The Microstrip Patch Antenna Computer Science Essay

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Microstrip antennas are used in various applications and these are used extensively because of light weight, conformability and low cost. These antennas can be incorporated with printed strip-line feed networks and active devices. This is a relatively innovative field of antenna engineering. The radiation properties of micro strip structures have been recognized since the mid 1950's. The application of this type of antennas started in early 1970's when conformal antennas were needed for missiles in defense service. Rectangular and circular micro strip resonant patches have been used widely in different array configurations. By developments in large scale integration a major contributing factor for recent advances of microstrip antennas is the current revolution in electronic circuit miniaturization. Generally conventional antennas are bulky and expensive part of an electronic system but the micro strip antennas (based on photolithographic technology) are lighter and cheaper so it seen as an engineering breakthrough.

1.1.1 Introduction [11]

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In its most elementary form, a Microstrip Patch antenna comprises a radiating patch on one side of a dielectric substrate, on the other side which consist a ground plane as shown below in the figure1.1

Figure 1.1 Structure of a Microstrip Patch Antenna

The patch is usually made of conducting material such as copper or gold and any possible shape can be adopted by this patch. The feed lines and radiating patch are generally photo etched, this etching is on the dielectric substrate.

For analysis and prediction of performance, the patch is usually square, rectangular, circular, triangular elliptical or any other common shape as exposed in the Figure 1.2. For a rectangular patch, the length L of the patch is usually 0.3333λ0< L < 0.5 λ0, where λ0 is the free-space wavelength. The patch is chosen to be very thin such as t << λ0 (where t denotes thickness of the patch). The height h of the dielectric substrate is usually 0.003 λ0≤h≤0.05 λ0. The dielectric constant of the substrate (εr) is typically in the range 2.2 ≤ εr≤ 12.

Figure 1.2 Common Shapes of Micro-Strip Patch Elements

Microstrip patch antennas radiate primarily due to the fringing fields between the patch edge and the ground plane. For excellent performance of antenna, a thick dielectric substrate with low dielectric constant is needed because this provides better efficiency, larger bandwidth and better radiation. However, by using this configuration the size of antenna becomes larger. To design a compact Microstrip patch antenna, substrates with higher dielectric constants must be used which are less efficient and have narrower bandwidth. Hence a trade-off must be recognized between the antenna dimensions and its performance.

1.1.2 Benefits and Drawbacks

Microstrip patch antennas are rising in popularity for use in wireless applications because of their low-profile structure. Therefore they have good compatibility for embedded antennas in handheld wireless devices for example cellular phones, pagers etc. The telemetry and communication antennas used on the missiles should to be thin and conformal and these are frequently in the form of Microstrip patch antennas. In Satellite communication they have been used successfully.

1.1.2.1 Advantages of Microstrip Patch Antenna [2]

Some of their main advantages are as follows:

Light weight and low volume.

Low profile planar configuration which can be easily made conformal to host surface.

Low fabrication cost, hence can be manufactured in large quantities.

Supports both, linear as well as circular polarization.

Can be easily integrated with microwave integrated circuits (MICs).

Capable of dual and triple frequency operations.

Mechanically robust when mounted on rigid surfaces.

1.1.2.2 Disadvantages of Microstrip Patch Antenna [2]

Microstrip patch antennas face some serious drawbacks as compared to conventional antennas. Some of their major drawbacks are as follows:

Narrow bandwidth

Low efficiency

Low Gain

Extraneous radiation from feeds and junctions

Poor end fire radiator except tapered slot antennas

Low power handling capacity.

1.1.3 Feeding Techniques [11]

Microstrip patch antennas can be fed by number of techniques. These techniques can be categorized as contacting and non-contacting feeding techniques. In the contacting method, the RF power can be fed directly to the radiating patch by using a connecting element like as microstrip line. In the non-contacting method of feeding, electromagnetic field coupling is done to transfer the power between the microstrip line and radiating patch. There are four most popular feed techniques are used for feeding are micro strip line, coaxial probe (both contacting schemes), aperture coupling and proximity coupling (both non-contacting schemes).

1.1.3.1 Microstrip Line Feed

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In microstrip line feed technique, a conducting strip is directly connected to the edge of the Micro strip patch as exposed in Figure 1.3. The width of conducting strip is smaller in size having comparison with the patch. This kind of feed arrangement has the benefit that it provides a planar structure by etching the feed on the same substrate.

Figure 1.3 Microstrip Line Feed

For impedance matching of the feed line with the patch without the need for any additional matching element the inset cut in the patch is made. By properly controlling the inset position this is obtained easily. Therefore this is a trouble-free feeding technique, because it offers easy fabrication and simplicity in modeling and good impedance matching. On the other hand the thickness of the dielectric substrate increases therefore surface waves and some spurious feed radiation also increases, cause of this the bandwidth of the antenna also increases. Undesired cross polarized radiations are also caused by feed radiation.

1.1.3.2 Coaxial Feed

Usually the Coaxial feed or probe feed is used for feeding the Microstrip patch antennas. As exposed in the Figure 2.4, the coaxial connector's inner conductor expands through the dielectric and this inner conductor is soldered to the radiating patch. The outer conductor of coaxial connecter is connected to the ground plane.

Figure 1.4 Probe fed Rectangular Microstrip Patch Antenna

The major advantage of this type of feeding scheme is that to match with its input impedance the feed can be placed at any desired location inside the patch. This feed method is easy in fabrication and has low spurious radiation. However, a major drawback is that it gives narrow bandwidth and complexity in modeling because drilling a hole in the substrate is needed and the connector protrudes outside the ground plane, thus it is not creating it fully planar for thick substrates (h > 0.02λ0). Also, for thicker substrates, the the input impedance are made inductive by increased probe length which creates matching problems. It is observed above that for a thick dielectric substrate; the microstrip line feed and the coaxial feed suffer from several drawbacks as well as large bandwidth. The non-contacting feed techniques can be used to solve these issues.

1.2 Rectangular Patch Antenna [2]

Microstrip antennas are among the most widely used types of antennas in the microwave frequency range, and they are often used in the millimeter-wave frequency range (below approximately 1 GHz, the size of a microstrip antenna is usually too large to be practical, and other types of antennas such as wire antennas dominate). These are also known as patch antennas, microstrip patch antennas consist of a metallic patch of metal that is on top of a grounded dielectric substrate of thickness h, with relative permittivity and permeability εr and µr as shown in Figure 1.5(a) (usually µr=1). The metallic patch may be of different shapes, with rectangular and circular being the most common, as shown in Figure 1.5(b) and Figure1.5(c).

Fig. 1.5 Rectangular & Circular Patch Antenna

Majority of the discussion in this section will focused on the theory rectangular patch. However the basics are the similar for the circular patch. If the circular patch is modeled as a square patch of the same area number of the CAD formulas given will be applied approximately for the circular patch. Number of methods can be applied to feed the patch as discussed below. The substrate is fairly thin so generally these are low profile structure; this is one of the benefits of the microstrip antenna. If the substrate is thin enough then antenna is "conformal" in the sense that the substrate be capable of to be bent to conform the antenna and it actually becomes to a curved surface (such as a cylindrical structure). 0.02 λ0 is the usual thickness of the substrate. By using a photolithographic etching process the metallic patch is typically made-up or by a mechanical milling process. This makes the construction relatively easy and low-cost (only the substrate material cost is counted). The microstrip antenna is usually light weight for thin substrates and reliable, this is another advantage included with this type of antenna. Microstrip antenna is usually narrowband that is the bandwidths of a few percent this is one of the major disadvantage associated with microstrip antennas. However some methods to increase the bandwidth will be discussed later. Also, the radiation efficiency of the patch antenna is also lesser as compare to the some other types of antennas. Its radiation efficiency exits between 70% and 90% being typical.

1.2.1 Basic Principles of Operation

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A resonant cavity is essentially created by the metallic patch, where the ground plane is on the bottom of the cavity, at the top of the cavity the patch exists, and the sides of the cavity are formed by edges. The patch take actions just about as a cavity with perfect conductor of electricity on the top and bottom surfaces and works as ideal "magnetic conductor" on the sides because the edges of the patch act approximately as an open-circuit boundary condition. For the analysis of the patch antenna this conducting behavior is very useful, it is also useful in understanding its behavior. The electric field is essentially in z direction and independent of the z coordinate inside the patch cavity. Therefore modes of the patch cavity can be explained by a double index (m, n). For the (m, n) cavity mode of the rectangular patch the electric field has the form

1.1

Where the patch length is denoted by L and the patch width is denoted by W. The patch is usually operated in the (1, 0) mode and in the y direction the field is essentially constant. The surface current on the bottom of the metal patch is in the x direction and it is given as:

1.2

For the operation in this mode the patch can be observed as a wide microstrip line having width W, resonant length is L; this length L is around one-half wavelength in the dielectric. At the centre of the patch (at x = L/2), the current is maximum while the electric field is maximum at the two "radiating" edges. To increase the bandwidth the width W is normally choosen to be larger than the length (W = 1.5 L is typical), as the bandwidth is proportional to the width. The width must be kept lesser than the two times the length, though to keep away from excitation of the (0, 2) mode. At first sight, it might appear that when the substrate is electrically thin the microstrip antenna will not be an effective radiator, since the patch current in (2) will be effectively shorted by the close proximity to the ground plane. If the modal amplitude A10 is constant, in fact the radiated field strength would be proportional to h. However, the increment in the Q is observed with cavity as decrement in h (because the radiation Q is inversely proportional to h). Therefore, it is concluded that the amplitude A10 of the modal field at resonance have inverse relationship with h. Consequently, the radiated field strength from a resonant patch is basically independent of h, if losses are not considered. The resonant input resistance is also almost not dependent on h. This make clears why a patch antenna can be work as an efficient radiator still for very skeletal substrates, even if the bandwidth will be smaller.

1.2.2 Resonant Frequency [2]

The resonance frequency for the (1, 0) mode is given by

1.3

Where c is the speed of light in space. The effective length Le to overcome the fringing effect of the cavity fields at each side of the edges of the patch length, the effective length Le is chosen as Le = L + 2ΔL. The Hammerstad formula for the fringing extension is given as:

1.4

Where: 1.5

1.3 Methods of Analysis [11]

There are different model for the parameter analysis of microstrip antenna which are listed below:

Approximate Model

Electromagnetic Simulation Model

Artificial Neural Network Model

Approximate model is based on numerical solution based on empirical formula (such as transmission line and cavity model). Electromagnetic simulation model is based on full wave such as method of moment and also with IE3d simulator. Artificial Neural Network Model uses neural model for the analysis.

Earlier the transmission line model, cavity model and full wave model (method of moment) are the preferred models for analyzing the microstrip patch antenna. Among them, the simplest technique model is transmission line model and it also providing the excellent physical insight but having less accurate. The cavity model provides the good physical insight and also more accurate as compare to transmission line model but a great complexity is associated with it. The accuracy of the full wave model are much more and versatile and it can be treated as single element, finite and infinite arrays, arbitrary shaped elements and coupling. Full wave model provides less physical insight as compare to the transmission line and cavity model and also this model are more complex in nature.

1.3.1 Transmission Line Model [18]

This model presents the micro strip patch antenna by using the two slots of height h and the width W which are apart from a transmission line of length L. The two non homogeneous lines of two dielectrics (the substrate and the air) are the fundamental part of microstrip as shown in Figure 1.6.

Figure 1.6 Microstrip Line Figure 1.7 Electric Field Lines

From the above figure we can observe that a great no. of electric field lines exist within the substrate itself and some parts of line reside in the air. Since the phase velocities of air and the substrate is differs from each other therefore this model is not supported the pure transverse-electric magnetic (TEM) mode of transmission. The transmission line model supports the quasi-TEM mode for the transmission. An effective dielectric constant (εreff) is necessary to calculate for the explanation of fringing and propagation of wane in line. The value of εreff is slightly lesser than εr because around the border of the patch the fringing fields are not confine in the dielectric substrate but these are also spread over the air shown in above Figure 1.6. The equation for the effective dielectric constant (εreff) is given by Balanis [2] as:

1.6

Where: εreff= Effective dielectric constant

εr = Dielectric constant of substrate

h = Height of dielectric substrate

W = Width of the patch

Figure 1.8 Top View of Antenna Figure 1.9 Side View of Antenna

From the Figure 1.8, we can conclude that the normal electric field components at the two edges are kept in opposite direction along the width of microstrip antenna and become out of phase. Since the patch is λ/2 long and electric field components is out of phase so in broadside direction they cancel the each other. The in phase tangential components provides the maximum radiated field normal to the surface structure by combining the resulting in phase components (as shown in Figure 1.9). These two edges can be treated as the two radiating slots alongside the width, that are separated by λ/2 and excite in phase are radiating in the half space above the earth plane. The fringing fields alongside the width of microstrip patch can modeled as radiating slots. Electrical dimensions of the microstrip patch looks larger than its physical dimensions. The extended dimensions ΔL on the each side along the patch length can be given by Hammerstad [3] as:

1.7

The effective length of the patch Leff now becomes:

1.8

The effective length for a given resonance frequency f0 is given by:

1.9

The resonance frequency fo for any rectangular microstrip patch antenna, for any TMmn mode can be given by James and Hall [14] as:

1.10

where m and n are modes along length L and width W respectively.

The width W for efficient radiation can be given by Bahl and Bhartia [15] as:

1.11

1.3.1.1 Limitation of Transmission Line Model

The basic limitation of transmission line model is it yields the least accurate results and it lacks the versatility. However, it does shed some physical insight. It also ignores field variations along the radiating edges.

1.3.2 Cavity Model [20]

Although the transmission line model is easy to use in practical approach but it has some inherent disadvantages associated with it. Specifically, it is useful for designing the rectangular patch and it also not insists in the field variation alongside the radiating edges (as in transmission line model). By using the cavity model these disadvantages can be overcome. A small introduction of the cavity model is presented below. In the cavity model, a cavity is bounded with the electric walls on the top and bottom and the interior region of the dielectric substrate which is modeled as the cavity.

The basis for this assumption is the following observations for thin substrates (h << λ).

Since the substrate is thin, the fields in the interior region do not vary much in the z direction, i.e. normal to the patch.

The electric field is z directed only, and the magnetic field has only the transverse components Hx and Hy in the region bounded by the patch metallization and the ground plane.

This observation provides for the electric walls at the top and the bottom.

Figure 1.10 Charge distribution and current density creation on the microstrip patch Consider

Consider Figure 1.10 shown above. When the power is provided to microstrip patch, the upper and lower surfaces of the patch a charge distribution is observed and also at the bottom of the ground plane. This charge distribution is controlled by two mechanisms as an attractive mechanism and a repulsive mechanism are discussed by Richards. In the attractive mechanism is applied in between the opposite charges on the bottom side of the patch and the ground plane, which helps in keeping the charge concentration intact at the base of the patch while the repulsive mechanism is applied between the same charges on the base surface of the patch, it causes pushing of some charges from the bottom, to the top of the patch. Because of this charge movement, currents flow at the top and bottom surface of the patch. The cavity model assumes that the height to width ratio (i.e. height of substrate and width of the patch) is very small and as a result of this the attractive mechanism dominates and causes most of the charge concentration and the current to be below the patch surface.

1.12

QT is total antenna quality factor and has been expressed by:

1.13

Qd represents the quality factor of the dielectric and given as:

1.14

Where: denotes the angular resonant frequency.

WT denotes the total energy stored in the patch at resonance.

Pd denotes the dielectric loss.

tan δ denotes the loss tangent of the dielectric.

Qc represents the quality factor of the conductor and is given as:

1.15

Where: Pc denotes the conductor loss.

Δ denotes the skin depth of the conductor.

h denotes the height of the substrate

Qr represents the quality factor for radiation and given as:

1.16

Where: Pr denotes the power radiated from the patch.

Substituting equation (1.13), (1.14), (1.15) and (1.16) in equation (1.12), we get

1.17

1.3.3 Electromagnetic Simulation Model [22]

Electromagnetic simulation model is based on full wave analysis such as method of moment and also with IE3d simulator. Electromagnetic simulation model give more accurate analysis of microwave patch antenna parameters - such as S-parameters, radiation patterns,etc. -compared to the approximate models, such as transmission line model, cavity model but suffer from the drawback of time-consuming intensive computations compared to the approximate models which are less accurate but faster.

CHAPTER 2

This chapter deals with Artificial Neural Network and their various types, backpropagation algorithm, working principle of backpropagation and different types of training used by backpropagation.

CHAPTER 2

ARTIFICIAL NEURAL NETWORK

2.1 Introduction To Artificial Neural Network [13]

Neural network has been motivated from the human brain because the human brain is an extremely composite, nonlinear and parallel computer. In the human brain it is expected that there are 10 billion neurons. In the human mind the neurons (which are the basic part of human brain) are structured in a manner so that human brain does any tasks much time faster than the any fastest digital computers which are in existence till today. On the time of birth, the human brain posses great structure and having ability to develop its own decision with the help of "experience". Any neural network consist of simple processing unit which have the natural phenomena of storing knowledge depending upon experience and makes this knowledge to available whenever it need to use. Neurons reassemble the human brain in two manners. Firstly, the knowledge is collected by the neurons form its neighboring environment with use of learning process and secondly this knowledge are stored with the help of interneuron connection strengths, commonly known as synaptic weights.

2.2 Network Architectures

The way by which the neurons of a neural network are organized with the help of synaptic weights is very well inked with the learning algorithm for trained to neural network. We may therefore speak of learning algorithms (rules) used in the design of neural networks as being structured.

We can categorize the different classes of neural network in two fundamental classes such as:

2.2.1 Single-Layer Feedforward Networks [13]

In neural network, neurons are arranged in layered form. In the simplest form of a layered network, we have an input layer of source nodes that projects onto an output layer of neurons (computation nodes), but not vice versa as shown in Figure2.1. Such a network is called a single- layer network, with the designation "single-layer" referring to the output layer of computation nodes (neurons).

Input layer Output layer

of source nodes of neurons

Fig. 2.1 Feedforward or Acyclic Network with a Single Layer of Neurons

2.2.2 Multilayer Feedforward Networks [13]

Another class of a feedforward neural network differs to the previous feedforward network because of one or more no. of hidden layers of neuron that are presented in the network. The computational unit are called as hidden neurons or hidden node. The basic aim of hidden neurons is to make an interconnection in between the external input and the output of the network in useful manner. The advantage of adding one or more hidden layers of neuron to the network is to make the network capable of extract the high order information as shown in Figure 2.2. In the input layer of network, the basic nodes of provide respective information of the input vector (activation pattern), which constitute the input signals applied to the neurons (computation nodes) in the second layer (i.e., the first hidden layer). The out coming signals from the second layer are treated as the input information for the third layer, and this process carry on for rest of network. Typically the neurons in each layer of the network have as their inputs the output signals of the preceding layer only. The overall performance of the network is determined by the output data sets of neurons of last layer which has the activation data pattern supplied with the help of source nodes on the input layer of the network.

Input layer of Layer of Layer of

source nodes hidden neurons output neurons

Fig. 2.2 Fully Connected Feedforward or Acyclic Network with One Hidden Layer and One Output Layer.

2.2.3 Multilayer Perceptrons[13]

This type of network uses a no. of source node on the input layer, one or more hidden layers of neurons as computation nodes and a layer of computation node at the output. The input signal from the source node is propagated to output layer through hidden layers on layer by layer in forward direction in the network. This arrangement of the neural network, which is a generalized version of single layer perceptron is commonly known as multilayer perceptron as shown in Figure 2.3.

Multilayer perceptrons uses the vastly popular algorithm known as the back-propagation algorithm for solving some of the different and difficult problems by supervised training used by back-propagation algorithm.

The three unique characteristics of the Multilayer perceptrons are listed below:

In the network, the each neuron includes a nonlinear activation function.

The one or more layers of hidden neurons presented in the network are not the part of either the input layer or the output layer of the network. These layers of hidden neuron in the network are used to learn the complex tasks with extracting more information from the source node vectors.

Changes in the population of synaptic connections or their weights are required whenever a change in the connection of the network is needed because the network has high degrees of connectivity which is determined by the network synapses.

Input layer First hidden layer Second hidden layer Output layer

Fig. 2.3 Architectural Graph Of A Multilayer Perceptron With Two Hidden Layers.

2.3 The Backpropagation Algorithm [13]

In the layered feedforward artificial neural network (ANN), the backpropagation algorithm is used for the training of network. This shows that the neurons of ANN are arranged in a layered way in which the signals or data are sends in forward direction towards the output with the help of hidden layer neurons. The error is calculated at the output which is propagated backward direction for the correction. There may be one or more intermediates layer of hidden neurons. The supervised learning is used by the backpropagation algorithm that means we provided a set of data with the example of input and output. The output of the network is comparing with given output and the error is calculated (difference between actual and expected results) and fed it back to the network. The main aim of the backpropagation algorithm is to reduce these errors to a desired value defined in the training. The training in backpropagation algorithm is starts with the random weights of neurons and algorithm goal is set to them a value so that the error becomes minimal.

The ith backpropagation algorithm is a weighted sum (the sum of the inputs x multiplied by their jith respective weights w) of the activation function of ANN.

2.1

For a linear neuron the output function should be identical (that means output=activation).Linear function neuron has the some limitations. The most common output function is signmoidal function and given as:

2.2

The sigmoidal function is identical to one for a large positive no., and zero for the smaller values. So this types of the function shows the smooth transition between the low and high values output of the neuron.

The basic goal of the backpropagation algorithm is to achieve the desired output for a certain inputs parameter by the proper training. The minimization of the error can be achieved if we adjust the weights of neurons because the error (difference between the actual and desired output) is depends on that weight. The error function at the output of the network can be defined as:

2.3

We take the error function as square because it will be always positive and also if the difference between the actual and desire output is large, the error is greater and if the difference is small, the error is lesser. The total error at the output of the neural network is basically the sum of errors of all the neurons presented at the output layer and given as:

2.4

Now we calculates that in which manner the error depends on the input, output and weights in the backpropagation algorithm and after getting this result ,we adjust the weights with the help of gradient decent method as given in equation (2.5)

2.5

In general the following formula can be written as: the change (adjustment) in the weight (w) will be negative of the dependence of the previous weight on the error of network, which is the derivative of E in respect to w and multiplied by a constant known as eta (η). The size of the adjustment will be depending on the constant eta (η) and also the amount of weight which contribute to the error function. That means the how large the amount of weight is used in error function, the adjustment will be much large and the adjustment will be small if weight contribute a less in to error function. The adjustment of weight is carry on until we find the correct weight for each of neuron so that the error reduces to its minimal value.

The goal of the backpropagation algorithm is to find the derivative of E with respect to weight w. Since we need to achieve this in backward direction so firstly we need to calculate that how much error depends on the output of the network which can define as derivative of E with respect to O.

2.6

In last we calculate that the amount of output that depends on the activation, which depends on the weight w as:

2.7

with the help of above equations, we get

2.8

the adjustment to each weight will be given as:

2.9

The two layered ANN can be trained with the above equation.

But to train the ANN with large (greater than two) layers, we have to make some considerations such as if we need to adjust the ikth weights of the previous layer then firstly we have to calculate that how the error depends on that weight as the input from the previous layer.

2.10

where:

2.11

and, assuming that there are inputs u into the neuron with v

2.12

On adding another layer we calculate the same by calculating that how the error depends on the inputs and weight associated with the first layer.

2.3.1 Different Training Models for Backpropagation Algorithm

There are several backpropagation training model which are listed below and categorized under three different section based on their training speed.

Gradient Descent backpropagation

Gradient Descent with momentum backpropagation

Variable Learning Rate backpropagation

Resilient backpropagation

Scale Conjugate Gradient backpropagation

Quasi Newton backpropagation

Levenberg Marquardt backpropagation

The first two training models are come in category of slow training model which are too slow for the practical problems. The last four training models come in category of fast training model which are further divided in two section one is based on heuristic techniques, which were developed from an analysis of the performance of the standard steepest descent algorithm (Variable Learning Rate and Resilient backpropagation) while the second uses standard numerical optimization techniques (Scale Conjugate Gradient, Quasi Newton and Levenberg Marquardt backpropagation).

In our thesis, we basically deal with fast training models which use the standard numerical optimization techniques.

2.3.1.1 Scale Conjugate Gradient Training

The basic of the entire backpropeagation algorithm to set the weights of all the neurons in the steepest down direction (negative to gradient) that is in the direction in which the performance function decay more quickly. The performance function does not essentially generated the best ever convergence even though the function falls most quickly with the negative of the gradient. In SCG algorithm a search is perform along conjugate directions that provided usually quicker convergence than steepest drop directions.

In most of training models a learning rate is used to decide the span of weight update (step size). In the conjugate gradient algorithm generally the weight update is done at each iteration. A investigate is done alongside the conjugate gradient direction to decide the weight update that minimize the performance error function alongside to line.

2.3.1.2 Basic step of Scale Conjugate Gradient

The first iteration of each conjugate algorithm is usually begun with a search within the steepest down direction (negative to gradient).

After that a sequential search is done to decide the best possible distance to move all along the present search direction as:

2.13

The new search direction is decided in order, that it is conjugate to the preceding search directions. The new steepest drop direction is combining with the preceding search directions in order to decide the new search direction:

2.14

All the conjugate gradient algorithms are differs from each other in a manner that how the constant βk is decided. The βk for the Fletcher-Reeves update is given as:

2.15

2.3.1.3 Quasi Newton Training

For the fast optimization, an option to the conjugate gradient algorithm is Quasi-Newton's algorithm. The step of Quasi-Newton's algorithm is given as:

2.16

where is defined as the Hessian matrix of second derivatives of concert index with respect to present value of weights and biases. The main advantage of Newton's algorithm over the conjugate gradient algorithm is that it often convergence more rapidly than the conjugate gradient algorithm. But it is too difficult and costly to calculate the Hessian matrix for any feedforward ANN. A group of algorithms which belongs to the Newton algorithm does not need to calculate the second derivatives. These types of algorithm are known as quasi Newton methods. Such a algorithm, at the each iteration are used to revise the Hessian matrix and the revise is calculated as a function of the gradient.

2.3.1.4 Levenberg Marquard Training

LM algorithm was also designed to get high order training speed without evaluating the Hessian matrix. Whenever the concert function has in the form of sum of squares as usually in all training feedforward ANN, then the Hessian matrix can be written as: H=JTJ 2.17

and the gradient can be given as:

g=JTe 2.18

where J is Jacobian matrix which have first derivatives of the network errors with respect to the weights and biases. The Levenberg-Marquardt method uses this Hessian matrix approximation as:

Xk+1=Xk-[JTJ+ µI]-1 JTe 2.19

When the scalar µ is zero it behaves like Newton's and when µ is large it behave like gradient descent training with small steps.

CHAPTER 3

This chapter deals with limitation in existence, objective of thesis, used methodology and data sets provided to the Artificial Neural Network for the training of network.

CHAPTER 3

PROBLEM FORMULATION

3.1 Limitation in Existence

Theoretically it is very difficult to calculate the output resonant frequency of large data sets. But with using ANN the process to calculate the resonant frequency is so easy, once the neurons are trained after that it gives the output very fast with very less error about 0.7%. Neural network is employed as a tool in design of the micro-strip antennas. In this we will trained the neurons on the basis of input to find the resonant frequency of rectangular micro-strip antenna.

3.2 Objective

Artificial neural network (ANN) models have been built usually for the analysis of micro-strip antennas in various forms such as rectangular, circular, and equilateral triangle patch antennas.

In this work, rectangular micro-strip antennas are the ones under consideration. The patch dimensions of rectangular micro-strip antennas are usually designed so its pattern maximum is normal to the patch. Because of their narrow bandwidths and effectively operation in the vicinity of resonant frequency, the choice of the patch dimensions giving the specified resonant frequency is very important. The analysis problem can be defined as to obtain resonant frequency for a given dielectric material and geometric structure. However, in the present work, the corresponding synthesis ANN model is built to obtain patch dimensions of rectangular micro-strip antennas (W,L) as the function of input variables, which are the height of the dielectric substrate (h), dielectric constants of the dielectric material (εr,εy ) and the resonant frequency (fo). This synthesis problem is solved using the electromagnetic formulae of the micro-strip antennas. In this formulation, 2 points are especially emphasized: the resonant frequency of the antenna and the condition for good radiation efficiency. Using reverse modeling, an analysis ANN is built to find out the resonant frequency immediately for a given rectangular micro-strip antenna system.

3.3 Methodology

Figure 3.1 shows the methodology that we have used for this thesis.

Literature survey

Study of micro-strip rectangular patch antenna

Study of artificial neural network

Study of back propagation algorithm & ANN tool box on Matlab

Generation of data set using formulas

Training of neural network using generated data set

Testing of neural model

Result

Fig 3.1 Methodology of Project

3.4 Calculation of resonant frequency of rectangular micro-strip antenna

In this project, almost all works have been done by choosing the dielectric substrate to be in an isotropic structure. So in this work, the ANN model is capable of giving results for both isotropic and anisotropic structures of the dielectric substrate. For an anisotropic substrate, the spacing parameter h is replaced by the effective spacing he, and the geometric mean εg is used for the dielectric constant εr:

3.1

3.2

The effective dielectric constant of the dielectric material is given in (3.2):

3.3

The actual length of the patch is given as:

3.4

where;

where ΔL is the extension of the length due to the fringing effects and is given by:

3.5

3.5 Table of input data set and corresponding output resonant frequency

h(m)

Єr

w(m)

L(m)

Fr

0.0032

2.33

0.057

0.038

2.4595

0.0032

2.45

0.057

0.038

2.4038

0.0032

2.33

0.059

0.038

2.4574

0.0032

2.33

0.0445

0.038

2.4729

0.0032

2.33

0.0455

0.038

2.4713

0.0032

2.33

0.0465

0.0315

2.9307

0.0032

2.33

0.0455

0.0305

3.0191

0.0032

2.43

0.0312

0.0305

2.9983

0.0033

2.41

0.0321

0.0366

2.5461

0.0033

2.43

0.0321

0.0356

2.602

0.0095

2.43

0.0321

0.0366

2.2975

0.0095

2.43

0.0312

0.0366

2.3009

0.0095

2.43

0.0312

0.0195

3.6842

0.0095

2.55

0.0195

0.0195

3.7426

0.0097

2.55

0.0195

0.0195

3.7312

0.0045

2.61

0.0875

0.0152

4.8602

0.0048

2.61

0.0875

0.0152

4.808

0.0048

2.61

0.0753

0.0152

4.8301

0.0048

2.61

0.0793

0.0152

4.8222

0.0048

2.66

0.0793

0.0186

4.0876

0.0048

2.66

0.0977

0.0186

4.063

0.0048

2.66

0.0987

0.0186

4.0619

0.0055

2.66

0.0987

0.0162

4.3887

0.0055

2.66

0.0986

0.0162

4.3888

0.0055

2.66

0.0989

0.0162

4.3884

0.0097

2.55

0.0169

0.0112

5.3838

0.0097

2.55

0.0199

0.0112

5.2987

0.0097

2.55

0.0174

0.0112

5.3682

0.0097

2.55

0.0174

0.0125

5.0324

0.0097

2.55

0.0142

0.0125

5.1331

0.0041

2.55

0.0123

0.0145

5.6301

0.0042

2.55

0.0125

0.0145

5.604

0.0042

2.52

0.0125

0.0145

5.6297

0.0042

2.58

0.0125

0.0145

5.5787

0.0042

2.58

0.0797

0.0145

5.1507

0.0042

2.58

0.0797

0.0144

5.1783

0.0045

2.58

0.0797

0.0144

5.1041

0.0045

2.58

0.0875

0.0144

5.0903

CHAPTER 4

This chapter deals with implementation of Neural Network, Specification of ANN, Neural Model, training and testing data sets, result and their discussion.

CHAPTER 4

RESULT AND DISCUSSION

4.1 Implementation of Neural Network

The implementation of data set on neural network is shown in flow chart Figure.4.1

Epochs calculation of resonant frequency of micro-strip antenna

Creating the feed forward network

Enter the pattern & target

Run for weight & biases

Set initial weight & biases

Set the training parameter

net. trainParam.time=07

Train the network

Simulate the network

4.2 Specification of ANN

The architecture of neural network used is [4x5x5x1], i.e. this neural network contains four input parameters εr(Permittivity in the x), L (Length of the patch), W (Width of the patch), h (Height of the dielectric substrate), two hidden layers of 5 neurons each and one output neuron. For training this neural network we have used "Levenberg-Marquardt optimization algorithm" (LM) algorithm. LM algorithm is a fast algorithm for training neural network. This network takes 2844 no. of epochs to get trained for given data set.

'purelin'= Pure linear

'tansig',= Ten sigmoidal

'trainlm'= Train data with Levenberg-Marquardt algorithm

[5'10'1]= 5 and10 hidden neurons, 1 output neuron and corresponding to this 4 input neurons.

p= Values of input data sets.

t= value of frequency from input data sets.

W= Weights.

b= Bias.

Training Condition

No. of epochs

500000

Training parameter goal

0.0000001

No. of input parameters

4

No. of hidden layers

2

No. of hidden neurons

10

No. of output prarmeter

1

No. of training condition

2

Table 4.1 Training condition for neural network

4.3 Neural Model

On the basis of 4 input neurons, 5 & 5 hidden neurons and 1 output neuron the neural model is shown in fig. 4.2.

h

εr

f0

w

L

Input layer First hidden layer Second hidden layer Output layer

Fig.4.2 Neuron Model with Two Hidden Layers with Five Neurons Each, Four Neurons In Input Layer And One Neuron In Output Layer.

4.4 Training & Testing Data Set

The training and testing data set are given in table 4.2 and table 4.3.

4.4.1 Data Set for Training

For the training of neural network the data set are given bellow:

h(m)

Єr

w(m)

L(m)

Fr(GHz)

(Th)

0.0032

2.33

0.057

0.038

2.4595

0.0032

2.45

0.057

0.038

2.4038

0.0032

2.33

0.059

0.038

2.4574

0.0032

2.33

0.0445

0.038

2.4729

0.0032

2.33

0.0455

0.038

2.4713

0.0032

2.43

0.0312

0.0305

2.9983

0.0033

2.41

0.0321

0.0366

2.5461

0.0033

2.43

0.0321

0.0356

2.602

0.0095

2.43

0.0321

0.0366

2.2975

0.0095

2.43

0.0312

0.0366

2.3009

0.0095

2.43

0.0312

0.0195

3.6842

0.0045

2.61

0.0875

0.0152

4.8602

0.0048

2.61

0.0875

0.0152

4.808

0.0048

2.61

0.0793

0.0152

4.8222

0.0048

2.66

0.0793

0.0186

4.0876

0.0048

2.66

0.0977

0.0186

4.063

0.0048

2.66

0.0987

0.0186

4.0619

0.0055

2.66

0.0987

0.0162

4.3887

0.0055

2.66

0.0989

0.0162

4.3884

0.0097

2.55

0.0169

0.0112

5.3838

0.0097

2.55

0.0199

0.0112

5.2987

0.0097

2.55

0.0174

0.0112

5.3682

0.0097

2.55

0.0174

0.0125

5.0324

0.0042

2.55

0.0125

0.0145

5.604

0.0042

2.52

0.0125

0.0145

5.6297

0.0042

2.58

0.0797

0.0145

5.1507

0.0042

2.58

0.0797

0.0144

5.1783

0.0045

2.58

0.0797

0.0144

5.1041

0.0045

2.58

0.0875

0.0144

5.0903

Table 4.2 Training Data Set

4.4.2 Data Set for Testing

For the testing of neural network the data set are given bellow:

h(m)

Єr

w(m)

L(m)

fr(GHz)

Th.

0.0032

2.33

0.0455

0.0305

3.0191

0.0095

2.55

0.0195

0.0195

3.7426

0.0048

2.61

0.0753

0.0152

4.8301

0.0055

2.66

0.0986

0.0162

4.3888

0.0097

2.55

0.0142

0.0125

5.1331

0.0042

2.58

0.0125

0.0145

5.5787

0.0041

2.55

0.0142

0.0125

6.2695

0.0097

2.55

0.0195

0.0195

3.7312

0.0032

2.33

0.0465

0.0315

2.9307

0.0041

2.55

0.0123

0.0145

5.6301

Table 4.3 Testing Data Set

4.5 Results

To analysis the parameter of microstrip antenna with FFBP-ANN using different training technique .An 30 input output training patterns are used for training of 5-10-1 ANN structure with performance goal Mean Square error (MSE)=1e-007 and maximum number of epochs set is 500000. With the learning method of MLFFBP-ANN based model it obtained that 2844 number of epochs are required to achieve to reduce the MSE level to 1e-007.While when these training patterns are applied to SCGFFBP-ANN model it takes near a 45000 epochs to achieve the same results.

The achievement of performance goal (MSE) has been done with LMFFBP-ANN with lesser number of epochs as compare to SCGFFBP-ANN. So LMFFBP-ANN model is more accurate and fast for the analysis of microstrip antenna.

4.5.1 Result Of ANN After Training

In table 4.4 the percentage value of error between resonant frequency (Theoretical) and resonant frequency (neural network) after training was given. The absolute error is about 0.0038GHz.

h(m)

Єr

w(m)

L(m)

Fr(GHz)

Fr(GHz)

Error

(Th)

(NN)

(GHz)

0.0032

2.33

0.057

0.038

2.4595

2.4595

0

0.0032

2.45

0.057

0.038

2.4038

2.4038

0

0.0032

2.33

0.059

0.038

2.4574

2.4573

1E-04

0.0032

2.33

0.0445

0.038

2.4729

2.4727

0.0002

0.0032

2.33

0.0455

0.038

2.4713

2.4716

0.0003

0.0032

2.43

0.0312

0.0305

2.9983

2.9983

0

0.0033

2.41

0.0321

0.0366

2.5461

2.5461

0

0.0033

2.43

0.0321

0.0356

2.602

2.602

0

0.0095

2.43

0.0321

0.0366

2.2975

2.2986

0.0011

0.0095

2.43

0.0312

0.0366

2.3009

2.2998

0.0011

0.0095

2.43

0.0312

0.0195

3.6842

3.6842

0

0.0045

2.61

0.0875

0.0152

4.8602

4.8602

0

0.0048

2.61

0.0875

0.0152

4.808

4.8083

0.0003

0.0048

2.61

0.0793

0.0152

4.8222

4.822

0.0002

0.0048

2.66

0.0793

0.0186

4.0876

4.0876

0

0.0048

2.66

0.0977

0.0186

4.063

4.0631

0.0001

0.0048

2.66

0.0987

0.0186

4.0619

4.0617

0.0002

0.0055

2.66

0.0987

0.0162

4.3887

4.3887

0

0.0055

2.66

0.0989

0.0162

4.3884

4.3884

0

0.0097

2.55

0.0169

0.0112

5.3838

5.3838

0

0.0097

2.55

0.0199

0.0112

5.2987

5.2987

0

0.0097

2.55

0.0174

0.0112

5.3682

5.3682

0

0.0097

2.55

0.0174

0.0125

5.0324

5.0324

0

0.0042

2.55

0.0125

0.0145

5.604

5.604

0

0.0042

2.52

0.0125

0.0145

5.6297

5.6297

0

0.0042

2.58

0.0797

0.0145

5.1507

5.1507

0

0.0042

2.58

0.0797

0.0144

5.1783

5.1783

0

0.0045

2.58

0.0797

0.0144

5.1041

5.1042

1E-04

0.0045

2.58

0.0875

0.0144

5.0903

5.0902

1E-04

0.0038

Table 4.4 Result of ANN After Training

Fig. 4.3 Performance goal met after training with LMBP-ANN

Capture 2 with scale cojuate Fig.4.4 Performance goal met after training with SCGBP-ANN

Figure (4.3) shown above shows a graph between MSE(Mean Square error) and no. of epochs for the LM (Levenberg Marquard) training models and it notice that after a 2844 no. of epochs this training model get the performance goal while SCG(Scale Conjugate Gradient) training model could not met these performance goal as shown in fig.(4.4).

4.5.2 Result of ANN After Testing

In table 4.5 the percentage value of error between resonant frequency (Theoretical) and resonant frequency (neural network) after testing was given. The absolute error is about 0.7813GHz.

h(m)

Єr

w(m)

L(m)

fr(GHz)

Fr(GHz)

Error

Th.

(NN)

(GHz)

0.0032

2.33

0.0455

0.0305

3.0191

2.8978

0.1213

0.0095

2.55

0.0195

0.0195

3.7426

3.8366

0.094

0.0048

2.61

0.0753

0.0152

4.8301

4.8286

0.0015

0.0055

2.66

0.0986

0.0162

4.3888

4.3888

0

0.0097

2.55

0.0142

0.0125

5.1331

5.1061

0.027

0.0042

2.58

0.0125

0.0145

5.5787

5.5532

0.0255

0.0041

2.55

0.0142

0.0125

6.2695

6.5822

0.3127

0.0097

2.55

0.0195

0.0195

3.7312

3.8197

0.0885

0.0032

2.33

0.0465

0.0315

2.9307

2.8294

0.1013

0.0041

2.55

0.0123

0.0145

5.6301

5.6396

0.0095

0.7813

Table 4.5 Result of ANN After Testingfig 2 with LmBP Method

Fig.4.6 Target-Output Variation Result With LMBP-ANN

Capture 2 with scale cojuate

Fig.4.7 Target-Output Variation Result With SCGBP-ANN

Fig.(4.6)and fig.(4.7) shows a graph variation between the output generated after training with Neural Network and target output for LM and SCG training model respectively. From the above, we can say that in certain condition LM training model is superior than SCG training models.

CHAPTER 5

This chapter deals with the conclusion of the thesis and their advancement in the future.

CHAPTER 5

CONCLUSION AND FUTURE SCOPE

5.1 CONCLUSION

In this work, the neural network is working as a tool in design of the microstrip antennas. In this design method, synthesis is defined as the forward side and then analysis as the reverse side of the problem. For that reason, one can obtain the geometric dimensions with high accuracy, which is the length and the width of the patch in our geometry, at the output of the synthesis network by inputting resonant frequency, height and dielectric constants of the chosen substrate. Furthermore, in our work, the synthesis can also be applied into anisotropic dielectric substrate. In this work, the analysis is considered as a final stage of the design procedure, therefore the parameters of the analysis ANN network are determined by the data obtained reversing the input-output data of the synthesis network. Thus, resonant frequency resulted from the synthesized antenna geometry is examined against the target in the analysis ANN network. Finally, in this work, a general design procedure for the microstrip antennas is suggested using artificial neural networks and this is demonstrated using the rectangular patch geometry.

5.2 FUTURE SCOPE

The future scope of work revolves around increasing the efficiency and decreasing the run time of the Neural Network by using other MLP algorithms like "Variable Learning Rate (VLR) backpropagation algorithm" and "Resilient backpropagation algorithm" and also with Radial basis function (RBF) network.