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Analysis of Finite Ground Plane Effects on Antenna
Performance Using Discrete Green’s Function
Salma Mirhadi, Mohammad Soleimani, and Ali Abdolali
Iran University of Science & Technology, School of Electrical Engineering
Tehran, Iran,, and
Abstract—In this paper, the application of the discrete Green's
function (DGF) method in the monopole antenna modeling with
finite and infinite ground plane is presented. In the infinite
ground plane case, the current distribution of antenna is
obtained using image theory. While, in the finite ground plane
case, the infinite free space DGF is used to obtain the current
distribution on the antenna and ground plane. It is shown that
the finite ground plane affects the properties of antenna,
particularly the radiation pattern.
Keywords-component; current distribution, discrete Green’s
function; finite ground plane; monopole antenna.
should be mainly concerned with more application of the DGF
method, especially, in antenna modeling. Therefore, the aim of
this paper is to apply DGF method to understand the effects of
the finite ground plane on the antenna performance. Without
loss of generality of this method, the monopole antenna has
been studied. The current distribution along the monopole
antenna as well as on the finite ground plane is determined
using DGF method. We show that the radiation pattern for the
monopole antenna is strongly affected by a finite sized ground
plane while the impedance of a monopole antenna is
minimally affected.
The initial concept of Discrete Green’s Function (DGF)
was developed by Vazquez and Parini in 1999 [1]. They
derived the analytical closed form of the DGFs for infinite free
space by the multidimensional z-transform of FDTD equations
in time and spatial domains. They also showed that the field
response of an arbitrary current source can be obtained as the
convolution of the impulse response of the FDTD equations
(DGF) and the current density. Then, they applied those
expressions in antenna modeling by satisfying of the boundary
conditions on the scatterers [2]. In fact, in this procedure, the
induced currents on the antenna are updated instead of
updating the fields in FDTD methods. Therefore, this
formulation does not require absorbing boundary conditions or
the computation of the free space nodes around the scatterers.
Other references, but not so much, presented further
investigation into this area. In [3], Kastner described the need
for an FDTD-compatible discrete Green’s function instead of
discretizing the continuous Green’s functions directly. He also
obtained another analytical expression for frequency and time
domain of DGF. By modeling of a Yagi-Uda array antenna
using DGF, Weili Ma et al. demonstrated considerable saving
in computing time and memory storage compared with the
traditional FDTD method [4]. Furthermore, Jeng derived new
closed form expressions for the dyadic discrete Green’s
function in free space using the ordinary z-transform along
with the spatial partial difference operators [5]. He claimed
that the extracted expressions are easier than those in [1].
At this point, the DGFs have been fully derived and its
property has been studied in detail. The next stage of work
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In this section, discrete Green's functions equations, derived
in [1-2, 4], have been briefly mentioned. The Green's functions
of the vector wave equation can be obtained through the
Green's function of the scalar wave equation by applying the
relationship between them. The discrete version of the scalar
wave equation with Kronecker delta excitation can be
expressed as:
g in, +j ,1k − 2 g in, j ,k + g in, −j ,1k
c ( Δt )
g in+1, j ,k − 2 g in, j ,k + g in−1, j ,k
g in, j +1,k − 2 g in, j ,k + g in, j −1,k
( Δy ) 2
( Δx ) 2
g in, j ,k +1 − 2 g in, j ,k + g in, j ,k −1
( Δz ) 2
= δ in−−i′n, ′j − j′,k − k ′
The solution of (1) can be achieved using the
multidimensional z transform versus Jacobi polynomials as [1]:
⎛n − m⎞
⎟⎟.∑ (n − 2m; p x , p y , p z )
g in, +j ,1k = ∑ ( −1) m ⎜⎜
m =0
⎝ m ⎠
∏ α sps J p( ls,−l )l (ξ s )( R2 s − R1s ) ps −l
t =i , j ,k
s= x , y ,z
⎛ Δx 2 + Δy 2 + Δz 2 ⎞
α −1
⎟⎟, β s = s
, R1, 2 s = β s ± β s − 1
R + R2 s
(n − 2m )!
ξ s = 1s
, ( n − 2m; p x , p y , p z ) =
R1s − R2 s
px! p y ! pz!
αs =
c 2 Δt 2
Δs 2
Image theory, infinite ground plane
finite ground plane, xg=landa/4
finite ground plane, xg=landa/2
finite ground plane, xg=landa
Amplitude of Current
Figure 1.
Current distribution of monopole antenna with finite ground plane
in DGF analysis.
The vector Green's functions take the form of a matrix. For
example, the vector Green's function relating the electric field
to the electric current is as:
[G ]
ej i , j ,k
⎡G xx
G xy
G yx
Δt ⎢
⎢G zx
G yy
G zy
G xz ⎤
G yz ⎥ g in, j ,k
G zz ⎥⎦
Where the matrix element are computed as:
[G ]
ejxx i , j ,k
= g in, j ,k − g in, −j 1,k − α x ∑ g il−−11, j ,k − 2 g il,−j1,k + g il+−11, j ,k
l =0
[G ]
= α x ∑ g il,−j1+1,k − 2 g il−−11, j +1,k + g il−−11, j ,k − g il,−j1,k
[G ]
= α x ∑ g il,−j1,k +1 − 2 g il−−11, j ,k +1 + g il−−11, j ,k − g il,−j1,k
ejxy i , j ,k
ejxz i , j ,k
l =0
ejzz i , j ,k ,k ′
n (Time Steps)
= Gejzz
i , j ,k − k ′
+ Gejzz
i , j ,k +k ′
In this case, it is only necessary to compute the semiinfinite DGF along the antenna. Then, the update equation for
the current on the antenna can be used as [2]:
dyadic DGF on the antenna and the finite ground plane. In
other words, in the infinite ground plane case, the current on
the monopole antenna in one dimension (Jz) has been
calculated. While, in the finite ground plane case, the one
dimension current on the antenna and two dimension currents
on the ground plane (Jx and Jy) have been calculated.
The monopole antenna with the length of λ / 4 at 300 MHz
has been studied. The spatial and time increments have been
selected as Δx = Δy = Δz = λ / 40 , cΔt = 0.5Δx respectively.
The antenna is excited by the incident Gaussian electric field.
The temporal responses of current at the feed point are shown
in Fig.2. As we can see, the smaller the ground plane, the
sooner the current change compared to the infinite ground
plane case due to the fact that the reflection of the ground
plane edge occurs faster.
[G _ image ]
l =0
n +1
z i , j ,k inc
n −1
− E z i , j ,k inc − ∑∑ J z i , j ,k ′ G _ imageejzz
n′= 0 k ′
i , j ,k ,k ′
However, in the finite ground plane case, we use the
infinite free space DGF and compute the all components of the
Impednace (Ohm)
In this section, modeling of monopole antenna with infinite
and finite ground plane has been done using DGF. The mesh
discretization and the current distribution of the monopole
antenna and the finite ground plane are shown in Fig.1.
In the infinite ground plane case, the infinite free space
DGF has been modified as (4) using image theory:
J z i , j ,k = −
Figure 2. Temporal Response of Current at the feed point to the Gaussian
incident field, xg( the side length of the square ground plane).
Other elements can be obtained similarly.
Imag−Infinite Ground
Real−Infinite Ground
Imag−finite xg=landa/4
Real−finite xg=landa/4
Real−finite xg=landa/2
Imag−finite xg=landa/2
Real−finite xg=landa
Imag−finite xg=landa
Frequency (MHz)
α + β = χ.
Figure 3. Imedance of monopole antenna with various groun plane size.
The real and imaginary part of impedance for different
plane sizes is shown in Fig.3. The impedance curve for finite
ground plane does not dramatically change compared to
infinite ground plane case. It can be seen from the figure that,
at 300 MHz, the imaginary part of impedance can be negative
for the finite ground plane.
The far field radiation pattern of antenna can be
determined from the magnetic vector potential obtained
through the integration of current. As we can see in Fig. 4, the
radiation pattern is strongly affected by the size of ground
plane. The direction of peak-radiation has changed from the xy plane to an angle elevated from that finite ground plane.
There is also back radiation that reduced with increasing
ground plane size.
Figure 4. Radiation pattern of monopole antenna with various groun plane
J. Vazquez and C.G. Parini, “Discrete Green’s function formulation of
FDTD method for electromagnetic modelling,” ELECTRONICS
LETTER, vol. 35, No. 7, pp. 554–555, April 1999.
J. Vazquez and C.G. Parini, “Anetnna modelling using discrete Green’s
function formulation of FDTD method,” ELECTRONICS LETTER, vol.
35, No. 13, pp. 1033–1034, June 1999.
R. Kastner, “ A Multidimensional z-transform evaluation of the discrete
finite difference time domian Green’s function,” IEEE Trans. Antennas
Propag., vol. 54, No.4, pp. 1215-1222, April 2006.
W. Ma, M. R. Rayner, and C. G. Parini, "Disctere Green's Function
Formulation of the FDTD Method and Its Application in Antenna
Modelin," IEEE Trans. Antennas Propag., vol. 53, No.1, pp. 339-364,
January 2005.
S. K. Jeng, "An Analytical Expresion for 3-D Dyadic FDTD-Compatible
Green's Function in Infinite Free Space via z-Transform and Partial
Difference Operators," IEEE Trans. Antennas Propag., vol. 59, No.4,
pp. 1347-1355, April 2011.
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