Study and Analysis of Wire Antenna using Integral Equations: A MATLAB Approach

2016 International Conference on Micro-Electronics and Telecommunication Engineering Study and Analysis of Wire Antenna using Integral Equations: A MATLAB Approach 1 Shekhar, 2 Taimoor Khan, 3 Abhishek Singhal, 4 Santosh Prasad Singh, 1 Department Of ECE, Dewan VS Institute of Engineering & Technology, Meerut 2 Department Of ECE, National institute of Technology,Silchar, Assam 3 Department Of ECE, SRM University, NCR Campus, Modipuram 3 Department Of EIE, SCRIET, CCS University Campus, Meerut Abstract- An Optimization numerical techniques for a very thin wire to find accurate current distribution, input impedance and radiation pattern using Pocklington s integral equation and Hallen s integral equation. However generally the current distribution is usually assumed to be of sinusoidal form finite diameter wires the sinusoidal current distribution is representative accurate. In this techniques both integral equation and some mathematical formulation is being done and shown with help of MATLAB. If we know the voltage at the feed terminals of a wire antenna and find the current distribution, the input impedance and radiation pattern can then be obtained. Keywords: Wire Antenna,Current distribution, Radiation Pattern,input impedance and MATLAB I. INTRODUCTION In a conductor the movement of the free electrons on the conductors represents the transmission line and the antenna. In order to illustrate the creation of the current distribution on a wire(dipole), and its subsequent radiation, let us first begin with the geometry of a lossless two-wire transmission line. The movement of the charges creates a traveling wave current, of magnitude Io/2, along each of the wires. When the current arrives at the end of each of the wires, it undergoes a complete reflection (equal magnitude and 180 phase reversal). The reflected traveling wave, when combined with the incident traveling wave, forms in each wire a pure standing wave pattern of sinusoidal form as shown in figure Figure 1 The current in each wire undergoes a 180 phase reversal between adjoining half-cycles. This is indicated in figure by the reversal of the arrow direction. Radiation from each wire individually occurs because of the time-varying nature of the current and the termination of the wire. In the two-wire balanced (symmetrical) transmission line, the current in a half Figure 2 half cycle of one wire is of the same magnitude but 180 out-of-phase from that in the corresponding half-cycle of the other wire. If in addition the spacing between the two wires is very small (s << ), the fields radiated by the current of each wire are essentially cancelled by those of the other. The result is an almost ideal (and desired) non radiating transmission line. II. FINITE LENGTH DIPOLE The techniques that were developed previously can also be used to analyze the radiation characteristics of a linear dipole of any length. To reduce the mathematical complexities, it will be assumed in this paper that the dipole has a negligible diameter (ideally zero). This is a good approximation provided the diameter is considerably smaller than the operating wavelength. The Current distribution for a very thin dipole (ideally zero diameter), the current distribution can be written, to a good approximation, as This distribution assumes that the antenna is center-fed and the current vanishes at the end points (z = ±l/2). Experimentally it has been verified that the current in a center-fed wire antenna has sinusoidal form with nulls at 978-1-5090-3411-6/16 $31.00 2016 IEEE DOI 10.1109/ICMETE.2016.126 94 93

the end points. For l = /2 and /2 < l < the current distribution is plotted as. Usually we are limited to the far-field region, because of the mathematical complications provided in the integration of the vector potential A. Since closed form solutions, which are valid everywhere, cannot be obtained for many antennas, the observations will be restricted to the farfield region. As the number of subdivisions is increased, each infinitesimal dipole approaches a length dz. For an infinitesimal dipole of length dz; positioned along the z- axis at z, the electric and magnetic field components in the far field are given, as Figure-3 Summing the contributions from all the infinitesimal elements, the summation reduces, in the limit, to an integration. Thus Figure-4 The input impedance was defined as the ratio of the voltage to current at a pair of terminals or the ratio of the appropriate components of the electric to magnetic fields at a point. The real part of the input impedance was defined as the input resistance which for a lossless antenna reduces to the radiation resistance, a result of the radiation of real power. In the previous section, the radiation resistance of an infinitesimal dipole was derived. The radiation resistance of a dipole of length l with sinusoidal current distribution, was also explained. By this definition, the radiation resistance is referred to the maximum current which for some lengths (l = /4, 3 /4,, etc.) does not occur at the input terminals of the antenna. i.e. Radiated Fields for the current distribution closed form expressions for the E- and H-fields can be obtained which are valid in all regions (any observation point except on the source itself). In general, however, this is not the case. III INTEGRAL EQUATION TECHNIQUE- MOMENT METHOD A very thin wires, the current distribution is usually assumed to be of sinusoidal form finite diameter wires (usually diameters d of d > 0.05 ), the sinusoidal current distribution is representative but not accurate. To find a more accurate current distribution on a cylindrical wire, an integral equation is usually derived and solved. Previously, solutions to the integral equation were obtained using iterative methods presently. it is most convenient to use moment method techniques. If we know the voltage at the feed terminals of a wire antenna and find the current distribution, the input impedance and radiation pattern can then be obtained. Similarly, if a wave impinges upon the surface of a wire scattered it induces a current density that in turn is used to find the scattered field. Whereas the linear wire is simple, most of the information presented here can be readily extended to more complicated structures. The impedance of an antenna depends on many factors including its frequency of operation, its geometry, its method of excitation, and its proximity to the surrounding objects. Because of their complex geometries, only a limited number of practical antennas have been investigated analytically. For many others, the input impedance has been determined experimentally The impedance of an antenna at a point is defined as the ratio of the electric to the magnetic fields at that point; alternatively, at a pair of terminals, it is defined as the ratio of the voltage to the current across those terminals. There are many methods that can be used to calculate the 95 94

impedance of an antenna Generally, these can be classified into three categories: (1) the boundary-value method, (2) the transmission line method, and (3) the Poynting vector method. In this chapter the integral equation method, with a Moment Method numerical solution, will be introduced and used first to find the selfand driving-point impedances, and mutual impedance of wire type of antennas. This method casts the solution for the induced current in the form of an integral (hence its name) where the unknown induced current density is part of the integrand. Numerical techniques, such as the Moment Method can then be used to solve the current density. In particular two classical integral equations for linear elements, Pocklington s and Hallen s Integral Equations, This approach is very general, and it can be used with today s modern computational methods and equipment to compute the characteristics of complex configurations of antenna elements, including skewed arrangements. For special cases, closed-form expressions for the self, driving-point, and mutual impedances will be presented using the induced emf method. This method is limited to classical geometries, such as straight wires and arrays of collinear and parallel straight wires. Pocklington's integral Equation technique, with a Moment Method numerical solution, will be acquainted and utilized first with locate the self-and driving-point impedances, and mutual impedance of wire sort of antennas. This strategy casts the solution for the instigated current in the form of an integral (henceforth its name) where the obscure induced current density is a piece of the integrand. Numerical strategies, for example, the Moment Method can then be utilized to explain the current density. In particular two classical integral equations for linear elements, Pocklington s. integral equation and This approach is very general, and it can be utilized with today's modern computational techniques and equipment to compute the characteristics of complex configurations of antenna components, including skewed courses of action. For uncommon cases, closed-form expressions for the self, driving-point, and mutual impedances will be exhibited utilizing the induced emf technique. This strategy is limited to classical geometries, for example, straight wires and arrays of collinear and parallel straight wires. Figure 5 The Pocklington s integral equation is Appeared in Figure 1, and it is alluded to as the incident electric field Ei (r). At the point when the wire is a antenna, the incident field is created by the feed source at the gap, as appeared in Figure 5. Part of the incident field impinges on the wire and actuates on its surface a linear current density Js (amperes per meter). The induced current density Js reradiates and creates an electric field that is alluded to as the scattered electric field Es(r). In this manner, anytime in space the total electric field Et(r) is the sum of the incident and scattered fields, or Pocklington's integral equation to determine the proportional filamentary line-source current of the wire, and therefore current density on the wire, by knowing the incident field on the surface of the wire. In the event that we expect that the wire is thin ( a<< ) such that equation reduces to 96 95

Equation can also be expressed in a more convenient form as distinctive points. This is alluded to as point- matching (or collocation). Doing this, and Where for observations along the center of the wire ( = 0) In matrix form, IV. MOMENT METHOD SOLUTION Pocklington's Integral Equations (3), has the form of where F is a known linear operator, h is a known excitation function, and g is the response function. F is an integrodifferential operator while for and it is a integral operator. The target here is to decide g once F and h are specified. While the opposite issue is often intractable in closed form, the linearity of the operator F makes a numerical solution possible. One technique, known as the Moment Method requires that the obscure response function be extended as a linear combination of N terms and composed as The unknown coefficients an can be found by solving using matrix inversion techniques, or V. RESULT USING MATLAB According to the expression given for current distribution, radiation pattern input impedance utilizing Pocklington integral Equation Moment Method, distinctive plot will satisfy the equation with MATALB program. Each a will be an obscure consistent and each gn(z' is a referred function usually referred to as a premise or extension function. The area of the gn(z') functions is the same as that of g(z'). Substituting (6) into (5) and utilizing the linearity of the F operator diminishes (5) to The fundamental functions gn are picked so that each F(gn) in above can be assessed conveniently, ideally in closed form or at the very least numerically. The main assignment staying then is to discover the an obscure constants. Expansion of above equations. prompts one condition with N unknowns. Only it is not adequate to decide the N unknowns a (n = 1, 2,..., N) constants. To determine the N constants, it is important to have N linearly independent equations. This can be accomplished by assessing (7) (e.g., applying boundary conditions) at N A program in MATLAB has been composed for magnetic-frill and delta gap. There are couple of presumptions which has been considered as :- Number of Sub Division =11 Radius of Dipole<Wave lengths> =0.02 Total Dipole Lengths<wave lengths>= 0.25, 0.50, 1.25. CASE :1 LENGTH = 0.2500 (WLS) RADIUS OF THE WIRE = 0.0200 (WLS) NUMBER OF SUBSECTIONS = 11.00 CURRENT DISTRIBUTION:- 97 96

INPUT IMPEDANCE:- Z= 403.0 -j 61.7 (OHMS) RADIATION PATTERN:- CASE :3 LENGTH = 1.2500 (WLS) RADIUS OF THE WIRE = 0.0200 (WLS) NUMBER OF SUBSECTIONS = 11.00 CURRENT DISTRIBUTION:- INPUT IMPEDANCE:- Z= 21.2 -j 123.1 (OHMS) CASE:2 LENGTH = 0.5000 (WLS) RADIUS OF THE WIRE = 0.0200 (WLS) NUMBER OF SUBSECTIONS = 11.00=> CURRENT DISTRIBUTION:- RADIATION PATTERN:- INPUT IMPEDANCE: Z= 72.7 -j 123.5 (OHMS) RADIATION PATTERN:- 98 97

CONCLUSION Current distribution is typically assumed to be of sinusoidal structure limited measurement wires the sinusoidal current distribution is representative accurate. To locate a more accurate current conveyance on a barrel shaped wire, pocklington's integral equation is generally derived and illuminated. For the wire antenna by already, answers for the the integral equation were acquired utilizing iterative strategies presently; it is most advantageous to utilize moment strategy techniques. Some numerical definition is being done and appeared with help of MATLAB. In the event that we know the voltage at the feed terminals of a wire antenna and locate the current distribution, the input impedance and radiation pattern can then be acquired ACKNOWLEDGEMENT The authors acknowledge interesting discussions with Prof. Harish Parthasarthey whose method of solving Pocklington s equation motivated the study. [7] H. C. Pocklington, Electrical Oscillations in Wire, Cambridge Philos. Soc. Proc., Vol. 9,1897. pp. 324 332. [8] L. W. Pearson and C. M. Butler, Inadequacies of Collocation Solutions to Pocklington-Type Models of Thin-Wire Structures, IEEE Trans. Antennas Propagat., Vol. AP-23, No.2, March 1975, pp. 293 298. [9]. C. M. Butler and D. R. Wilton, Analysis of Various Numerical Techniques Applied to Thin-Wire Scatterers, IEEE Trans. Antennas Propagat., Vol. AP-23, No. 4, July 1975, pp. 534 540. [10].D. R. Wilton and C. M. Butler, Efficient Numerical Techniques for Solving Pocklington s Equation and their Relationships to Other Methods, IEEE Trans. Antennas Propagat., Vol. AP-24, No. 1, January 1976, pp. 83 86. [11] Constantine A. Balanis Antenna Theory: Analysis and Design, 4th Edition, Wiley Publication. 2016 REFERENCES [1]Constantine A. Balanis, Antenna Theory : Analysis and design, John wiley & sons New Delhi, 2008 [2] G.S. N. Raji, Antennas and Wave Propagation Published by Pearson Education, 2004 ISBN 10: 8131701840 / ISBN 13: 9788131701843 [3] Comparison between solution of POCKLINGTON S and HALLEN S integral equations for Thin wire Antennas using Method of Moments and Haar wavelet., International Journal of Innovation and Applied Studies ISSN 2028-9324 Vol. 12 No. 4 Sep. 2015, pp. 931-942 2015 Innovative Space of Scientific Research Journals http://www.ijias.issr-journals.org/ [4] Electromagnetism and relativity theory, Application of advance signal an analysis, Harish Parthasarthy, (I K International publication) [5] Advanced signal analysis and its applications to mathematical physics, Harish Parthasarthy, (I K International publication)2008 [6] R. F. Harrington, Field Computation by Moment Methods, Macmillan, New York, 1968 99 98