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NBS TECHNICAL NOTE 1040

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

—I i

i ii jiiiJiiwiMiaAWiwuaiBtiaflaaBBB^^ jtam^msmmmmmm
NATIONAL BUREAU OF STANDARDS

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'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. 'Some divisions within the center are located at Boulder, CO 80303.
A Method of Determining

tlie Emission and Susceptibility Levels of

Electrically Small Objects Using a TEM Cell

fCATlOWAX- BURBAI' A or *»A»DARD8

Ippalapalli Sreenivasiah LIBRARY

David C. Chang^ Kj i .-'^V^VU '^^J^J^ /rio-r^.

JUL t 1981

Mark T. Ma**

no.ib</c>

Electromagnetics Laboratory mi

Department of Electrical Engineering c~-^

University of Colorado

Boulder, Colorado 80309

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

J/0

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued April 1981
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1040

Nat. Bur. Stand. (U.S.), Tech. Note 1040, 52 pages (April 1981)

CODEN: NBTNAE

US, GOVERNMENT PRINTING OFFICE WASHINGTON 1981

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

Price $3.25 (Add 25 percent for other than US. mailing)
TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS iii

FOREWORD i V

1 . INTRODUCTION 1 2.0 FIELDS AND RADIATED POWER DUE TO A LOCALIZED SYSTEM OF CURRENTS 2 3.0 MULTIPOLE EXPANSION OF FIELDS GENERATED BY A LOCALIZED

DISTRIBUTION OF CURRENTS INSIDE A WAVEGUIDE WITH UNIFORM CROSS-SECTION 4 4.0 EXPERIMENTAL PROCEDURE FOR DETERMINING THE TOTAL POWER RADIATED

BY THE EUT 5 5.0 DETERMINATION OF THE SUSCEPTIBILITY LEVEL OF A RECEIVING OBJECT 8 6.0 SOME EXPERIMENTAL RESULTS FOR THE RECEIVING CASE 11 7.0 CONCLUSIONS 18

8.0 ACKNOWLEDGMENTS 18

REFERENCES 19 APPENDIX 1 20 APPENDI X II 21 APPENDIX III 25 APPENDIX IV 27 APPENDIX V 33

APPENDIX VI 35

LIST OF ILLUSTRATIONS

1. A conceptual NBS TEM cell 36

2. Test set-up for determining the total power radiated,

in free space, by an electrically small EUT 37

3a. Orientation of the EUT in step (b), section 4.2 38

3b. Orientation of the EUT in step (c), section 4.2 38

3c. Orientation of the EUT in step (d), section 4.2 39 3d. Orientation of the EUT in step (f), section 4.2 39 3e. Orientation of the EUT in step (g), section 4.2 40

3f. Orientation of the EUT in step (h), section 4.2 40

4. Equivalent circuit representation of the EUT as a receiving antenna system 41

5. Test set-up for determining the average susceptibility level

of an electrically small EUT 42

6. Normalized transverse electric field distribution inside TEM cell 43

7. Monopole antenna, crystal detector, and fiber optic transmitter

combination 44

8. Receiving object inside a TEM cell 45

-Hi-
FOREWORD

/ This report describes theoretical and experimental analyses developed by staff of the

University of Colorado at Boulder in collaboration with the Electromagnetic Fields Division

of the National Bureau of standards (NBS), under a contract sponsored by NBS. Professor

David C. Chang heads the University team. Dr. Mark T. Ma of NBS serves as the technical

contract monitor. The period covered by this report extends from July 1979 to July 1980.

The work described in this report represents a further aspect of establishing a theoretical basis for the technical analyses of transverse electromagnetic (TEM)

transmission lines. The general purpose of pursuing theoretical studies is to evaluate the

use of TEM cells for (1) measuring the total rf radiated power by a device inserted into the

cell for test, or (2) performing necessary susceptibility tests on a small electronic

device.

The particular problem of characterizing emission properties of electrically small

radiating sources by tests inside a TEM cell was addressed previously [1]. Basically, an unknown small radiating object was modeled as an equivalent dipole system consisting of

three orthogonal electric dipoles and three orthogonal magnetic dipoles, each excited with

arbitrary amplitude and phase. Systematic measurement procedures to be taken inside a TEM

cell were outlined for determining the individual dipole moments and the corresponding total

power that would be radiated by the object in free space. The theory was well supported by

the measurement results performed for a self-contained battery-operated spherical dipole. A basic assumption was then made that a constant TEM field distribution existed in the space

occupied by the spherical dipole.

The same theory is tested for the susceptibility purpose in this report by representing

the test object as an equivalent receiving antenna and by defining the susceptibility level

as the per-unit power dissipated in a particular load across some terminals of the object

illuminated by a specified incident electromagnetic field. This per-unit dissipated power

is further expressed as the mismatch loss factor of the test object in free space, which in turn depends on the equivalent antenna and load impedances. During the course of this

study, it is found that that electric quadrupole terms coupled with possible variations of

the field distribution around the test area also play an important role in determining

accurately the mismatch loss factor. For this reason, the electric quadrupole terms are added to the previous formulation where only electric and magnetic dipoles were considered. Again, necessary measurement procedures for determining the unknown

susceptibility characteristics of the test object are described together with some experimental results.

Previous publications under the same effort include:

Tippett, J.C. and Chang, D.C., Radiation characteristics of dipole sources located

inside a rectangular coaxial transmission line, NBSIR 75-829 (January, 1976).

Tippett, J.C, Chang, D.C., and Crawford, M.L., An analytical and experimental

determination of the cut-off frequencies of higher-order TE modes in a TEM cell, NBSIR

76-841 (June 1976).

Tippett, J.C. and Chang, D.C., Higher-order modes in rectangular coaxial line with

infinitely thin inner conductor, NBSIR 78-873 (March 1978).

-iv-
Sreeni vasiah, I. and Chang, D.C., A variational expression for the scattering matrix of

a coaxial line step discontinuity and its application to an over moded coaxial TEM cell, NBSIR 79-1606 (May 1979).

Tippett, J.C. and Chang, D.C., Dispersion and attenuation characteristics of modes in a TEM cell with a lossy dielectric slab, NBSIR 79-1615 (August 1979).

Sreeni vasiah, I., Chang, D.C., and Ma, M.T., Characterization of electrically small

radiating sources by tests inside a transmission line cell, NBS Tech Note 1017

(February 1980).

Wilson, P.P., Chang, D.C., and Ma, M.T., Excitation of a rectangular coaxial

transmission line due to a vertical electric Hertzian dipole, NBS Tech Note 1037

(March 1981).

-V-

A METHOD OF DETERMINING THE EMISSION AND SUSCEPTIBILITY LEVELS

OF ELECTRICALLY SMALL OBJECTS USING A TEM CELL

I. Sreenivasidh, D.C. Chang and M.T. Ma

An electrically small radiating source of arbitrary shape may, to

a first order, be modeled by an equivalent dipole system

consisting of three orthogonal electric dipoles and three

orthogonal magnetic dipoles each excited with arbitrary amplitude

and phase. Determination of the individual electric dipole

moments and the cross-components of such an equivalent dipole

system for the emission case by tests inside a TEM cell, was described in an earlier work by Sreeni vasiah, Chang and Ma,

assuming a constant TEM modal field distribution in the space

occupied by the test object. A method of accounting for the

first-order variation in the TEM modal field distribution is presented in this report. This involves the inclusion of electric

quadrupole terms in the modeling of the test object. Using the

reciprocity principle , the same method is extended to the

determination of susceptibility levels of electrically small test objects. Some experimental results for the susceptibility test,

demonstrating the importance of the quadrupole terms, are presented.

Key words: Analysis; dipole moments; electrically small objects;

emission; quadrupoles; measurements; susceptibility; TEM cell.

1.0 INTRODUCTION

An electrically small radiating source of arbitrary shape may, to a first order, be

modeled by an equivalent dipole system consisting of three orthogonal electric dipoles and

three orthogonal magnetic dipoles each excited with arbitrary amplitude and phase. In an earlier work [1,2] we described a measurement procedure for determining the individual

electric/magnetic dipole moments and the cross-components of such an equivalent dipole

radiating system by tests inside a TEM cell (figure 1). This method enables us to determine

the free space emission characteristics of an equipment under test (EUT).

A more precise modeling of an arbitrary radiating object requires, in principle, the

inclusion of higher order multi-pole moments in its equivalent representation. Even though

these multi-pole moments may have only a negligible effect on the total power radiated by

the EUT in free space, they can still affect the measurements taken inside a TEM cell if the

TEM modal field distribution is not uniform in the test space occupied by the EUT. In order

to take into account the variation of the field in the test region, it is necessary to include the electric quadrupoles in the equivalent representation of the EUT.

In the next section we present required mathematical expressions for the total power

radiated by a localized system of sinusoidally varying currents, taking the electric

quadrupoles into account. In section 3.0 we obtain related expressions for the fields due

to a system of dipoles and quadrupoles placed inside a uniform section of waveguide. Based

upon these expressions we present a measurement scheme in section 4.0 which enables us to determine the total power radiated by an object from tests taken inside a TEM cell. In section 5.0 we use the reciprocity principle and extend the test procedure to the
determination of the susceptibility level of an EUT. Some experimental results for the

receiving case demonstrating the nmportance of the quadrupole terms are presented in section

6.0, followed by concluding remarks in section 7.0.

2.0 FIELDS AND RADIATED POWER DUE TO A LOCALIZED SYSTEM OF CURRENTS

In most practical situations an arbitrary radiating object may be represented by an equivalent localized system of currents. To study the radiation characteristics of such a system of currents, we may restrict ourselves to a sinusoidal time dependence since an arbitrary time variation may be handled by its frequency components- by means of Fourier

analysis. In this section we start with the vector potential due to a volume distribution

of currents and expand it in terms of the dipole and multi-pole moments. The resulting

expression will be used to obtain the far fields and the total power radiated by the source

under consideration.

The solution for the vector potential in the Lorentz gauge is given by [3]

u^ p-jkir - r' /

V J (f'') ^|r - r' A (F) = ^ /J (P) ^ "_''

dv' (2.1)

where m^ is the free-space permeability, k = u/c is the wave number, w is the radian

frequency, c the velocity of light in free space, J the current density vector, r the field

position vector, r' the source position vector, and v is the volume occupied by the source. A time dependence in the form of e-^'^^ is assumed and suppressed. The electric and

magnetic fields are then given, in terms of the vector potential, by the following

equations:

joip^e^E = V X B (2.2)

where

B = V X A (2.3)

and Gq is the free-space permittivity.

In the far zone we may approximate the exponential in (2.1) by

|r - r' I =: r - n • r' and the inverse distance may be replaced by r, giving

g-jkr

lim A (?) ^ \-— / (?') eJ^" • »" dv' (2.4)

kr-x» v

where r = |r| and n is a unit vector in the direction of r. If the source dimensions are

small compared to a wave length, we can expand the integral in (2.4) in powers of k and

write
-jkr

A '(r) = ^^4^7— I -^ / J (r') (n • -r'f dv'. (2.5) n=o V

Keeping only the first two terms we obtain

-jkr

Mr) = '%^^ / [J + jkJ (R.P)]dv'

V -jkr

= ^°

/ {J + VzJk [(F' X a) X n + (n.r') J + (nO) P']} dv' (2.5)

V

where we used the vector identity ax(bxc) = (a«c)B - (a«B)c to write the second term as a sum of an a nti -symmetric and a symmetric terms in r' and J. We see that the first term corresponds to an electric dipole, the anti -symmetric term

which is the first inside the square brackets corresponds to a magnetic dipole and, as shown

in Appendix I, the symmetric term consisting of the second and third inside the square

brackets corresponds to an electric quadrupole. Thus, we write

A (r) = 4^ {P - Jk n X M + j I (Q-n)} e-^*"^ (2.7)

where P and M are the electric and the magnetic dipole moment vectors given by

P = / 3 (?') dv' (2.8a)

V

R = V2 / (r' X D) dv', (2.8b)

V

Q is the electric quadrupole tensor whose components are given by

V

and

r' = X x' + y y' + z z' , x,y,z = unit vectors. (2.8d)

Using the expression (2.7) for the vector potential, we can derive the electrtc and the

magnetic fields in the far zone in accordance with (2.2) and (2.3) and the total power P^ radiated by the source via the Poynting vector formulation. The final expression for Pq as derived in Appendix II is given by

Pq = 10 k^ {|P|^ + k^ |M|^ +I0 (^ ^1 "" ^2)> (2.9a)

where

Qi = IQ.^l' - \Qy/ ^ \Q,/ (2.9b)
Q-j = IQ r + IQ 1 + IQ I - R (Q Q* + Q Q* + Q Q* )• (2.9c) W2 i^yyI i^vv\ '^zz' e XX ^yy ^yy^zz zz^xx' ^ '

The above equations for the toftal power radiated are equivalent to those obtained by Jackson

[4] except for the difference In units and the different manner in which P and Q are defined.

3.0 MULTIPOLE EXPANSION OF FIELDS GENERATED BY A LOCALIZED

DISTRIBUTION OF CURRENTS INSIDE A WAVEGUIDE WITH UNIFORM CROSS-SECTION

The electric and magnetic fields generated by a localized -distribution of currents

inside a waveguide with uniform cross-section may be expanded in terms of the ortho-normal

modes corresponding to the waveguide cross-section. If the walls of the waveguide are perfectly conducting, the coefficients of such an expansion may be obtained by an application of Lorentz's reciprocity principle [5]. When this is done, the amplitudes a^ and b^ for the positive and negative going fields corresponding to n^" mode are found to be

given by

(b)

- -W O-Ej^^dv' (3.1)

where J is the volume current density, E^ ' is, respectively, the backward and forward

progressing ortho-normal electric field distribution function corresponding to the n''" mode,

and the integration is extended over the volume occupied by the source.

If the size of the radiating source is so small that E may be assumed to be constant

over the volume occupied by the source, the coefficients a^ and b^ are simply given in terms

of the electric and/or magnetic dipoles characterizing the source. If E is not constant

over the volume occupied by the source we may expand E in a Taylor series about a suitably

chosen origin and approximate E by the following

E,(u) « E^(0) + D'.vE^(O) (3.2)

so that a^ and b^ are now given by

(b^

= -\ {E^^^O) . / J dv' + / [GWe^^^O)] . J dv'} (3.3)

n V V

where u' is the position vector in a localized coordinate system, which may or may not be

the same as r'. In Appendix III we reduce (3.3) to the following form, using a method

similar to that of Collin [5]:

(') = -\ (EJ^^O) . P - jcou^ nj^)(0) . R +1/2 VE^^)(0) : Q} (3.4)

where H^ is the corresponding ortho-normal magnetic field distribution of the n^^ mode,

P, R and Q are given by (2.8a-c) and the double dot product between two tensors C and 5 is
given by

S:D =

I C^^D^g, a,B = x,y,z. (3.5) a,

If we restrict ourselves to a waveguide or TEM cell with dimensions such that at the

operating frequency all the modes, except the fundamental TEM mode, are under cut-off, the

output from both ports of the TEM cell consists of only TEM fields with coefficients a^ and

b corresponding to n=0 in (3.4). Further, E and H corresponding to the TEM mode are related through

H^-^ = + 2 X E^-^c (3.6a)

and

ll - -e^ e^ J^^ (3.6b)

where z is a unit vector in the propagation direction, Cg is the free space characteristic

impedance, and e is the normalized transverse electric field.

Using (3.6a and b) we can write a^ and b^ as

(b)

= -V2 {[P ± jk (R X z)] . e^ +1/2 VE^-'^O) : Q} . (3.7)

Thus ttie nonuniformity of the field distribution at the cell's center, i.e., vE*' '(0), gives

rise to an electric quadrupole contribution to the output fields. If we take more terms in

the Taylor series expansion (3.2), we will find that still higher order multi-pole moments

contribute to the output fields. However, for many practical purposes it is sufficient to retain terms up to electric quadrupoles. Based upon the relationship (3.7), we describe, in the next section, an experimental procedure for determining the total power radiated by an EUT, in free space, by tests inside a TEM cell.

4.0 EXPERIMENTAL PROCEDURE FOR DETERMINING

THE TOTAL POWER RADIATED BY THE EUT

4.1 Principles Underlying the Measurement Procedure

We consider the experimental setup shown in figure 2, where the EUT is placed at a point (0,yQ,0) along the y-axis in the transverse plane midway of the cell. The output

yields from the two ports of the cell are added in or out of phase and the resulting power

is measured by means of an output indicator. Assuming that the end transitions are perfectly matched, we find from (3.7) that the powers P^ and P^, corresponding to summing

the outputs from the two ports in and out of phase respectively, are given by
2 . _ -, _~ _"• = , 2 Ps = 1^0 ^\^l = l^-^o ^^/4 (vE, + vE^) : Q| (4.1a;

d- 1^0- '^ol' Ph = |a. - bj' -= Ij^(f^-^^) |jk(M X z) •

. %^V4 i, +1/4 (vE^-vE^^; (vE, - vEJ : ='q| (4.1b;

v

= e,-y

^'oy^ ay

where e and vE are evaluated at the test point (OjYq.O). For a rectangular co-axial TEM cell, with a symmetrically located center septum, it is known [5] that e has only y- component and VE has only three non-vanishing components, namely

(vE ) , (VE ) , and (vE ) , at any point (0,y„,0) along the y-axis of the cell. These

^ XX yy o yz \ •>jq-> i are given by

- (vE^) = [it) = e' (4.2a)

^ o'xx ^ o'yy oy ^ ' 3EK)yz=^=^J% (^-^^^

where

(4.2c)

(4. 2d)

Using these relations we can write P^ and V^ as

P = IP e - V? (Q - Q ) e' P (4.3a)

s ' y oy ^ ^^xx ^yy' oy' ^ '

P, = k2 |(M - V? Q ) e |2. (4.3b)

d ' ^ x ^ ^yz' oy ' ^ '

Note that when the quadrupole terms ^.tq. not important, (4.3a) and (4.3b) will reduce to

the same results obtained previously [1]. The above two equations will form the basis for

the experimental procedure that we describe in this section.

Even though (4.3a) and (4.3b) represent a special case, ten unknowns, (P , Q^^, Qyy,

Qw^j M^. df'e all complex numbers in general) are still involved. They may be determined, in

principle, by making ten measurements and then solving the resulting simultaneous equations,

from which the detailed radiation pattern of the unknown emitter in free space may be

computed. This exercise is, however, still not simple because of the presence of non-linear

equations with high-order terms. Fortunately, from the practical point of view, we may be

only interested to know the total radiated power in free space, P^, rather than the detailed

radiation pattern. If this is the case, the measurement data taken in accordance with the

procedure described later will be sufficient to enable us to determine Pq. Before

describing the measurement procedure, a remark about the coordinate frame is in order.

We note that P^, Q^^, Q in (4.3a) and M^ and Q in (4.3b) refer to the dipole and

quadrupole moments of the EUT with respect to (x,y,z) frame of reference where x,y,z-axes

^r^ parallel to x,y,z-axes of the TEM cell with origin 0' at the test point (0,yQ,0) as
shown in figure 2. Let (x',y',z') be a frame of reference fixed with respect to the EUT and

denote the dipole and quadrupole moments of the EUT with respect to this (x',y',z') frame of

reference by small letters, i.e., p, m, q „, etc. When measurements are made with the EUT oriented in a specific direction we must substitute for P , Q^^^, Q , M^, and Q in (4.3a)

and (4.3b) in terms of p , m , q „ (a,e = x,y,z). Thus different orientations of the EUT

a a ap result in different equations for the output powers. We make use of this feature to obtain

the desired quantities for determining P^.

In what follows we describe a step-by-step measurement procedure and give an expression

for Pq in terms of the measured powers. The lengthy but straightforward analytical details,

which justify the following measurement procedure, are given in Appendix IV.

4.2 Measurement Procedure

a. Remove the phase shifter from the circuit shown in figure 2 so that the output

indicator measures the sum powers corresponding to in-phase addition of the fields from the

two output ports of the TEM cell. Ensure that the electrical lengths of the cables

connecting the output ports to the summing network dre equal. All the measurements will be

done with the EUT at a fixed test point (0, y^, 0) but with different orientations, and all the rotations are assumed to be counter-clockwise.

b. Position the EUT at the test point such that x' ,y' ,z'-axes of the EUT coincide with

x,y,z-axes respectively, and measure the output power and denote it by P . Now rotate the

EUT about its z'-axis in seven equal steps of 45° and measure the corresponding output

powers P^^^/4, P^^^/g' Pz,3Tr/4' ^z,.' Pz,5^/4' Pz,37t/2' Pz,77t/4- ^^^^ that the second

subscript on P corresponds to the total rotation (e in figure 3a) of the EUT about its z'- axis, from its starting position.

c. Position the EUT at the test point such that y' ,z' ,x'-axes of the EUT coincide

with x,y,z-axes respectively, and measure the output power P^ q. Rotate the EUT about its

x'-axis in seven equal steps of 45° and measure the corresponding output powers P^ ^,aj

Px,Tr/2' Px,3Tr/4' Px,T,' Px,5Tr/4' Px,37t/2' Px,77t/2- '"^'9"^^ ^b shows a typical orientation of

the EUT in this step.

d. Position the EUT at the test point such that z' ,x' ,y'-axes of the EUT coincide with

x,y,z-axes respectively, and measure the output power P„ „. Rotate the EUT about its y'-

y »^ ^—

axis in seven equal steps of 45° and measure the corresponding output powers Py ^z^, Py ^/o,

^y,3Tr/4' ^,71' Py,57T/4' Py,3Tr/2' ^y,7i^/A (^ee figure 3c).

e. Insert the 180° phase shifter in the circuit so that the output fields are now added out of phase. In practice this may be accomplished by a cable with one half wave

length.

f. Position the EUT at the test point (0,yQ,0) such that its x' ,y' ,z' -axes coincide

with x,y,z-axes respectively, and measure the output power and denote it by M^ .. Now rotate the EUT about its x'-axis in three equal steps of 45° and measure the corresponding

output powers M^^^/^, M^^^/^* ^,3n/4 (^ee figure 3d).

g. Position the EUT at the test point such that its y' ,z' ,x'-axes coincide with

x,y,z-axes respectively, and measure the output power M.. ^. Rotate the EUT about its y'-

y »^

7
with

axis in three equal steps of 45° and measure the corresponding output powers M ^^4, My ^^2'

^y,3Tr/4 ^^^^ figure 3e). / h. Position the EUT at the test point such that its z' ,x' ,y'-axes coincide with

x,y,z-axes respectively, and measure the output power M^ . Rotate the EUT about its z'- axis in three equal steps of 45° and measure the corresponding output powers M^ ^/4, M^' ^i^i

^z,3ti/4 (^^^ figure 3f). As shown in Appendix IV, the total power P^ that would be radiated by the EUT in free

space is given by

^%y 2 (4.4a)

fi = V2 I (M^^^ . ^.,/2)/e^oy ^''''^

^2 = V? y (M /. + M , ,.)/e2 , (4.4c) ^ V ^ ^ a,TT/4 a,3iT/4' oy' ^ ' a

where the summation in (4.4b) and (4.4c) is over a = x,y,z and the sign ambiguity in (4.4a)

is resolved as explained in Appendix IV.

In the next section we consider the problem of determining the susceptibility

characteristics of an object. By making use of the reciprocity principle, we show that a similar test procedure may be used to determine the average power received by the EUT when

illuminated by an incident plane wave in free space, where the average is taken over G and ()> polarizations and all possible angles of the incident wave.

5. DETERMINATION OF THE SUSCEPTIBILITY LEVEL OF A RECEIVING OBJECT

5.1 Characterization of the EUT as a Receiving Antenna

When studying the susceptibility level of an EUT, we are interested in finding the

amount of dissipated power in a load placed across some terminals of the EUT which is illuminated by a specified incident electromagnetic field. In practice, the load under

consideration is some component of the EUT that is most likely to fail under the exposure

condition. One may wish to know the average power dissipated in the load <P|_> where the

averaging is taken over all possible orientations of the EUT and over both 6- and (^- polarizations of the incident electromagnetic field. To determine such an average

susceptibility level, we may conveniently consider the EUT as a receiving antenna system

with an antenna impedance Z^, delivering power to a load Z, connected across the terminals

aa' as shown in the equivalent circuit (figure 4). We know that the average receiving

cross-section of a matched electrically small receiving antenna in free-space is equal to
x2/4Tr where the average is taken over all possible angles of an incident plane wave and the

antenna is assumed to be matched to the load as well as to the incident polarization. Then,

from the equivalent circuit (figure 4) of the EUT we find that [7]

<Pl>o = ^ ii) Pin (5.1a)

where the subscript refers to free space,

n = 4 RgRL/K^L ^ ^a^l^ ^^•'^^^

is the 'mismatch loss factor' of the EUT in free space, p^-^ is the incident power density,

and we have multiplied the average receiving cross-section by a factor of one-half to account for both possible polarizations. Thus, to determine the threshold power density of

an incident wave that produces a fixed <P|_>n. all we need to know is the 'mismatch loss

factor' n. We note that n is dependent upon the transmitting properties of the EUT through

Rg = Re (Zg). The total power transmitted, P^, by the EUT in free space when driven by a current I at its feed terminals is given by

2 P^ = 1/2 |I| Rg (5.2)

This P^ should be the same as P^ in (2.9a) provided that |I| refers to the magnitude of the

terminal current responsible for P, R and t}. In devising a procedure for determining the

'mismatch loss factor' in free space of an electrically small EUT, we first determine, as shown in Appendix V, two dimensionless quantities A and B representing the power dissipated

in the load of the EUT per unit input power to the TEM cell corresponding to in-phase and

out-of-phase excitations, respectively,

A= (ri/8) {iP'e - V? (Q' - Q' ) e' |^}/P, (5.3a) ^ ' ' ' y oy '^ ^^xx jy oy ' t

B = (n/8) {k2|M; - 1/2 Q;j' ey/P^ (5.3b)

where P', M' , Q' • a = x,y,z refer to the equivalent electric dipole, magnetic dipole,

electric quadrupole moments of the EUT with respect to the TEM cell axes when it is in the

transmission mode with the terminal current I. These two equations will form the basis for devising an experimental procedure to obtain an expression for determining the 'mismatch loss factor' n needed to find <P|_>q in

(5.1a).

5.2 Measurement Procedure for Determining n

We use the experimental setup shown in figure 5 where the EUT is placed at a test point

(O.yQ.O) and the TEM cell is excited at both ends by means of a signal generator and a power

divider. The isolators prevent the incident power from re-entering the signal generator.
The 180° phase shifter provides a means of exciting the two ports of the cell in- or out-ofphase. The power dissipated Pj^ fn the load is measured by means of a detector and output

indicator combination connected across the test terminals of the EUT. By monitoring the

input power to the TEM cell, the normalized quantities A and B in (5.3a) and (5.3b) may be

obtained for different orientations of the EUT. We establish, as in section 4, two frames

of reference (x,y,z) and (x',y',z') fixed with respect to the TEM cell and the EUT,

respectively, and go through the same sequence of operation^ as those described in steps (a]

through (h) of section (4.2), except that at each step we now measure the normalized

dissipated powers A^

q

, B^ g in place of the output powers P^ q, M^ g, respectively, where

a = x,y,z and e takes on different values as described in the same measurement procedure.

As mentioned in Appendix V, P^. is given in terms of P', R' , and f}' in exactly the same manner as Pq is given in terms of P, I^ and Q by (2.9a-c). Now if we compare (5.3a,b) with

(4.3a,b) we find that A^ and B^ ^ have the same relation to P', R' and Q' as P a and M . have with P, R and Q except for the factor (n/8P^). Thus a weighted sum of A^ g and B^

g with identical weighting factors as those of P^ g and M^ g in (4.4a-c) will yield

P^(n/8P^) = n/8. This implies that n is given by

2 2 n = 20 k2 { (/ A7~7 + /ATTTT) + (/ A~~r ± /A^ ^ .„) Z,Tr/2 Z,3tt/2' ^ X,ir/2 X,3tt/2'

where

' (^\../2 ^ ^\.3./2) >/^^oy + 8k2 (6f; + 4f^), (^-^^^

n = '/2 I (B,,o ^ ^,./2)/^^oy'

(5.4b)

'2 - ^2 I (B,,,/4 - B 4)/e2 (5.4c)

a

and again the sign ambiguity in (5.4a) is resolved as explained in Appendix IV. Using the above equations we calculate n which, when substituted in (5.1a), determines

<PLVPinUnder the special condition that P' and Q' are the only non-vanishing components due

to the electric dipole and quadrupole and that no magnetic dipole moment exists, the

normalized dissipated power may be expressed as follows, in accordance with the details

given in Appendix VI:

A = -oS-lp' e + V? Q' e'

1^ (5.5a)

z,o 8P^i y oy '-^ ^yy oy' ^ ^

2

t

and

A^ = pK-lP' e - V? Q' e' I (5.5b

z,TT 8P, ' y oy '-^ ^ yy oy' ^

10
^.9 = 0' (5.5c)

where

From (5.5a) and (5.5b) we obtain

or

1 + -V a2

z.TT (e - A e )^

• ^ oy oy'

where

e„ (/A „

- /A ) A = Q' /2 P' = -^ ^ ^^ > (5.6c)

yy y g' (/a + /a ) oy ^ z,o z.tt' can be determined by measuring A^ q and A^ ^ and referring to the field distribution curve

given in figure 6. When, in addition, Q' = o, we have A = o. A simple measurement of either A^

o

^'^ ^z it will then be sufficient to determine n given by

n

= 80 k2 A /el [y). (5.7) z,o oy^-'o' ^ '

6. SOME EXPERIMENTAL RESULTS FOR THE RECEIVING CASE

To verify the theoretical basis presented in previous sections, some measurements were

made using a symmetric rectangular co-axial TEM cell of dimensions 1.20 m x 1.20 m x 2.4 m

with a center conductor of width 0.992 m. The cell has matched transitions at both ends and

is partially loaded with absorbing material to suppress the higher order modes at high

frequencies. Earlier measurements indicated that the characteristic impedance of the cell

remained unaltered at 50 n after placing the absorbing material. However, field

measurements using a high impedance calibrated probe showed that the normalized field

distribution e is modified from the ideal theoretical distribution corresponding to an unloaded cell with perfectly conducting walls [1]. A comparative plot of the theoretical

and the measured field distribution along the y-axis of the cell is given in figure 6. Measurement results for the transmitting case when a self-contained battery-operated

spherical dipole is placed inside the TEM cell have been used to compare with the results

measured in an open space. This comparison indicates that effects of the quadrupole terms

in this special case are apparently negligible. Thus, the results reported previously

11
without considering the quddrupole terms remain good [1], and are not repeated here.

In our experiment for the receiving case, monopole antennas of different lengths were

used as the test objects and the 'mismatch loss factor' n for each antenna was obtained from

measurements taken at different test points inside the TEM cell. These antennas were made

simply by soldering a gauge-10 copper wire to the center conductor of an N-type panel

connector. A 100 MHz signal source with 1000 Hz square wave modulation was used to excite

the two ports of the TEM cell by means of a Tee-connector and cables of equal length. Outof-phase excitation was accomplished by means of an additional cable with an electrical

length of 1.5 m corresponding to one half wave length. Identical 20 dB attenuators were

used in place of isolators. An additional 10 dB pad was used as a buffer between the signal source and the power divider. The output from the signal generator was monitored by means

of a directional coupler and power meter combination (see figure 5). The test antenna was connected to a crystal detector and the detected output was measured using a VSWR meter. An attempt to use the standard BNC cable to bring out the detected output proved to be

impractical. Slight movement of the cable resulted in fluctuations of several decibels in

the output indicator, thus rendering the output readings meaningless. This problem was avoided by using a fiber optic transmit-receive link (FOL). The test antenna and crystal

detector combination was directly attached to the input end of the transmitter which was battery operated. The output from the transmitter was brought out through a fiber optic

cable and fed to the receiver placed outside of the TEM cell and the output from the

receiver was fed to a VSWR meter using standard BNC cable. This provided a stable and

reliable means of measuring the output from the test antenna. The test antenna, crystal

detector, and the fiber optic transmitter combination was enclosed inside a styrofoam box to facilitate the positioning and orientation of the antenna. Initial calibrations were performed to determine (i) the power going into the TEM cell with a known indication on the

power meter monitoring the output from the signal generator, and (ii) the input power to the

crystal detector, required to produce a known deflection on the VSWR meter. The calibrated

results were then the normalized quantities (power dissipated in the load per unit power

input to the TEM cell) A and B corresponding to in-phase and out-of-phase excitations,

respectively. Thus the crystal detector and the fiber optic link constituted the load to the test antenna. A sketch of the test antenna, crystal detector, and the fiber optic

transmitter combination is shown, along with a coordinate frame of reference (x',y',z') in figure 7. The z'-axis is assumed to be pointing outwards from the paper so that x',y',z'

constitute a right-handed coordinate system. The geometry of the test object shown in

figure 7 suggests that the EUT should physically have only a y'-component of electric dipole

(Py) and no magnetic dipole components. This was verified by noting the absence of any

measurable output under the following conditions:

i) out-of-phase excitation of the cell with arbitrary orientation of the test antenna.

ii) in-phase excitation of the TEM cell with the y'-axis of the test antenna in the xz plane of the TEM cell. We may obtain the 'mismatch loss factor' n by measuring the

normalized output power A^^q or A^ ^ in accordance with the simplified version (5.7) where

^z,o corresponds to orienting the EUT such that x' ,y' ,z'-axes coincide with x,y,z-axes,

12
TABLE I

Measured mismatch loss factor n by neglecting the quadrupole term

and taking the feed point as the reference point

Monopole Length 5 cm 10 cm 15 cm

n n n ^0 eoy(yo) in units of 10"^ in units of 10"^ in units of 10"^ cm V/m

9 = 6 = TT e = e = TT 9 = 9 = TT

20.0 13.80 38.82 217.2 508.9

22.5 13.30 38.10 217.0 504.7

25.0 12.72 35.57 226.6 513.0

27.5 12.28 73.02 36.73 281.3 228.8 576.6 518.0

30.0 11.90 72.26 35.88 277.8 233.5 575.6 518.8

32.5 11.43 72.41 34.33 277.7 240.6 582.0 537.8

35.0 10.90 74.02 286.9 595.5

37.5 10.63 71.61 280.2 592.2

40.0 io.;^5 72.38 287.9 611.4

Mean 54.76 254.6 552.9

Maximum 74.02 287.9 611.4

Minimum 34.33 217.0 504.7

(Max-Mi

)/Mean 72.48% 27.85% 19.30%

9=0 means that n is obtained by measuring A, .

e = TT means that n is obtained by measuring A, ^

13
respectively, provided that the n^agnetic dipole and the quadrupole terms are indeed

absent. The measured results for/ monopole antennas of lengths 5 cm, 10 cm, and 15 cm are tabulated in Table 1 as a functibn of the test point position (0,yQ,0).

Since n is characteristic of the EUT, it should be, in principle, invariant to the

position of the test point y and also we should obtain the same value for n regardless of

whether it is calculated by measuring A, . or A, ^. However, the results in Table I show

considerable variation in the values of n, particularly so for the case of 5 cm monopole

antenna where the minimum value of n is 37.3% below the mean and the maximum value of n is

3b. Z% above the mean, thus showing a spread of 72.5%. The discrepancy in the measured

results may be explained if we recognize that, in obtaining these results, we considered the

feed point F in figure 7 to be the reference point where the crystal detector is connected

to the antenna and the dipole moment is defined. However, the current distribution on the

EUT, consisting of the monopole antenna, crystal detector, and the fiber optic transmitter

combination, is highly asymmetric about this point, particularly for the case where the

monopole length is 5 cm. Hence, the quadrupole terms will have significant effect on the

measured results. To minimize the discrepancy in the measured results, we should either

choose the reference point such that the quadrupole moments are negligible with respect to the reference point, or include quadrupole terms in the calculation. In Table II we tabulated results obtained by choosing the geometric center G in figure 7 to be the test

point. From the geometry of the EUT it was felt that the current distribution should be

essentially symmetric about the geometric center so that the quadrupole moments with respect

to this point will be wery small. From the results we note a dramatic improvement where the

values of n for the 5 cm antenna show a spread of only 19.4% below the mean value and 11.8%

above the mean. But the results are still not quite satisfactory. Furthermore, in practice

the nature of the current distribution on the test object is generally unknown and the

selection of a suitable reference point would be difficult. To overcome this difficulty, we need to account for the quadrupole terms in the measurement and calculation procedure. Let

us now consider the results obtained by accounting for the dominant quadrupole term q . If we keep only this term in addition to the contribution form p , we will need both

measurements A^ ^ and A^ ^ to obtain n (see (5.6a,b) or Appendix VI). The measured results

including the quadrupole term are presented in Tables III and IV corresponding to taking the

points F and G (figure 7) as the reference points, respectively. The improvement in the

results is immediately evident. Comparing the n values for the monopole of 5 cm in length

in Table II and III, we see that the 'spread factor', which we defined as the difference

between the maximum and the minimum expressed as a percentage of mean value, is now reduced

from 31.2% to 10.9% by taking the quadrupole term into account. Comparing the results in

Tables III and IV, we do not see appreciable difference between the two. This shows that

the results obtained after including the quadrupole term are quite insensitive to the choice

of the reference point on the EUT, whereas a comparison of Tables I and II shows that the

results obtained by neglecting the quadrupole term are highly sensitive to the choice of the

reference point on the EUT. Also, as noted before, the results obtained by neglecting the

quadrupole terni are sensitive to the location of the EUT inside the cell unless the

14
TABLE II

Measured mismatch loss factor n by neglecting the quadrupole term

and taking the geometric center as the reference point

Monopole Length 5 cm 10 cm 15 cm

n n n

>o eoy(yo) in units (Df 10-6 in units of 10"^ in units of 10"^ rm V/m

e = e = TT 9 = e = TT = 6 = TT

20.0 13.80 57.82 51.34 248.2 576.2

22.5 13.30 57.85 50.83 239.8 246.5 548.2

25.0 12.72 58.46 50.37 243.1 255.6 537.0 551.8

27.5 12.28 58.46 49.02 240.6 254.5 540.6 550.3

30.0 11.90 57.14 47.58 240.6 258.8 536.7 551.6

32.5 11.43 58.21 46.53 242.4 264.1 541.5 562.7

35.0 10.90 59.20 46.62 254.6 287.4 563.5 591.6

37.5 10.63 58.04 44.96 248.8 278.2 568.6

40.0 10.25 57.86 42.69 254.8 608.2

Mean 52,.94 253.1 552.0

Maximum 59 .20 278.4 608.2

Minimum 42,.69 239.8 536.7

(Max-Min )/Mean 31 .19% 15.25% 12.95%

0=0 means that n is obtained by measuring A^

q

6 = IT means that n is obtained by measuring A., _

L ,71

15
TABLE III

Measured mismatch loss factor n by considering the quadrupole term

and taking the feed point as the reference point

Mcjnopole Length 5 cm 10 cm 15 cm

n n n

^0 ' cm

-oy'^o^

V/m

leoy^^o)!

V/m^

in units. of 10-6 in units. of 10-6 in units, of 10-6

9 = 9 = TI 9 = 6 = Tt 9 = 9 = TI

20.0 13.80 17.5 - 52.74 - 233.1 - 530.9

22.5 13.30 19.0 - 53.90 - 234.9 - 529.1

25.0 12.72 19.0 - 54.06 - 246.2 - 538.8

27.5 12.28 19.0 54.10 53.60 259.3 249.4 548.4 545.1

30.0 11.90 19.0 53.03 53.03 255.3 255.3 547.1 547.2

32.5 11.43 17.6 53.70 50.00 256.0 262.2 553.9 566.1

35.0 10.90 15.8 55.83 - 265.7 - 568.5 - 37.5 10.63 15.4 54.03 - 259.6 - 565.7 - 40.0 10.25 14.5 54.96 - 267.2 - 684.6 -

Mean 53.58 253 .7 552.1

Maximum 55.83 267 .2 584.6

Minimum 50.00 233 .1 529.1

(Max-Min)/Me an 10.88% 13 .44% 10.05%

6=0 means that n is obtained using A, „

8 = TI means that n is obtained using A,

The quadrupole correction term (Qyu/Py) is obtained by using (5.6c) at y^ = 30 cm.

16
TABLE IV*

Measured mismatch loss factor n by considering the quadrupole term

and taking the geometric center as the reference point

Monopole Length 5 cm 10 cm 15 cm

Yq eoY^-^o^ 1^' (-y ) I ^" units of 10"^ in units of 10"^ in units of 10"

cm V/m V/m^

9 = 9 = TT 9 = 9 ^ 11 9 - 9 = TT

20.0 13.80 17.5 53.84 55.28 - 241.2 - 570.1

22.5 13.30 19.0 53.39 55.26 247.8 238.7 - 541.6

25.0 12.72 19.0 53.76 54.97 251.6 247.1 537.0 544.9

27.5 12.28 19.0 53.48 53.67 249.3 245.8 547.7 543.2

30.0 11.90 19.0 52.25 52.25 249.6 249.6 544.0 544.2

32.5 11.43 17.6 53.40 50.92 251.1 255.1 548.6 555.4

35.0 10.90 15.8 54.57 50.75 263.2 269.4 570.5 584.4

37.5 10.63 15.4 53.51 48.94 257.2 269.2 575.6 -

• 40.0 10.25 14.5 53.44 46.37 263.2 - 615.5 -

Mean

- 52 .78 253.1 558.8

Maximum 55 .28 269.4 615.5

Minimum 46 .37 238.7 537.0

(Max-Min)/Me<in 16 .88% 12.13% 14.05%

*See footnotes under Table III

17
reference point on the EUT is so\ chosen that the quadrupole moments are "^ery small with

respect to the reference point. ^Even after including the quadrupole term, our results still

show 10 to 15% spread. This maV be attributed to a number of factors, the most important

one being that we are looking at the results for a very wide range of y^ (the position of

the test object). In this range (20 cm to 40 cm) even the derivative of e^ is not constant

(see figure 6). If we restrict y^ to a narrower range, the variation in measured results

would be considerably smaller. The other factors that might result in systematic errors in the measurement are:

1. The loading of the TEM cell with absorbing materials implies that the field

distribution may not be that of a perfect TEM mode. There may be a small axial component of

the electric and/or the magnetic field.

2. In our computation we included only the dominant component, q^y, of the electric

quadrupole. Inclusion of all the components of the quadrupole tensor should improve the

accuracy of the results.

7. CONCLUSIONS

The theoretical analysis and the measurement procedures described in this report should

enable one to obtain the free-space radiated power of an arbitrary but electrically small

radiating object by measurements taken inside a TEM cell. The method presented is more accurate compared to our previous method [1,2] since the electric quadrupole terms are also

included in modeling the test object. We have also extended the analysis and the test procedure to the determination of the susceptibility levels of electrically small objects.

Our experimental results for the receiving case clearly demonstrate the following:

i) The nonuniformity of the TEM modal field distribution inside a TEM cell could

result in considerable error in the interpretation of the measurement results if the

electric quadrupole terms are not included in modeling the EUT.

ii) If the quadrupole terms are not included, the measured results are highly

sensitive to the location of the EUT inside the TEM cell and also to the choice of the

reference point on the EUT. A poor choice of the reference point may result in errors of

the order of 70%.

tii) If the quadrupole terms are included in modeling the EUT, the measured results

are not only more accurate, but also much less sensitive to the location of the EUT inside

the TEM cell and to the choice of the reference point on the EUT.

8. ACKNOWLEDGMENTS

The authors are grateful to Dr. M. Kanda for his technical discussions during the course of this work. The authors dire indebted to M.L. Crawford for his constant cooperation

in making the TEM cell experimental facility available to them. They also wish to thank

C.K.S. Miller and F.X. Ries for their continuous interest and support.

18
REFERENCES

[1] I. Sreenivasiah, D.C. Chang, and M.T. Ma, "Characterization of electrically small

radiating sources by tests inside a transmission line cell," NBS Tech Note 1017, Feb. 1980.

[2] I. Sreenivasiah, D.C. Chang, and M.T. Ma, "Emission characteristics of electrically

small radiating sources from tests inside a TEM cell," to be published by IEEE Trans,

Electromag. Compat.

[3] C.T.A. Johnk, Engineering Electromagnetic Fields and Waves , John Wiley & Sons, New

York, 1975, Ch. 11, pp. 592-608.

[4] J.D. Jackson, Classical Electrodynamics , John Wiley & Sons, New York, 1975, Ch . 9, pp. 391-401.

[5] R.E. Collin, Field Theory of Guided Waves, McGraw-Hill Book Co., New York, 1960, Ch.

5, pp. 200-204 and Ch. 7, pp. 285-294.

[6] J.C. Tippet and D.C. Chang, "Radiation characteristics of electrically small devices

in a TEM transmission cell," IEEE Trans, on Electromag. Compat., vol. EMC-18, no. 4, pp. 134-140, Nov. 1976.

[7] R.E. Collin and F.J. Zucker, Antenna Theory , Part I, McGraw-Hill Book Co., New York,

1969, Ch. 4, pp. 93-103.

[8] J. A. Stratton, Electromagnetic Theory , McGraw-Hill Book Co., New York, 1941, Ch. 1.14, pp. 38-44.

19
\ APPENDIX I

/ REDUCTION OF SYMMETRIC TERM IN (2.6) DUE TO AN ELECTRIC QUADRUPOLE

We examine the symmetric term in (2.6) in its component form

/ {(fi-r') D + (n-D)r'} dv'

-

/ {(n^x' + n^' + n^z') (x J^ + y J^ + z J^)

+ (n^J^ + n^J^ + n^J^) (x x' + y

y' + z z')} dv

- X / {2n X' J + n (x' J + y' J ) + n (z' J + x' J )} dv' •' X x y^ y-'x' z x z'

+ yJ / {2n y' J + n (y' J + z' J ) + n (x' J + y' J )} dv'

} ^ ^

J y z^-' z y' x^ y •' x'

+ z / {2n^z' J^ + n^(z' J^ + x' J^) + n^(y' J^ + z' J^)} dv'

= ^-n = 0(n) (Al.l)

where Q is an electric quadrupole tensor with the components

V

and the vector Q(n) has the components given by

Q =

I Q fln.. (A1.3)

20
APPENDIX II

POWER RADIATED BY A COMBINATION OF AN ELECTRIC DIPOLE,

A MAGNETIC DIPOLE AND AN ELECTRIC QUADRUPOLE

The expressions (2.2) and (2.3) for the electric and the magnetic fields take the

following simpler form in the far zone:

E - jco nx(n X A), (A2.1a)

B - -j k n X A (A2.1b)

and the time-averaged power radiated per unit solid angle is given by

P (e,4>) = V2 Rg (r^ n.(E x R*)} (A2.2)

which, upon using (A2.1a,b) may be written as

P (p,^)

-^ In X A|^ (A2.3)

where c,q = 120tt is the free space characteristic impedance. Substituting for A from (2.7] we can write the total power radiated in free space Pq as

Pq = ^^/ In X (P - j k Rx^! + j J

q-nll ^ d^

= ^1^ / {jnxPl + k2 |nx(R X R)|% |^Rx(O.R)|^ dS^ (A2.4)

4tt

since we can show that the following integrals vanish identically.

Ii = / (nxa) • [nx(nxB)] df^ (A2.5a)

I2 = / (nxa) • [nx(c«n)] dQ (A2.5b)

13=/ {nx(nx5)} • {nx(c'n)} dn (A2.5c)

where a, 5 are arbitrary constant vectors and c is an arbitrary symmetric tensor.

Using the vector identity

ax(Bxc) = (a-c)5 - (a-B)c (A. 2. 6)

and noting that n-n = 1 we write the first integral in (A2.5a) as

Ii = / (nxa) • {n(n-b) - 5} da

21
= / n . ^Bxa) df^ = (A2.7)

since nxa • n = and (

/ n dJ^ = 0; a - x,y,z. (A2.8)

Using the vector identity

(ax5) . (cxa) = (a-c)(B'a) - (a-a)(5.c) (A2.9)

we write the second integral in (A2.5b) as

I2 = / {a-(c-n) - n-(c-n)(n-a)} df^ = (A2.10)

which follows from (A2.8) and the vanishing of the following integrals,

/ n n„n dn = a,B,Y= x,y,z. (A2.11)

Using (A2.9) we reduce I3 to

13=/ (fixb) • (c'fi) dn

= - / b'{nx(c'n)) dfi

/ 5*{x (nc -nc)+y(nc -nc)+z(nc -nc)}dn (A2.12;

^ yz zy'-'Wx xz' xy yx'

where

^a " ^ ^ag^B'

"'^ " ^,y>^- (A2.13)

3

Using the relation

/ n^n^ dQ = ^ 6^^; a,B=x,y,z (A2.14)

and noting that c is symmetric, we find that K is identically zero. Now we consider the integral in (A2.4) term by term. The first term becomes

/ |nxP|2 dn =

/ (nxP) • (nxP*) dn

= / (|P|^ - |n-P|^ dQ = -^ |P|^ (A2.15)

where the property (A2.14) is again used. Hence the power radiated by the electric dipole

22
is given by

P = 10 k2 |P|2. (A2.16;

With the aid of {A2.9), the second term in (A2.4) may be written as

/ {nx(nxm)} • {nx(nxm*)} do.

=

/ (nxm) • (nxm*) dn = -^ |m| (A2.17)

and the power radiated by the magnetic dipole is given by

P = 10 k^ |R|2. (A2.18)

m

Finally, the third term in (A2.4), the quadrupole radiation, involves the integral,

/ In x Q (n) i^ dn

= / {|Q (n)|2 - In-Q {n)|2} dfi

=

/ I Q 00*0^0" dfi - / y Q oQ*;." n„n n. dQ (A2.19]

a,e,Y a,B,Y,5

where the summation is over x,y,z. Using (A2.14) and the result

/ n n n n^ dn = i^ (6 „6 . + 6 5„. + 5 .6^ ) (A2.20] -' a 3 Y 5 15 a3 Yi5 aY 3<S a6 By

we write

/ lnxq(n)|2 df^ -^ ^ IQ |2_4^( ^ Q ^ Q* + 2 J I IQ J2} (A2.21]

\ H\ I I 3 ^ '^aB 15 ^ aa '^ ^YY ^^ aB a,B a Y a,

SO that the power radiated by the quadrupole term is given by

Pn = ^ ^ 3 I IQ J2 - i ); Q |2}. (A2.22]

q 4 ^ „ a, B aa ^ ' a, B a

Adding (A2.16), (A2.18), and (A2.22) we obtain

k2

^o~- ^e^ ^m^ ^a" ^^ ^' ^'^l' ^ ^' I'^l'^lo ^^ ^ l^aB'' " ' ^ ^aal'^^" ^^2.23)

a, B a

Using the symmetry property of Q we write Pq as

P^ = 10 k2 {|P|2 + k2 |M|2 + ^ (3 Qi+ Q2) ((A2.24a)

23
where

Qi = w IQ 1^ + IQ 1^ + IQ |2

'"^xy' '^yz' '^zx' :A2.24b)

Q2 = IQ,J^ ^ IQ,,I^ ^ IQ,,|2 - R^ (Q^^Q*^. Q^^Q*^. Q^^Q*^). (A2.24c)

24
APPENDIX III

EXPRESSIONS FOR a^ and b^ IN TERMS OF DIPOLE

AND QUADRUPOLE MOMENTS

We consider the second integral on the right hand side of (3.3) and write the integrand

in its component form as

.. 3E _ 9E ^^ 3E

(G'.VE)O = (X u; + y u; + z u^) . (XX -^ + xy -g^ + xz -g|

.. 3E _ 9E .. 9E

.. 3E _ 3E _ 3E

3E 3E 3E

X ^ X 3x y 3x z 3x'

3E 3E 3E y ^ X 3y y 3y z 3y^

3E 3E 3E

+ u' (J ^+J ^+J -^) (A3.1)

z ^ X 3z y 3z z 3z' ^ '

where we dropped the subscripts, superscripts, and argument on E

_ and used u^,

III u^. and u^ to denote the x,y,z-components of u'. Note that u' is identical to r' when the coordinate

origins of the EUT and TEM cell frames coincide with each other.

Next we use the relation

VxE = - juM R (A3. 2)

to write the expression V2 J'^p (u'xJ)«H in its component form as

3E 3E

- 1/2 >.^ (u'x:3).n = 1/2 (u; j^ - u^ j^)(-^ - -^)

^ '/2 (^;jx-";^z)(^-Tt)

' V2 (u: J - u' JJ(^-^). (A3.3;

X y y x" 3x 3y

Adding (A3.1) and (A3. 3) and making some rearrangement we can write

(u'-vE)-J = -

I/2 jwp (u'xJ)-H

9E 9E

+

I/2 (u' J + u' J ){-— + -T^)

X y y x" 3y 3x' 3E 9E

y z z y''^ 9z 9y'

25
+ i/2\ (u; J, + u: jj(^ + -^) Z "X "X "Z' ^ 9X 3Z'

3E 3E 3E.

X "X "33^ ^y ^y "ly ^z ^z ^z' + u'i J., -K^ + u'. J .-K^ + u' J, -TT^. (A3. 4]

From the definition of the double dot product between two tensors we note that

a, 3

and using the definitions (2.8b and c) we can write

/ (u'-vE) . D dv' = - jwM R-R + 1/2 vEiQ (A3. 6] V

which leads to

(+) (+) (+)

\l)

= - \ {E^ • P - ^m^\ • M + V2 VE^ : 01. (A3. 7]

26
APPENDIX IV

DERIVATION OF AN EXPRESSION FOR Pq IN TERMS OF THE MEASURED POWERS USING A TEM CELL

1. Equations for Powers Corresponding to 1n-Phase Measurement .

In order to interpret the measured results using a TEM cell, we write (4.3a) in terms

of the dipole and quadrupole moments, p , q ., (a,B = x,y,z), of the EUT for different

a dp

orientations of the EUT. A general expression for the output power for arbitrary

orientation of the EUT will be too involved. However, in all the measurement steps that we described in section IV, the orientation of the EUT is so specified that one of the

coordinate axes of the EUT is made coincident with one of the axes of the TEM cell and EUT

is rotated in a plane transverse to this fixed axis. Thus, the rotation is always performed

in x'y'-, y'z'- or z'x'-plane of the EUT only. Let us examine step (b) in the measurement

procedure where a typical orientation of the EUT is shown in figure 3a. For this choice of

the orientation of the EUT, we note that

x = x'cose - y'sine {A4.1a)

y = x'sine + y'cose (A4.1b)

J^ = J' cose - j; sine (A4.1c)

J = J' sine + J' cose. (A4.1d)

So that

P = p sine + p cose (A4.2a)

and

V2 (Q - Q ) = / (x J - y J ) dxdydz '- ^^xx ^yy' ' ^ X y

= / {(x'cose - y'sine)(j'cose - J'sine)

X y

[x'sine + y'cose)(J'sine + J'cose)} dx'dy'dz'

X y

= / [(x' J' - y' J')(cos2e - sin2E

2 (x' J' + y' J' ) sine cose] dx'dy'dz'

y X

V2 (q^x " "^^^^ '^"^^^ " ^xy

^'"^^ (A4.2b;

27
where we have made use of the fact that dxdydz = dx'dy'dz' [8]. As we noted in section IV,

p and q g (a, 3 = x,y,z) are the components of the electric dipole and quadrupole of the

EUT, referred to its own frame of reference (x',y',z'). Equation (4.3a) may be rewritten as

I (p sine + p cose) e - { V? (q - q ) cos2e - q sin2e} e' p = p (A4.3a)

1 \Kj^ ^y ' oy ^ ^^xx 7y' ^xy * oy ' z,e ^ '

A cyclic change of the coordinate axes will yield the following equations corresponding to steps (c) and (d) of the measurement procedure.

|(p sine + p cose) e - ( V? (q - q )cos2e - q sin2e} e' p = P „ (A4.3b) I vi^y 1^2 oy ^ ^^yy ^zz' ^yz oy' x,e ^ '

I (p sine + p cose) e - { V? (q - q ) cos2e -

I vK^ q sin2e} e' P = p ^. (A4.3c) Kx ' oy "^ ^^zz ^xx' ^zx oy' y,e ^ '

Note that in (A4.3a) to (A4.3c) we have used the notations P^ q, P^^ q, and P g to denote the

measured sum powers for different EUT orientations in accordance with steps (b), (c) and (d)

described in section IV. What we now need is a manipulation of these equations in a manner

that |p|2 = p2 + p2 + p2 ig given in terms of the measured powers, preferably as a weighted

X y z sum of the measured powers. Before doing so we derive equations for powers corresponding to out of phase measurement.

2. Equations for Powers Corresponding to out-of-Phase Measurement . Consider step (f) of the measurement procedure involving out-of-phase measurements. We now need to express (4.3b) in terms of the dipole and quadrupole moments of the EUT for

different orientations. A typical orientation of the EUT in this step is illustrated in

figure 3d. The coordinate transformation relations are given by

y = y'cose - z'sine (A4.4a)

z = y'sine + z'cose (A.4.b)

J = J' cose - J' sine (A4.4c)

y y z

J = J' sine + J' cose (A4.4d)

z y z

which result in the following expression for Q in terms of q , q^^, and qy^:

^yz

= ^ (y ^z ^ ^ ^y) "^^"^^"^^ ^^/2 (q^y " ^^^) sin2e + q^^ cos2e. (A4.5a)

It is an easy matter to verify that M^ is unaltered under a rotation in the yz-plane. That

is

M^ = m^. (A4.5b)

28
We may now write (4.3b) as

k2 e2 |m - Va (q - q ) sin26 -V? q cos2e|2 = m (A4.6a'

A cyclic change of coordinate axes results in the following two equations:

k2 e2 Im - Va (q - q ) sinZe -V? q cos2e|2 = m ^

k2 e2 Im - V4 (q - q ) sin2e -V? q cos2e|2 = M ^. oy I z

'^ ^^xx ^yy' ^ ^xy ' z,6

(A4.6b)

(A4.6c)

We note that when 6 is a multiple of tt/4, each of the equations (A4.6a)-(A4.6c) involves

either a diagonal or an off-diagonal element of q. Because of this, the expression for the

power radiated due to a combination of magnetic dipole and electric quadrupole of the EUT

takes on a simpler form which we derive next. Also note that we have used the notations

V[ Q, My 9

. and M^ in (A4.6a) to (A4.6c) to denote the measured difference powers for

different EUT orientations in accordance with steps (f), (g) and (h) described in section

IV.

3. Power Radiated, P^ + P , due to the Magnetic Dipole and Electric Quadrupole

Moments of the EUT

By a simple substitution we can derive the following equations from (A4.6a)- (A4.6c)

k2 e2 {Im |2 + 1/4 |q |2} = V? (M + M ,„)

oy ' x' "ti^yzi 'c \ x,o x,tt/2'

k2 e2 { Im |2 + Va Iq |2} = V? (M + M ,„)

oy

'

' y' '"• '^zx' '^ ^ y,o y,-^/2'

k2 e2 {Im |2 +V4|q P} = V? (M + M .,) oy ^xy z,o z,tt/2^

k2 e2 {|m |2 +y^ Iq - q |2} =1/0 (M ,, + M

oy yy zz X,tt/4 X,3Tr/4'

k2 e2 {|m |2 + — Iq - q |2} =1/0 (M ,, + M ^ ,J

oy ' y' 1^ '^zz ^xx' '^ y,n/4 y,3w/4^

k2 e2 {|m |2 + -y^ |q - q |2} = l/2 (M ,. + M . ,,

)

oy ' z' 1^ '^xx yy ' "^ z,tt/4 z,3Tr/4

where m , q . (a, 6 = x,y,z) dre complex in general.

Ot Otp

(A4.7a)

(A4.7b)

(A4.7c)

(A4.7d)

(A4.7e)

(A4.7f)

Taking the sum of (A4.7a)-(A4.7c) we obtain

k2 (|m|2 +V4qi) - h (A4.8a)

where

Im 1 2 + I 111 1 2 + |ni I ' x' ' y ' ' z' (A4.8b;

29
(A4.8c)

(A4.8d)

Similarly, taking the sum of (A4.7d)-(A4.7f) we obtain

k2 (|m|2 +-^ q^) = f2 (A4.9a)

where

a

a,TT/4 a,3TT/4' oy'

q2 = lq,J2 + |q^^|2 . |q^J2 . Re (q^^qj^ + q^^q*^ + q^^q*^) (A4.9c)

and Re stands for 'real part of. From (2.9a) we note that power radiated by the magnetic

dipole m and electric quadruple q is given by the sum P^^ + P ,

P^ + P^ = 10 k'^ (|m|2 +^q^ +^ q^). (A4.10)

From (A4.8a) and (A4.9a) we obtain

if^ +^f2 = k2 (|m|2 +^q^ ^^q^) (A4.11)

so that

P + P = 2 k2 (3 f, + 2 fo). (A4.12) m q ^ ^ ^' ^ '

where f-^ and f^ are given in (A4.8d) and (A4.9b) in terms of the measured difference powers,

M , M ,/i, M ,0, and M -, //,, a = x,y,z. This is the precise reason why steps (f), (g) a,0 a,TT/4 a,it/2 a,3Tr/4 •' ^ j v \ 1

> \3/

and (h) in section IV are proposed.

4. Power Radiated, P^, by the Electric Dipole Moment of the EUT

To derive an expression for Pg we first write, using {A4.3a), the following eight

equations corresponding to e = O.tt, tt/2, 3tt/2, tt/4, 3tt/4, 5Tr/4, and 7Tr/4, respectively.

|(p - q )|2 = P n/e^ (A4.13a) \\y^j ^zz'' z,0' oy ^ '

|(p + q )|2 = P /e2 (A4.13b) '^^y ^zz" z,Tr oy ^

i(p + q )|2 = P ,„/e2 (A4.13c)

'^*^x ^zz'i z,Tr/2' oy ^ '

|(p - q )|2 = P . ,o/e2 (A4.13d)

'^*^x ^zz'' z,37r/2' oy ^ '

30
|-(p //2 + p //2 + q )|2 = P ../e2 (A4.13e)

(p //7 - p //7 - q )|2 = P , ../e2 (A4.13f)

^*^x ^y ^xy'' z,3tt/4' oy ^

'

(p //2 + p // 2 - q )|2 = P ^ ../e2 (A4.13g)

(p //7 - p //7 + q )|2 = P ^ ../e2 (A4.13h) ^^x ^y' ^xy'' z,7Tr/4' oy

^ '

where q and q Are defined as ^xy ^zz

q = q e' /e (A4.14a)

^xy ^xy oy oy ^

'

q =V? (q - q ) e' /e . (A4.14b)

^zz ^ ^^xx 7y' oy oy

^ '

The radiation from a typical EUT arises from the surface current distribution over the body

of the EUT. Hence, for an electrically small EUT, it is fair to assume that the equivalent

electric dipoles and quadrupoles d^re in phase. Then v^e can verify the following by a simple

substitution:

p2 + q2 =1/^ (p + p )/e2 (A4.15a)

^x ^zz '^ W,Tr/2 z,3ir/2" oy ^

'

p q =1/4 (P /o - P o /o)/e2 (A4.15b) ^x^zz ^ ^ z,Ti/2 z,3tt/2'' oy ^

'

p q =1/4 (P - P n)/e2 (A4.15c)

p p = 1/4 [P /. + P c /^ - P o //, - P 7 /J/e2 . (A4.15d) ^x^y ^ z,tt/4 z,5tt/4 z,3ti/4 z,7tt/4-" oy ^

'

Using the last three equations we write q2 as

(P - P )(P - P ) ^2 ^'^/^ Z.37t/2^; Z,7T z,0^ ... .^. ^zz

~- T-^~{-^ ~? —p —p )

• (^4.16)

oy ^ z,tt/4 z,5w/4 z,3^/4 z,77r/4^

There are a number of ways of obtaining p2 . A direction substitution of (A4.16) into

(A4.15a) yields one value for p2 . Substitution of (A4.16) into the square of (A4.15b)

yields another. A third way is to take the square root of the product of (A4.13c) and

(A4.13d), giving

p2 - q2 = + /p ,, P ., ,0 /e2 , (A4.17)

^x ^zz z,Tr/2 z,3Tr/2 ' oy

^

which can be combined with (A4.15a) to obtain

31
1 n2 = -5-^ (p /o + p o ,., + 2 /p ttp r~r)

^X 4et,„ Z,Tr/2 Z,3ir/<^ Z,tt/2 Z,3tt/2' oy

1 (/p ,„ + /p.

4e2 ' 2,^/2 - z,3tt/2' oy

4e^ z,Tr/2 z, 3-17/2 x zz

oy 1

= XT- (^P ^ - ^P o /o) if p2 < q2 . (A4.18)

4e^ z,tt/2 z,37r/2 x zz oy

In (A4.18), the relative values of p^ and q^ can be confirmed by comparing (A4.15a) and

(A4.16). The values for p2 obtained by different substitutions may be used as a check to see (i) whether or not the measured data (P^ g) are accurate, and (ii) whether or not the

assumption of no phase difference between p^ and q is reasonable.

^ zz Similarly, the following formulas for p^ and p2 can also be derived:

y 4e^ x,tt/2 x,3Tr/2

oy

p2 = 7-^ (/P ~ ± /P , .J (A4.20)

"^z 4e2 y,iT/2 y,3Tr/2

oy

where the sign in (A4.19) is determined by the relative values of p^ and q^ , and that in (A4.20) is determined by the relative values of p^ and q^ . The expressions for

q2 and q^ can be obtained by replacing the subscript z in (A4.16) by x and y,

xx yy

respectively.

Even though p^ , p2 or p^ may appear to involve only two measured powers as evidenced by

x y z (A4.18) to (A4.20). The choice of "+" or

"-" sign, however, is determined by

q2 , q2 or q^ , and all of these three latter quantities involve eight different

XX 7y ^zz ^ measurements. This explains the reason why steps (b), (c) and (d) in section IV are proposed.

The radiated power, P , due to the electric dipole moment, is given by

Pg = 10 k2 (p2 + p2 + p2). (A4.21)

The total equivalent power radiated by the EUT in free space is then

P = P + P + P (A4.22)

e m q

which after substituting for P , P , and P , yields (4.4).

32
APPENDIX V

POWER RECEIVED BY A RECEIVING ANTENNA INSIDE A TEM CELL

To derive an expression for the induced open circuit voltage at the terminals of a receiving antenna inside a TEM cell vje consider two sets of fields (E^.H^) and (£2,^2)

corresponding to the following cases:

1. El ,Hi are produced by exciting the antenna at its feed terminals by a voltage source.

The feed terminals are assumed to be centered at a point (xQ,yQ,0) inside the cell as shown

in figure 8. Let J^ be the corresponding volume current density distribution on the

antenna.

2. E2 ,H2 are maintained by feeding the TEM cell with external sources.

The fields (E^ ,Hi ) and (£2,^2) satisfy the following identity in accordance with the

Lorentz reciprocity principle [5],

/ (E1XH2 - E2xHi)-nds = / E2-Ji dv (A5.1)

S V

where S is any surface enclosing the antenna, v is the volume enclosed by S, and n is the

outward directed normal to S. When the EUT is in the receiving mode, the right-hand side of (A5.1) represents VI,

where V^^ is the induced open circuit voltage across the antenna terminals of the equivalent

circuit shown in figure 4, and I is the input current. When the EUT is in the transmitting

mode and the TEM cell and frequency are such that only the dominant TEM mode exists,

a^ = 0, bj = (see figure 8). Under this condition, the left-hand side reduces to

-2 (a 32 + b b2), which can certainly be expressed in terms of the measured powers given in (4.3a) or (4.3b). In particular, when x = (EUT is positioned at the cell center), and

the cell is measured with two output ports connected in phase (a2 = b2), we have

IV^^I |I| = 2|a2||a„ + b„|,

' oc ' ' ' ' ^ ' '

'

or

|V ^|2 |i|2 = 4|,T|^ ip'

e .1/ (Q' _ Q' )e' 1^. (A5.2)

' oc' '' '21 I y oy '^ ^^xx yy' oy' ^ '

When the cell is measured with two output ports connected out of phase (a2 = - b2 )

, we have

|V„"| |I| = 2|a2||a„ - b„| I oc ' ' ' ' ^ I

I

'

or

|V 'P |I|2 = 4|a2|%2|M' -1/2 Q' |2 e2 . (A5.3)

' oc ' ' '

' '^ ' ' x '^ 7z ' oy

33
The total power, P|. , that would be radiated in free space by a transmitting EUT corresponding to an excitation terminal current I is given by

IV P^ =1/2 |I|2 R^, (A5.4)

where R, is the radiation resistance or the real part of the antenna impedance. Note that

P^ can be expressed in terms of P', M' and Q' in exactly the same manner as Pq is given in terms of P, M and Q by (2.9a).

The power, P|_, dissipated in the load of a receiving EUT (see figure 4) is given by

IV ^|\

Pl = Vz ly-^^yrV (A5.5)

where l^ is the load impedance with Rr as its real part, and Z' is the equivalent antenna

'- '- a impedance of the EUT inside a TEM cell.

In general, Z' is different from the corresponding impedance Z, in free space.

However, most of the equipment to be tested for susceptibility purposes are inefficient when

viewed as receiving antennas. Thus, Z and V both have very large reactive parts and small

a a resistive parts. Since Z and Z' differ mainly in their resistive parts, we may assume that

a a

IZ^.Z; IHZl-ZJ. (A5.6)

Substituting (A5.2) or (A5.3) into (A5.5) and using (A5.4), we obtain, for the in-phase

case,

R R, |aj|2

P| = D i7 ^ 7 12 |P' e -V? (Q' - Q' )e' l^ (A5.7a)

L P. |Z, + Z K ' y oy ^ ^^xx 7y' oy' ^ ' t L a

and for the out-of -phase case,

R R. |aj|2 \- pJz'.Zj'^ ^^l^; -VZ %,\' %' (A5.7b)

Defining the dimensionless quantities A and B as the power dissipated per unit input

power to the TEM cell, we have

A E P^/2|a2|2 \_i =^ IP' e -V? (Q' - Q' ) e' P (A5.8a)

\ i\ 8P ' y oy ^ ^ XX ^y^ oy' ^

and

B = P"/2|a2|2 = oK-k2|M' -V? Q' |2 e2 , (A5.8b)

L ' 2 1 8P, ' X ^ yz ' oy

where n is the mismatch loss factor defined in (5.1b).

34
APPENDIX VI

EXPRESSION FOR n IN TERMS OF A^ ^ and A^ ^

When Py and qyy are the only non-vanishing components of the electric dipole moment and

electric quadrupole moment, the radiated power P^^ is given by

Using (5.3a) and (A4.3a) with the assumption that p and q are of the same phase, we write

A = (n/8)(p e + V? q e' )2/P^ (A6.2a) z,o ^ '^^'^y oy '^ ^yy oy' t ^ '

A = (n/8)(p e - V2 q e' )^/P.. (A6.2b)

z,ir ^ '^^y oy "^ ^yy oy' t

^

'

Since q is in general smaller than p , and e' is negative according to figure 6, we have

yy q e ' ^ ^y A > A and V? I-^^^-^^^—^|<1. We then have

Z,7r z,o "^ ' p e ' ^y oy

e /A - /A

A = q /2p = -^ ( -^^

z==:) >0. (A6.3)

^y y ^oy /A + /A -^ Z,0 Z,Tr which can be determined by measuring A, _ and A, „ and the field distribution curve given in

figure 6. Once A is known, n can be obtained by substituting for P^ from (A6.1) in (A6.2a,b) and the result is

1 +4^^^

n = 80k2 A . —, j-TT • (A6.4) z,o (e + A e' )2

TT oy oy

35
Figure 1. A conceptual NBS TEM cell

36
Figure 2. Test set-up for determining total power radiated,

in free space, by an electrically small EUT

37
Z, 2'

*• X

z,x'

Figure 3. (a) Orientation of the EUT in step (b), section 4.2

(b) Orientation of the EUT in step (c), section 4.2

38
2. y

x.x'

Figure 3. (c) Orientation of the EUT in step (d), section 4.2

(d) Orientation of the EUT in step (f), section 4.2

39
x.y'

x,z'

Figure 3. (e) Orientation of the EUT in step (g), section 4 (f) Orientation of the EUT in step (h), sectlSn 4.'

40
V oc

Figure 4, Equivalent circuit representation of the EUT

as a receiving antenna system

41
ISOLATOR

180°

PHASE

SHIFTER

POWER

DIVIDER

ATTENUATOR

DIRECTIONAL

COUPLER

r

ISOLATOR

POWER

METER

©SIGNAL

GENERATOR

Figure 5. Test set-up for determining the average susceptibility

level of an electrically small EUT

42
THEORETICAL DISTRIBUTION CORRESPONDING TO UNLOADED

CELL

MEASURED DISTRIBUTION IN THE LOADED CELLIDOHED PART

OF THIS CURVE IS EXTRAPOLATED)

Figure 6. Normalized transverse electric field distribution

inside a TEM cell

43
Crystal detector

Geometric center G

Fiber optic

transmitter

Figure 7. Monopole antenna, crystal detector and

fiber optic transmitter combination

44
i • o

Figure 8. Receiving object inside a TEM cell

45
U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1, PUBLICATION OR

REPORT NO. NBS TN-1040

2. Performing Organ. Report No. 3. Publication Date

April 1981

4. TITLE AND SUBTITLE

A Method of Determining the Emission and Susceptibility Levels

of Electrically Small Objects Using a TEM Cell

5. AUTHOR(S)

I. Sreenivasiah, D. C. Chang and M. T. Ma

6. PERFORMING ORGANIZATION (If joint or other than NBS. see In struct/on s)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covers

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

10. SUPPLEMENTARY NOTES

1 j Document describes a computer program; SF-185, FIPS Software Summary, is attached.

!l

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

bi bliography or literature survey, mention it here) An electrically small radiating source of arbitrary shape may, to a first order, be

modeled by an equivlenet dipole system consisting of three orthogonal electric dipoles

and three orthogonal magnetic dipoles each excited with arbitrary amplitude and

phase. Determination of the individual electric dipole moments and the cross-

_

components of such an equivalent dipole system for the emission case by tests inside

a TEM cell, was described in an earlier work by Sreenivasiah, Chang and Ma, assuming

a constant TEM modal field distribution in the space occupied by the test object.

A method of accounting for the first-order variation in the TEM modal field distribution is presented in this report. This involves the inclusion of electric quadrupole

terms in the modeling of the test object. Using the reciprocity principle, the same method is extended to the determination of susceptibility levels of electrically

small test objects. Some experimental results for the susceptibility test, demonstrating the importance of the quadrupole terms, are presented.

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

Analysis; dipole moments; electrically small objects; emission; quadrupoles;

measurements; susceptibility; TEM cell. •

13. AVAILABILITY

[Vj Unl imited Q For Official Distribution. Do Not Release to NTIS

iXX Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

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NBS TECHNICAL PUBLICATIONS

PERIODICALS

JOURNAL OF RESEARCH—The Journal of Research of the National Bureau of Standards reports NBS research and develop- ment in those disciplines of the physical and engineering sciences in which the Bureau is active. These include physics, chemistry,

engineering, mathematics, and computer sciences. Papers cover a broad range of subjects, with major emphasis on measurement

methodology and the basic technology underlying standardization. Also included from time to lime are survey articles on topics

closely related to the Bureau's technical and scientific programs. As a special service to subscribers each issue contains complete

citations to all recent Bureau publications in both NBS and non- NBS media. Issued six times a year. Annual subscription; domestic

$1.1; foreign $16.25. Smgic copy, $} domestic: $3.75 foreign. NOTE: The Journal was formerly published in two sections: Section A "Physics and Chemistry" and Section B "Mathematical

Sciences." DIMENSIONS/NBS—This monthly magazine is published to in- form scientists, engineers, business and industry leaders, teachers, students, and consumers of the latest advances in science and

technology, with primary emphasis on work at NBS. The magazine

highlights and reviews such issues as energy research, fire protection, building technology, metric conversion, pollution abatement,

health and safety, and consumer product performance. In addition, it reports the results of Bureau programs in measurement

standards and techniques, properties of matter and materials, engineering standards and services, instrumentation, and

automatic data processing. Annual subscription: domestic $11;

foreign $13.75. NONPERIODICALS

Monographs—Major contributions to the technical literature on

various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with in- terested industries, professional organizations, and regulatory

bodies. Special Publications—include proceedings of conferences sponsored by NBS, NBS annual reports, and other special publications

appropriate to this grouping such as wall charts, pocket cards, and

bibliographies. Applied Mathematics Series—Mathematical tables, manuals, and

studies of special interest to physicists, engineers, chemists,

biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a worldwide program coordinated by NBS under

the authority of the National Standard Data Act (Public Law

90-396).

NOTE: The principal publication outlet for the foregoing data is the Journal of Physical and Chemical Reference Data (JPCRD)

published quarterly for NBS by the American Chemical Society

(ACS) and the American institute of Physics (AIP). Subscriptions,

reprints, and supplements available from ACS, 1 155 Sixteenth St., NW, Washington, DC 20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research results,

test methods, and performance criteria related to the structural and

environmental functions and the durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of

the Code of Federal Regulations. The standards establish nationally recognized requirements for products, and provide all concerned interests with a basis for common understanding of the

characteristics of the products. NBS administers this program as a supplement to the activities of the private sector standardizing

organizations. Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the con- sumer. Easily understandable language and illustrations provide

useful background knowledge for shopping in today's technological marketplace. Order the above NBS publications from: Superintendent of Documents. Government Printing Office, Washington. DC 20402. Order the following NBS publications—FIPS and NBSIR'.t—from

the National Technical Information Services, Springfield. VA 22161

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Federal Information Processing Standards Publications (PIPS

PUB)—Publications in this series collectively constitute the Federal information Processing Standards Register. The Register

serves as the official source of information in the Federal Govern- ment regarding standards issued by NBS pursuant to the Federal Property and Administrative Services Act of 1949 as amended.

Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717(38 FR 12315, dated May II, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or

final reports on work performed by NBS for outside sponsors

(both government and non-government). In general, initial dis- tribution is handled by the sponsor; public distribution is by the National Technical Information Services, Springfield, VA 22161,

in paper copy or microfiche form.
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