NATL INST. OF STANDi *.TECH Reference

NBS

PUBLICATIONS

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AlllDb 3MD351

*^eAo of NBS TECHNICAL NOTE 1081

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Possible Estimation

Methodologies for

Electromagnetic Field

Distributions in Complex

Environments

M. Kanda

J. Randa

N. S. Nahman

-QC

100

.U5753

\io . 1031

1985
NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU PF STANDARDS

LIBRARY 1^~ MBS

Possible Estimation Methodologies .^^3

for Electromagnetic Field Distributions <-

'

in Complex Environments M?r

M. Kanda

J. Randa

N. S. Nahman

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

f ¥ %

\ XJ/

X kl"£^ kcit/iutoJ /Ko-kU.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued March 1985
National Bureau of Standards Technical Note 1081

Natl. Bur. Stand. (U.S.), Tech Note 1081, 52 pages (Mar. 1985)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1985

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402
CONTENTS

Page

1

.

Introduction 1 2. Statistical Approach 2

2.1 Rayleigh Distribution. 2

2.2 Examples 5

2.3 Other Relevant Statistics 6

3. Finite-Element Action Approach 8

3.1 Introduction and Motivation 8

3.2 General Formulation 11

3.3 Simple Example 14

3.4 Problems and Prospects. 19

4. Scanning Techniques 21

4.1 Cylindrical Scanning. 21

4.2 Spherical (Directional) Scanning 26

5. Summary. 29

6References 30

Appendix 31

m

Possible Estimation Methodologies for Electromagnetic

Field Distributions in Complex Environments

M. Kanda, J. Randa, and N. S. Nahman

Electromagnetic Fields Division

National Bureau of Standards

Boulder, Colorado 80303

The problem of measuring and characterizing complicated multiplesource, multiple-frequency electromagnetic environments is becoming

more important and more difficult as electrical devices proliferate. This paper outlines three general approaches to the problem

which are currently under investigation at the National Bureau of

Standards. The three approaches are: 1) a statistical treatment of

the spatial distribution of electromagnetic field intensities, 2) a numerical computation using a finite-element (or lattice) form of

the electromagnetic action functional, and 3) use of a directional

probe to scan a volume. All three methods are still in the development stage, but each appears promising.

Key words: action, directional scanning, environment characterization, field levels, finite element, hazard assessment, multiple

source, statistical approach.

1. Introduction

There has recently been a rapid increase in higher powered, multifrequency, electromagnetic (EM) radiation sources which complicate greatly the

environment in which modern electronic equipment, both military and civilian,

must operate. At the same time, there has been an even more dramatic increase

in the quantity of electronic equipment, such as minicomputers and microprocessors, composed of semiconductor devices, and hence more sensitive to

interference. Today there is a complex matrix of electronic equipment trying

to operate in a complex EM environment. Therefore, the estimation of the maximum electromagnetic (EM) field strength is becoming very important in many

interference problems. For example, electromagnetic waves penetrate into

buildings which house sensitive electronic and ordnance items. There has been

great interest in determining the electromagnetic environment inside buildings.

Heretofore, it has been customary to make several spot measurements of

the electromagnetic field strengths produced by existing transmitters. However, it is obviously impossible to completely determine the field existing in
an enclosure using a manageable number of spot measurements. In order to

rigorously determine the EM field distributions in a volume, systematic mea- surements of electric field amplitude and phases have to be made at points at

most half a wavelength apart (at the highest frequency present).

There is a need for a general method of extracting the maximum amount of

useful information about the EM field distribution within an enclosure from

the minimum amount of measurement effort. In this paper we discuss three possible approaches to this problem which are currently being explored and developed at the National Bureau of Standards (NBS) . We first examine a statistical approach to the estimation of EM field distributions. In a previous paper

by the first author [1], five types of time and amplitude statistics were used

in order to unravel the complexities involved in an EM environment. Here we

shall discuss the statistical distribution of scattered EM fields and illustrate its usefulness using experimental and simulated examples. The second

method is a finite-element-action calculation using a small number of measurement points. The true field cannot be determined, but it may be possible to

reconstruct the smoothest configuration or the "most probable" one. The third

and final approach discussed is the most conventional, using directional

scanning to bound or approximately determine fields within a given volume.

2. Statistical Approach

2.1 Rayleigh Distribution

When the configuration of the scatterers is random and sufficiently dispersed to give a wide (and hence, equivalently uniform) phase distribution,

the field configuration, E, typically caused by the reflectivity of various

multipath types of scatterers, both stationary and moving, is given by

E =

I A.ej *i, (2.1)

i=l

1

where A^ and <j>^ are random amplitudes and phases of scattered fields.

Resolving E into its real and imaginary components, Re E and Im E, from the

central limit theorem, it follows that Re E and Im E are normally distributed,

as long as N is large (> 10). In particular, when the A.,- and ^ are uncor-
related and hence independent random variables, it can be shown that Re E and

Im E have the same variance with zero mean values. The amplitude distribution

r r p F (r) is then found to be the Rayleigh distribution [2] p r D (r) = —~ exp (- -r—5)

E a do

for r>0.

The mean <E>, mean square <E^> and standard deviation (s.d.) are respectively given as

<E> = ] rp F (r) dr = / jr a (2.2)

L 2

<E 2> = 2a2 (2.3)

and v

s.d. = /<E 2> - <E> 2 = /2 - - o. (2.4)

The cumulative probability distribution is defined as the probability that the

random variable E is equal or less than the value E , and can be obtained by

integrating the Rayleigh probability density function, i.e.,

E

o c 2

2 P(E < E ) - J 4 ex P (" Tl) dr " 1 " exP C" TT)- ( 2 - 5 ) q a co co

Hence, its complement is

Eo 2

p (E > E ) = exp(- y^i) (2.6)

The median of E, E m

, is, therefore,

E 2

P (E <Em

) = 1 - exp

(- JJJ2)

= 0.5 (2.7)

E m

s 1.18 a. (2.8)

or

The average crossing rate (ACR) presents the average number of times the

EM field strength crosses various levels and is usually given as positive

crossing per second versus EM field strength. The average level crossing rate

at a given level E Q is [3]

00 n(E = E ) = / p p

E (E ;r) dr (2.9)
where p F (Eo,r) is the joint probability distribution of E, and p = -rp gives

the slope of the signal envelope E. For the case of the Rayleigh distribution, the level crossing rate becomes

E F 2 v °

° n (E = E ) = —- -^ exp(- -^) (2.10)

where* *^n

a

2 **\>\r 2ai.

v

2 = <E 2> . (2.11)

The average field strength can be determined by averaging the random

trial samples of instantaneous field strength measurements recorded over a certain path length in distance. The variation in the average field strength

is typically caused by the relatively small scale variations along the propagation path.

There are situations where a signal is received in two different ways,

one via many widely spaced scatterers or reflectors, the other by a quite different mechanism, e.g., directly from a transmitter. The analysis of the am- plitude distribution of the resultant signal can be achieved as follows. Consider the sum

•a N

E = A e

J<l>+ I A.eJ<|,i. (2.12)

i=l

The first term represents the direct-path propagation through the walls of

houses and other objects. The second term represents the reflected and scattered waves due to randomly oriented objects such as houses, buildings, walls,

and overhead wires, and is therefore, Rayleigh distributed. Since the directpath propagation is attenuated through each penetration of a wall, etc., the

amplitude of this direct wave will be

M

Ao tt exp [- I

6. d.] (2.13)

i=l

] ^

where d^ is the thickness and 5j is the attenuation constant of each of the

walls. When M is large, the central limit theorem indicated that the sum of

the random term 6.,-d.j will be normally distributed. Hence, the amplitude distribution of A is lognormal, i.e.,
P n -(r) =

1

r (Inn - y)^

(2.14)

r a /2tt

where y is the mean and a

2 is the variance of the exponent in eq (2.13).

The probability density function of the total field given by the sum of a random plus a Rayleigh phasor is [2]

P(r) = 2r

a a /2ir °

/ A" exp[-

(*n A n - y)

2 r2 + A2 2r A,

2 a

2 ] -In (" ) dA

(2.15)

where a is the mean-square value of the scattered components.

For the cases of large and small values of E,

1 2 -

exp [- ( jm r - y) /2a2

] for r > /2a

P E

(r) =

<

ra/2ir

exp (- r 2/2o2

) for r < /2a

where y is the mean and a

2 is the variance of E.

(2.16)

(2.17)

Equations (2.16, 2.17) indicate that E is lognormal for large values and

Rayleigh distributed for small values.

The Rayleigh distribution has one parameter a to be determined which follows simply from the estimated mean a given as

1 N

and therefore

P= N i=

Z

i

E i' (2.18)

a = / — y. (2.19)

2.2. Examples

A number of electromagnetic field measurements have been made on the

propagation of radio waves at VHF and higher frequencies in the presence of

buildings, trees and other obstacles [1,4,5,6]. As an example, the cumulative

amplitude probability distribution and the average crossing rate of the EM

noise measured in a coal mine [1] are shown in figures 2-1 and 2-2,
respectively. Figure 2-1 indicates that, when the field strength is low

(approximately 90% of the total sample), the Rayleigh distribution is a good

approximation of the EM noise in a mine. On the other hand, when the EM field

strength is high (less than 10% of the total sample), EM noise data show a departure from the Rayleigh distribution. The statistical properties of the

peak values of EM field may be close to the log-normal distribution as indicated in eq (2.16)

.

Measurements of attenuation due to buildings, trees, and other obstacles

at long propagation distances (e.g., ranging from 100 to 500 m for 800 MHz

propagation), have also been performed [6], As an example, figure 2-3 shows

the cumulative amplitude probability distribution of the attenuation data. A

straight line on the coordinates in figure 2-3 represents a log-normal distribution of signal levels. The measured distribution is approximated by a lognormal distribution which is discussed in eq (2.16).

To simulate these experimental results, numerical studies of the cumulative amplitude probability distribution for a two-dimensional cavity model

with random noise sources have been made [7], Figure 2-4 shows the cumulative

amplitude probability distribution for EM fields measured inside the rectangular cavity. This figure is obtained from 100 data sets which are generated by

using random noise source positions and random wavelengths. The results follow a lognormal distribution except for the low probability region where most

electromagnetic interference data show a departure from a lognormal distribution. The deviation from a lognormal distribution in the low probability

region indicates an accurate estimation of the maximum field strength for the

measured data may be very difficult.

2.3 Other Relevant Statistics

In its most general form the characterization of a stationary electromagnetic environment is based on observations of a three-dimensional, random

variable. As such, the electromagnetic field is represented by the electric

and magnetic field vectors E and TT, respectively, where both of these vectors

may be stationary random functions of their spatial coordinates.

If the field vectors are not stationary, the probability distributions of
the electromagnetic field vectors are dependent upon time. The process

required for the environment to change from one stationary state to another is

defined as the transient state and is not considered here.

For stationary electromagnetic excitation the applied fields or sources whose collective effects establish the EM environment within a volume are defined here as steady-state fields. Thus, the random nature of the field is

established by the random spatial distribution of the sources, and the random

boundaries of objects within the volume or the randomness of the boundary

itself.

There are various statistical approaches which may be useful for characterizing this class of electromagnetic environments; two approaches are previously discussed; i.e., cumulative probability distribution and averaging

crossing rate.

Three other approaches are mentioned here as having possible merit.

A. Interpulse Spacing Distribution

The interpulse spacing distributions give the probability distribution

for the spacing between successive pulses in the received noise process.

These distributions are, of course, functions of the noise amplitude level.

B. Pulse Duration Distribution

The pulse duration distributions give the probability distribution for

the pulse widths and are given in terms of the percentage of pulses which

exceed various widths in seconds.

C. Two Sample Variance Analysis

It is essential to know how much data to gather when dealing with statistical quantities. Therefore, in any measurement of a statistical phenomenon

the minimum length of time over which the phenomenon is observed should be

determined. Two sample variance analysis can be used to accomplish this

determination. The basic idea to be discussed briefly below has been implemented often in the discussion of frequency stability.
A record of the phenomenon under consideration, y(t), is divided into a number of equal time segments of length t, and the average value of y(t), y^,

of each segment is calculated by

VT y

= ± j y(t) dt, (2.20)

k

t

where y^ is the k th segment average starting at time t^. Next, the sample

variance (sample size two) o

2 (2,t), of successive averages is calculated.

That is

where

0kM2,,)= "? (yn->"k)

2 =|(Vk+ r^)

2

- < 2 - 21 > n=k

1 k+1

yk 4 e yn (2.22)

n=k

is the average of the two successive segment averages y^ and y^. The two

sample variance, a

2 (2,t), for this special case (sample size two) is then

defined to be

a

2 (2,t) = <a2 (2,t)>, (2.23)

where the brackets represent the average of a

2 (2,t) over all pairs of suc- cessive yk constructed from y(t). The preceding calculation is repeated for

various values of averaging period, t. For a given maximum allowable deviation in y(t) the minimum averaging time can then be determined.

Many examples of these time and amplitude statistics for the time dependent, EM noise are given elsewhere [1],

3. Finite-Element Action Approach

3.1 Introduction and Motivation

In this section we describe an approach which in effect attempts to solve

Maxwell's equations within the volume of interest. The type of problem in

which we are interested, however, differs fundamentally from those problems

for which Maxwell's equations are usually solved, and it resists the use of

standard methods. The most fundamental difference is that we do not know the
sources. In addition there are the complications that the geometry and the

time dependence are not simple. The information available includes boundary

conditions at conducting walls or dielectric interfaces within the volume, and

the measured values of the fields at some number of measurement points. We are free to specify the number and location of the measurement points, but we obviously want the number to be small —or at least "reasonable." The general

qualitative idea is that given the geometry and boundary conditions, knowledge

of the fields at a few points should enable one to extract some information

about global properties such as average or maximum electromagnetic energy density.

Without knowledge of the external sources radiating into the volume of

interest, and with only a few measurement points, there will not in general be

a unique solution to Maxwell's equations. Consequently, it will be impossible

to actually determine the field everywhere. The best one can hope for is that

the approximate values and positions of the maximum field intensities will be

the same for all allowed solutions. Failing that, one wants a method which

consistently finds the one solution of all those possible which has some property of interest, such as the smallest maximum power density.

An approach which appears to hold some promise of fulfilling these rather

demanding (perhaps impossible) requirements is based on a finite-element

treatment of Hamilton's principle applied to electrodynamics. For a classical

field theory, Hamilton's principle states that if one considers the quantity

t 2 t 2

S[*a

] =

J dt L = / dt |d3x t(*

a

(x,t)), (3.1)

ti t x where £.(\|> ) is the Lagrangian density of the system depending on the independent fields ij> a(x,t), then the values assumed by the fields for the correct

physical solution are such that S is stationary with respect to small variations of the fields, V*(x,t). The functional S[^

a

] is called the action, and

for electrodynamics the stationarity requirement yields Maxwell's equations

[8], as we shall outline in the next subsection. It is generally assumed that

the stationary point of the action is in fact a minimum, and general conditions are known for which this is so [9], Unfortunately, these conditions are generally not met in the cases of interest to us, and so finding the
stationary point of S is not necessarily as "simple" as minimizing it. This

will be discussed further in a later subsection.

Finite-element calculations minimizing the action have been used before

in electromagnetics, particularly for waveguide and cavity problems [10,11],

and the connection to other variational calculations has been pointed out

[11]. Previous approaches, however, have obtained a set of linear equations

by using finite elements and setting the appropriate derivatives equal to zero. The system was then solved by numerically inverting the matrix. Such

tatics will not work here because there is not a unique solution, and hence

the matrix would be (very) singular. Our preference is to work directly with

the action and search numerically for its stationary points, rather than working with its derivatives. This approach is probably closer in spirit to

methods currently popular in quantum field theory [12] than to the traditional

differential equations approach.

In the finite-element approach we replace the continuous variables x and

t by a four-dimensional grid of points on which the fields are defined. The

integral in the action then becomes a sum, and the field values at each

(unmeasured) point are varied until a stationary point of the action is

found. The field values at measurement points are set equal to their measured

values and not allowed to vary. When a stationary point is found, one has a solution of Maxwell's equations for which the fields take on their measured

values at measurement points. Which of the many possible such solutions one

finds will depend on the initial field configuration (before we vary the

fields) and on the method used to find the stationary point. In the present

work we are using a smooth starting configuration and a gradient "minimization" procedure, which should find the solution which has the smoothest field

configuration.

The remainder of this section is devoted to a more detailed account of

the progress made in formulating this approach and the (foreseeable) remaining

problems. We shall first present the general framework, then digress on the

question of whether the stationary point is a minimum. Next a simple example

is given, and finally we discuss remaining obstacles and the direction of

future work.

10
3.2 General Formulation

The action for electromagnetism is given by [8]

S[A,*] = ] dt ] d

3x {i (V<|>(x,t) +.|j-A(x,t)) • e(x,t) . (v*(x,t)

1

1 + -^-A(x,t)) - (V x A(x,t)) • u

_1

(x,t) • (V x A(x,t)J]

+ A(x,t) • J(x,t) - p(x,t) (x,t)} , (3.2)

where we have assumed all media are lossless, and e and y are the permittivity

and permeability tensors. A and <|> are the usual vector and scalar potentials, __ _3_ __ __ E(x,t) = -V<fr - — A , B(x,t) = V x A . (3.3)

ot

The time integral in eq (3.2) is typically taken from -» to + °°, and similarly

the volume integral usually extends over all space--al though one can also con- sider a finite volume or time range, subject to conditions we shall note

below. J and p are the current and charge densities respectively. We have

written the action in terms of the potentials rather than E and H, because in

varying the action we want to take variations only with respect to independent

variables, whereas E and H are related through eq (3.3). In fact, eq (3.3)

implies the two homogeneous Maxwell's equations,

V-B = , VxE+ — B = 0. (3.4)

3t

Although there are four independent potential functions (A,<|>) appearing in eq

(3.2), only three are independent, since the gauge choice will provide an extra relation.

If one considers variation of the action due to a small variation of

<t>(x,t) (6<j)(x,t)) one finds (assuming the permittivity tensor is symmetric)

11
t

2 t2

6 S = - / dt / d

2

x n • D 6<j>(x,t) + / dt / d

3x (V • D - p) 6<j>, *i s l

i (3.5)

5 - e • (-V* --JJ-A),

where the surface integral is over the surface enclosing the entire volume.

If we require that <J> not vary on the boundary surface, but otherwise let

6cj>(x,t ) be arbitrary, then the first term in eq (3.5) vanishes, and requiring

.that the action be stationary yields

6 S = / dt j d

3x (V • D - p) 6<j> =

*l (3.6)

* V • D(x,t) - p(x,t) = .

In a similar manner, requiring that S be stationary with respect to small

variations of A(x,t) leads to

6S=0 + — 6-VxH + J = 0, (3.7)

A dt

where A(x,t) must be held fixed not only on the boundary surface at all times

but also throughout the volume at times tj and t2» What this means in practice is that in order to use this in a calculation, one must specify the

fields on the boundary surface at all times and throughout the volume at the

initial and final times. In any real application, such a superabundance of

information will not be available for the volume of interest. In order to

exploit the stationarity of the action we must expand the volume considered

beyond just the region of interest, out to distances where the fields can be

assumed to be negligible. The same applies to the time; tj is chosen before

the fields are turned on and t2 after they are turned off.

Having extended the volume under consideration to virtually all of spacetime, we must restrict the problem to manageable size. We envision the division depicted in figure 3-1. The volume marked Vj is the region of interest;

it is assumed to be free of primary sources, but it may contain conductors

with induced currents. Volume V

B is a buffer zone between Vj and V

s

, which is

the rest of space, wherein are located any primary sources. The general idea

12
is to divide the action into one piece from the integral over volumes Vj and

Vg and another piece from V<;. Because V^ contains unknown sources, we do not

try to determine the fields there, which means that we also will not know the

fields on the surface between V

s and Vg. The fields on that surface will be

allowed to vary or will be fixed by a reasonable guess. Obviously, near the

boundary between V<j and Vg the solution obtained will be very sensitive to the

choice for the fields on that boundary, and therefore it will not be reliable. As one moves away from the outer boundary of Vg the values of the

fields should be influenced more by the measurement points and less by the

values on the surface between V$ and Vg. For points far enough away from that

surface--i .e. within Vj —it should be possible to obtain reliable solutions,

given enough measurement points. The hope is that "far enough away" and

"enough points" are not so large as to render the method impractical for most

applications. The positions of the measurement points will clearly affect the

size required for Vg; it may well be advantageous to make a few measurements

on the perimeter of Vg. It would probably also be advisable to choose the

boundary of Vg to coincide with conducting walls when it is feasible, in order

to constrain the fields on the boundary as much as possible.

The quantity we shall consider then is a reduced action, S, which is

defined as in eq (3.2) but with the integral restricted to V. + V

R

. (Note

that in the general multiple-frequency case the various volumes are fourdimensional space-time volumes.) For definiteness and simplicity we choose a gauge which will be used in the remainder of the section. The choice is

<|>(x,t) = 0, so that E = -3A/3t . The next step is to discretize the expression

for the reduced action, converting the volume integral into a summation which

approximates it. There are any number of ways to do so; we are not interested

in their relative (dis)advantages at this time. The discretized reduced

action will have the general form

S = / dt I AV (i(^A(t)) • I v(!rA(t))

a.B.YeVjfcVg ( 3>8 )

- (V x A(t)l • y"

1

• (V x A(t)) } ,

where a number of points require explanatory comment. The discrete indices

a,3,Y label the spatial points, on which are centered the volume elements

13
AV „ . We have left the time variable continuous for now, anticipating the

<*3y r 3 single-frequency example below, Section (3.3), but for the general case time

too would be made discrete. The quantities (8A/8t) and (V x A) will in

a3Y a&Y

general depend on the values of the field A at a number of points on (or

within) the surface bounding AV _ . A concrete realization of this discretiza-

* 3 cx3Y

tion of S will be given in the example below. Imposition of the constraints

required by the known values of E and/or H at measurement points can be rather

complicated, but a simple case illustrates the idea. If only one frequency is

present then for our gauge choice E « A , and a measurement of the eleca3Y a3Y

trie field at a point fixes A at that point. A similar comment applies to

a3Y

boundary conditions at perfectly conducting walls; for the single-frequency

case they can be imposed with relative ease.

The calculation then proceeds as follows. A grid is defined within

Vg^Vj, and A is fixed at measurement points and appropriate components are

set equal to zero at conducting walls. All other A 's are varied, and one a3Y

searches numerically for a stationary point of S in eq (3.8). When (if) a stationary point is found, then that set of A constitutes an approximate

cx3y

solution to Maxwell's equations within Vg ® Vj which is consistent with measured field values and boundary conditions. The entire procedure, including

the numerical search method, is greatly clarified by an example, to which the

next subsection is devoted. The question of whether the stationary point is a maximum, minimum, or neither is addressed in the Appendix.

3.3 Simple Example

Having presented the general ideas of this approach, we now attempt to

implement it in a simple example—a rectangular waveguide with perfectly con- ducting walls. This is obviously not supposed to be a practical application,

and many of the difficulties and nuances of the general case are absent. It

is a practice problem to demonstrate the idea and to provide a basis on which

to build toward solution of real problems. The rectangular waveguide is

chosen because it has simple known solutions, because the boundary conditions

are easily imposed, and because it reduces to a two-dimensional problem,

thereby reducing the computational exercise.

14
We continue to use the 4>(x,t) = gauge. The known time and longitudinal

dependence are imposed by writing the vector potential and current induced in

the walls as

A(x,t) = A(x

±

) cos(o$t - kz) = A(l) cos(wt - kz),

(3.9)

/(J)2 IT'' J(x,t) = J(i) cos(ut - kz),j k ~~ » To ~ ~'

c a

where c is the speed of light in the waveguide medium, x = (x,y) is the twodimensional transverse position vector, and where A(x,t) and A(i) are real.

Substitution of eq (3.9) in eq (3.2) yields

a+A b+A 2 2 S = C J dx j dy ^ [A 2 (l) + A

2(l) ] + ^ A

2 (l)

-A -A y

(3.10)

- [(3

y

A

z (l)) 2 + (3 x Az (l)) 2 + (3 x A

y

(l) - 3yA

x (l)) 2

]

+ 2y 3(1) • A(l)},

C = 4- ] dt / dz cos 2 (ut - kz) « y~ j dt / dz sin 2 (wt - kz).

In writing eq (3.10) we have assumed that the material in the waveguide is

isotropic and that the range of the t and/or z integration(s) is either very

long or an even number of cycles. The induced current term and the limits of

the transverse integrations require explanation. In order that the stationary

point of S yield Maxwell's equations, the variations of A must be zero on the

boundary, which in our two-dimensional case here means that we must specify

A on the transverse boundary. In order to be able to specify all components

of A we choose the boundary to lie a few skin depths within the conducting

walls where the field can safely be assumed to vanish. Referring to figure 3-

2, the x integration goes from x = -A to x = a + A, and the y integration from

-A to b + A, where A = N6, some suitable number of skin depths. Then, however, the currents induced in the walls are contained within the integration

volume. Fortunately, we can show that the contribution to the action from the

volume within the conductor is negligible, and we can write

15
S - C J dx / dy \g [A

y

2(l) + A2(i)] +^A|(l) - [(3 A

z (l))

2

2 2

(3 " U)

+ [3X

A

Z (D) + ty A

y

(l) - 3y

a

x (i)) ]} .

We still need to impose E. =0 and B = at the conductor walls,

tan norm

which is accomplished by the requirements that

A

z (0,y) = Az (x,0) = Az (a,y) = A

2

(x,b) = 0,

A

x (x,0) = Ax (x,b) = 0, (3.12)

A

y

(0,y) = A

y

(a,y) = 0.

The action is then discretized by breaking up the waveguide into a rectangular

grid as in figure 3-3, with area elements centered on crosses and field values

defined on dots. The spacing between dots is Ax = a/N , Ay = b/N . The field

x y

value for an area element is given by the average of the values at the four

corners of the element. For derivatives, the average of th.e two appropriate

differences is used, e.g.,

d A

hnr) H> j 4) 2a7 tVi+1 > j +1

> - V1

' ^

(3.13)

+ A

x (1+l,j) - A

x (i,j)].

The action then takes the form

N N

S - C Ax Ay j[ I {(|-)

2 DUa.e) + A(a-1,3) + A (a, 3-1) + A (a-1,3-1)]

a=l 3=1

TM

y y y y

+

(|iT

)2 [A

x( a ' e ) + A

x (a-1,3) + A

x (a,3-D + A

x (a-1,3-1)]

2 + (^)

2

[A

z (a,3)

+ A

z (a-1,3) + Az (a,3-D + Az

( a-1 , 3-1)]

2

- [j^f [A^a.3) - A

z (o,H)

+ A

z (a-1, 3) - Az

( a-1,3-1) ] -{-^) [Az (a,3) - A

z (a-1,3) ( 3 - 14 )
+ A fo,H) - A 7 (a-1,&-1)]

-

-

[-25J- (A

y

(a,3) - A

y

(a-1,3) + A

y

(a,3-1) - A

y

(a-1,0-1))

- J^ (Ax (a,3) - A

x (a,3-1) + Ax (*-1,3) - Ax (a-1,3-1) )]

2

} .

We next wish to fix the field values at a small number of measurement

points and vary all the fields not fixed by measurement or boundary condition

until a stationary point is found. Because of our gauge choice and the fact

that we are only considering a single frequency, a measurement of E is a direct measurement of A. The numerical calculation then proceeds as follows. The fields A are fixed at measurement points, and the appropriate

components are set equal to zero at the boundaries (eq (3.12)). These are not

allowed to vary during the computation. A starting field configuration is

generated according to the prescription

A

x (x,y)

A

y

(x,y)

A

z (x,y)

B (y) m — I A (X ,Y ) x n n

1

R(x) n=l

B (x) m

x - X B (Y )

' x n

R(x) n=l

1 A (X ,Y ) y n n

J x - X B (X )

' y rr

N B (y) B (x) m — "

I A (X ,Y ) R(x) n=l

z n n

1

(3.15)

B (Y ) B (X )

' x n y n

where R is just a normalization factor and the B functions enforce the boundary conditions,

N R(x) = I n=l

1

x - X

B M =

( i . e2

^

/?

)(l - e2 - 3 (^' /5

) , (3.16)

B ( X ) = (1 - e2 - 3x/5

)(l - e

- 2 - 3(a - x)/e

) ,

17
with £ chosen to be A/4. The vectors X = (X ,Y 1 are the measurement points,

n

v n nJ r ' and Nm

is the number of measurements. The initial configuration given by eqs

(3.15) and (3.16) is an arbitrary choice, but it does have the desirable properties that it smoothly interpolates between measurement points, obeys the

boundary conditions, and is not the correct answer. The last point is important because the question is whether the computation can find the correct so- lution, not whether it recognizes the answer if given it.

Having generated initial values for all the A(i,j), we go through the

grid setting the values of A at each point (i,j) equal to the values required

to make 3S/3A (i,j) = given the current values of A at neighboring points,

a Such a procedure would drive us to a nearby extremum if only there were one.

In either the continuous (3.11) or discrete (3.14) form, however, the action

can be shown to have no (finite) extremum. The numerical procedure will then

run away, given enough time. (This is true for any grid size; it just takes

longer for finer grids.)

There is a way to locate the stationary point nevertheless. At a stationary point, S should change very little as we pass through the grid

changing the A(i,j)'s, and so we plot S as a function of the number of passes

through the grid. Figure 3-4 shows the result for a = 0.8X, b = 0.4A. The

program was told A at nine points, (X

n ,Y

n

) with X

n = a/4, a/2, 3a/4, and Y

n =

b/4, b/2, 3b/2. A 40 x 20 grid was used. The slope of the curve is quite

small from N - 40 to N == 90; the approximate stationary point is somewhere in

this range. The exact point doesn't matter much since the fields do not

change much in this range of N. Figure 3-5 plots the fields as a function of

x for a few values of y for N = 65. The correct solution is the TE^q mode,

A

x

= A

z = °> A ( x »y) = si n ~ (solid line in figure). The crosses represent

A , the circles A

x

. The computed A

z is zero for all x and y and is not

plotted. The results are clearly very good in this admittedly simple case,

suggesting that the method holds some promise for practical applications.

We have also performed tests in which the number and positions of the

measurement points were varied. Qualitatively, the results are about what one would expect. As fewer measurement points are taken, or when they are clumped

together or all far from the center, the computed solution deteriorates. A

worst case is only one measurement point (which only sets the scale of A ) far

18
from the center. For waveguide and grid as above, but with one measurement

point at (X,,Y,) = (0.1,0.1) x, the action behaves as in figure 3-6. Again the

general location of the approximate stationary point is apparent. Taking it

to be at N = 55, one obtains the field configurations of figure 3-7. Although

the agreement with the correct solutions is not so good as it was with nine

measurements, it is still recognizable and would be quite useful if we only

wanted an estimate of the maximum field within the enclosure.

This simple example may not have been a very demanding test, but the

finite-element action approach did work; and it worked well enough to bolster

our hope in the eventual practical applicability of the procedure.

3.4 Problems and Prospects

Although the waveguide example given above is a simple problem, the success of the finite-element action approach is significant nonetheless. The

program did find a solution to a simple two-dimensional problem without

knowing the source(s). Furthermore, many of the complexities of practical

problems are little more than technical details in this approach. The extension to three spatial dimensions, for example, requires more computational

time, but no new concepts or techniques. Also, because the symmetry of the

waveguide has not been used, extension to irregular geometries should be no more difficult than specifying the geometry for the computer (provided the

geometry is not such that it requires a hopelessly large number of grid

points). Pieces of dielectric material can be included simply by storing the

permittivity at each grid point, e(i,j), and restoring e(i,j) to eq (3.15).

The biggest unforeseen difficulty so far, that the stationary point is a saddle point, has been overcome. There are probably more elegant and/or efficient ways of finding the stationary point, but we have demonstrated that at

least there is_a_way to find it. It is possible that the procedure can be

modified so that the stationary point is a minimum, but we do not know how at

this time, and it is not necessary as long as saddle points can be located.

All this is not to say that we are home free. It is possible that in

practical problems the functional structure of S renders the stationary point

much harder to locate. In addition, there are a number of technical hurdles

19
to be overcome before this technique becomes an everyday tool, but none of

them is obviously insuperable. In a practical case boundary conditions could cause some trouble. Irregular boundaries are no problem in principle, but

they will entail computational complications. A more serious difficulty is

the handling of open boundary conditions. In the example given, the volume

was completely enclosed, but one component was effectively free at each

wall. When all three components are free over some part of the boundary

surface, it is likely that some measured points will need to be on or near that surface to get good results. In the waveguide case we did try fixing the

value of all components of A on the boundary, and the only major effect was very close to the walls. This suggests that when open boundaries are present

the buffer zone (cf. fig. 3-1), wherein we solve for the fields but do not

believe the results, will not need to be too large.

There could also be difficulties associated with the measurements in

practical problems. If more than one frequency is present, then a measurement

of E is not a direct measurement of A, but rather of 3A/3t. Consequently

not A but rather its derivative must be held fixed in the computation.

Similarly, a measurement of the magnetic field would fix a combination of

derivatives of A, not A itself. Another difficulty is that it is much easier

to measure the peak or average E than it is to measure E at one time t, but

the latter quantity is the easier to include in the calculation. And, of

course, eventually we need to confront the fact that the measurements are not

perfect, and uncertainties must be included.

Other than the measurement complications just mentioned, complicated time

dependence should not raise major difficulties. It increases the dimension of

the problem—real problems will be four-dimensional as opposed to the twodimensional example we did—but it is handled in much the same way as the

spatial dimensions. It increases the computing time required, of course, but

it requires no new developments.

As for the computation itself, no great effort has been made as yet to

make it fast or efficient. For less simple geometries a different (e.g.,

triangular) gridding system would be more versatile, and the discretization of

the integral (3.11-14) could also be improved. Such refinements and sophistications, however, fall in the fine-tuning category. The more immediate task

20
is to advance to more realistic problems and try to handle the concomitant

complications. It appears that this action-based approach may be a viable

practical method. If that proves to be the case we would have a very powerful

method indeed; but considerable work, and perhaps even some ingenuity, is re- quired before we reach that point.

4. Scanning Techniques

4.1 Cylindrical Scanning

The approach that is presented in this subsection involves cylindrical

scanning. The ease with which a complex EM environment can be scanned by

moving a highly directional probe along a single axis for different azimuthal

orientations makes cylindrical scanning very attractive. The theoretical development given here is based on the source-scattering matrix [13], which is

very analogous to the well-established plane-wave matrix approach [14].

A highly directional probe antenna and its coordinate system fixed to the

probe antenna are shown schematically in figure 4-1. Unknown radiating

sources are located outside a cylindrical volume, p > p . The EM fields in

the source-free region (p < p ) will be expanded in a complete set of cylindrical eigenfunctions,

00 00 E(p,4>,z) = I 1 [b}(Y) M

n (p,M) n ny

f|= — 00 —00

+ b

2 (y) N (p,<M)] dY,

n nY

H(p,<j>,z) = — VxE (4.1)

00 00 m T~ I 1 My) N (P,<I>,Z) + b

2 M (p,<J>,z)] dY

Z n n nY n nY U D=_co -oo

I = /uTTT « 377 a,

where the M and N functions are given by

M

nT

(p,*,z) = Vx(J

n (K P ) e

-jn<1)e-JYZ

e

z

)

21
= p^J(Kp) e - KJ'(Kp) eJe"jn VJYZ

L p n p n <j) J

N nY(p,*,z) = £ Vx M^ (4.2)

4 [-J* J>P) e

p -f=On (Kp)e; + K

2

J

n (K P )ez

] e"H"^.

All quantities with a hat ( *) over them denote unit vector; e

Ja) (w>0) time

dependence, and the rationalized mks systems are used throughout. The freespace wave number k is defined by k = w/y e = "~ = "T" » tne axia ^ P art °f tne

C A wave propagation vector is denoted y and K = A 2 - y

2

. If one can use a probe

antenna which measures the transverse (to eJ components of the electric

field, then the b (y) can be determined directly. Namely, using the orthogonality relationship for the M and N functions, i.e.

00 2 IT „«> 2 IT

1 I (I xH. ,)«e dHz s 0, J 1 (R nn x N. , )

• e d* dz =

J n ^ ny n y ' p ' hK nr n r; p

Y - ""

(4.3)

L

S

o [R

ny

xRn'Y^ -%d+dz=^Jn (K P ) J>p) 6_n>n

,6( T+ y).

By crossing N into eq (4.1), one gets the b*(y), i.e.

bn^> ' 4^" (Kp ) J

1 (Kp ) > Jfl x E(p„.*.z)} • e* d* dz

j J [-21 E (p ,4>,z) + K*E ( Po ,4,,z)]e

jn *ejrZ d* dz.

Similarly, cross M into (4.1) to yield the b

2 (y),

b

n ^

4,^0 (K Po) ' "/ £'.••*> *in ^* ^

^ n

u -oo u

Equations (4.4,4.5) give the modal coefficients in terms of the measured

transverse electric field. The efficient computation of the double integrals

in eqs (4.4,4.5) can be carried out by use of the sampling theorem and a fast

Fourier transform.

22
Consider a more general probe antenna whose receiving functions R (y) are known with respect to a cylindrical coordinate system fixed in the probe and

centered on itself. The output of the probe is given by

b'o = I I J R* (Y) a;

s

( Y) d Y , (4.6)

s=l n=-°° -°°

where a' are the modal coefficients of the J (Kp) modes which are excited by sources existing outside the probe in the cylindrical coordinate

( p' , <f>' ,z') system fixed in the source.

Specifically, the a field in cylindrical coordinates (p'j^'jZ 1 ) is given

by

00 00 E'(p'.V.z') - I / [a;i( Y ) ^(p'.^'.z') + a'2( Y) ^(p'.V.z'JldY.

(4.7)

00 00 R'(p\*',z') --±-

i / [b;MY) ^(p'.f.z 1 ) + b;i( Y) ^(p'^'.z-jjdY

" o n=_oo —oo '

'

Now M„ is defined as before,

ny

M^Ufcz) = VxJ n (K P )e-jn VJTZ

e

z

= [^n J

n (K P ) - u; (K P ) e^]e-J' n V^

R^U.fcz) = 17 V x M

(1)

ny k ny

(4.8)

4 U* J n^ e>^J n^% + K2j n^ eJe'^V^.

Now, regardless of whether the probe antenna or EM source coordinate system is

used, the EM field at every point in space must be the same.

E' Cp' »'

z< ) = E(p.4>»z)

(4.9)

H'tp'^'.z' ) = R(p,«|.,z]

23
when (p',<t>

, ,z

l ) and (p,<)>,z) refer to the same point in space. But as the

probe antenna scans the radiation source, the source coordinate system is

merely rotated through an angle <t> , about the z-axis, and translated a distance z along the z-axis, i,e.

p' = p, <j>' = <|>-<j> and z' = z-z (4.10)

and eq (4.9) becomes

E' (p, <t>-<J> , z-z ) = e (p><M)

(4.11)

H' (p,<l)- <|> , z-z ) = H(p,<j),z)

i i when (E,H) and (E,H) are written explicitly in terms of the linearly independent cylindrical waves in eqs (4.1) and (4.7). The orthogonality relations eq

(4.3) show that the only way eq (4.11) can be satisfied is if

*

ln *(y) = b^( Y)e-jn Vjnz o, s-1,2. (4.12)

Thus the probe receiving equation becomes

• 00 00 b'Uo.Zo) =

I I J R^(y) b^Y)e-jn<t>0 " jz

° dy . (4.13)

s=l n=-°<> -»

c Let us discuss how to evaluate some characteristics of b (y). In principle, as the probe antenna scans the EM environment, the output b'(<j>o,z ) of the probe

antenna is recorded for

< <|> < 2tt and -~ < z < °°.

That is, the amplitude and phase of b'(<j> ,z ) are measured for all values of

<t> and z . In practice, the z Q scan can be limited to some finite scan

length, since b' ( 4>o» z o) ^ s assumed negligible outside this region. Also, data

need be sampled and recorded only at a finite number of measurement points.

Having measured b* ( <t> ,z )

» the Fourier series and integral of eq (4.13) can be

immediately inverted to yield the solution

24
I R

s

( Y ) b* ( Y) = [Rj (y) bj (y) + R

n

2 (y) b* ( Y)]

„ . n n n n n n S=l

2-rr 1

r r ut, . _ n J n *n JYZ ( = 4^2 / / b' U ,z ) e

J "^eJ,t o d* dz e I ( y) 4lT -co n (4.14)

1 2 Since eq (4.14) involves two unknowns b (y) and b (y), two linearly independent scans are necessary to account for the polarization of the EM

fields. The second scan produces a second equation to complement eq (4.14),

i.e.,

2

's s I R

n

S (Y) b* (y) = R^(y) bi(Y) + R

n 2 (Y) b2( Y) S=1

(4.15)

1 °° 2X jn<j) j'yzq

= —p / / b' U ,z ) e e d<j> dz = I_' (y)

TV _oo U

where R (y) and bo(<j> ,z ) are the receiving characteristic and the output of

reoriented probe, respectively.

Assuming for the moment that the receiving function (R 1

, R' 2

) of the

probe antenna is known, eqs (4.14) and (4.15) can be solved immediately for

the source characteristic b(Y), i.e.

b

X (Y) = [R n

2 (Y) I n (Y) - R

2 (Y) I n (Y)]/A

n (Y)

b

2 (Y) = [R

n

X (Y) i;(Y) - R^(Y) I n (Y)]/A

n

( Y)

(4.16)

where

A

n (Y) = R; 2

(Y) R

nMY) - R

2 (Y) R'^Y) . (4.17)

Equation (4.16) indicates that the cylindrical wave expansion coefficients

b

1 and b2 for the EM complex field environment can be determined provided the

n n

r r receiving characteristics of the probe antenna are known. There are several

ways to determine the receiving characteristics of a general probe antenna in

terms of the scalar cylindrical waves. Since it is beyond the scope of this

paper, it will be discussed in a future paper. But it is very easy to show

that, if a probe antenna is an ideal electric dipole, i.e., a probe that measures the transverse components of the electric field, then eqs (4.4) and

(4.5) emerge immediately from (4.16) and (4.17). For more directional probes,

25
a different expansion may be more useful, and we next turn our attention to

this possibility.

4.2 Spherical (Directional) Scanning

If one has a probe which is highly directional, this feature can be used

for directional or spherical scanning at one point, a suggestion due to Chang

and Maley [15]. The directional features are most directly exploited by a plane-wave expansion. Assuming a single frequency, the electric field is

written as

E(x,t) = E(x) e jwt

,

3 - - (4.18)

Zfo = /77T3 e

"Jk 'X

(Ztt) 3 l

^

If E is to satisfy Maxwell's equations, we must have k 2 = <d 2/c 2

, allowing us to write (for large volumes)

E(Jc) = e

k

(k) 2w«(|- k),

It is necessary at this point to set forth the notational conventions for

this subsection since many different angles will be encountered. There are two coordinate systems of interest, one fixed with respect to the volume of

interest (the "lab" coordinate system), and one fixed with respect to the

probe (the "probe" coordinate system), cf. figure 4-2. The two coordinate

systems have a common origin, the location of the probe (assumed to be

small). The z, axis of the probe system is described by angles e , <j> in the

lab system; the £ axis of the probe system is chosen to lie in the z x z, plane. In the course of the scanning e and $$ vary, of course. The direction of incidence of a plane wave with respect to the probe system will be

denoted 6',<j>'. This same direction as seen in the lab system is called

8,<j). The angles 0,<j> depend not only on 6',$' but also on 9q

»<I>q. And con- versely e' = e' (8,<t>,e ,<j> ), y = <j>' ( e, cj>,

, 4>o) • A straightforward exercise

26
with rotation matrices yields the relations

cos 0' = sin e sin 8 cos( (|>- 4> ) + cos 8

o cos e » (4.20)

cot <j>' = cot( <j»-<t>o) cos e - sin e cot csc(<t>-<t> ).

For the sake of simplicity and clarity in this presentation, we limit

ourselves to the case of a scalar field. The response of the probe to a single plane wave incident at angles 0',<j>' in the probe coordinate system will

be the product of the probe acceptance at that angle (A), the amplitude per

solid angle of the wave (eM, and the solid angle,

dR = A(e' ,V ) ej,(0',<t>') da'. (4.21)

But the plane wave component from 0' ,<j>'in the probe system is the same as the

component from 0( 0' ,<(>', 0o»<j>o) > <K e ' » ' » 9 o» H) in tne ^ aD system,

e'

k (0\<j>') = e

k (0(0' ,<t»' ,0 o ,<t> o ), 4>(e' ,4>' ,e ,<|>o)). (4.22)

Using this fact and integrating the differential response (4.21) over all incident angles in the probe system, we obtain for the total response

1 TT R( ©o» 4> ) = / d cos 0' / d<j>' A( 8'

,

4)' ) (4.23)

-1 -TT e^ (0( 0'

,

<j>' , 0o> <J>o) > 4»( e ' s^

1 »©o»4>o) JFor purposes of numerical inversion, we find it more convenient to rewrite eq

(4.23) as an integration over lab-system angles,

1 TV R(e »*o) = / d cos / dcf. A(e' ( e, <j>, e , <t> ) ,

<t>' ( e, <j>,

e

, <j> ) ) e

k

( e,<t>). (4.24)

-1 -TT

If the probe is perfectly directional A is just a product of delta functions

of -

O and 4) - <j> ; and the response at o ,<j>o measures e. (

e

, <j> ) directly. For imperfect directional probes, a little more work is required. The

probe response is recorded for some number N of scan angles r r meas

27
9,-»*:»i = ^Nmeas* We must invert (4.24) to obtain e. (6,9), which then determines the field throughout the volume from eq (4.19).

To invert eq (4.24) we use a pulse expansion of e

k (cos0,<j>) and point

matching. Assume for simplicity that the scanning was done at regular intervals 6 n in cose and 6 in <t>. Let

o <p

_ N

e \

e,(e,<j>) =

I I enm n(cos e,<t>),

'k nm

n=l m=l

N

e =2/ 6

e> %= 2 ^V

(4.25)

where n (cos e,<b) is the unit pulse function centered at nm , r

, cos e = -1 +(n - 2-)6

Q

, <j> = (m - j)6

n (cose, 9) =1 -1 + (n - 1)6 Q < cose < -1 + n6,

and (m-l)6 A < <t> < m5 x, (4.26)

9 9

= otherwise.

For each of the Nmeas measurement directions we get an equation of the form

N

e Nd> R(V»i) - I I e

k m An >i> M>

n=l m-1

(4#2?)

-l+n6 mS

A

nn, (9, f «. )= / u d cose J* d<t> A (e'(e, ^e. , $ .), <>• ( 0, <j>.e,,*_•)).

1 ' -l+(n-l)6

e (m-1) 6^

n '

Because the measurement angles coincide with the centers of the pulses, it is

a simple matter to relabel indices and force eq (4.27) into the form of a simple matrix equation. The m and n indices are combined into one index

running3 from 1 to N = N n xN,,

meas e 9'

N +1

a

i = eii a

2 = e

i2 a

9 = p 21

9

k " ek

' 9

k " e

k 9

k e

k

, ... ,

a\ = ^ l (Qv ^),af = A

12^.,*.), ... , (4.28)

28
6- = R(e. , $. ).

Then eq (4.27) takes the form

Nmeas

*_1

(4.29)

g

Jk = I (a

1

)

' 1 r

The e. 's are then substituted into eq (4.25) to yield the approximation for

e

k

(e,<t>), which in turn is substituted into eq (4.19) to yield the field E(x).

We thus can obtain an estimation of E(x) even with only a few scanning

angles, though of course the approximation becomes better as more measurements

are made. It remains to be seen how large the angular increments in the scan- ning can be in practical applications. The expansion of eq (4.19) will only

be valid for |x| less than the distance to the first source or scatterer, but

the method still offers the promise of an estimate of the field throughout a volume, from a limited number of measurements at just one point.

5. Summary

As more sources contribute to ambient electromagnetic fields, and as the

electronic devices operating in these ambient fields become more numerous,

sensitive, and important, the problem of efficiently measuring and characterizing complicated electromagnetic environments is becoming increasingly

acute. We have discussed three different approaches to the problem, outlining

the foundation, present status, and direction of future development of each.

Because there is so little previous work on characterization of complicated

electromagnetic environments, any new method tends to require completely new development beginning from the basics, and as a result progress can be rather

slow.

Each method discussed is quite promising in the sense that each appears

to have a reasonable chance of actually working, and each will be very useful

if it does work. However, each method also requires further development and

work. That work is in progress.

29
6. References

[1] Kanda, M. Time and amplitude statistics for electromagnetic noise in

mines, IEEE Trans, on Electromagnetic Compatibility. Vol. EMC-17, No. 3:

122-129, August 1975.

[2] Beckmann, P. Probability in communication engineering. New York, NY:

Harcourt, Brace and World, Inc.; 1967.

[3] Lee, W.C.Y. Mobile communications engineering. New York, NY: McGrawHill Book Company; 1982.

[4] Wells, P. I.; Hill, D. A.; Longley, A. G.; FitzGerrell, R. G.; Haidle, L.

L. and Glen, D. V. An experiment design for the measurement of building

attenuation, OT Technical Memorandum 75-199, May 1975.

[5] Hoffman, H. H.; Cox, D. C. Attenuation of 900 MHz radio waves propagating into a metal building, IEEE Trans, on Antennas and Propagation,

Vol. AP-30, No. 4, 808-811, July 1982.

[6] Cox, D. C; Murray, R. R.; Norris, A. W. Measurements of 800 MHz radio

transmission into buildings with metallic walls. The Bell System

Technical Journal, Vol. 62, No. 9, 2695-2717; November 1984.

[7] Ishimaru, A.; Sigelmann, R. A.; Kuza, Y. Private communication.

[8] Landau, L. D.; Lifshitz, E. M. The classical theory of fields, Chapter 4,

Reading, MA: Addi son-Wesley, 1962.

Morse, P. M.; Feshbach, H. Methods of theoretical physics, Chapter 3, New

York, NY: McGraw-Hill, 1953.

[9] Whittaker, E. T. Treatise on the analytical dynamics of particles and

rigid bodies, Chapter 9, New York, NY: Dover, 1944.

[10] Ferrari, R. L.; Maile, G. L. Three-dimensional finite-element method for

solving electromagnetic problems, Electron. Lett. Vol. 14, No. 15, 467-

468; 1978.

30
Webb, J. P.; Maile, G. L.; Ferrari, R. L. Finite-element solution of

three-dimensional electromagnetic problems, IEE Proc. Vol. 130H, No. 2,

153-159; 1983.

Rahman, B. M. A.; Davies, J. B. Finite-element analysis of optical and

microwave waveguide problems, IEEE Trans, on MTT, Vol. MTT-32, No. 1, 20-

28; 1984.

[11] Morishita, K.; Kumagai , N. Unified approach to the derivation of variational expression for electromagnetic fields, IEEE Trans, on MTT, Vol.

MTT-25, No. 1, 34-40; 1977.

[12] Ramond, P. Field theory: A modern primer, Reading, MA: Benjamin/Cummings,

1981.

Itzykson, C; Zuber, J. B. Quantum field theory, New York, NY:

McGraw-Hill, 1980.

[13] Yaghjian, A. D. Near-field antenna measurements on a cylindrical surface: A source scattering-matrix formulation. Nat. Bur. Stand. (U.S.)

Tech. Note 696; 1977 September.

[14] Kerns, D. M. Plane-wave scattering-matrix theory of antennas and an- tenna-antenna interactions: formulation and applications. J. Res. Nat.

Bur. Stand. (U.S.). 80B: 1, 5-51; 1976 January-March.

Kerns, D.M. Plane-wave scattering-matrix theory of antennas and antennaantenna interactions. Nat. Bur. Stand. (U.S.). Monogr. 162, 179 p., June

1981.

[15] Chang, D. C. and Maley, S. Private communication.

Appendix

In order to locate a stationary point of the action we need to know

whether we are looking for a maximum, a minimum, or neither. In addition, if

it is an extremum, we would like to know whether it is a global or only a local extremum. It is often assumed that it is a minimum, but our experience

indicates that exceptions are neither so rare nor so pathological as one might

expect.

31
For mechanical systems the problem is addressed in Whittaker's classic

treatise [9], (Note that Whittaker uses the word "action" for a different

quantity than we do. It is, however, a difference of nomenclature not substance.) The type of stationary point which S has depends on the "kinetic

focus" of the original point, defined as follows. Consider a point A on an actual trajectory; and let another actual trajectory pass through A, at some

small angle with respect to the first. If the two trajectories intersect

again, say at point B, then the kinetic focus of A is the limiting value of B as the angle between the two trajectories at A goes to zero. The relevant

result then is that the stationary point of the action S is a minimum if the

final point in the integration occurs before the kinetic focus of the initial

point; if it occurs after the kinetic focus the stationary point is neither a minimum nor maximum but rather a saddle point.

To make sense of the preceding verbiage, an example is in order.

Whittaker considers a particle on a sphere with no forces acting. For

variety, and because it is a little more relevant for us, we consider a simple

harmonic oscillator such as a mass (m) on a spring (spring constant k). The

action is

S =f J dt [x(t) 2 -£x(t)2], (A.l)

t

i where x(t) is the position at time t. The kinetic focus for a given

x. = x(t.) is easily determined from our knowledge of the solutions. Since

solutions of different amplitude all have the same period, if two trajectories

(solutions) are both at x^ at time t^, then one half period later both will be

at x = -x.. Consequently, the kinetic focus of t^ occurs at

t = t. + T/2 = t. + ir/rn/k . We therefore expect the stationary point of S will

be a minimum for t

f

< t. + T/2 and a saddle point if t- > t. + T/2.

We can verify this expectation by direct calculation. If we vary x(t)

and set the first variation of S equal to zero we just obtain the equation of

motion. Considering the second order variation of S, we have

§<JS- j dt [<*)*-*<*)*]

*1 (A.2)

32
tf

= [6x 6x]^

f

- / dt 6x [Sx + - Sx],

, t.

The first term vanishes because 6x(t) = at the end points t = t., t

f

. This

fact also allows us to expand 6x(t) in a discrete Fourier sine series,

00 6x(t) =

I

Sx sin [n ir(t-t.)/At] , n_1

(A.3)

At = t

f

- t . .

Substituting eq (A.3) in eq (A. 2) yields

I

SY2

S =#" I [(-Tt)

2 "-] Sx 2

. (A.4) m x 2

L

n lK t& J m

J n

v ' n=l

Since the original 6x(t) was arbitrary, so too are all the Sx

n 's. Therefore,

6 2

S will always be positive only if [(mr/At) 2 - k/m] > for all n. This re- quires

At < * ^ = I

T * (A * 5)

For At > T/2, 6 2S can be positive or negative depending on the choice of

6x(t). Therefore, as expected, the stationary point is a minimum if tf is

earlier than the kinetic focus, and a saddle point if tf is after the kinetic

focus. (If tf occurs at_ the kinetic focus, 6 2

S > 0.)

Transferring these results to the electromagnetic problems of interest

requires further work; but there are reasons, both heuristic and empirical,

for expecting the stationary point to^be a saddle point. One heuristic reason

is that we are formally considering J dt L, so that tf must be later than the

kinetic focus provided it exists. Secondly, in the mechanical case the transition from minimum to saddle point is related to the nonuniqueness of the

solution. For the harmonic oscillator example, for t

f

< t. + T/2 specifying

t., t

f

, x(t.)» and x(t f

) specifies a unique solution (i.e. amplitude and

phase). However, for tf = t^ + T/2 there is a continuum of solutions possible

for a given t-j, tf, x(t.)» x(t f

). This is no coincidence since the kinetic

focus occurs at the (limit of the) intersection of real trajectories passing

through the initial point. It is then reasonable to infer that the stationary

33
point ceases to be a minimum when tf exceeds the first time at which it is

possible to have more than one solution. The nonuniqueness of the solutions

in the electromagnetic problems we are attacking was noted earlier, and so we have another qualitative reason to expect a saddle point.

Finally, there is the empirical evidence: in the example problem presented in subsection (3.3), the stationary point is shown to be a saddle

point. This is a potential disaster computationally, but it will prove possible to locate the saddle point.

34
v>S

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Unto r by -i'°8,o( -Ipp)

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-40-. llttt ILL.-H

IT it, TT

i ft

-soLll n

MOUMDI J 5 I 5 20 30 40 50 GO 70 10 85 90 95

Percent of Time Ordinate is Exceeded

98 99

Figure 2-1. Cumulation amplitude probability distribution of EM noise in a mine.

35
I0,000 r

-60 -50 -40 -30 -20 -»

MAGNETIC FIELD STRENGTH, H (46 RELATIVE TO I MICROAMPERE PER METER)

Figure 2-2. Average crossing rate of EM noise in a mine,

36
COu

LUQ

Z

Z

O

t—<

z

10 -

2 5 10 20 30 40 50 60 70 80 90 95 98 99 PERCENT PROBABILITY THAT ATTENUATION > ORDINATE

Figure 2-3. Cumulative amplitude probability distribution of the building's

attenuations.^-' The straight line represents a log-normal distribution.

37
Field Strength (dB)

o

o

en

ro

en

2P _j.

CD

-^ O

O

CD

"3 I\3 >-* O

O CO —

O

3"

CD

o

eno

O—

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CD o

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m o

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en

CD

Q.

CD CD

Q. oo

CD

CD

CD

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00

CD

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CD

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CD Figure 2-4. Cumulative amplitude probability distribution of fields in a rectangular cavity.'- 7 ^ The straight line represents a lognormal distribution.

38
Sources

Fields = on surface

Figure 3-1. Schematic division of total volume into volume of interest (Vj),

buffer volume (V B

), and remaining volume containing all sources (V$).

y ^ f~ (a+A, b+A)

^

(-A, -A

_ J

Figure 3-2. Rectangular waveguide dimensions and axes. Also depicted is the

boundary of the volume over which the action integral extends.

i : i XI X I X I <^-* —*--f-

„ x

i xl x

Figure 3-3. Lattice of points for finite-elment action,

39
300

N

Figure 3-4. Action as a function of N, the number of passes through the grid

in the numerical computation, for nine measurement points.

40
1.0

0.9 yr ^^Sc ° A* yX Jk X Ay 0.8 / V

/ V y

= o.2a

0.7 / \ 0.6 / \ 0.5 / \ 0.4 / \

0.3 ~ f \ "

0.2 - / \ -

0.1 _/ \-

01 / o o000 0o° ° o°°\

oo g ° v0oo o 00o *°o u (

•0.1 -a I I I I I I I 3) -

x/A

Figure 3-5. Fields A

x and A as functions of x at fixed y, after 65 passes

through the grid in the numerical computation with nine measurement

points. Solid line is correct result for Ay

. Correct result for A

x is

zero, a) y = 0.2A b) y = O.lA c) y = 0.05A

41
0.7 0.8

42
Figure 3-6. Action as function of N, for one measurement point.

43
1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

-0.1

x x « x X

~os

qQOOOOOOOOqqq 9 o e ww °ooooooOO°

I.U

0.9

I I

I ^^~"T""^\ '

I

I o Ax X Ay 0.8 y = 0.1A

0.7 X yXXXXXw * * X \\

X x X X

X

X

X

0.6

0.5

0.4 / x

f X

X X

X

X

X X \

0.3

0.2 X

X

X \

X \

X \ - x\

0.1

'% x\ -

o: ft o o o o oooooooo OAArirL ,-°°°\

9 © 00odo °0oooooo ° i

-0.1

I I

I I I I , b»

0.1 0.2 0.3 0.4

x/A

0.5 0.6 0.7 0.8

Figure 3-7. Computed fields as functions of x at fixed y, after 55 passes

through the grid, with one measurement point. Solid line is correct result

for Au

. Correct result for A is zero, a) y = 0.2X b) y = 0.1X

J x c) y = 0.05X

44
xx xx x y = 0.05a _

ftOftOOOOPOpft o-o-e

c)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

x/X

45
Figure 4-1. Coordinate system for probe antenna

46
Figure 4-2,

ni ng.

Coordinate-system conventions for spherical directional scan47
HBS-114A IREV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBS TN-1081

2. Performing Organ. Report No 3. Publication Date

March 1985

4. TITLE AND SUBTITLE

Possible Estimation Methodologies for Electromagnetic Field Distributions in

Complex Environments

5. AUTHOR(S) M. Kanda, J. Randa and N. S. Nahman

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

10. SUPPLEMENTARY NOTES

"] Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

The problem of measuring and characterizing complicated multiple-source, multiplefrequency electromagnetic environments is becoming more important and more difficult

as electrical devices proliferate. This paper outlines three general approaches to

the problem which are currently under investigation at the National Bureau of

Standards. The three approaches are: 1) a statistical treatment of the spatial

_

distribution of electromagnetic field intensities, 2) a numerical computation using

a finite-difference (or lattice) form of the electromagnetic action functional, and

3) use of a directional probe to scan a volume. All three methods are still m the

development stage, but each appears promising.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

action, directional scanning, environment characterization, field levels, finite

element, hazard assessment, multiple source, statistical approach

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