4.4 Microstrip dipoleBasic theoryMicrostrip antennas are frequently used in today's wireless communication systems. Thanks to their low profile, they can be mounted to the walls of
buildings, to the fuselages of airplanes or to the reverse sides of mobile phones. Moreover, microstrip antennas are fabricated using the same
technology as producing printed circuit boards. Therefore, the fabrication is relatively simple and well reproducible. Finally, microstrip antennas
can be simply integrated directly to microwave circuits, which are based on microstrip technology, and therefore, no special transmission lines,
symmetrization circuits or connectors are needed on the contrary to classical antennas
[8].
Nevertheless, there are several disadvantages of microstrip antennas. Narrow operation band is the main disadvantage. Due to this property, the
design of microstrip antenna arrays exhibiting sufficiently low level of sidelobes is a really hard nut to be broken. Even the
parasitic radiation of the feeding microstrip network, which can deform the
directivity pattern
[8].
One of the most frequently used types of microstrip antennas, the
patch antenna,
is depicted in fig. 4.4A.1a. The antenna consists of a conductive rectangle of the dimensions A × B,
which is etched on a dielectric substrate. The antenna is fed by the
microstrip transmission line
(fig. 4.4A.1, the microstrip goes from the front edge of the substrate crossways from the left). The second side of the
substrate (on fig. 4.4A.1a depicted as the bottom one) is continuously electroplated. The electroplated side plays the role
of a reflector in the sense of zero potential (from the point of view of feeding) and in the sense of limiting radiation in the direction
behind the reflector. Further, we call the electroplated side the
ground plane.
The microstrip antenna
that is fed by the
microstrip transmission line
(fig. 4.4A.1a) can be considered as an open (nonshielded) openended transmission line, which is significantly widened
at its end. If electromagnetic wave propagates along such transmission line, electromagnetic energy is primarily radiated into surrounding at the
nonhomogeneities (spontaneous widening of the microstrip at the border of the feeding line and the antenna element and the open end of this element)
of the transmission line. The structure therefore behaves as a transmitting antenna. Moreover, if the length of the microstrip antenna element equals
to the half of the wavelength on this widened transmission line, then
input impedance
of such an antenna is purely real [8]. Then, the antenna is said to be in resonance.
Radiation of a
microstrip antenna
can be explained in different ways. We can come out of the current distribution on the antenna element, which can be consequently understood as a
wire antenna
consisting of a very wide and a very thin antenna conductor. Next, we can come out from the line of electric intensity at the front
side and at the back side of the antenna element (from the point of view of the feeding transmission line) and we can explain the radiation as an
effect of a strong horizontal component (i.e. of the component, which is oriented in parallel with the ground plane) of electric field intensity vector
at those edges.
As an alternative to the microstrip feeding of the microstrip antenna, a feeding by coaxial probe can be used (fig. 4.4A.1b).
Whereas the outer conductor of the coaxial cable is connected to the grounding plane, the inner conductor goes through a gap in the grounding plane
and continues through the substrate to the microstrip antenna element, where it is fixed (fig. 4.4A.1c).
Significant reduction of the parasitic radiation of the feeding transmission line is the main advantage
[8].

a)  b)  c) 
Fig. 4.4A.1  A microstrip antenna consisting of a single rectangular microstrip element.
a) feeding by microstrip transmission line b) feeding by coaxial transmission line c) cut through antenna b) in the plane of inner conductor of coaxial line 

On the other hand, the coaxial feeding losses the main advantages of the microstrip feeding  a simple realization of a feeding system when
microstrip antenna elements are grouped into arrays.
Now, turn our attention to the mathematical modeling of
microstrip antennas.
Thanks to the popularity of microstrip antennas, several types of models of those antennas have been developed. Numerical models are of different
validity domains (some are valid for lower microwave frequencies, some are valid for a limited interval of dimensions, etc.). Numerical
models significantly differ even in their CPUtime demands (some models are based on closedform expressions, which leads to low computational
requirements, and some models are based on numerical methods, which leads to high CPUtime demands). The methods significantly differ even in the
reached accuracy.

a)  b) 
Fig. 4.4A.2  Microstrip dipole plus reflector.
a) global view b) discretization net for x component of current density 

In the next, we concentrate on modeling
microstrip antennas
by the
method of moments,
which was described in the paragraph 4.1. As already said, the method comes out of Maxwell equations
in the integral form, and therefore, we compute voltages and currents on the microstrip antenna element instead of searching for the distribution
of electromagnetic field in the antenna surrounding.
In this paragraph, we are going to simplify the analysis. Instead of analyzing a
patch antenna,
we deal with a
microstrip dipole
(fig. 4.4A.2) only. At the dipole, a small width of the antenna microstrip B is assumed. Then, only longitudinal
currents (direction x) need to be considered and onedimensional problem is going to be solved, which is an analogy to the solution of a
wire dipole
(paragraph 4.1)
Analyzing the antenna, we assume an infinitely wide substrate, infinitely small thickness of metallic layers t ≈ 0,
perfect electric conductivity
of all metallic parts, lossless dielectrics and vacuum in the antenna surrounding.
Assume a plane wave
of an angular frequency ω, which impinges the
microstrip dipole.
The electric field intensity vector E^{I} (the upper index I denotes the incident wave) describes the incident wave.
This wave induces conductive currents in the antenna wire, which are described by the current density vector J. Since the current has to be
zero at the ends of the microstrip dipole, a charge described by the charge density ρ is accumulated here. In the following halfwave,
the orientation of conductive currents is changed and the accumulated charge leaves the dipole
[9].
The effects of currents are described by the
vector potential

$\mathbf{A}\left(\mathbf{r}\right)={\displaystyle \underset{S}{\iint}\left\{{G}_{A}\left(\mathbf{r}{\mathbf{r}}_{0}\right)\cdot \mathbf{J}\left({\mathbf{r}}_{0}\right)\right\}d{S}_{0}}$

( 4.4A.1a )

and the effects of charges are described by the
scalar potential

$V\left(\mathbf{r}\right)={\displaystyle \underset{S}{\iint}\left\{{G}_{V}\left(\mathbf{r}{\mathbf{r}}_{0}\right)\rho \left({\mathbf{r}}_{0}\right)\right\}d{S}_{0}}$.

( 4.4A.1b )

Value of the vector potential A(r) in the observation pint r on the
microstrip dipole
can be computed by the consecutive addition of contributions of components of the current density vector J from all the points of the
microstrip antenna element r_{0} whereas the current density vector J is multiplied by the respective column of the
dyadic Green function
G_{A}. The observation point, where the value of the vector potential is computed, is determined by the vector r.
Summation of contributions of all the currents flowing on the surface of the antenna element to the vector potential A(r)
in the point r is performed by the integral over all the surface of the microstrip dipole S. The source points, where currents
contributing to the vector potential to the point r are consecutively determined by the vector r_{0}. The argument of the
dyadic Green function rr_{0} shows the contribution of the current in r_{0} the potential in r.
Dealing with the computation of the scalar potential, the
dyadic Green function
G_{A} is replaced by the
scalar Green function
G_{V}, and instead of the current density vector, the scalar charge density r appears. Except of this, eqn.
(4.4A.1b) is identical with eqn. (4.4A.1a).
As already said, current density and charge density on the
microstrip dipole
are mutually associated. Therefore, eqns. (4.4A.1a) and (4.4A.1b) have to be completed by the
continuity theorem
given by

$j\omega \rho =\nabla \cdot \mathbf{J}$.

( 4.4A.1c )

Eqn. (4.4A.1c) expresses the fact that current flowing from a closed region (see the current divergence as the
righthand side) has to equal to the charge reduction in this region within one second (see time derivative at the lefthand side).
If both the vector potential and the scalar one on the microstrip dipole are expressed, then the electric field intensity, which is radiated
by the antenna, can be computed

${\mathbf{E}}^{S}=j\omega \mathbf{A}\nabla V$.

( 4.4A.1d )

Time derivative of the
vector potential
jωA describes a dynamic contribution of electric charge on the dipole to the transmitted wave (electric conductive currents,
i.e. charges in motion, are sources of vector potential). Gradient
scalar potential
gives a static contributionof electric charge on the dipole to the transmitted wave (static charges, which are concentrated
this moment at the ends of the dipole, are sources of scalar potential).
The final problem, which has to be solved, are
boundary conditions.
Since all the metallic parts of the antenna are perfectly electrically conductive, components of electric field intensity, which are tangential
to the antenna surface, have to be zero on this surface

${\mathbf{n}}_{0}\times {\mathbf{E}}^{S}={\mathbf{n}}_{0}\times {\mathbf{E}}^{I}$.

( 4.4A.1e )

In the abovegiven relation, E^{S} denotes electric intensity of a wave, which is transmitted by the antenna, and
E^{I} is electric intensity of a wave, arriving to the antenna from outside. The vector n_{0} is unitary vector,
which is perpendicular to perfect electrically conductive surfaces.
As already explained in the paragraph 4.1, analytical solution of (4.4A.1) is unknown, and therefore, we utilize the
moment method
for its solution.
In the first step of the analysis of the microstrip dipole, the antenna is placed to
Cartesian coordinate system
(fig. 4.4A.2). Next, vector equations (4.4A.1) are rewritten in the elected coordinate system to the scalar
form. Moreover, we assume a special case when moving on the surface of the
microstrip dipole
(thanks to the
boundary conditions,
value of the tangential component of electric field intensity is known, which can be used further).
We start at eqn. (4.4A.1a), which describes dependency of
vector potential
on current density vector. The mutual relation between those two quantities is described by
dyadic Green function.
In a fact, dyadic Green function is a matrix 3 × 3 which elements are functions describing dependency of components x, y, z
of vector potential on the components of current density vector. A single scalar element G_{A}^{st} of this dyadic Green function
can be understood as sth component of vector potential, which is supplied by an elementary electric dipole (described by a constant
current density vector) in the direction t. Obviously, only the component xx of dyadic Green function is nonzero (assuming a very small
width of the microstrip dipole, ycomponent of current density vector is zero and the only nonzero J_{xx} can be source of
xcomponent of vector potential A_{x}; other components of vector A are of zero value).
If x component of
vector potential
(i.e., the component oriented along the dipole) is going to be computed, then (4.4A.1a) can be rewritten to

${A}_{x}\left({x}_{m},{y}_{n}\right)={\displaystyle \underset{S}{\iint}\left\{{G}_{A}^{xx}\left({x}_{m},{y}_{n}{x}^{\prime},{y}^{\prime}\right){J}_{x}\left({x}^{\prime},{y}^{\prime}\right)\right\}d{x}^{\prime}d{y}^{\prime}}$.

( 4.4A.2 )

Here, (x_{m}, y_{n}) are coordinates of a point on the surface of the dipole, where vector potential
A_{x} is computed. Coordinates (x', y') specify the position of xcomponent of current density, which supplies
xcomponent of vector potential. During the integration, coordinates (x', y') walk through all the points of the antenna surface
S. The symbol J_{x} denotes xcomponent of current density vector, the symbol G_{a}^{xx}
represents x diagonal term of
dyadic Green function.
That way, (4.4A.1a) is adopted for the case of the analyzed antenna, and therefore, the attention is turned to the
rearrangement of (4.4A.1b). Since (4.4A.1b) is a scalar equation containing scalar quantities only,
the adoption consists in considering the introduced coordinated system

$V\left({x}_{m},{y}_{n}\right)={\displaystyle \underset{S}{\iint}\left\{{G}_{V}\left({x}_{m},{y}_{n}{x}^{\prime},{y}^{\prime}\right)\rho \left({x}^{\prime},{y}^{\prime}\right)\right\}d{x}^{\prime}d{y}^{\prime}}$.

( 4.4A.3 )

Again, scalar potential V is computed on the surface of the dipole in the point (x_{m}, y_{n}).
Computing this potential, the product of
scalar Green function
G_{V} and charge density r is integrated over the whole surface of the dipole. The movement on the surface during integration
is done by changing coordinates (x', y').
Next, we turn our attention to
continuity equation
(4.4A.1c). Since only xcomponent of current density vector is nonzero (conductive currents can flow in the direction
of dipole axis only), the relation can be rewritten to the form

$j\omega \rho \left(x,y\right)=\frac{\partial {J}_{x}\left(x,y\right)}{\partial x}$.

( 4.4A.4 )

Considering (4.4A.4), charge density r can be expressed as a function of current density J_{x}
and can be substituted to (4.4A.3). That way, charge density is elliminated from (4.4A.3) and both
potentials are expressed as functions of components of current density vector

$V\left({x}_{m},{y}_{n}\right)\frac{1}{j\omega}{\displaystyle \underset{S}{\iint}\left\{{G}_{V}\left({x}_{m},{y}_{n}{x}^{\prime},{y}^{\prime}\right)\left[\frac{\partial {J}_{x}\left({x}^{\prime},{y}^{\prime}\right)}{\partial x}\right]\right\}d{x}^{\prime}d{y}^{\prime}}$.

( 4.4A.5 )

Further, vector potential
(4.4A.2) and
scalar potential
(4.4A.5) are substituted to (4.4A.1d), which enables us to compute electric field intensity of the
radiated wave. Since both vector potential (4.4A.2) and scalar one (4.4A.5) are functions of an
unknown current distribution on the surface of the dipole, even electric field intensity is a function of this current distribution

${E}_{x}^{S}\left({x}_{m},{y}_{n}\right)=j\omega {A}_{x}\left({x}_{m},{y}_{n}\right)\frac{\partial V\left({x}_{m},{y}_{n}\right)}{\partial x}$,
${E}_{x}^{S}\left({x}_{m},{y}_{n}\right)=j\omega {\displaystyle \underset{S}{\iint}\left\{{G}_{A}^{xx}\left({x}_{m},{y}_{n}{x}^{\prime},{y}^{\prime}\right){J}_{x}\left({x}^{\prime},{y}^{\prime}\right)\right\}d{x}^{\prime}d{y}^{\prime}}+\frac{1}{j\omega}{\displaystyle \underset{S}{\iint}\frac{\partial}{\partial x}\left\{{G}_{V}\left({x}_{m},{y}_{n}{x}^{\prime},{y}^{\prime}\right)\left[\frac{\partial {J}_{x}\left({x}^{\prime},{y}^{\prime}\right)}{\partial x}\right]\right\}}d{x}^{\prime}d{y}^{\prime}$.

( 4.4A.6 )

Magnitude of electric field intensity on the surface of microstrip dipole can be determined, assuming perfect electric conductivity, from
boundary condition
(4.4A.1e). Current distribution J_{x} is the only unknown in (4.4A.6).
And the equation (4.4A.6), which contains the unknown function J_{x}(x, y), is going to be
solved by
moment method
The way of obtaining a piecewiseconstant approximation of
current distribution
on the basis of (4.4A.6) was described in the paragraph 4.1. Therefore,
the approach is here reminded only.
 The region, where the solution of the integral equation is going to be found, has to be discretized (the surface of the dipole is divided to
subregions, which do not overlap on one hand and which totally cover the whole analyzed dipole on the other hand (see fig.
4.4A.2b).
Performing discretization,
boundary conditions
have to be kept in mind. In our situation, xcomponent of current density J_{x} has to be zero at edges
x = 0, x = A because ends of the dipole can be understood as open ends of the microstrip transmission line.
Exploiting piecewise constant approximation of current density components,
boundary conditions
can be met simply. The discretization mesh is extended behind the end of the dipole for one half of the discretization segment. Then, we enforce
those extended segments to represent zero value of current (see fig. 4.4A.2b).
 The sought function J_{x}(x, y) is approximated exploiting known
basis functions
(they are of unitary value over surface of a single discretization element and of zero value over the rest of elements) and unknown approximation
coefficients

${\tilde{J}}_{x}^{\left(n\right)}\left(x,y\right)={\displaystyle \sum _{nx=1}^{Nx}{J}_{x}^{\left(nx\right)}{\Pi}^{\left(nx\right)}\left(x,y\right)}$.

( 4.4A.7 )

In the abovegiven relation, J_{x}^{~(n)} denotes approximation of xcomponent of current density over
nth discretization element, J_{x}^{(nx)} is a sample of exact value of this component in the middle of
nth element, Π^{(nx)} denotes
basis function,
which is unitary over the element nx and which is zero elsewhere, and Nx is total number of elements in the discretization mesh.
 Approximation (4.4A.7) is substituted into the initial equation (4.4A.6).
Since the approximation (4.4A.7) does not meet the initial equation (4.4A.6) exactly, we have to respect
this fact adding the
residual function
R_{x}(x, y) to (4.4A.6) together with the approximation (4.4A.7)

${R}_{x}\left(x,y\right)={E}^{S}\left(x,y\right){\displaystyle \sum _{nx=1}^{Nx}\left\{{J}_{x}^{\left(nx\right)}{\displaystyle \underset{S}{\iint}\left[{G}_{A}^{xx}\left(x,y{x}^{\prime},{y}^{\prime}\right)+{\overline{G}}_{V}\left(x,y{x}^{\prime},{y}^{\prime}\right)\right]d{x}^{\prime}d{y}^{\prime}}\right\}}$.

( 4.4A.8 )

In these relations, G_{A}^{xx} is x diagonal term of
dyadic Green function
and using G¯_{V}, partial derivative of
scalar Green function
is expressed.
Since basis functions
Π are unitary over a respective element and are zero elsewhere, they do not have any representation in the abovegiven relations. Next,
integration and summation were swapped, and the approximation coefficients J_{x}^{nx} were moved in front of the integral
thank so their constant character.
 Residual function R_{x}(x, y) is going to be minimized. Lower values of the residual function are, closer our
solution to the exact solution is. Minimization is done by the
method of weighted residuals

$\underset{S}{\iint}\left\{{W}_{x}\left(x,y\right){R}_{x}\left(x,y\right)\right\}dxdy}=0$.

( 4.4A.9 )

Here, R_{x} denotes the residual function computed according to (4.4A.8). The symbol W_{x}
represents properly elected
weighting functions.
Weighting is done by
Dirac pulses
in order to eliminate one of integrations thanks to the filtering property

$\underset{S}{\iint}\left\{\delta \left(x{x}_{m},y{y}_{n}\right){R}_{x}\left(x,y\right)\right\}dx\text{\hspace{0.05em}}dy}={R}_{x}\left({x}_{m},{y}_{n}\right)$.

( 4.4A.10 )

Accuracy of the method, which performs weighting by
Dirac pulses,
cannot be very good because the error is not minimized globally in the whole analyzed region but only in points, where Dirac pulses
are of nonzero value (we operate with moments of residual function only).
 Using the same number of
weighting functions
as the number of unknown approximation coefficients is, we obtain the set of N linear equations for N unknown coefficients. Solving
this set of equations, we obtain unknown values of approximation coefficients, and therefore, approximation of the current distribution on the
microstrip dipole can be composed. Considering known current distribution, the desired technical parameters of the antenna
(input impedance,
gain
or
directivity pattern)
can be computed.
The whole algorithm described by the above points is given in detail in the layer B and in
[10], [11].
Replacing all the derivatives by
central differences
in the abovegiven algorithm, we obtain a matrix equation

${\mathbf{U}}_{x}={\mathbf{Z}}_{xx}{\mathbf{I}}_{x}$.

( 4.4A.11 )

In this equation, U_{x} is column vector of voltages in the direction x on discretization elements. This voltage is computed
by multiplying xcomponent of electric field intensity by xsize of the discretization element

${U}_{x}\left(m,n\right)={E}_{x}\left(m,n\right)a$.

( 4.4A.12 )

The symbol a denotes the height of the discretization element (i.e. the size in the direction x).
Since the
microstrip dipole
is supposed to be fabricated from perfect electric conductor (voltage on this conductor is zero), the vector of voltages is filled in by zeros only
except of elements relating to the excitation gap.
Next, I_{x} is column vector of currents in the direction x, which is unknown this moment for us. Elements of
I_{x} are related to the component of current density J_{x} as follows

${I}_{x}\left(m,n\right)={J}_{x}\left(m,n\right)B$

( 4.4A.13 )

(B is the width of the dipole, and consequently the width of the discretization element). Impedance matrix Z_{xx}
describes contribution of currents I_{x} and contribution of charge densities ρ (expressed from
continuity equation
(4.4A.4) using xcomponents of current density J_{x} on elements) to voltages U_{x}
on those elements. Elements of the impedance matrix Z_{xx} are known (see the layer B)

${Z}_{xx}\left(m,n\right)=\frac{j\omega a}{B}{\Gamma}_{A}^{xx}\left(m,n\right)+\frac{1}{j\omega aB}\left[{\Gamma}_{V}\left({m}^{+},{n}^{+}\right){\Gamma}_{V}\left({m}^{},{n}^{+}\right){\Gamma}_{V}\left({m}^{+},{n}^{}\right)+{\Gamma}_{V}\left({m}^{},{n}^{}\right)\right]$.

(4.4A.14 )

In order to evaluate the impedance matrix Z_{xx}, values of integrals of Green functions Γ_{A}^{xx}
and Γ_{V} over the surface of the discretization element have to be computed for various distances between the source elements
(over its surface, current distribution and charge one are integrated) and the observation one (on its surface, electric intensity is computed.
Description of the numeric computation of those integrals in Matlab is given in the layer D.
Matlab program, which performs analysis of the
microstrip dipole
by the
moment method,
is described from user's point of view in the layer C. Here, we provide only illustration
results obtained by the program.

Fig. 4.4A.3  Current distribution on halfwavelength symmetric microstrip dipole with planar reflector in the distance of one quarter of wavelength. Distribution computed for 40 cells. 

For simplicity, the substrate between the dipole and the reflector is assumed to be of the same parameters as vacuum. If the dipole length equals
to one half of wavelength and if the dipole width is B = λ/1000, following values of input impedance are obtained (in the first row of
the table, the number of discretization elements to which antenna is subdivided is given):
Tab. 4.4A.1  Input impedance for different number of discretization elements 

N 
10 
20 
30 
40 
R_{vst} [Ω] 
95.2 
97.3 
98.0 
98.4 
X_{vst} [Ω] 
72.4 
73.6 
74.7 
75.5 

The results show the method to exhibit good stability with respect to the number of discretization elements. Moreover, the input impedance
of the microstrip dipole with the reflector computed by
moment method,
is close to the results of analytical computations (sinusoidal current distribution J_{x} a wire dipole over an infinite planar
reflector assumed). For a single wire dipole,
radiation resistance of the antenna
(related to the input) equals to R_{Σ}= 85.6 Ω.
Investigating approximation of the distribution of xcomponent of the current along the microstrip dipole, we can show that depicting
approximation coefficients from the vector I_{x} into a chart. That way, we obtain a course which is close to a sinusoidal current
distribution (see fig. 4.4A.3).
