Circuits and Systems, 2011, 2, 330-337
doi:10.4236/cs.2011.24046 Published Online October 2011 (http://www.scirp.org/journal/cs)
Artificial Intelligence in the Estimation o f Pa tc h
Dimensions of Rectangular Microstrip Antennas
Vandana Vikas Thakare1, Pramod Singhal2
1Department of Electronics and Instrumentation Engineering, Anand Engineering College, Keetham, Agra, India
2Department of Electronics Engineering, Madhav Institute of Technology & Science, Gwalior , India
Received July 13, 2011; revised August 8, 2011; accepted August 15, 2011
Abstract
Artificial Neural Network (ANNs) techniques are recently indicating a lot of promises in the application of
various micro-engineering fields. Such a use of ANNs for estimating the patch dimensions of a microstrip line
feed rectangular microstrip patch antennas has been presented in this paper. An ANN model has been devel-
oped and tested for rectangular patch antenna design. The performance of the neural network has been com-
pared with the simulated values obtained from IE3D EM Simulator. It transforms the data containing the di-
electric constant (εr), thickness of the substrate (h), and antenna’s dominant-mode resonant frequency (fr) to
the patch dimensions i.e. length (L) and width (W) of the patch. The different variants of back propagation
training algorithm of MLFFBP-ANN (Multilayer feed forward back propagation Artificial Neural Network)
and RBF-ANN (Radial basis function Artificial Neural Network) has been used to implement the network
model. The results obtained from artificial neural network when compared with simulation results, found
satisfactory and also it is concluded that RBF network is more accurate and fast as compared to different
variants of back propagation training algorithms of MLPFFBP. The ANNs results are more in agreement
with the simulation findings. Neural network based estimation has the usual advantage of very fast and
simultaneous response of all the outputs.
Keywords: Microstrip Antenna, Bandwidth, Simulation, Modelling, Neural Networks, CAD
1. Introduction
Microstrip antennas are used in a wide range of mobile
communication applications which demands multi band
and/or wideband frequency operations, high power gain
omni directional radiations patterns etc. Therefore design
of printed antennas to meet the requirements of multiple
operational services becomes a difficult task. This war-
rants in the very high accuracy of the calculation of
various design parameters of microstrip patch antennas.
Patch dimensions of a rectangular microstrip antenna is a
vital parameter in deciding the performance and the utility
of an antenna. In the present work, microstrip line feeding
is taken as a preferred method of feeding the input power
to the antenna. The calculation of exact patch dimensions
of rectangular microstrip patch antenna becomes ex-
tremely important where the antenna size is drastically
small. A number of papers have been appeared on the
calculation of patch dimension of microstrip antennas
[1-3]. However, these papers suffer considerable deviation
in the calculated value of patch dimensions compared to
theoretical and simulation findings. In this paper, an at-
tempt has been made to exploit the capability of artificial
neural networks to calculate the length (L) and width (W)
of microstrip patch antenna over a ground plane with a
substrate thickness h and dielectric constants εr. The results
are in good agreement with the simulation findings.
Neural networks have recently gained attention as a fast
and flexible vehicle to EM /Microwave modeling, simu-
lations and optimization. Recently CAD approach based
on neural networks has been introduced in the microwave
community for modeling of passive and active microwave
component. Number of research papers [4-14] indicates
how ANN can be used efficiently to calculate different
design and performance parameters of microstrip anten-
nas. However the literature shows that only three layer
MLPFFBP has been preferred to prove the utility of ANN
in the area of microstrip antenna design. In this work, the

V. V. THAKARE ET AL.331
authors extend the work on the use of the artificial neural
network (ANN) technique taking into account different
variants of back propagation training algorithm with
MLPFFBP and RBF ANN model is stressed upon in place
of conventional numerical techniques for the microstrip
antenna design.
2. Design and Data Generation
The rectangular microstrip antennas are made up of a
rectangular patch with dimensions width (W) and length
(L) over a ground plane with a substrate thickness h
having dielectric constant εr. There are numerous sub-
strates that can be used for the design of microstrip an-
tennas, and their dielectric constants are usually in the
range of 2.2 < εr < 12. Thin substrates with higher di-
electric constants are desirable for microwave circuitry
because they require tightly bound fields to minimize
undesired radiation and coupling, and lead to smaller
element.
The software used to model and simulate the proposed
microstrip patch antenna is Zeland Inc’s IE3D software.
IE3D is a full-wave electromagnetic simulator based on
the method of moments. It analyses 3D and multilayer
structures of general shapes. It has been widely used in the
design of MICs, RFICs, patch antennas, wire antennas,
and other RF/wireless antennas. It can be used to calculate
and plot the S11 parameters, VSWR, current distributions
as well as the radiation patterns.
As an example microstrip line feed rectangular patch
microstrip antenna is designed to resonate at 8 Ghz fre-
quency with dielectric constant (εr) = 2, substrate thick-
ness h = 1 mm, L = 12.6 mm, W = 15.3 mm. The length
and the width of the patch are calculated by the given
relationships. (1), (2), (3) and (4) mentioned in [15,16].
rr
2
W2f 1




v (1)
L
2f



rreff
v2L
(2)
where v

is the free space velocity of the light.


W
0.3 0.264
Lh
0.412 W
h0.258 0.8
h










reff
reff

(3)
where is extension in length due to fringing effects
and effective dielectric constant is given by
L
1/2
rr
11 h
112
22 W






 




reff (4)
The transmission line model is applicable to infinite
ground planes only. However, for practical considerations,
it is essential to have a finite ground plane. It is known
that similar results for finite and infinite ground plane can
be obtained if the size of the ground plane is greater than
the patch dimensions by approximately six times the
substrate thickness all around the periphery. Hence, for
this design, the ground plane dimensions would be given
as:
Lg = 6 (h) + L = 6(1) + 12.6 = 17.6 mm (5)
Wg = 6 (h) + W= 6(1) +15.3 = 21.3 mm (6)
With the calculated values of various design parameters
the patch antenna is designed for 8 GHz resonating fre-
quency. The exact position of feed point can be deter-
mined by using IE3D Electromagnetic Simulator. The
width Wo of microstrip line taken as 0.5 mm and the feed
length is 2 mm. The patch is energized electromagneti-
cally using 50 ohm microstrip feed line. The geometry of
the example antenna is as shown in the Figure 1.
IE3D software has been used to calculate the return loss
(S11) & hence the dominant resonating frequency of the
antenna and the data is generated in the form of fr (reso-
nating frequency) for different specified range i.e. 1.5_ εr
_ 3.5, 1 mm _h _5 mm, 11 mm _ L _ 16.5 mm and 13 mm
_W _19 mm and has been used to train the various ANN
model. The generated data were then arranged in five
matrices. The feed coordinates and dimensions of micro-
strip line are kept constant and corresponding resonating
frequencies are recorded and this data has been used as a
training data and test data for MLFFBP and RBF ANN.
The three matrices containing the values of εr, h and fr are
used as the input to the network. The other two matrices
containing the corresponding values of L and W i.e.
length and width of the patch are the outputs of the neural
network.
Figure 2 shows the return loss (S11) vs. frequency
curve for the given physical dimension for the example
antenna indicating that antenna is resonating at 8 GHz.
Figure 1. Microstrip line feed rectangular patch antenna.

V. V. THAKARE ET AL. 332
Figure 2. The return loss (S11) in dB vs. resonating fre-
quency of microstrip antenna.
The length, width, substrate thickness h and dielectric
constant (εr), were varied for the specified range to see
the effect on the microstrip antenna bandwidth. It was
observed that antenna performance could be controlled
by varying these parameters to a large extent.
3. Network Architecture and Training
For the present work three layer network MLFFBP [17,18]
with 6 different training algorithms and RBF network is
preferred to model the microstrip line feed patch antenna.
3.1. Multi Layer Feed Forward Back
Propagation (MLFFBP)
MLP networks are feed forward networks trained with the
standard back propagation algorithm as shown in Figure
3. They are supervised networks and also they required a
desired response to be trained. With one or two hidden
layers they can approximate virtually any input output
map. The weights of the network are usually computed by
training the network using the back propagation algo-
rithm.
In the present work three layers multilayer perceptron
feed-forward back propagation artificial neural network
with one hidden layer and trained by six different variants
of back propagation training algorithms is used and
compared to design microstrip patch antenna.
Back propagation [17] was created by generalizing the
Window-Hoff learning rule to multiple-layer networks
and nonlinear differentiable transfer functions. Standard
back propagation is a gradient descent algorithm, in which
the network weights are moved along the negative of the
gradient of the performance function.
There are many variations of the back propagation al-
gorithm. The simplest implementation of back propaga-
tion learning updates the network weights and biases in
the direction in which the performance function decreases
most rapidly—the negative of the gradient. Six different
variants of back propagation algorithm have been used to
train the MLPFFBP network model developed for the
proposed design.
Scaled Conjugate Gradient (SCG) This belongs to
the class of conjugate gradient methods which show super
linear convergence on the most problems. This was de-
signed to avoid the time consuming line approach. The
algorithm is an implementation of avoiding the compli-
cated line search procedure of conventional conjugate
gradient algorithm
Bayesian Regularization algorithm (BR) This algo-
rithm updates the weight and bias values according to the
Levenberg-Marquardt optimization and minimizes a lin-
ear combination of squared errors and weight. It also
modifies the linear combination so that at the end of
training the resulting network has good generalization
quality.
Levenberg-Marquardt optimization algorithm (LM)
This is a least square estimation method based on the
minimum neighborhood idea and does not suffer from the
problem of slow convergence. The LM method combines
the best feature of the Gauss Newton technique and the
steepest decent method but avoid many of their limita-
tions.
Quasi Newton algorithm (QN) This is based on
Newton’s method but doesn’t require calculation of sec-
ond derivatives; an approximate Hessian Matrix is up-
dated. At each iteration of the algorithm the update is
computed as a function of gradient Conjugate.
Gradient of Fletcher Reeves algorithm (CGF) In this
algorithm a search is performed along the conjugate di-
rections which produces generally faster convergence
than steepest descends direction. The algorithm updates
weight and biases values according to the formulas pro-
posed by Fletcher and Reeves.
Adaptive G radient Decent algorithm (AGD) It trains
any network as long as its weight, net input, and transfer
functions have derivative functions. Back propagation is
used to calculate derivatives of performance with respect
to the weight and bias variables. Each variable is adjusted
according to gradient descent.
ANN structure i.e. number of layers, number of neu-
rons in each layer, neurons activation function, learning
algorithm and training parameters is not known in ad-
vance. Hence the network model is analysed with differ-
ent number of hidden layers in the structure and also the
numbers of processing elements are also varied to acquire
the accuracy. Hence it is concluded that three layer MLP
with one hidden layer and 20 processing elements in the
hidden layer is the optimum network structure for the

V. V. THAKARE ET AL. 333
Figure 3. Three layer feed forward artificial neural network.
proposed problem. The network is trained with six dif-
ferent training algorithms to achieve the required degree
of accuracy and hence compared for network perform-
ance.
In the network there are three input neurons in the input
layer, 20 hidden neurons in the hidden layer and two
output neurons in the output layer. The various inputs to
the network are εr, h and fr and the outputs of the neural
network are L and W i.e. length and width of the patch.
The training time is 35 seconds and training performs in
361 epochs. In this work, different algorithms are used for
training the proposed neural model and a comparative
evaluation of relative performance is carried out for es-
timating patch dimensions L and W of microstrip antenna.
In order to evaluate the performance of proposed
MLFFBP-ANN based model for the design of microstrip
antenna, simulation results are obtained using IE3D
Simulator and generated 120 input-output training pat-
terns and 30 inputs-output test patterns to validate the
model. The network has been trained for different εr, h and
fr values of the proposed example but in a specified range.
During the training process the neural network auto-
matically adjusts its weights and threshold values such
that the error between predicted and sampled outputs is
minimized. The adjustments are computed by the back
propagation algorithm. The training algorithm most
suitable is trainlm .The error goal is 0.001 and learning
rate is 0.1. The other network parameters used were noise
factor of 0.004 and momentum factor of 0.075. The
transfer function preferred is tansig in the architecture.
3.2. RBF Networks
Radial basis function network [17,18] is a feed forward
neural network with a single hidden layer that use radial
basis activation functions for hidden neurons. RBF net-
works are applied for various microwave modeling pur-
poses. The RBF neural network has both a supervised and
unsupervised component to its learning. It consists of
three layers of neurons—input, hidden and output. The
hidden layer neurons represent a series of centers in the
input data space. Each of these centers has an activation
function, typically Gaussian. The activation depends on
the distance between the presented input vector and the
centre. The further the vector is from the centre, the lower
is the activation and vice versa. The generation of the
centers and their widths is done using an unsupervised
k-means clustering algorithm. The centers and widths
created by this algorithm then form the weights and biases
of the hidden layer, which remain unchanged once the
clustering has been done. A typical RBF network struc-
ture is given in Figure 4.
The parameters cij and λij are centers and standard de-
viations of radial basis activation functions. Commonly
used radial basis activation functions are Gaussian and
Multiquadratic. Given the inputs x, the total input to the
ith hidden neuron i

is given by Equation (7).
2
1
,1,2,3,, 

 





njij
i
jij
xc iN
(7)
where N is the number of hidden neurons. The output
value of the ith hidden neuron is zij = σ (i

) where σ (i

)
is a radial basis function. Finally, the outputs of the RBF
network are computed from hidden neurons as shown in
Equation (8)
0
N
kki
i
yw

ki z (8)
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V. V. THAKARE ET AL. 334
λ
ij
＞c
ij
Figure 4. Radial basis function (RBF) artificial neural net-
work.
where wki is the weight of the link between the ith neuron
of the hidden layer and the kth neuron of the output layer.
Training parameters w of the RBF network include wk0,
wki, cij, λij, k = 1,2, ···, m, I = 1 ,2, ···, N, j = 1,2, ···, n.
In the RBF network, the spread value was chosen as
0.01, which gives the best accuracy. The network was
trained with 120 samples and tested with 30 samples. In
the structure there are 3 inputs and 2 outputs are chosen
for the present synthesis ANN Model. RBF networks are
more fast and effective as compared to MLPFFBP for this
antenna design example. The RBF network automatically
adjusts the number of processing elements in the hidden
layer till the defined accuracy is reached. In the present
work the RBF ANN model the network is choosing 26
processing elements in the hidden layer. The training time
is 25 seconds and training performs in 212 epochs and the
training algorithm is unsupervised k-means clustering
algorithm.
4. Results and Conclusions
4.1. Results
It has been established from Table 1 that the Leven-
berg-Marquardt algorithm with structure (3-20-2) is the
suitable model to achieve optimal values of speed of
convergence and accuracy in case of MLPFFBP. It has
been observed that total number of 125 epochs as shown
in Figure 5 is needed to reduce MSE level to a low value
1.6e - 027 for three layers MLPFFBP. With Levenberg-
Marquardt (LM) training algorithm and tansig as a
transfer function achievement of such a low value of
performance goal (MSE) indicates that trained ANN
model with trainlm as a training algorithm is an accurate
model for designing the Microstrip patch antenna. The
Figure 5. Number of epochs to achieve minimum mean
square error level with Levenberg-Marquardt (LM) as a
training algorithm in case of MLPFFBP.
maximum absolute error at each value of L and W (patch
dimensions) antenna is estimated for the random values of
input parameters but in specified range i.e. the range for
which network is trained. Various transfer functions are
used for training the network and average minimum MSE
on training and CV data is measured. It is observed that
tansig as shown in Figure 6 is most suitable transfer
function for the present work. The MLP neural network is
trained using learning rules namely Levenberg-Marquardt
(LM), Scale Conjugate Gradient Back propagation (SC-
GBP), Gradient of Fletcher Reeves algorithm (GFR),
Quasi Newton algorithm (QN), Bayesian Regularization
Algorithm (BR) and Adaptive Gradient Decent (AGD).
Minimum MSE and maximum absolute error is measured
on training and test data and is indicated in Table 1 and
Figure 7. It is concluded that Levenberg- Marquaradt is
most suitable learning rule for our neural network with
3-20-2 structure. For generalization the randomized data
is fed to the network and is trained for different hidden
layers. It is observed that MLP with single hidden layer
gives better performance as shown in Figure 8. The
number of Processing Elements (PEs) in the hidden layer
is also varied. The network is trained and minimum MSE
is obtained when 20 PEs are used in hidden layer as shown
in Figure 9.
As the work signifies RBF ANN is also used to model
the microstrip line feed patch antenna. It is established
from Table 1 and Figure 7 that RBF is giving results not
only more accurate but fast also. The presented RBF
network has performed training in only 74 epochs to the
MSE value of 6.79e-28 as shown in Figure 10. So it is
concluded that RBF architecture is better from MLPFFBP
to the accuracy of 99.69% and quite faster comparatively.
4.2. Conclusions
In this paper, an attempt has been made to indicate to the

V. V. THAKARE ET AL. 335
Table 1. Comparison of different variants of back propagation training algorithm for the design of microstrip line feed rec-
tangular patch Antenna using artifici al ne ural network.
Training algorithms Number
of epochs
Mean
square error
Estimation of length (L)
Maximum Absolute Error
Training data Test data
Estimation of (W)
Maximum Absolute Error
Training data Test data
Levenberg-Marquardt (LM) 125 1.6e - 027 0.0141 0.1813 0.0227 0.1765
Scale Conjugate Gradient Back
propagation (SCGBP) 450 9.20e - 25 0.0784 0.6221 0.1334 0.4190
Gradient of Fletcher Reeves
algorithm (CGF) 640 2.65e - 13 0.1519 1.6140 0.1724 1.9992
Quasi Newton algorithm (QN) 800 8.03e - 19 0.1290 1.7645 0.1645 1.3004
Bayesian Regularization
Algorithm (BR) 590 3.67e - 20 0.1721 1.8999 0.1205 1.4143
Adaptive Gradient Decent(AGD) 485 1.77e - 026 0.0667 0.3215 0.0814 0.5216
RBF 74 6.79e - 28 0.0008 0.0748 0.0016 0.1027
Table 2. Comparisons of results of IE3D EM simulator and RBF ANN for the calculation of patch dimensions of microstrip
antenna.
S. No. εr h mm fr GHz W (IE3D) mm W (RBF ANN) mmL (IE3D) mm L (RBF ANN) mm
1 2 1 8.0 15.3 15.23 12.6 12.54
2 2.2 1.4 7.51 15.6 15.56 12.8 12.72
3 2.5 1.7 6.82 15.9 16.0 13.1 12.99
4 2.7 1.9 6.45 13.9 14.01 13.3 13.31
5 2.9 1 7.12 13.7 13.72 13.5 13.48
6 2.6 1.8 6.73 13.2 12.99 12.2 12.30
7 3 1.2 6.84 14.4 14.38 12.4 12.43
8 2.4 1.7 8.83 15.4 15.50 13.4 13.21
9 2.1 1.5 8.79 13 13.10 11.0 11.10
10 2.8 1.6 6.39 15.2 15.13 13.2 13.10
11 2.9 1.7 7.34 15.4 15.51 13.4 13.31
0
0.02
0.04
0.06
0.08
0.1
0.12
hardlim tribas logsig purelin satlin tansig
Transfer functions
Average Minim u m M S E
FOR TRAINING DATAFOR TEST DATA
Figure 6. Graph showing variation of average minimum
MSE on Training and Test data set for different transfer
functions in the neural network.
reader’s one of the approach to model the patch antenna
using MLFFBP and RBF-ANN. The results obtained with
the present technique were closer to the simulation results
generated by simulating the example antenna in IE3D
Simulator. Table 1 shows the comparison of results of
different variant of back propagation algorithm of
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
LM
SCGBP
CGF
QN BR
AGD
RBF
Traini ng a l gorithms
A ver ag e ab so l u te er r o r
FOR TRAINING DATAFOR TEST DATA
Figure 7. Graph showing variation of average absolute er-
ror on training and test data for different training algo-
rithms
MLPFFBP ANN and RBF ANN. The paper concludes
that results obtained using present ANN techniques are
quite satisfactory and also that RBF ANN is giving the
best approximation to the design as compared to
MLPFFBP ANN. Table 2 respectively depicts the com-
parison of results between RBF ANN and simulated values
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V. V. THAKARE ET AL. 336
0
0.02
0.04
0.06
0.08
0.1
0.12
12345678910
Number of hidden la
y
ers
Average Minimum MSE
FOR TRAINING DATAFOR TEST DATA
Figure 8. Graph showing variation of average minimum
MSE on Training and Test data set for different no. of the
hidden layers in the neural network.
0
0.02
0.04
0.06
0.08
0.1
0.12
2610 1418
No. of neurons in the hidden la
y
er
Averag e M inimum MSE
FOR T RAI NING DAT AFOR TEST DATA
Figure 9. Graph showing variation of average minimum
MSE on Training and Test data set for different no. of hid-
den layers in the networ k.
Figure 10. Number of epochs to achieve minimum mean
square error level with RBF ANN.
for those 11 input combinations which are not included in
the set of training data and found satisfactory.
A neural network-based CAD model is developed for
the design of a rectangular patch antenna, which is robust
both from the angle of time of computation and accuracy.
A distinct advantage of neuro computing is that, after
proper training, a neural network completely bypasses the
repeated use of complex iterative processes for new cases
presented to it. The developed network structure can
predict the results for patch dimensions provided that the
values of εr, fr and h are in the domain of training values.
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