Time Domain Analysis of the Multiple Wires Above a Dielectric Half-Space

1. D.Poljak et al.: Time Domain Analysis.... 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]Department of Electronics, University of Split, R.Boskovicaa bb, 21000 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, 53300 Kuala Lumpur, Malaysia

2. 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

3. D.Poljak et al.: Time Domain Analysis.... • 1INTRODUCTION • This workdeals 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.

4. D.Poljak et al.: Time Domain Analysis.... • 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

5. 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 Eincxm denotes the incident field on m-th wire, In is the transient current induced on the n-th wire, Nw is the total number of wire elements, and r(θ,t) is the space-time reflection coefficient

6. 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 F0A(t) and FLA(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).

7. D.Poljak et al.: Time Domain Analysis.... • 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 charge distribution along the wire can be determined from the relation: where q is the linear charge distribution along the wire configuration. The H-field energy is represented by the following relation: while the E-field energy is measured by the integral over squared charge:

8. 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:

9. D.Poljak et al.: Time Domain Analysis.... The solution in time on the i-th space segment can be expressed: where Iik are the unknown coefficients and Tk 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 Ngdenotes the total number of global nodes Aji are the global matrix terms, gjl* is the whole right-hand sidecontaining the excitation and the currents at previous instants.

10. 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 Mand Ntdenotes the total number of segments and time steps, respectively.

11. D.Poljak et al.: Time Domain Analysis.... 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,..., Nt and the total energy measure is given by sum of WIand Wq. ICEAA 2005, Turin,Italy, September 2005

12. D.Poljak et al.: Time Domain Analysis.... 5 COMPUTATIONAL EXAMPLES • a single wire in free space operating in antenna mode - the wire dimensions: L=1m, a=2mm - excitation: - parameters: V0 = 1.0V g=2 109 s-1 t0= 2ns Time domain energy variation for dipole excited by a Gaussian voltage pulse

13. D.Poljak et al.: Time Domain Analysis.... • a single wire in free space operating in scatterer mode - the wire dimensions: L=1m, a=2mm - excitation: - parameters: E0 = 1.0V g=2 109 s-1 t0= 2ns Time domain energy variation for scatterer excited by a Gaussian incident plane wave field

14. D.Poljak et al.: Time Domain Analysis of the Energy Stored.... • 2-wire array above a PEC ground plane The coupled wires are located above PEC ground at height h=0.25m. - the wire dimensions: L=1m, a=2mm, d=0.5 m. - excitation: - parameters: V0 = 1.0V, g=2 109 s-1, t0= 2ns Active wire Passive wire

15. D.Poljak et al.: Time Domain Analysis of the Energy Stored.... • 3-wire array above a dielectric half-space 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. - excitation: - parameters: V0 = 1.0V, g=2 109 s-1, t0= 2ns Active wire Passive wires Transient current induced at thecenter of the active and passive wire

16. D.Poljak et al.: Time Domain Analysis.... • 3-wire array above a dielectric half-space 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. - excitation: - parameters: V0 = 1.0V, g=2 109 s-1, t0= 2ns Active wire Passive wires The H-field (WI) E-field (Wq) andtotal energy (Wtot) energy measures as afunction of time for the active and passive wires, respectively

17. 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: E0=65kV/m, a=4*107s-1, b= 6*108s-1. Central wire Side wires Transient current induced at the centerof the central and side wires, respectively

18. 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: E0=65kV/m, a=4*107s-1, b= 6*108s-1. Central wire Side wires The H-field (WI) E-field (Wq) andtotal energy (Wtot) energy measures as afunction of time

19. 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.

20. More or less, that’s it! Thank you