Finite difference time domain calculation of transients in antennas with ...

N92-19748 Finite

Difference

Time

of Antennas

Domain

Transients with

Calculation

in

Nonlinear

Loads

by

Raymond Department

Luebbers, John of Electrical The Pennsylvania University

Park, and

Kent Department

of

Beggs, and Karl Kunz and Computer Engineering State University PA

16802

by

Chamberlin

Electrical

University Durham,

and

Computer

Engineering

of New Hampshire New Hampshire

July

1991

Abstract Determining nonlinear involve

transient loads is calculating

electromagnetic a

challenging frequency

fields

in

antennas

problem. Typical domain parameters at

methods a large

with used number

of different frequencies, then applying Fourier transform methods plus nonlinear equation solution techniques. If the antenna is simple enough so that the open circuit time domain voltage can be determined independently of the effects of the nonlinear load on the antenna current (an infinitesimal dipole, for example), time stepping methods can be applied in a straightforward way. In this paper transient fields for antennas with more general geometries are calculated directly using Finite Difference Time Domain methods. In each FDTD cell which contains a nonlinear load, a nonlinear equation is solved at each time test case the transient current in a long dipole nonlinear load excited by a pulsed plane wave is this approach. The results agree well with measured results previously published. The extends the applicability of the FDTD method involving scattering from targets including materials, and to coupling between antennas loads. It may also be extended to propagation materials.

step. antenna computed

As a with a using

both calculated and approach given here to problems nonlinear loads and containing nonlinear through nonlinear

INTRODUCTION Calculating and

the

scatterers

containing

difficult

problem.

frequency

domain

frequencies.

scattering

linear

achieved

approach,

the

the

extended

to

of

nonlinear

materials

as

For voltage

the on

the

and

open

nonlinear

the

circuit

an

Finally, approach difficult In approach

model using

to to this will

of time

This

excellent

thin

paper

the

more

extended

of

applied

straight to

be

marching

wire. general

Finite to

this from

the and cannot regions

nonlinear

open

the

other

domain

dispensed input

by

in

domain his

a problem

applied

Kanda

work

with,

to

resulting methods

used

circuit

independently

be

the

However,

[4], this

to who

area.

method

of

moments

method

appears

geometries.

Difference include

at

bulk

domain

can as

time

obtaining

approach

frequency

previous a

to

tedious

determined

with

was

used

antenna

time

used

marching.

However,

the

or

approach

[5]

apply

be

time

this

be

quite

the

here.

Tesche

load,

review

Schuman a

and

the

be

the

time

through

load

can

to

will

be

the

can

Liu

voltage

circuits.

gives

of

portions,

using

And

where

Tesche

for

by

presented

nonlinear

approach

directly

approach

and

nonlinear

scattering

terminals the

Liu

complicated may

in

determine

paper.

propagation

situation

of

circuit

nonlinear

FDTD

antenna

effects of

solved

the

to

harmonic approach

[3]

resources.

or

simpler

the

portion the

can

a

involving

materials,

apply

been

this

in

involved

situations

to

portion

this

for

computer

has

measurements

in

frequencies

significant

be

from

of

and

presented

results

a

characteristics

nonlinear

obtained

is

to

transformed

results

material

loads.

linear

with

antennas

this

nonlinear

agreement

domain

number

involve

the

results

frequency

large

solved

their

used

domain

problem,

in

of

analysis

into

the

good

and

validate

also

then

number

[i]

frequency

of

and

large

with

or

approach

series

problem

wideband

domain, They

Volterra

the

portion

a

Weiner

antennas

separated

calculated

and

a

from

at

fields

loads

traditional

methods

with

electromagnetic

nonlinear

The

Sarkar

combination

[2,3]

transient

Time nonlinear

Domain lumped

(FDTD) loads.

The

reader is literature

assumed to have some familiarity with is extensive, with some representative

in the following scattered field

references formulation

[6-8]. of [8],

this method. The papers included

The authors are using but with linear time

the

differencing. APPROACH To

illustrate

example

of

length

z

model

in

in

The

load.

in

series

with

using

diode

the

a

model

from

of

I.

As

at

consider A

wire

0.81

mm

The the

ohm

in

at

capacitance

the of

for

accurate

results

[3],

to

describe

the

the

loaded

half

at

located

its parallel

wire

is

used

this

load

is

two

actual

of

pF

with

the

(the

0.5

at

is

of

base

specific

dipole

is

center

resistor

diode

the

dipole

described

I00

single

us

[3].

diameter

cell

a

junction

let

Figure

FDTD

lumped

made

the

a

the

diodes

total

and

shown

axis.

method, taken

meters as

the

were

interest,

0.6

midpoint, to

the

a

measurements

monopole).

must

to

also

be

frequencies

The included

contained

in

pulse. In

order

circuit

in

FDTD,

approximating parallel

with

let

us

a

linear

a

resistor

first lumped

approach

used

consider

an

load

to

approach

consisting

(conductance)

in

model

of an

FDTD

the

diode

to a

capacitor cell.

in Starting

with

VX

where

H,

E,

_

permittivity, approach y=JAy, field

for z=KAz, in

a

and

a

and

the

are

the

particular

(I)

= E

@E -at

space becomes, FDTD

+ aE

magnetic

conductivity,

discretizing eq.

H

cell,

and

electric

and and for

following

time, the

(i)

z

with component

fields,

the

the

[6]

t=nat,

Yee x=I_x,

of

electric

n+l n E z -E z (_7 X H n+1/2) z = E

n+1 Ez

+ a

(2)

At

where

6

the

and

a

pertain

superscripts

component. E _I

in

curl

H

consider

total

field

through

the

cell current

direction.

Using are

equation

E

the

z

the

the

next

lumped

The the

term

ax,

elements,

H _,

with

H

cell

surrounding

involving

E

and

density the

the

cell

in

the

and

Az,

and

rewrite

voltages,

z

assuming

the

and

the

flowing

is

can

the

term

term

&y

for

Instead,

aE _I

through

we

solved

curl

current The

cell,

is

differences.

in

dimensions

E z and

particular

E n and

terms.

direction.

cell

of

of

flowing

flowing

the

of

equation

displacement

across

terms

the

The

density

constant

in

of

the

this

finite

density

is

in

conduction

fields

meanings

location of

values

spatial

component.

of

reference

time

using

cell

FDTD

previous

current

derivative

time

applying

physical

electric

time

the

particular

the

in

computed

the the

the

of

term

the

denote

Usually terms

gives

to

above

currents

as

n+l n E z -E z _I.xAy(V

X H n÷I/2)z = C

+ G

AZ

AZ

E_ ÷'

(3)

6t

where

now

cell,

C

and

G

Note

the is

is

Clearly

the

the

that

first lumped lumped

one a

can

lumped

appropriate

value

dimensions,

and

the

cell.

Thus

capacitor

term

and

total

"parallel

identify

of

the

in

_z

similarly lumped

of

cell

values

and

are

interchangeable

can

a

load

be

the

_

and in

a terms

4

through of

the

a

parallel can

be

solving

cell,

the

cell.

setting

the

an

cell and

the

a

combination modeled

appropriately. of

the

capacitance.

to on

the

across

resistance

(resistance) of

voltage

based

lumped is

with

equivalent

cell

that

flowing

capacitance

parallel as

the

for

conductance

the

E _I C

c

current

plate"

conductance

setting (3)

the

capacitance

a a

is

simply

Equations for

E _I.

of of

a by

(2)

However, one warning is that the cell permittivity cannot be set too low. FDTD in the form presented in this paper cannot in general model materials with an epsilon that is too small (much less than free space) without becoming unstable. Such materials can be modeled using FDTD modified for frequency dependent materials [9], but with additional computational effort. If the conductance G (conductivity a) is great enough so that the displacement current term in eq. (2) (or (3)) can be neglected then this term may be dropped. Indeed, if (2) or (3) is solved for E_I and G (or equivalently o) is allowed to go to infinity, the a)

correct small

result enough

of so

E_I=0

that

the

the

displacement

current

but

nevertheless

neglecting

result the

in

capacitance

of

elements

in

Now

the

current

the

free

us

in

simplifies the

diode

diode, cell,

can

plus

for

solve

the

the

the

C

figure

id

space,

term,

will

is

that

than

the

adding

to the

junction

lumped

conduction

resistance

the

the

but

the

circuit

to

does

diode this not

sizes current current

of R

measure

capacitance

capacitance

of

element

the

used

represents

and

circuit

by

approach

resistor,

total

for

space

than

free

this

lower

oscilloscope

i d represent

(3)

id

the

derivation

Letting also

this

the

for

(or

capacitance.

capacitance

with

following

is

In of

junction

results

which we

cell

free

with

made

G

smaller

current

be

this

capacitance

parallel

consideration.

extend

resistance The

cannot

cell

2.

filled

argument

with

the

Figure

FDTD

the

physical

filled

to

cell

making

is

displacement

cell

with

in

[3].

The

actually

cell

current

the

this

FDTD

proceed

input

space

diode.

affect

the

shown the

an

However,

conduction

A

parallel

let

interest, models

of

obtained.

through

instabilities.

capacitance

is

of is

of

the

not

approximation appreciably under through through

the the

obtaining

= _x_y(VxHn.I/2)

z -

5

C_Z At

r n+1 n ,E z -Ez)

(4)

Once i d is determined from obtained from the equation

(4),

the diode

voltage

vd can be

n+l

2 Vd + id R

Since

the

diodes

nonlinear

are

in

the

= v c = _Z

circuit,

i d = 2.9xi0

id

(6)

as

are

given

Raphson

the

i d must

in

both

which

taken

in

initial

also

satisfy

the

for

diode

(6a,b)

method

can

be

applied

required The

Newton-Raphson

the

very

good

[3].

equal,

the

E,

NewtonE _I

approach since

E n,

starting

Since

the

fast

of

a

for

was

value

a

in

solve

was

with

be

This

previous

iteration

given

to

equality.

(6b)

Vd_0

must

convergence

of

(6a)

equations

and

E _I

Vd