4.5 Microstrip antennaBasic theoryWe become familiar with the
patch antenna
in the paragraph describing the analysis of the
wire dipole
(par. 4.4) already. Nevertheless, we left this antenna and concentrated on the onedimensional
moment method
for the analysis of a dipole. Since patch antennas play a significant role in today's wireless communication systems, we pay our
attention to its analysis in this special part of the textbook.
In the introductory part, architecture and properties of
patch antennas
are described. Then, two ways of the antenna analysis are explained. Finally, accuracy and computational demands of those two methods
are compared. In the appendix, an overview and basic features of commercial programs for the analysis of patch antennas are given.
I. Introduction
First notes on
patch antennas
appeared in the literature at the beginning of fifties of 20^{th} century. This time, planar antennas, which can meet the surface
of airplanes and military equipment, were demanded. Later, patch antennas were intensively exploited even in the civic are (in communication especially).

a)  b) 
Fig. 4.5A.1  Classical patch antenna. a) aperturefed patch antenna
b) feeding microstrip at the bottom side of the substrate is drawn in red, apertures in the ground plane in green. 

These days, a plenty of various types of patch antennas are
at our disposal. These antennas differ in the shape of antenna elements (rectangular, circular), in
polarization
they operate with, in feeding (in par. 4.4, microstrip and coaxial feeding were described), etc. In our description, we concentrate on
rectangular patch antennas fed by a microstrip transmission line. We subdivide path antennas into two groups depending if the feeding microstrip
is placed on the same side of the substrate as the antenna element (classical
patch antenna,
fig. 4.5A.1a) or not. The second (newer) case is going to be investigated more intensively.
Within last 15 years, indirect feeding of patch antennas is frequently used. For this purpose, small coupling slots in the ground plane are utilized.
These slots are usually positioned in the center of antenna elements. Below the slot, where the feeding microstrip is placed, a certain (inductive)
impedance can be observed. This impedance physically represents an antenna element, which exhausts energy from the microstrip transmission line.
The microstrip crosses the slot and continues further (see overlaps in fig. 4.5A.1) in order to compensate the reactive
part of the abovementioned impedance by the shunt. Antennas, which are fed by the described way, are for simplicity called
aperturefed patch antennas.
Both groups of antennas exhibit some similarities and some differences (see tab. 4.4A.1).
Small impedance bandwidth is a common property of patch antennas. For
classical patch antennas,
the bandwidth is about 2 to 3%, for
aperturefed antennas
about 4 to 6%. High quality factor of patches is the reason. Patches behave as a resonator of a
quality factor
of several tens.
Antenna efficiency
is another important parameter.
Radiation efficiency
(computed for lossless antenna) is given by the ratio of radiated power (integrating Poynting vector in the far region over the whole halfspace)
and real power on the input port of the antenna (element).
Energy, which is not radiated by the antenna, is taken away in the form of surface waves along the infinite dielectric substrate (in real situation,
the substrate is limited, of course).
From the practical point of view, the total efficiency is important. Except of losses in dielectrics,
surfacewave
losses contribute to its value. Surfacewave losses are extremely undesired. Lossy dielectrics guiding surface waves can be imagined as a
lossy transmission line. Exploiting a limited substrate, standing waves are formed in the substrate. Higher losses are related to higher standing
wave ratio. Increase of losses due to the increase of standing wave ratio is called
surfacewave losses.
The total efficiency of patch antennas is for 1 to 2 dB worse that the efficiency of
reflector antennas.
Patch antennas exhibit very good
crosspolarization.
This parameter is not important for antennas operating linearly polarized waves. Designing a dual antenna exhibiting high polarization isolation,
this parameter is of high importance.
Tab. 4.5A.1  Overview of parameters of patch antennas 


classical antenna 
aperturefed antenna 
note 
impedance bandwidth 
2 to 3 % 
4 to 6 % 
in % of resonant frequency 
radiation efficiency 
80 % 
80 % 
surfacewave losses only 
total efficiency 
40 to 60 % 

considering losses in dielectrics 
crosspolarization 
35 to 40 dB 


patch coupling 
indispensable 
small 

feeder radiation 
middle (*) 
small 
(*) reducible by feeding topology 

II. Analysis of standalone antenna element
There are many methods for the analysis of a standalone antenna element. We are going to describe three methods, which are of the most frequent use.
Two of the are going to be explained in detail.
The first method of the analysis understands the antenna element as a wide microstrip transmission line. As the radiation source, fringe electric
fields are considered (fig. 4.5A.2a).
The second method of analysis consists in replacing the space below the antenna element by a lossy resonator. Then, input impedance can be computed
using the theory of cuboidal
cavity resonators
[14]. In that case, zero normal component of electric field intensity is
assumed on sidewalls of the resonator (see fig. 4.5A.2). Hence, sidewalls can be understood as perfect magnetic conductors.
Numerical analysis of the antenna element is the third method. The numerical analysis is based on the
moment method,
which is formulated in the
spectral domain
or in the spatial one (this formulation was introduced in par. 4.1,
par. 4.3 and par. 4.4). Even here, we concentrate on the moment
method in the spatial domain.

a)  b)  c)  d) 
Fig. 4.5A.2  Semantics of models for the analysis of a standalone element (top wall and bottom one are perfectly electrically conductive, sidewalls are perfectly magnetically conductive).
a) transmission line method b) cavity resonator c) moment method (small basis functions) d) substitive model 

II.I. Analysis of a standalone element by the transmission line method
The method excels in simplicity on one hand, and enables to reach a good accuracy on the other hand. Accuracy of computing
input impedance
of the antenna element depends of the choice of antenna model. In this paragraph, the simplest model considering radiating slots between
two neighboring antenna elements only is assumed (tangential fringe components of electric field intensity are considered as a single source of
radiation). The other effects (mutual coupling among radiation sources, radiation of marginal edges) are neglected.
Designing the antenna, the admitance of the radiating slot Y_{S}
(arrows in fig. 4.5A.2) has to be known [14]

${Y}_{S}=\frac{\pi W}{{\lambda}_{0}{Z}_{0}}\left\{1+j\left[1\mathrm{ln}\left(2\right)\mathrm{ln}\left({k}_{0}w\right)\right]\right\}$

( 4.5A.1 )

Here, W denotes length of the slot [m], w width of the slot [m] (it equals to the height of the substrate d approximately),
λ_{0} denotes freespace wavelength [m], k_{0} is freespace wavenumber
(k_{0} = 2π/λ_{0}) [m^{1}] and Z_{0} is freespace impedance
[Ω].
In the relation, influence of the permittivity of the substrate is not included because the constant magnitude of electric field intensity is
considered along the whole slot.
Designing the antenna, the length of the antenna element is chosen (0.48 ÷ 0.49) λ/2, where λ = c/[f
(ε_{eff})^{1/2}] and where the
effective permittivity
of the substrate ε_{eff} is computed using
quasistatic relations
or numerically (computing effective permittivity or
characteristic impedance
of microstrip transmission line, a proper computer program has to be chosen (several programs do not properly model the
dispersion
of the transmission line on higher frequencies; hence we recommend Txline from AWR, or TRL85 in Serenade 8.5SV).
The length of the antenna element does not equal exactly to the half of the wavelength because the capacitive impedance of the righthand slot has
to be transformed to the inductive impedance so that the capacitive impedance of the first slot on the frequency f can mutually compensate the
inductive one.
In the second step, such width W of the antenna element is chosen so that the desired
resonant resistance
of the antenna element is obtained. This resistance can be expressed as R_{rez} = (2 G_{S})^{1}, where
G_{S} is the real part of the slot admitance

${G}_{S}=\mathrm{Re}\left\{{Y}_{s}\right\}=\frac{\pi W}{{\lambda}_{0}{Z}_{0}}$

( 4.5A.2 )

Substituting freespace wavelength λ_{0} and freespace characteristic impedance Z_{0}, we get

${R}_{rez}=120\frac{{\lambda}_{0}}{2W}$.

( 4.5A.3 )

Choosing W = λ_{0}/2, the
resonant resistance
equals to 120 Ω.
The described model of the antenna element is valid for electrically thin substrates only (we assume d / λ
≈ 0.01, where d is the height of the substrate).
If the resonant resistance is required to be higher than 120 Ω, the length of the antenna element W has to be smaller than one half
of the wavelength. Resonant resistance cannot be significantly increased because its maximum value cannot be (respecting feasibility of the
feeding microstrip) higher than 150 Ω. If
impedance transformer
is exploited, then the antenna element of the impedance about 240 Ω can be used.
Optimizing the width of the antenna element by extending its length is not recommended due to the spatial reasons and functional ones (parasitic
resonance and parasitic lobes can appear).
Satisfactorily small radiation resistance can be achieved by feeding the antenna element near its center instead of the edge feeding.
An example of the design of a standalone antenna element is given in the layer C including user's guide
of an accompanying Matlab program.
II.II. Analysis of a standalone element by the moment method
Applicability to the limited number of shapes of the antenna element and to the
electrically thin substrates
is drawback of the
transmissionline method.
In order to analyze elements of a general shape and elements on electrically thick substrates, the
moment method has to be used.
The planar antenna element (twodimensional one in contrast to par. 4.4) is assumed to be fed by a
microstrip transmission line.
Computing the input impedance of the antenna in the position of the microstrip input is the aim of the analysis.

Fig. 4.5A.3  The mesh of current cells and charge ones 

On the antenna element, there is a given current and charge
distribution (in reality, a current is distributed over the antenna element). In order to compute the input impedance, currents and charges have to be
computed first. Then, another quantity (electric field intensity, magnetic field intensity,
input impedance,
etc.) can be computed.
Since the moment analysis was in detail described in par. 4.1, par. 4.3
ane par. 4.4, a brief overview is presented here only:
 The analyzed structure is divided to
charge cells and
current cells.
The shape of cells is chosen as simple as possible (rectangular one). Next, both current and charge are assumed being constant over the cell
(their value can differ from a cell to a cell). We concentrate on current cells because the charge can be computed using
continuity equation.
The mesh of current cells is shifted for one half of a cell with respect to charge cells.
 We formulate a relation among electric field intensity in the center of ith current cell and currents over all elements (including
ith one). Tangential components of electric field intensity in the center of elements are required to be zero except for source elements
(the structure is assumed to be fed by flowing ideal voltage sources). Explanation of the connection of the source is given in point 4.
 As a result, a set of simultaneous linear equations is obtained. Solving it, currents over all cells are obtained.
 On the basis of computed current distribution over the antenna element,
input impedance
can be computed. Input impedance cannot be computed directly because the structure is fed by sources in the horizontal direction, which contacts the
feeding microstrip by the first pole, and the auxiliary shunt by the second pole. The described voltage source is placed to the slot, which is
denoted by one in fig. 4.5A.3. The shunt is depicted by a dotted line. The ratio of a voltage and a current in the slot
does not equal to the correct value of input impedance, and several additional steps have to be performed (we call them
deembedding,
see layer B).
A detailed description of the analysis is rather complicated, and therefore we refer to the layer B.
Further information is given in [15]. An example of moment analysis of 2D
structures is given in the layer C.
Finally, we provide an overview of commercial programs, which are based on
moment method.
Tab. 4.5A.2  Commercial programs based on moment method 

Product 
IE3D 
Ensemble 
Momentum 
Sonnet 
Microwave Office 
Producer 
Zeland Software 
Ansoft 
Agilent Technologies 
Sonnet 
Applied Wave Research 
Version 
v. 7 (2000) 
v.7 (2000) 
ADS (1999) 
v. 6 
v. 3.22 (1999) 
Domain ^{2}) 
3D/spatial 
2.5D/spatial 
2.5D/spatial 
2.5D/spectral 
2.5D/spectral 
Cells 
Triangle/Rectangle 
Triangle 
Triangle/Rectangle 
Rectangle 
Rectangle 
Equidist. mesh 
NO 
NO 
NO 
YES 
YES 
Current approximation 
Linear 
Linear (RWG) 
Linear (rooftops) 
Linear (rooftops) 
Linear (rooftops) 
G. function for PEC box 
NO (v. 8 YES) 
NO 
NO 
YES 
YES 
Modeling open problems 
Exact 
Exact 
Exact 
Approximate 
Approximate 
Aperture feeding 
YES 
YES 
YES 
NO 
NO 
Conductor thickness 
YES 
NO 
NO 
YES 
YES 
Limitations 
   
   
   
^{1}) 
^{1}) 

^{1}) The transition between layers can be modeled by a single current layer only (i.e., it has to be much shorter than a wavelength).
^{2}) Determination whether the program is 2.5dimensional or 3dimensional is rather problematic. Therefore, a simplified
criterion is applied here: if the analyzed structure can contain conductors, which are oriented in an arbitrary angle with respect to the ground plane,
then the program is denoted as 3dimensional. 
In some aspects, the abovegiven programs are similar, in the other ones, they differ. A planar formulation of the moment method in frequency
domain (conductive facets are discretized) is a common feature. The basic difference consists in openness or closeness of the formulation
(structure in a conductive box). Formulation of a closed problem is simpler because
Green functions
for a conductive box filled by a layered medium can be composed more easily compared to open structures.
Exploitation of basis functions, which are much smaller than wavelength, is another common feature of programs. In order to imagine how memory
demands [MB] for the storage of matrices of coefficients rise, the following table was built:
Tab. 4.5A.3  Memory demands [MB] for different number of unknowns 

Unknowns 
100 
1000 
10000 
IE3D, Ensemble, Momentum 
0,16 
16 
1600 
Sonnet, Microwave Office 
0,16 (0,08)* 
16 (8)* 
1600 (800)* 
 * lossless structure 
A significant difference among programs is hidden in the type of the domain, where the problem is formulated. Whereas the programs, which are
based on the spatial version of the moment method, can use rectangular cells, the spectraldomain programs allow a homogeneous mesh of rectangular
elements only, which leads to an enormous size of resultant matrices. Therefore, Sonnet and Microwave Office utilize the technique
of subdomains, which are formed by summing contributions of elementary cells. Programs Sonnet and Microwave Office exploit FFT algorithm
to accelerate double summations, which replace computation of double integral in the spectral method. The difference between Sonnet and
Microwave Office is in the type of FFT: Microwave Office admits the total number of elements to be a product of prime numbers,
Sonnet is limited to the power of two due to the use of standard FFT.
Programs Sonnet and Microwave Office are advantageous (compared to spacedomain programs) in real matrices of coefficients
for lossless structures (twice larger problems can be solved, and moreover, the problem is solved twice more quickly thanks to the use of real
arithmetic).
All the abovedescribed programs provide similar possibilities from the point of view of
fullwave analysis.
Of course, there are differences is the speed of programs. In general, programs based on the
spatial moment method
exhibit higher speed than those based on the
spectral version
of the method. More complicated development of programs is a price which has to be paid for that.
Simple and wellthroughout user's interface is another very important feature of programs. From that point of view, Microwave Office
is very popular.
