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]

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

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


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.


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,

z

x

y

AB

A'B'

h

r o o

o o

a

L

d1d2

C

C'h

d1d2

V (t)g z

x

y

r o o

o o

a

(x01, y01, z01)

(x0M, y0M, z0M)

(xL1, yL1, zL1)

(xLM, yLM, zLM)(x02, y02, z02)

(xL2, yL2, zL2)

EincHinc

Multiple wires above a dielectric half-space: a) antenna mode, b) scattering mode


W

N

n

nm

nmnt L

L

nm

nmn

excxm

Nm

ddxR

cRtxIr

dxR

cRtxI

tcxt

E w

,...2,1,

'4

)/,'(),(

'4

)/,'(

1

1

*

*

0

0

2

2

2

2

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

- tn+1 n

n2n=1

,

4 er( ,t)= A (t)+ (-1 n ( t) ,) A It1-

| x - |x = arctg ,2h


The corresponding Hallén equation set can be readily derived from the Pocklington equation performing a straightforward convolution:

w

LexcxmLmm

N

n

t LL

Nmdxc

xxtxE

c

xLtF

c

xtF

ddxddxR

cRtxIrdx

R

cRtxIw

,...2,1,')'

,'()()(

''4

)/,'(),('

4

)/,'(

0

0

1 00

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).


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.

t

dtx

txI=q

0 '

,'

The H-field energy is represented by the following relation:

while the E-field energy is measured by the integral over squared charge:

dxtxI4

=W ,2

L

0

0I ),( '

'' dxtxq4

1=W ),(2

L

0

q


4 SOLUTION OF THE SPACE TIME INTEGRAL EQUATION

The local approximation for unknown current can be expressed in the form:

IftxI T)','(

The space boundary discretization of Hallen equation set results in the local equation system:

*

2

*2

0

0

1 '

4

cos sin 1'

4cos sin

( ) ( )

'1( ', ) '

2

Rt

cj i

Rt

j i c

j j

j i

T

j il l

T r r

j il l r r

Lj jl l

excx j

l l

f f dx dx IR

f f dx dx IR

x L xF t f dx F t f dx

c c

x xE x t f dx dx

Z c


The solution in time on the i-th space segment can be expressed:

tN

k

kkii tTItI

1

)'()'(

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:

jj

timesretardedalljiji

N

iiji

j A

gIAIA

Ic

Rkt

g

c

Rkt

kt

*

1

*)(1*1

where Ng denotes 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.


5 COMPUTATIONAL PROCEDURES FOR ENERGY MEASURES

First, the charge distribution is obtained by the solution of integral:

dtIfx

q i

T

i

M

i

N

k t

t

k

1 1 '

which can be carried out analytically:

1 11 1

1 1

1

2

tNMm m m mi i i i

i m

q I I I Ic

where M and Nt denotes the total number of segments and time steps, respectively.


ICEAA 2005, Turin,Italy, September 2005

The H-field energy measure is obtained by evaluating the integral:

M

ii

T

i

t

I dxIfx

Wk

1

20 ''4

The solution is available in the closed form and is given by:

IW = 10 7 x

3I i

k 2 I i

kI il

k I il

k 2 il

M

, k 1, 2,, N t.

The E-field energy is obtained from the integral:

210

1'

4 'k

MT

q i ii t

W f q dxx

for which the solution is then:

2 2

1 110

1,

4 3

Mk k k k

q i i i ii

xW q q q q

k=1,2,..., Nt

and the total energy measure is given by sum of WI and Wq.


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:

V0 = 1.0V

g=2 109 s-1

t0 = 2ns


2 20( )

0( ) g t tv t V e

a single wire in free space operating in scatterer mode

Time domain energy variation for scatterer excited by a Gaussian incident plane wave field


L=1m, a=2mm

- parameters:

E0 = 1.0V

g=2 109 s-1

t0 = 2ns

- excitation:


2 20( )

0( ) g t tE t E e

Active wire Passive wire

The coupled wires are located above PEC ground at height h=0.25m.


L=1m, a=2mm, d=0.5 m.

2-wire array above a PEC ground plane

- excitation:2 2

0( )0( ) g t tv t V e

- parameters:

V0 = 1.0V, g=2 109 s-1, t0 = 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 wire Passive wires

The array is located above dielectric medium (r =10) at height h=1 m.


L=1m, a=2mm, d=0.5 m.

3-wire array above a dielectric half-space

- excitation:2 2

0( )0( ) g t tv t V e

- parameters:

V0 = 1.0V, g=2 109 s-1, t0 = 2ns


The array is located above dielectric medium (r =10) at height h=1 m.


L=1m, a=2mm, d=0.5 m.

3-wire array above a dielectric half-space

- excitation:2 2

0( )0( ) g t tv t V e

- parameters:

V0 = 1.0V, g=2 109 s-1, t0 = 2ns

The H-field (WI) E-field (Wq) and total energy (Wtot) energy measures as a function of time for the active and

passive wires, respectively

Active wire Passive wires


3-wire transmission line above a PEC ground

The wires are located above PEC ground at height h=5m.


L=30m, a=cm, d=3m

- excitation:

0( ) at bte eE t E - parameters:

E0=65kV/m, a=4*107s-1, b= 6*108s-1.

Transient current induced at the center of the central and side wires, respectively

Central wire Side wires


3-wire transmission line above a PEC ground

The wires are located above PEC ground at height h=5m.


L=30m, a=cm, d=3m

- excitation:

0( ) at bte eE t E

- parameters:

E0=65kV/m, a=4*107s-1, b= 6*108s-1.

The H-field (WI) E-field (Wq) and total energy (Wtot) energy measures as a function of time

Central wire Side wires


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.


More or less, that’s it!

Thank you

Documents

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]