⋅dl→=−∬s∂∂tB→(r→,t)⋅ds→,(2)∮cH→(r→,t)⋅dl→=∬s{∂∂tD→(r→,t)+J→(r→,t)}⋅ds→,(3)∯sD→(r→,t)⋅ds→=0,(4)∯sB→(r→,t)⋅ds→=0. These equations are discretized by
first defining a grid cell complex G as
(5)G={Vi,j,k∈R3∣Vi,j,k:=[xi,xi+1]×[yj,yj+1]×[zk,zk+1]},i=1,…,I−1,j=1,…,J−1,k=1,…,K−1. Faraday’s law of (1) is then
discretized on a cell Vi,j,k of G according to
(6)e⌢x(i,j,k)+e⌢y(i+1,j,k)−e⌢x(i,j+1,k)−e⌢y(i,j,k)=−ddtb⌢⌢z(i,j,k), where e⌢x(i,j,k)=∫(xi,yj,zk)(xi+1,yj,zk)E→⋅dl→ is the voltage along one edge of surface Sz(i,j,z) and b⌢⌢z(i,j,k)=∫sz(i,j,k)B→⋅ds→ is the magnetic flux through the cell face Sz(i,j,z) as shown in Figure 2. The above
procedure is carried out over all grid surfaces of G to obtain the
matrix equation:Figure 2
Electric voltages and magnetic flux allocations on a cellVi,j,k.(7)Ce⌢=−ddtb⌢⌢, where C is the discrete curl
matrix, e⌢ is the vector containing electric voltages,
and b⌢⌢ is the magnetic flux vector. The magnetic
Gauss’s law of (4) is then discretized by performing the surface integral over
the six faces of the cell shown in Figure 2 to obtain
(8)−b⌢⌢x(i,j,k)+b⌢⌢x(i+1,j,k)−b⌢⌢y(i,j,k)+b⌢⌢y(i,j+1,k)−b⌢⌢z(i,j,k)+b⌢⌢z(i,j,k+1)=0. Upon applying the above discretization
to all grid volumes of G, the following matrix equation is obtained:
(9)Sb⌢⌢=0 with S being the discrete
divergence matrix.Ampere’s law of (2)
and the electric Gauss’s law of (3) are discretized in a similar manner using a
dual grid G˜ with foci of G as its grid points to
obtain the corresponding matrix equations:
(10)C˜h⌢=ddtd⌢⌢+j⌢,S˜d⌢⌢=q. As the final step,
the following equations involving constitutive parameters are used to relate
the integral of the voltage and flux state variables allocated on G and G˜:
(11)d⌢=Mεe⌢+p⌢⌢,j⌢⌢=Mκe⌢,h⌢=Mνb⌢⌢−m⌢, where Mε,Mκ,andMν are the permittivity, conductivity, and
reluctivity matrices, respectively, and p⌢⌢andm⌢ are permanent electric and magnetic
polarizations. Additionally, the time derivatives in (7) and (10) are discretized through
application of a simple forward difference scheme as
(12)ddtb⌢⌢z(i,j,k)=b⌢⌢zn−1/2(i,j,k)−b⌢⌢zn+1/2(i,j,k)Δt, where Δt is the time step
chosen to satisfy the Courant stability criterion.


## 3. Results

We first illustrate
the accuracy and effectiveness of our numerical implementation by calculating
the radiation pattern andS11 parameter of a 9 × 9 periodic planar patch antenna and comparing the results
with their corresponding measured values. Antenna parameters are selected to
obtain a broadside pencil beam pattern. Referring to the antenna geometry shown
in Figure 1, the following parameters are chosen: a=0.54λ,b=0.12λ,w=0.04λ,l=0.5λ,h=0.5333λ,εr=1.0, and an operating frequency of fr=4GHz. The leaky waves
are excited by an x-directed infinitesimal dipole antenna of length ld=0.1λ located at the middle of the substrate and the E-plane radiation
pattern (φ=0°)
and the S11 parameter are calculated and compared with the
measured results. The comparisons are seen in Figure 3 and indicate excellent
agreements.(a)
E-plane pattern calculated and measured at 4 GHz(b)
A comparison of the calculated and measuredS11After the accuracy
of our numerical implementation is verified, we first investigate the effect of
changing the operating frequency on the radiation property of the antenna with
its parameters described above. The frequencies of 4.0 GHz, 4.618 GHz, and 5.08 GHz are considered. As seen in Figure4, the resulting radiation patterns are
broadside for fr=4.0GHz and scans to 25° and 40° at the two higher frequencies.Figure 4
E-plane pattern for f=4.0, 4.618, and 5.08 GHz.The next study
focuses on the variation of the strip lengthl. As seen in Figure 5, the
patterns become broader as l decreases from 3.75 cm to 3.0 cm and 2.25 cm (corresponding to 0.5 λ, 0.4 λ, and 0.3 λ) supporting the fact that the
patch arrays are not confining the leaky waves and the patterns become closer
to the pattern of a simple short electric dipole.Figure 5
E-plane pattern for l=2.25, 3.0, and 3.75 cm.In order to study
the effects of changing the periodicity of the arrays in they direction
on the patterns, we compute patterns for b of 0.9 cm, 2.4 cm, and 3.6 cm
(corresponding to 0.12 λ, 0.32 λ, and 0.48 λ). Results are shown in Figure 6
and indicate that the main beam scans from broadside to 15° and 33°,
respectively, illustrating the scanning capability of the structure.Figure 6
E-plane patterns for b=0.9, 2.4, and 3.6 cm.We then investigate
the effect of substrate height on the radiation property of the antenna ath of 3.99 cm, 4.63 cm, and 5.7 cm. It is seen in Figure 7 that the resulting
radiation pattern is broadside for h=3.99cm, and then it scans to 25°,
and 37° as h varies from 4.63 cm to 5.7 cm.Figure 7
E-plane patterns for h=3.999, 4.63, and 5.7 cm.The last study involves varying the
substrate permittivity from air,εr=1 to 1.1, and 2.2 and calculating the far-field patterns of the antenna. The
simulation results are shown in Figure 8, where it is observed that as the
permittivity increases from 1 to 1.1, the main lobe scans from broadside pencil
beam to 12°. However, for εr=2.2, the pattern is degraded which is consistent with the results reported in
[9].Figure 8
E-plane patterns for εr=1.0, 1.1, and 2.2.


## 4. Design Rules

In this section, we use our numerical
results presented above to obtain empirical equations that can serve as
convenient design tools for finite arrays of leaky-wave patch antennas.


### 4.1. Substrate Height

Based on the results shown in Figures6 and 7, we have found that for a finite structure of 9 × 9 patch antenna, the
substrate thickness h is related to the scan angle θp by
(13)hλ0=0.58εr−sin2θp, where λ0 is the free-space
wavelength and εr is the substrate relative permittivity. For
the infinite structure operating in the order 1 mode, the theoretical value of
the coefficient in (13) is 0.58 instead of 0.5 [9].


### 4.2. Operating Frequency

According to (13), with a fixed
substrate height h, a desired scan angle θp can be obtained by varying the
frequency. The results in Figure 4
show that as the frequency increases, the
scan angle increases as predicted by (13).


### 4.3. Maximum Scan Angle with a Single  Beam

It is known that for moden=2 to radiate at broadside, the substrate height h should be [9]
(14)hλ0=1.0εr. However, for this substrate
thickness, the scan angle for n=1 mode can be written based on (13) as
(15)θp=sin−11.973εr1.724=sin−1[0.814εr]. Therefore, (15) should provide the
maximum possible angle for single mode operation without degradation from the
higher-order modes. Based on (15), for an air-filled substrate, the maximum
scan angle should be 54°. When this value is used in (13), a substrate
thickness of 0.987 λ0 is obtained. To illustrate the accuracy of
(15), we calculated the E-plane pattern of the antenna with its
parameters given as a=0.54λ,b=0.12λ,w=0.04λ,l=0.5λ,h=0.987λ0,εr=1.0, and an operating
frequency of fr=4GHz. The results are seen in Figure 9
which show a nondegraded beam that has scanned to θp=52°. The results for the
broadside radiation (h=0.533) are also included for illustration
purposes.Figure 9
E-plane patterns for εr=1.0, h=0.533λ, and h=0.987λ.


## 4.1. Substrate Height

Based on the results shown in Figures6 and 7, we have found that for a finite structure of 9 × 9 patch antenna, the
substrate thickness h is related to the scan angle θp by
(13)hλ0=0.58εr−sin2θp, where λ0 is the free-space
wavelength and εr is the substrate relative permittivity. For
the infinite structure operating in the order 1 mode, the theoretical value of
the coefficient in (13) is 0.58 instead of 0.5 [9].


## 4.2. Operating Frequency

According to (13), with a fixed
substrate height h, a desired scan angle θp can be obtained by varying the
frequency. The results in Figure 4
show that as the frequency increases, the
scan angle increases as predicted by (13).


## 4.3. Maximum Scan Angle with a Single  Beam

It is known that for moden=2 to radiate at broadside, the substrate height h should be [9]
(14)hλ0=1.0εr. However, for this substrate
thickness, the scan angle for n=1 mode can be written based on (13) as
(15)θp=sin−11.973εr1.724=sin−1[0.814εr]. Therefore, (15) should provide the
maximum possible angle for single mode operation without degradation from the
higher-order modes. Based on (15), for an air-filled substrate, the maximum
scan angle should be 54°. When this value is used in (13), a substrate
thickness of 0.987 λ0 is obtained. To illustrate the accuracy of
(15), we calculated the E-plane pattern of the antenna with its
parameters given as a=0.54λ,b=0.12λ,w=0.04λ,l=0.5λ,h=0.987λ0,εr=1.0, and an operating
frequency of fr=4GHz. The results are seen in Figure 9
which show a nondegraded beam that has scanned to θp=52°. The results for the
broadside radiation (h=0.533) are also included for illustration
purposes.Figure 9
E-plane patterns for εr=1.0, h=0.533λ, and h=0.987λ.


## 5. Conclusions

We have presented a complete analysis
of a 3-dimensional periodic planar array patch antenna. The finite integration technique
(FIT) was used to obtain theoretical results. The adequacy and accuracy of our
approach were verified by computing antennas radiation pattern andS11 parameter and comparing results with their corresponding measured values.
Excellent agreements were observed. We then studied the effects of different
antenna parameters such as substrate height and permittivity, array
periodicity, patch dimensions and frequency on both broadside, and scanned-beam
patterns. This study confirms the beam scanning properties of these antennas.
The substrate thickness and its dielectric constant play a major role in
determining the beam angle. Based on our analysis, some empirical design rules
are given which should be of interest to practical patch antenna designers who
would want to avoid extensive numerical analysis.



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*Source: 384792-2008-04-20.xml*