Time Domain Analysis of the Multiple Wires Above a Dielectric Half-Space Poljak [1], E.K.Miller [2], C. Y. Tham [3], S. Antonijevic [1], V. Doric [1] [1] [2] [3] [1] [1] Department of Electronics, University of Split, R.Boskovicaa bb, Split, Croatia [2] 597 Rustic Ranch Lane, Lincoln CA 95648, USA [3] Faculty of Engineering and Science, Tunku Abdul Rahman University Jalan Genting Klang, Setapak, Kuala Lumpur, Malaysia D.Poljak et al.: Time Domain Analysis....

CONTENTS Introduction Time domain integral equation formulation for thin wire arrays Time domain energy measures for transient response Computational examples Conclusion D.Poljak et al.: Time Domain Analysis....

1 INTRODUCTION This work deals with the energy aspect of transient radiation from wire arrays The time variation of the total energy of the field shows the character of the antenna energy loss by radiation. Determining the currents along the wires, by solving the Hallen integral equation set, TD energy measures are obtained by spatially integrating the square of the current and charge along the wire as a function of time. Computational examples related to the wire antenna array and multiple transmission lines are presented in the paper. D.Poljak et al.: Time Domain Analysis....

2 TIME DOMAIN INTEGRAL EQUATION FORMULATION FOR THIN WIRE ARRAYS An array with an arbitrary number of elements, operating in either antenna or scattering mode is considered, Multiple wires above a dielectric half-space: a) antenna mode, b) scattering mode D.Poljak et al.: Time Domain Analysis....

Space-time currents along the wires are governed by the set of the coupled Pocklington integral equations: where E inc xm denotes the incident field on m-th wire, I n is the transient current induced on the n-th wire, N w is the total number of wire elements, and r(θ,t) is the space-time reflection coefficient D.Poljak et al.: Time Domain Analysis....

The corresponding Hallén equation set can be readily derived from the Pocklington equation performing a straightforward convolution: The unknown functions F 0A (t) and F LA (t) are related to the multiple reflections from the wire ends. The Hallen integral equation set can be handled via time domain version of the Galerkin Bubnov Indirect Boundary Element Method (GB-IBEM). D.Poljak et al.: Time Domain Analysis....

The charge distribution along the wire can be determined from the relation: where q is the linear charge distribution along the wire configuration. 3 TIME DOMAIN ENERGY MEASURES FOR TRANSIENT RESPONSE The energy measures represented by the current and charge induced on an object yield insight into where and how much the object radiates as a function of time. The H-field energy is represented by the following relation: while the E-field energy is measured by the integral over squared charge: D.Poljak et al.: Time Domain Analysis....

4 SOLUTION OF THE SPACE TIME INTEGRAL EQUATION The local approximation for unknown current can be expressed in the form: The space boundary discretization of Hallen equation set results in the local equation system: D.Poljak et al.: Time Domain Analysis....

The solution in time on the i-th space segment can be expressed: where I i k are the unknown coefficients and T k are the time domain shape functions. Choosing the Dirac impulses as test functions, the recurrence formula for the space-time varying current can be written as: where N g denotes the total number of global nodes A ji are the global matrix terms, g jl * is the whole right-hand sidecontaining the excitation and the currents at previous instants. D.Poljak et al.: Time Domain Analysis....

5 COMPUTATIONAL PROCEDURES FOR ENERGY MEASURES First, the charge distribution is obtained by the solution of integral: which can be carried out analytically: where M and N t denotes the total number of segments and time steps, respectively. D.Poljak et al.: Time Domain Analysis....

ICEAA 2005, Turin,Italy, September 2005 The H-field energy measure is obtained by evaluating the integral: The solution is available in the closed form and is given by: The E-field energy is obtained from the integral: for which the solution is then: k=1,2,..., N t and the total energy measure is given by sum of W I and W q. D.Poljak et al.: Time Domain Analysis....

5 COMPUTATIONAL EXAMPLES a single wire in free space operating in antenna mode Time domain energy variation for dipole excited by a Gaussian voltage pulse - the wire dimensions: L=1m, a=2mm - excitation: - parameters: V 0 = 1.0V g=2  10 9 s -1 t 0 = 2ns D.Poljak et al.: Time Domain Analysis....

a single wire in free space operating in scatterer mode Time domain energy variation for scatterer excited by a Gaussian incident plane wave field - the wire dimensions: L=1m, a=2mm - parameters: E 0 = 1.0V g=2  10 9 s -1 t 0 = 2ns - excitation: D.Poljak et al.: Time Domain Analysis....

Active wirePassive wire The coupled wires are located above PEC ground at height h=0.25m. - the wire dimensions: L=1m, a=2mm, d=0.5 m. 2-wire array above a PEC ground plane - excitation:- parameters: V 0 = 1.0V, g=2  10 9 s -1, t 0 = 2ns D.Poljak et al.: Time Domain Analysis of the Energy Stored....

Transient current induced at the center of the active and passive wire Active wirePassive wires The array is located above dielectric medium (  r =10) at height h=1 m. - the wire dimensions: L=1m, a=2mm, d=0.5 m. 3-wire array above a dielectric half-space - excitation:- parameters: V 0 = 1.0V, g=2  10 9 s -1, t 0 = 2ns D.Poljak et al.: Time Domain Analysis of the Energy Stored....

The array is located above dielectric medium (  r =10) at height h=1 m. - the wire dimensions: L=1m, a=2mm, d=0.5 m. 3-wire array above a dielectric half-space - excitation:- parameters: V 0 = 1.0V, g=2  10 9 s -1, t 0 = 2ns The H-field (W I ) E-field (W q ) and total energy (W tot ) energy measures as a function of time for the active and passive wires, respectively Active wirePassive wires D.Poljak et al.: Time Domain Analysis....

3-wire transmission line above a PEC ground The wires are located above PEC ground at height h=5m. - the wire dimensions: L=30m, a=cm, d=3m - excitation:- parameters: E 0 =65kV/m, a=4*10 7 s -1, b= 6*10 8 s -1. Transient current induced at the center of the central and side wires, respectively Central wireSide wires D.Poljak et al.: Time Domain Analysis of the Energy Stored....

3-wire transmission line above a PEC ground The wires are located above PEC ground at height h=5m. - the wire dimensions: L=30m, a=cm, d=3m - excitation:- parameters: E 0 =65kV/m, a=4*10 7 s -1, b= 6*10 8 s -1. The H-field (W I ) E-field (W q ) and total energy (W tot ) energy measures as a function of time Central wireSide wires D.Poljak et al.: Time Domain Analysis....

5 CONCLUDING REMARKS The work deals with time domain energy measures describing the behaviour of multiple thin wires (operating in antenna or scattering mode) located in a half ‑ space configuration. The analysis of time domain energy measures makes possible to view the electromagnetic behaviour of wire array. The formulation of the problem is based on the corresponding set of the space- time Hallen integral equations. The integral equations are handled by the space-time Galerkin Bubnov scheme of the Boundary Integral Equation Method (GB-BIEM). Determining the currents and charges along the wires the time domain energy measures are calculated by spatially integrating the squared current and charge. The total energy dissipates more slowly when parasitic wires are present than for the case of the single wire excited by the same pulse. D.Poljak et al.: Time Domain Analysis....

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