4.4 Microstrip dipoleAdvanced theory
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a) | b) |
Fig. 4.4B.1 | Microstrip dipole plus reflector.
a) global view b) discretization net for x component of current density |
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As a
microstrip dipole,
we denote an antenna consisting of two narrow microstrip arms, which are fed by a symmetric source in the center. The antenna
is placed on an upper side of the dielectric substrate. The bottom side of the substrate is fully covered by a metallic layer and I of zero potential
(fig. 4.4B.1).
Effects of currents flowing on the antenna can be described by
vector potential
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( 4.4B.1a )
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Effects of charges on the antenna can be described by
scalar potential
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( 4.4B.1b )
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In the above-given equations, A(r) denotes vector potential in the point r, J is current density vector in the point
r0, GA is
dyadic Green function
and GV denoted
scalar Green function
(see the layer D). The parameter of Green functions r|r0
tells us that we compute the contribution of a current (charge) from the point r0 to potentials in the point r.
The symbol r denotes charge density.
Current density and charge one are mutually associated by the
continuity equation
If both the vector potential and the scalar one are expressed on the surface of the
microstrip dipole,
we can evaluate electric intensity of a wave, which is radiated by the antenna.
Applying (4.4B.1) to the
microstrip dipole
from fig. 4.4B.1 and substituting from
continuity equation
to (4.4B.1b), we obtain the following relations
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( 4.4B.2a )
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( 4.4B.2b )
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( 4.4B.2c )
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Here, Ax denotes x-component of
vector potential
and V is
scalar potential,
GAxx denotes x diagonal element of
dyadic Green function
and GV is
scalar Green function,
Jx is x-component of sought vector of
current distribution
and Ex is x-component of the radiated electric field intensity. Details are given in the
layer A.
Substituting vector potential (4.4B.2a) and scalar one (4.4B.2b) to the relation
(4.4B.2c), we obtain the initial equation for moment analysis of the dipole
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( 4.4B.3 )
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Magnitude of electric field intensity on the surface of the
microstrip dipole
can be computed using a
boundary condition
if perfect conductivity of metallic parts is assumed (except of the excitation gap, the intensity is zero). Therefore,
x-component of current density Jx is the only unknown in (4.4B.3).
In the first step, surface of the dipole is divided into discretization elements. Center of the first discretization element is denoted by
1, center of the second element by 2, etc. The upper bound of the discretization element is denotes by the same number completed
by superscript "+" (plus), the lower bound by superscript "-" (minus); see fig. 4.4B.1.
Next, points in the center of discretization elements are used for computing x-component of electric field intensity.
Since contribution of the current to the magnitude of electric field intensity is described by vector potential without presence of derivatives
(see eqn. 4.4B.2), vector potential is computed in the center of elements.
On the contrary, contribution of charges to electric field intensity is described by
scalar potential
performing two derivations according to x. In numerical computations, derivations are replaced by central differences. Values of scalar
potential V, which derivatives are used for determining contributions of charges to the electric intensity of the radiated wave, have to be
known at the borders of discretization elements so that the result of central differentiating appears in the center of the element. Values of
charge density are computed from
continuity equation
deriving components of current density in centers of discretization elements (derivatives are replaced by central differences again).
In order to obtain values of charge density on the border of elements, we have to differentiate components of current distribution in the center
of elements. This fact suits us very well because the computed values of current density are valid just in these points.
Finally, values of current density components are computed in the center of elements and values of charge density have to be evaluated on borders
of the elements. Therefore, value of components of
vector potential
and value of components of electric field intensity are valid for the center of elements, and value of
scalar potential
for border of elements.
In the next step, we substitute
piecewise constant approximation
of current distribution to initial relations and we replace partial derivatives by central differences. Exploiting
continuity equation,
charge density on the upper edge and on the low one of the discretization element is expressed
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( 4.4B.4a )
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( 4.4B.4b )
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Symbol a height of the discretization element (see fig. 4.4B.1),
Jx(mx, nx) corresponds to a constant value of x-component of current density vector
on the surface of the element containing the central point mx, nx) and ω is angular frequency.
On the basis of equations (4.4B.4a) and (4.4B.4b), we compute contribution of charges, which are
represented by charge density ρ, to the x-component of electric field intensity vector. Considering charge densities of the upper edge
ρ(mx+, nx) and on the low one ρ(mx-, nx),
scalar potential
on those edges can be computed. Moreover, substituting partial derivatives according to x by central differences, we obtain contribution of
charges to x-component of electric field intensity.
Now, charge densities on borders of discretization elements are known and we assume that those values are valid not only on the borders but too
over the whole surface of
charge elements
(they are of the same size as discretization elements but they are shifted so that borders of discretization elements can be in the center of charge
ones as depicted in fig. 4.4B.1). Then, charge densities can be described by the following
piecewise constant functions
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( 4.4B.5a )
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( 4.4B.5b )
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In these relations, Π(xm+, yn| x,y) denotes a function, which is
unitary in the rectangular region with the center in (xm+, yn), a is height of the element and
B is its width. The similar situation is for points (xm-, yn). Values of charge density
ρ(xm+, yn) and (xm-, yn) in the center
of this rectangular region are given by relations (4.4B.5).
If charge density on the
microstrip dipole
is known, we can substitute the distribution to
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and we can compute scalar potential on respective charge elements. For the charge element, which center lies on the upper edge of the element
(mx, nx), we get
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( 4.4B.6 )
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and similarly for V(mx-, nx).
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Fig. 4.4B.2 | Validity region of charge density value ρ(2x+, 1x)
i.e. validity region of scalar potential V(2x+, 1x). Validity region of scalar potential contributions to the magnitude of electric intensity of wave, radiated by antenna |
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Before continuing, let us discuss the relation (4.4B.6). Rearranging it, we swapped integration and summation, and the
integral of the product of the unitary pulse Π and the
scalar Green function
GV over the whole dipole was substituted by the integral of the single scalar Green function over the surface of this
charge element,
where π is non-zero.
Dealing with indexes, (m, n) determines position of the observation element, over which the value of scalar potential
is computed, and indexes (p, q) specifies position of the source element, whose charges contribute to the scalar potential
of the element (m, n).
Scalar Green function
GV is the only continuous function in (4.4B.6), and therefore, the integral has to be evaluated for this
function only. Evaluating this integral for various distances between the source element and the observation one, position of the observation element
is changed only, and the source element stays in the origin of the coordinate system. Therefore, integration limits stay the same in all the cases
(from -a/2 to +a/2 for the coordinate x' and from -B/2 to +B/2 for the coordinate y').
Now, constant values of scalar potential are known over all
charge elements.
Therefore, we can compute contribution of charges, which are represented by scalar potential, to the values of
x-component of electric field intensity
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( 4.4B.7 )
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Here, a denotes height of the element, B is the width of the element, and values of scalar potential V over charge elements
are given by the relation (4.4B.6).
As already said, we are aimed to express an approximation of electric field intensity of the radiated wave on the surface of the dipole as a
function of current density on this dipole. Hence, we substitute (4.4B.6) for
scalar potential
V on the border of elements, which enforces (4.4B.7) to be a function of charge density r
on edges of elements
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( 4.4B.8 )
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In the above-given relation, a denotes height of element and B is its width. Angular frequency ω corresponds to the
frequency, on which antenna is analyzed. The symbol VExS denotes a contribution of
scalar potential
V to the magnitude of x-component of the vector of electric intensity of the radiated wave. Finally, GV
represents integral of
scalar Green function
GV over the surface of the element
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( 4.4B.9 )
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As the next step in expressing electric field intensity as a function of current density, values of charge density from
(4.4B.5) are substituted to (4.4B.8)
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( 4.4B.10 )
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The relation describes the contribution to the component of the vector of electric field intensity using unknown values of the component of current
distribution vector Jx and known coefficients ΓV, given by (4.4B.9). From the point of
view of
scalar potential,
the aim was reached, and therefore, the attention is turned to the
vector potential.
In order to evaluate the contribution of the current to the electric field intensity, we have to compute
vector potential
substituting
piecewise constant approximation
of current density to (4.4B.2a)
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( 4.4B.11 )
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In the above-given relation,
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( 4.4B.12 )
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Next, a is height of the discretization element and B is its width. GAxx denotes x diagonal component of
dyadic Green function.
Function Π(px, qx| x',y') is unitary over an element with the center in the point
(px, qx) and is zero elsewhere. Values Jx represent piecewise constant current density in the
element mesh (px, qx).
During derivation, integration and summation were swapped, and integral over the whole dipole surface S was replaced by the integral over
a single element (due to multiplying by Π the integrand is non-zero over a single discretization element only).
Finally, substituting
vector potential
to (4.4B.3) and replacing derivatives of
scalar potential
by contributions (4.4B.10), we obtain the final equation
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( 4.4B.13 )
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Since eqn. (4.4B.13) is rather complicated, we rewrite it into a matrix form
Here, Ux is column vector of voltages in the direction x over elements. Voltages are computed by multiplying
x-component of electric field intensity by x-dimension of the discretization element
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( 4.4B.15 )
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The symbol a denotes height of the discretization element (i.e. the element dimension in the direction x).
Since the microstrip dipole
is assumed to be fabricated form perfect electric conductor, the vector of voltages consists of zeros only (except of excitation elements).
Next, Ix is column vector of unknown current in the direction x. Elements of Ix
are related to current density Jx by the equation
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( 4.4B.16 )
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(B is width of the dipole, and therefore, even the width of the discretization element). Impedance matrix Zxx
describes contributions of currents Ixx and contributions of charge densities r to voltages Ux
over elements. Single elements of the impedance matrix Zxx are obtained comparing (4.4B.13) to
(4.4B.16)
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( 4.4B.17 )
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In order to evaluate impedance matrix Zxx, we have to know values of integrals of Green functions over the surface
of the discretization element for different distances between source elements and observation ones. Numerical computation of those integrals is
described in the layer D.
The Matlab program for the analysis of the
microstrip dipole
by the
moment method
is described in the layer C from user's point of view.
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