NBS TECHNICAL NOTE 1017

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

Characterization of

Electrically Small Radiating Sources

by Tests Inside a

Transmission Line Cell

).1017
NATIONAL BUREAU OF STANDARDS

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KATIOIUX »Wt»>V

or BtAKDABBe

Characterization of mTrw

no-t Glu^- -Cue

Electrically Small Radiating Sources

by Tests Inside a

Transmission Line Cell

no. ion

Ippalapalli Sreenivasia

David C. Chang ^

MarkT. Ma ^

University of Colorado

Boulder, Colorado 80309

2

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

,/

i>^i

6,"'^EAU 0* ^

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued February 1980
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1017

Nat. Bur. Stand. (U.S.), Tech. Note 1017, 68 pages (Feb. 1980)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON; 1980

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

Stock No. 003-003-02157-1 Price $3.50 (Add 25 percent additional for other than U.S. mailing)
FOPFWOPD

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.

Mark '^ , Ma of NBS serves as the technical contract monitor. The

period covered by this report extends from August 1^78 to

June 1979.

The work described in this report represents a further

aspect of technical analyses of transverse electromagnetic ( TEM

)

transmission line cells developed at MBS. T^he purpose of this

effort 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.

Starting with the premise that an electrically small

radiating object of arbitrary nature can be characterized by a composite dipole system consisting of three orthogonal electric

and three orthogonal magnetic dipoles excited with arbitrary

amplitudes and phases, this report presents measurement

procedures and analytical derivations for determining the total

emission and free-space radiation pattern of the radiating

source. The relevant theoretical analysis is given first,

followed by a general measurement procedure. '^he ensuing

simplifications for determining just the total power radiated by

the equipment under test (EOT) are then discussed. When the FUT

is characterized by only electric or magnetic dipoles, the

measurement procedure is extended to cells that do not support

the TEM mode. The needed modifications in the measurement

procedure, when the transition regions of the cell are

mismatched, are also discussed.

The measurement procedure described in this report is simple

to carry out once the normalized field distribution corresponding

to the fundamental mode in the cell is determined. This can be

111
achieved from a known theoretical expression or by measurements

with a probe. Experimental results for some selected cases are

found in qood agreement with the theoretical prediction.

Previous publications under the same effort include;

Tippett, J. C. , and Chanq, D. C. , Radiation characteristics of

dipole sources located inside a rectangular coaxial

transmission line, NBSIR 75-829 (Jan. 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, NPSIR 78-873 (March 1978).

Sreenivasiah, 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 (Aug. 1979).

XV
TABLE OF CONTENTS

Page

1

.

INTRODUCTION 1

2. RADIATION CHARACTERISTICS OF DIPOLE SOURCES

LOCATED INS IDE A TEM CELL 2

3. DETERMINATION OF RADIATION CHARACTERISTICS

OF DIPOLE SOURCES BY TESTS INSIDE A TEM CELL 7

3 , 1 Measurement Procedure 7

3.2 Interpretation of the Results 8

3.3 Determination of the Normalized Field Strength,. 13

3.3.1 Using Theoretical Expression 13

3.3.2 Using a Calibrated Probe 14

3.3.3 Using a Standard Electric

or Magnetic Dipole of Known Strength 15

3.3.4 Using an Uncalibrated Probe

with Linear Response 15

4. DETERMINATION OF THE TOTAL EMISSION BY AN ELECTRIC

AND/OR MAGNETIC DIPOLE USING A SYMMETRIC TEM CELL.... 17

4.1 Electric Dipole Source 17

4.2 Magnetic Dipole Source 18

4.3 Composite Source Consisting of Electric

and Magnetic Dipoles 18

5. THE EFFECT OF MISMATCHED TRANSITIONS 19

5.1 Cell with Identical Transitions 21

5.2 Cell with Dissimilar Transitions 21

5.3 Measurement of Equivalent Circuit Parameters

of the Transitions 22

5.3.1 Cell with Identical Transitions 23

5.3.2 Cell with Dissimilar Transitions 24

6 . EXPERIMENTAL RESULTS 26

7 . CONCLUS IONS 29

8 . ACKNOWLEDGMENTS 29

REFERENCES 30

APPENDIX I 32

APPENDIX II 37

APPENDIX III 40

APPENDIX IV 4 3

APPENDIX V 47

V
Figure 1.

Figure 2.

Figure 3.

Figure 4a.

Figure 4b.

F i g u re 4c.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Table I.

Table II.

LIST OF ILLUSTRATIONS AND TABLES

Page

Diagram of an NBS TEM cell 51

An arbitrary current filament

in a uniform cylindrical guide 52

Test setup for determining radiation

characteristics of small electric

and/or magnetic dipole sources 53

One orientation of the equipment under test (EUT)

with respect to x-, y-, z-axes of the cell 54

Another orientation of the EUT with respect

to the cell axes 55

A third orientation of the EUT with respect

to the cell axes , 56

Cross-sectional views of a 1.20 x 1.20 x 2.40 m

symmetric rectangular coaxial TEM cell

developed at NBS 57

Absorber loading in the 'li'FM cell

shown in figure 5 58

Normalized field strength distribution

in the TEM cell shown in figure 5 59

Scattering matrix representation of a cell

with mismatched transitions 60

Equivalent circuit representation of a TEM cell

with mismatched transitions 61

Measured dipole moment m vs. the height y^

of the test point from the center conductor....... 28

Measured dipole orientation compared

with actual dipole orientation 28

V3-
CHARACTERIZATION

OF ELECTRICALLY SMALL RADIATING SOURCES BY TESTS

INSIDE A TRANSMISSION LINE CELL

Ippalapalli Sreenivasiah

David C. Chang

and

Mark T. Ma

An electrically small radiating source of

arbitrary nature may be modeled by an equivalent dipole

system consisting of three orthogonal electric dipoles

and three orthogonal maqnetic dipoles, each excited

with arbitrary phase. An experimental procedure for

determining the emission characteristics of such an

equivalent dipole system by tests inside a single-mode

transmission line cell is described in this report,

followed by some experimental results.

1. INTRODUCTION

Currently there is a considerable interest in developing

methods for measuring electromagnetic (EM) emissions and EM

susceptibility of electronic equipment [1] . Rectangular coaxial

transverse electromagnetic (TEM) cells (fig. 1) developed at the

National Bureau of Standards (NRS) [2-5] for this purpose are

quite satisfactory for establishing a standard test field

environment for such measurements. However, one must devise a method to relate measurements taken inside the cell to the freespace radiation characteristics of the object. In this report,

we consider a typical equipment under test (EUT) as consisting of

an electrically small box housing low-frequency ac circuits.

Leakage currents to the exterior surface may be replaced by

equivalent electric/magnetic dipoles with appropriate magnitudes

and phases, and all such dipoles may be combined to form a composite source consisting of three orthogonal electric and

three orthogonal magnetic dipoles v;ith appropriate moments and

excitation phases. The question then is to determine the unknown

moments and phases of the individual electric and/or magnetic

dipoles of which the composite source, characterizing the EUT, is

made. Total radiation power of the FJT^ and its far-field

characteristics in free-space are then determined from values of
these composite dipoles. In section 2 we give theoretical

expressions for the modal amplitudes of the fields radiated, in

the positive and negative directions of a uniform cylindrical

guide, by arbitrary dipole sources located inside the guide.

For a symmetrical cell, separation of the electric and

magnetic dipole sources is achieved by taking the sum and

difference of outputs from the two ends of the TEM waveguide

enclosing the sources. A measurement procedure, which enables

one to obtain the needed parameters for determining the freespace radiation pattern of and the total power radiated by a composite source consisting of orthogonal electric and orthogonal

magnetic dipoles, is then described in section 3.

In assessing the emission level of an EOT, we are often not

interested in knowing the complete free-space radiation pattern

but rather the free-space total radiated power. In such event,

the measurement procedure is considerably simplified, as

discussed in section 4, depending upon the geometry of the test

cell and the type of the source.

Generally, a test cell consists of a section of uniform

transmission line with matched transitions to SO-n coaxial

connectors on both ends. In section 5 we discuss the effect of

mismatched transitions and how to modify the measured results to

account for such a mismatch. A procedure for determining the

scattering matrices of the transitions is also described in the

same section. Experimental results are given in section 6,

followed by concluding remarks in section 7.

2. RADIATION CHARACTERISTICS

OE DIPOLE SOURCES LOCATED INSIDE A TEM CELL

The electric and magnetic fields generated in the positive

and negative z-directions by an electric and/or magnetic dipole

in a guide of arbitrary cross section (fig. 2) may be represented

in terms of individual modes inside the waveguide as
-(±) , ,a, - (±)

E = I (^) F^ ,2.1a

n n

i'"' = i (^ i

'"'

, (2.1b)

b n n n

where, with a time dependence of e ,

+ 13 z -(±) - - . n

E = (e ± e ) e (2.2a)

n n zn

-l±) - - ""^^n^

H = (±h + h ) e

'^' , (2.2b)

n n zn

e , h are transverse vector functions, e and h are

n n

' zn zn axial vector functions at any given cross section, and B^, is the

propagation constant of the same mode. For E modes, h is _ zn zero, while for H modes e is zero. Orthogonality between two

zn modes, each properly normalized, implies

/ / e X h • zds =6 , (2.3)

s m n mn

where s represents the cross-sectional area of the guide. Then,

an application of the Lorentz reciprocity theorem, assuming that

the waveguide walls are perfectly conducting, yields the

amplitudes a , b of each mode in the positive and negative z- directions away from the dipole source [6]

(^) =-\ /,3 . ,.';' 6. , (2.4)

n
where the vector J represents the total current flowing through

an inf initesimally thin current filament of a length described by

C. For a short electric dipole, the above expression may be

written as

1 -

2 e E

( + ) n (2.5)

where m = Jd£ is the strength of the dipole. If the current

filament forms a closed loop of small cross section s, one

obtains from eq (2.4)

':'

n ^ IJI / E

^^^

. d£ =

2 n C

-i IJI / / V .E^^'

- (i(iJM ) H • m

2 o n m

ds

(2.6)

where m = |J| s is the magnetic dipole moment, w is the angular

frequency, and ijq is the free-space permeability.

In general, both m and m^ are complex vector moments given

by

iij; ex

m

m =

em

X m e ex

m

+ v m e m

111; l\|;

ey ez

m

'• m

+ z m e ez

m

(2.7)

w ith X, y, and z representing unit vectors along the x-, y- , and
z-directions , respectively. Denoting the TEN mode by subscript

o, we write

- (±) / , ^

^ +ikz

E =(xe +ye)e , o ox oy

(2.8a)

- ±

" " +ikz

H =(xh +yh)e , o ox oy

(2.8b)

and note that

- (±) H=±zxE/c ^ - (±) , , o o o (2.9)

where k is the free-space wave number.

Using eqs (2.5) and (2.6), we may v/rite the amplitudes of

the radiated fields corresponding to a combined presence of

electric and magnetic dipoles as

a 1 - - - " - (,)=-— [m -e ±ikm '(zxe)] b 2 e o m o

TT (m ± i k M) • e

2 e o

(2.10a)

where

ri = m xz; e =(e x + e y) (2.10b)

m o ox oy

so that
a + b = - m . e (2.11a)

e o

a - b = - i k (mmox z) • e^ . (2.11b)

In obtaining eq (2.10a) we have assumed that only the

dominant TEM mode exists inside the cell. The subscript o has

been omitted to simplify the notation. The relations in

eqs (2.11a) and (2.11b) derived above are very important in

determining the strengths of dipole sources (electric and/or

magnetic) located inside a TFAi transmisison line. This will be

clearer if we consider the experimental setup shown in figure 3

where the test object is placed at a point (x , y , 0) v/ith

respect to a frame of reference, in x-, y-, and z-coordinates

,

which is fixed with respect to the cell. The outputs from the

two ends of the cell are added in phase or out of phase with each

other by means of a 180° phase shifter. In practice, one may

obtain the sum/difference at one port by short circuiting the

other port at distance of an odd/even number of quarter

wavelengths away from the test object (assuming that the end

transitions are perfectly matched). We note that the sum

power Pg = (a+b)(a+b)* is independent of the magnetic dipole

v;hile the difference power P^ = (a-b)(a-b)* is independent of the

electric dipole. Furthermore, only the components of m and

M that are along the direction of the transverse electric field

vector e^{x^, y^, 0) contribute to the output. In the following

sections, we describe some experimental procedures, which are

based on the preceding observations, for determining the

radiation characteristics of electric and/or magnetic dipole

sources by placing them inside a single-mode TEM cell.
3. DETERMINATION OF RADIATION CHARACTERISTICS

OF DIPOLE SOURCES BY TESTS INSIDE A TEM CELL

In this section we describe the experimental procedure for

tests taken inside a cell consisting of a section of TEM

transmission line, with perfectly conducting inner and outer

conductors, and perfectly matched transitions at both ends. As

shown in Appendix I, the free-space radiation chracteristics of

an EUT characterized by three orthogonal electric or magnetic

dipoles can be completely determined through the quantities

2 2 2 m , m , m , m m cos{\b -li; )/ m m cos ( iJ; -ij; ),

ex ey ez ex ey ^ex ey ey ez ey ez m m^ m m m-^ mm-' m-' m m-' m

and m m cos(4; -\i; ) . mmez ex mmez ^ex

3.1 Measurement Procedure

To obtain the twelve unknown quantities mentioned above, we

first establish two frames of reference (x,y,z) and (x',y',z')/

the former fixed with respect to the cell with the origin (0) at

the geometric center of the cell and the latter fixed with

respect to the EUT with the origin (0') at the center of the

EUT. As shown in figure 3, let the z-axis be along the length of

the cell, the y-axis be perpendicular to the center septum, and

the X-axis be along the width of the septum. Then we go through

the following sequence of measurement operations where all the

rotations are assumed to be in the counter-clockwise direction.

(1) Position the EUT at a convenient test point {x^,y^,0)

such that its x'-, y'-, and z'-axes (designated initially as X-, y-, and z-axes in figure 3) are along the directions of x-

,

y-, and z-axes, respectively.

(2) Rotate the EUT about its z'-axis b^^ an angle of e^ in
the counter-clockwise direction, 6^ being a convenient value

ranging from to 7r/2 . This angle 0^ and the test point

(x^^y^^O) shall remain fixed throughout the experiment. Measure

the sum power P -, and the difference power P^-, for this

orientation (fig. 4a) of the FUT. The sum power is obtained by

removing the 180° phase shifter (fig. 3) from the circuit, and

the difference power is obtained by placing the phase shifter in

the circuit.

(3) Rotate the EUT by 90° about its z'-axis from the

position in step (2). Measure the sum power Pg2 and the

difference power P^^o*

(4) Position the EUT at the test point such that x'-, y'-,

and z'-axes are along z-, x-, and y-axes, respectively. Rotate

the EUT by an angle 9^ about its x'-axis and measure the sum

pov/er P -^ and the difference power P^-, corresponding to this

orientation (fig. 4b).

(5) Rotate the EUT by 90° about the x'-axis from its

position in step (4) and measure the sum and difference powers

Ps4 ^^^ Pd4-

(6) Position the EITt^ at the test point such that x'-, y*-,

and z'-axes are along y-, z-, and x-axes, respectively. Rotate

EUT by an angle 9^ about its y'-axis and measure the powers P^^

and P^t^ corresponding to this orientation (fig. 4c).

(7) Rotate the EUT by 90° about the y'-axis from its

position in step (6) and measure the sum and difference powers

Ps6 ^"^ Pd6-

3.2 Interpretation of the Results

Noting that in eqs (2.11a) and (2.11b)

m f m » m

ex ey ez

m m'^ m
represent the respective components of the dipole moments along

X-, y-, and z-axes of the TEM cell, we obtain the following

matrix eauations for the unknown moments

m' , m' , m'

ex ev ez

m eym-^ m

along the x'-, y'-f and z'-axes of the EUT and the cross

components

p pi = p

e s (3.1a)

B M = P^/(k ) , m d

(3.1b)

where

B =

P''

q''

i2

g'

P'

p'

q'

2p' q'

-2p' q'

.2 2p'q' n .2 -2p'q'

,2 2p'q'

,2 -2p'q'

(3.1c)

M = [m* e ex

m m

m'

m

m'

ez

m

m'

exy

m

m'

eyz

m

m' ] ezx

m

(3. Id)
s si s2 s3 s4 s5 s6 (3.1e)

rp ^d ^^d2 ^dl ^d4 '^dS ^d6 ^d5^ (3. If)

p' = p cos 9 + q sin 9 (3.1g)

q'^ = q^ cos 9 - p sin 9 o ^ o (3.1h)

p = e (x , y ) • X

o o o

(3.1i)

^ = % ^^o' ^o^ • ^ ' (3.1J)

and we use the notation

m' = m' m' cos(ii; - ^ ) , etc,

exy ex ey ex ey

m mm mm

The solution to the previous equations is readily given by

10
and

where

and

[ m' m' m' ] = - C P / I e I , ex ey ez 2s o

(3.2a)

mx my mz 2 d

'^ ^ ' o ' (3.2b)

[ m' m' m' ]'^ = d P /(4 p' q' ) , exy eyz ezx g / \ t- a / ^ (3.2c)

[m' m' m' ]'^ = DP, /(4p'q'k^), mxy myz mzx d

'^ ''^ ^ ' (3. 2d)

C =

1

1

-1

(3.2e)

D =

r

1

-f

f

-1

-f

f

f

1

-f

f

-1

-f

-f

f

1

(3.2f)

f = (p' - q- ) / I e I (3.2q)

11
The far-field free-space radiation pattern of the EUT is

given, in terms of the dipole moments and the cross components,

by (Appendix I)

15Tr,2,2 2 .2, 2 2.2 2 P(e, (}) ) = — { A (cos e cos (j)+ sm cf) ) + A (cos 6 sm ^ + cos ij)

)

r X ^ 2.2 . 2 . + A Sin 9 - 2 A sin 6 sm* cos* - 2 A smS cos6 sin*

z xy ^ ^ yz ^

where

- 2 A sine cos9 cos* + (B - B ) cosG

zx ^ xy yx

+ (B - B ) sine cos* + (B - B ) sine sin*} (3.3a)

yz zy ^ zx xz ^'

2 2 2 2

A = m' + k m' ; a = x, y, z (3.3b)

a ea ma fir

2

A o = 1^' ^ + k m' „; a, 3 = X, y, z , (3.3c)

a3 ea3 ma

3

B ^ = 2km m „ sin(i|; - \b ^); a, 3 = x, y, z . (3.3d)

a3 ea m3 ^ea ^m3 ' '

' ^'

Note that when the EUT is characterized by three orthogonal

electric or magnetic dipoles, B „ vanishes. In any event, the

total radiated power by the EUT in free space, Prjn, is given by:

2

P^ = / P (e, cj)) dfi = ^^

(hi + aJ + hb . (3.4)

^^ ^x y z

12
We note that this method fails, in view of eqs (3,2c) and

(3. 2d), to give the cross components if p'q' is zero. However,

one can always choose the test point (x^, y ) and 6 such that

p'q' is nonzero. In a TEM cell possessing symmetry about the yz

plane, one would normally choose the test point to lie in this

plane (x^ = 0) in which case p = and for a particular choice of

the test point (0, y^) p'q' will be maximum if 6 is Tr/4 . With

Xq = and 9 = 7r/4, p' = q' = q / /2 and f = 0, which simplify

the expressions for B in eq (3.1c) and D in eq (3.2f).

To obtain the absolute values of the dipole moments using

the above method we need to know e which is the normalized field

o

strength, corresponding to the TEM mode, at the test point

(Xq, yQ) . In the next subsection, we describe four different

ways to obtain this normalized field strength defined through the

relation

-'fie xh .zds=l. (3.5)

s •' o o

3.3 Determination of the Normalized Field Strength

3.3.1 Using Theoretical Expression

For simple geometries like circular coaxial or symmetric

rectangular coaxial TEM transmission lines, e may be calculated

using the known theoretical expressions. For symmetric

rectangular coaxial TEM transmission line, e is given by [7] :

-i m' dn(m' z ) /z

e (x,y) = J— (3.6a)

° K(a') {P (z)}^2

^ o

'

13
P^(z) = {sn^(m'w) - sn^(m'z)} (3.6b)

m' = K(k)/a = K(k' )/b (3.6c)

a = sn(m'w,k) (3 .6d)

1/2

a' = (1 - a^) (3.6e)

z = X + iy , (3.6f

)

where 2a, 2b are the width and the height of the TEM cell; 2w

the width of the center conductor; and Z^, the characteristic

impedance of the line. The function K(k) and K(k') are complete

elliptic integrals of the first kind of moduli k and k'

,

respectively, where

k' = / 1 - k2

and sn and dn are Jacobian elliptic functions which may be

calculated using the algorithm given in [8] . Amplitude and phase

of e are tabulated in [7] for some typical cell sizes, together

with the computer program which generates these values.

3.3.2 Using a Calibrated Probe

If a calibrated field-strength probe is available, e may

simply be obtained by probing the cell at the desired point

(x ,y ) with unit power at the input port while the other port is

terminated with a matched load.

14
3.3.3 Using a Standard Electric or Magnetic Dipole

of Known Strength

With a standard electric dipole, we measure the power

outputs P„ and P.. at one port corresponding to the dipole X y

orientations along x- and y-directions , respectively, with the

other port terminated with a matched load. Then

e = e X + e y

o ox o\7

may be obtained through the relations

e^ = 4P / m^ , (3.7) ox X ^ se ' y y

where m is the known moment of the standard dipole.

With a standard magnetic dipole, we measure the power

outputs at one port P and P corresponding to the dipole

orientations along x- and y-directions , respectively, with the

other port being terminated with a matched load. Then e and

e^ are given by

e^ = 4P / (k m )^ , (3.8) ox X ^ ^ sm ' y y

where m^^^ is the known moment of the magnetic dipole.

3.3.4 Using an Uncalibrated Probe with Linear Response

We note that e" (x ,y ) represents the field strength at

{x^,y^) when unit power is flowing across the transmission line

and also that the power flow across a TEM transmission line is

given by

PteM = ( / E . dl)2 / z^ , (3.9)

15
where Z^ is the characteristic impedance of the line, E is the

electric field strength, and C is any path lying in the crosssectional plane and extending from the inner conductor to the

outer conductor. Using eq (3.9) we may write the normalized

field strength 'e^ix^fY^) as

which may be measured using an uncalibrated probe since the

calibration factor appears both in the numerator and

denominator. For doing so, the probe must have a linear response

(or else, a calibration of the probe is needed), and the

characteristic impedance, Z , of the line must be known.

The last expression [eq (3.10)] for "e is an important one

since it enables one to extend the measurement procedure to any

TEfl transmission line geometry for which the theoretical

expression for the TEM modal distribution is not available.

Though we started with the assumption of an ideal TEM

transmission line, we can use the same method even when the

transmission line is slightly lossy, as when the cell is lined

with absorbing material, or when the inner and outer conductors

are made up of imperfect conductors, provided that the TEM nature

of the cell is largely unaltered.

The measurement procedure described in this section enables

one to determine the magnitudes of the individual

electric/magnetic dipole moments along three orthogonal

directions and cross components required to obtain the free-space

radiation pattern of a composite dipole consisting of three

orthogonal electric or magnetic dipoles, each excited with an

arbitrary phase. Oftentimes, one is interested in knowing only

the total power radiated in free space by the dipole or the

EUT. In such a case, we need to take only six measurements,

three for the electric dipole and three for the magnetic

dipole. In the next section we discuss simple measurement

procedures that may be used in such a case,

16
4. DETERMINATION OF THE TOTAL EMISSION

BY AN ELECTRIC AND/OR MAGNETIC DIPOLE USING A SYMMETRIC TEM CELL

In all the measurement procedures that we are going to

discuss in this section, we assume that x-, y- , and z-axes refer

to a frame of reference fixed with respect to the test cell with

the origin at the geometric center, if one exists, of the cell.

The point (x ,y ,0) refers to the center of the test object with

respect to (x,y,z) frame of reference, and x'-, y'-, and z'-axes

refer to a frame of reference fixed with respect to the test

object with the origin at the center of the test object. In all

the cases we assume that the test cell consists of a section of a closed TEM transmission line, symmetric about the yz-plane with

perfectly matched transitions at both ends. It is also assumed

that the frequency of operation is such that only the dominant

TEM mode can propagate. The orthonormal modal function giving

the field distribution of the TEM mode is denoted by eQ(x,y).

4.1 Electric Dipole Source

Terminating one port of the cell with a matched load, we

measure the output powers P„„, Po^^ and P at the other port

corresponding to the orientations of the test object given by

(x'-»-y), (y'->-y)f and (z'-»-y), respectively. Since e has

only the y-component at the test point, the electric dipole

moments are given by

m'? = 4 P . / |e I^ ; i = x,y,z . (4.1) ei ei o ^ '

The total power Pm radiated in free space is given by

P„ = 40Tr^(m'^ + m'^ + m'^) / X^ , (4.2) Te ^ ex ey ez '

17
where X is the wavelength of operating frequency.

4,2 Magnetic Dipole Source

Terininating one port of the cell with a matched load, we

measure the power outputs Pj^^ ' ^mv' ^^'^ ^mz ^^ ^^^ other port

corresponding to the orientations of the test object given by

(x'->-x), (y'->-x), (z'-»-x), respectively. Then the magnetic

dipole moments are given by

m^l = 4 Pj^. / (k le^l)^ ; i = x,y,z . (4.3)

The total power Prji„ radiated in free space is given by

P^ = 160 Ti'^(m'^ + m'^ + m'^) / X^ . (4.4) Tm mx my mz

4.3 Composite Source Consisting of Electric and Magnetic Dipoles

When the source consists of electric and magnetic dipoles,

we need to make measurements of sum and difference powers as

shown in figure 3. We measure the powers P • and P^ir i = x,y,z

at the sum and difference ports corresponding to the orientations

of the test object given by (z'->x, x'->y), (x'^x, y'^y)r and

(y'->-x, z'>y), respectively. Then we have

m'? = P . / |e 1^ , i = x, y, z (4.5) ei si '^ o ' ' ^

r

(4.6)

2 .- . 2

m' = P^ / (k le 1)

mx dy o

?

The total radiated power Prp, in free space, is given by

Prp = Pnf. + Pmj^ where Pr^^ and P^j^ may be obtained from eqs (4.2)

and (4.4).

18
In all the cases discussed, we need the value of "e at the

test point. This may be obtained using any of the four methods

given in Section 3.3.

So far, we have assumed that the test cell consists of a section of TEM transmission line with perfectly matched

transitions at both ends. However, if the test cell is made of a section of waveguide that cannot support a TEM mode, the modal

function corresponding to the fundamental mode may also have a z- component in which case the simple relation of the form in

eg (2,8) is no longer true. As a result, it is not possible to

decompose the electric and magnetic dipoles using the sum and

difference technique that we discussed in this section and the

previous section. When the EUT can be characterized entirely by

dipoles of the electric or magnetic type but not of both types,

it is still possible to obtain the x-, y-, and z-components of

the dipole by a modified procedure that is described in

Appendix II.

5. THE EFFECT OF MISMATCHED TRANSITIONS

In the preceding sections we assumed that the test cell has

a section of uniform transmission line with perfectly matched

transitions at both ends. If the transitions are not matched, we

need to modify the measurement procedure to account for the

mismatches introduced by the transitions. In this section, we

describe such a measurement scheme under the assumption that the

transitions are lossless and reciprocal, which is true in almost

all the practical situations. First, we note that, as shown in

Appendix III, any lossless reciprocal two-port network may be

represented by an equivalent circuit consisting of two sections

of lossless lines coupled by an ideal transformer. A TEM cell

with mismatched transitions may thus be represented by the

equivalent circuit shown in figure 9 where the equivalent

circuits corresponding to the transitions are enclosed by dotted

lines. In this equivalent circuit, L represents the length of

the transmission line section of the cell excluding the

19
transitions. The output connectors are assumed to have the same

characteristic impedance as that of the transmission line section

of the cell. In Appendix IV, we derive the matrix equations

relating the complex amplitudes a and b given by eq (2.ina) and

the complex amplitudes a and b that would be measured using a cell with mismatched transitions. rising the relation of

eq (A4.5a) for the case when the radiator is located midway

between the cell (z = 0), we obtain the following equation using

straightforward algebraic manipulations:

la ± b|^ = |F_^|^ \ {a ± G b) \ ^ (5.1a)

where

n2 + 1 ikil' 1 - n2 -ikJi'

F^ = ( -2- ) (e -^^ ± e ^^) (5.1b)

± ' 2n

1 + n2b

n^(n2 + 1) ik(£2b-^2a^ , ^^^^b . ^ " "I "^^Hb,

G = -^ ^ e "^ "" (e ^"^ ± e "")

(5.1c

n, (n2 + 1) 1 + n2

b a a

2

ikJl' , 1 - n, -ikA

. / la , b la \ ^ (e ± ^ e ) 1 + n^b

SL\ = i, + Vo L (5. Id)

b b

and the circuit parameters ^i^/ ^Ib' ^?a' ^2b' "a' '^b'

and L are shown in figure 9. Based upon eq (5.1a), we describe

the methods of accounting for the mismatched transitions

depending upon whether the transitions are identical or not.

20
5.1 Cell with Identical Transitions

If the cell has identical transitions the quantity G in

eq (5.1c) reduces to unity. For a specific orientation of the

p test object, the true sum and difference powers s corresponding to matched transitions^are obtained in terms of the

P measured sum and difference powers s with mismatched

transitions, through the following relation

2 ~

P = IF I P (5.2)

with n = nj^ = 1 and i-^^ = l-^y^ = i-^ .

5.2 Cell with Dissimilar Transitions

If the cell has dissimilar transitions, simple sum and

difference power measurement does not give enough information to

obtain the true sum and difference powers corresponding to

matched transitions. To obtain the latter, we use the following

modified procedure:

We connect an attenuator and phase shifter in the negative

port (see fig. 9) before taking the sum and difference. The

attenuation in dB is given by -20 log \G\ and the required phase

shift by _/G. Then the true sum and difference powers are

expressed in terms of the measured sum and difference power

through the same relation in eq (5.2) with F being given by

eq (5.1b). If |C| > 1, then we connect the attenuator and the

phase shifter in the positive port with the subscripts a and b

interchanged

When making measurements with a cell having mismatched

transitions, we should note that it is not possible to obtain the

sum/difference power at one port by short circuiting the other

port at odd/even number of quarter wavelengths. Further, even

when the source is made of only electric or magnetic dipoles, we

cannot make measurements at one port with the other port

terminated by a matched load. We need to use the general

21
procedure outlined in section 3 with the modifications discussed

above. It is also obvious that we need to know the equivalent

circuit parameters of the transitions to account for their

effect. In what follows, we describe a measurement procedure to

determine these parameters.

5.3 Measurement of Equivalent Circuit Parameters

of the Transitions

The measurement of the equivalent circuit parameters of the

transitions is complicated by the fact that the transitions are

integral parts of the TEM cell and, in practice, it is not

possible to isolate them from the cell for the purpose of

measuring the circuit parameters of the individual transitions.

Any external measurements on the cell can only provide the

equivalent circuit parameters of the complete cell including both

transitions. As will be clear later, whereas the complete cell

may be characterized by only three independent real parameters

(two in the case of identical transitions), we need six

independent parameters (three in the case of identical

transitions) to individually characterize the transitions. Thus

it is not possible to obtain the equivalent circuit parameters of

the transitions by external measurements alone, by which we mean

without a probe or a standard radiator inside the cell. However,

it is possible to completely characterize the transitions by

making measurements with a standard radiator inside the cell and

combining these results with the equivalent circuit parameters of

the complete cell which may be obtained through external

measurements alone. In what follows we describe a step-by-step

procedure of obtaining the equivalent circuit parameters of the

transitions. The required analytic details are given in

Appendices IV and V. We treat the case of identical transitions

separately from that of dissimilar transitions.

22
5.3.1 Cell with Identical Transitions

If the cell has identical transitions, the parameters

ha = hh " h' ha = hb = h' ^"^ '^a = "b = " ^iH

completely characterize the transitions. To obtain these

parameters we proceed as follows:

1. Measure the cascading matrix R of the entire cell

using any standard method [6]. Because of the symmetry of the

cell, the cascading parameters ^^2' ^21' ^^'^ ^22 ^^® related to

R-^-, through

^12 " ^21 " ^^11 ^11 "

-^^"^^ (5.3a)

and

^22 " ^11 * (5.3b)

2. Terminate the negative port of the TEM cell with a matched load corresponding to the characteristic impedance of the

transmission line section of the cell and measure the complex

ratio of the voltages a(+g) and a(-g) across a matched

detector at the positive port with a standard electric dipole

placed inside the cell in transverse planes z = +g and z = -g,

respectively (fig. 9), where g is a convenient value. To obtain

maximum sensitivity, the dipole must be aligned with the vertical

E-fieid inside the cell.

3. Calculate the real parameters n and £, using the

following relation which is based upon eq ( A4 . 7 ) :

^_^2 -ikJi^ a^ e^^^^+1 .^^

( ""^ ) e = {i(—^

-r-^T——) sin(2kg) - cos(2kg)} e , 1 + n^ a^ e^^'^^-l (5.4a)

23
where

(5.4a)

^j. = a(+g) / a(-g) (5.4b)

After obtaining n, we proceed to calculate £2 using eq (A5.2a),

sin( 2kJi )

cos(2kii2 )

2n'

r

4 2

(1 + n ) sm kL' - 2n cos kL'

2n cos kL (1+n ) sin kL'

Re ( I'll'

Im'I'll' v

where

and

(5.5a)

A = 4n'^ cos'' kL' + (1 + n^")^ sin^ kL' , (5.5b:

L' = 2%^ + L (5.5c)

We note that the parameter %2 ^^ not needed in calculating

the correction factor, F. Thus it is not necessary to measure

the cascading parameter R-,-. of the entire cell unless one is

interested in obtaining the complete scattering/cascading matrix

of the transition,

5.3.2 Cell with Dissimilar Transitions

If the cell has dissimilar transitions, we essentially

follow the same procedure as discussed above except that we

repeat step 2 on both ports. The calculation procedure is also

slightly more complicated as described below.

1. Measure the cascading matrix R of the complete cell.

24
since the cell has dissimilar transitions, the relations in

eqs (5.3a) and (5.3b) are now replaced by the following more

restrictive relations which follow from the lossless and

reciprocal nature of the cell.

Rl2 = Rji (5.6a)

R^^ = ^^2 (5.6b)

^11 ^22 " ^12 ^21 " ^ • (5.6c)

2. Terminate the negative port of the cell with a matched

load and measure the complex ratio of the voltages a(+g) and

a(-g) across a matched detector at the positive port with a standard electric dipole placed inside the cell at z = +g and

z = -g, respectively.

3. Terminate the positive port of the cell with a matched

load and measure the complex ratio of the voltage b(-g) and

b(+g) across a matched detector at the negative port with the

standard electric dipole placed at z = -g and z = +g

,

respectively.

4. Calculate the real parameters, n^ , £, , n^, and ^-lu/

using the following relations which are derived in Appendix IV

1-n^ -i2kJl, b exp(i2kg)+l ., ^ e ^^ = {i(;7^ ) sin (2kg) - cos (2kg)} e^""

2 1+n b exp(i2kg)-l

El it (5.7a)

2 ~

1-n. i2k£,. a exp(i2kg)+l ., ^ e

^'^ = {i(:3^— —) sin(2kg) - cos(2kg)} e^"^^

l+n^^ a^ exp(i2kg)-l

(5.7b)

where

25
\ = a(+g) / a(-g) , (5.7c)

and

bj^ = b(-g) / b(+g) . (5.7d)

5. Having known n^, i , n,^, and ^lu^r ^e then calculate

coS ^^^^23 "^ ^2b^ ^"^ coS ^(^2a ~ ^2b^ using eq (A. 5), and

hence determine £_ and Jl_, uniquely.

ZS. 2,13

6. EXPERIMENTAL RESULTS

'T'o verify the theoretical analysis presented in previous

sections, some measurements were made using a symmetric

rectangular coaxial TEM cell of dimensions 1.20x1.20x2.40 m-^ as

shown in figure 5, and a center conductor of width 99.2 cm. The

cell has matched transitions at boths ends and is partially

loaded with absorbing material, as shown in figure 6, to suppress

the higher-order modes at high frequencies. A spherical dipole

radiator having a radius of 5 cm was used as the test source.

The radiator consists of two hemispherical shells which are fed

at the poles and held together by threading onto a dielectric

disc such that there is a gap of 3 mm between them. The

electronic circuitry feeding the dipole is enclosed within the

shells and consists of a battery-operated power supply and a 30-MHz crystal oscillator followed by an amplifier. T'hus, the

radiator is self-contained and no external connections are

needed. The spherical dipole radiator and the TEM cell were

developed at NBS by M. Crawford. Earlier measurements indicated

that the characteristic impedance of the cell remained unaltered

after placing the absorbing material. However, measurements with

a short probe indicated that the normalized field distribution is

different from the theoretical distribution corresponding to an

unloaded cell. In figure 7 we have plotted the theoretical

distribution using eq (3.6) and the measured distribution along

the vertical axis of the cell.

26
Two sets of measurements were made. In one experiment, the

dipole was placed such that it was aligned with the vertical axis

of the cell, and the dipole moment was measured using our

technique at various test points along the y-axis of the cell to

see if the dipole moment is sensitive to changes in the impedance

of the surrounding medium. Table I gives the computed values of

the dipole moment as a function of the vertical distance y of

the test point from the center conductor. As we can see from the

table, there is no appreciable change in dipole moment as the

test point is moved along the vertical axis.

In the second experiment, the x-, y-, and z-components of

the dipole were measured using our technique for various

orientations of the dipole radiator, and from these measurements

the (9,(t)) coordinates of the dipole vector were computed and

compared with the actual orientations of the dipole. The results

are tabulated in Table II. From these results we note that the

measured (9,<t)) coordinates of the dipole vector are close to

the actual values within ±1°. Also, the spread in the measured

dipole moment (Table I) is within ±2.5 percent, thus establishing

the usefulness and validity of our method for the case of an

electric dipole source.

At the time of writing this report, a magnetic dipole source

was not available, and hence it was not possible to test our

method for the case of a magnetic dipole source or a composite

source.

27
Table I. Measured dipole moment vs. the height of the test point

from the center conductor.

-Vo (cm)

15

20

25

30

35

40

45

50

"""Using theoretical e^CO^y^) corresponding to unloaded cell.

ImJ-'

Di pole Moment

Ne'"'

(Amp. Meter X 10"')

2.983 2.740

2.916 2.705

2.830 2.711

2.736 2.708

2.647 2.728

2.531

~ 2.645

2.418 2.606

2.333 2.625

++Using experimental ^^(0,7^) measured with high impedance probe

Table II. Measured dipole orientations compared with actual

dipole orientations.

Measured 6, 4) (deg.)

^o

(cm)

20

30

40

20

20

20

30

30

Actual 9 , (j) (e = 45° ) (e = OO ) (deg. ) o o

\ -3 '

62.94, 60.38 62.19, 59.36 61.4, 58.4

63.11, 58.61 60.00, 56.55 61.4, 58.4

63.10, 58.16 62.90, 60.37 61.4, 58.4

66.79, 69.73 65.28, 70.98 65.36, 72.19

73.77, 136.97 73.59, 136.32 72.81, 136.98

48.57, -55.35 49.27, -53.92 47.60, -54.19

44.21, 140.14 44.68, -144.70 42.67, -139.27

76.60, -40.88 80.59, -41.87 80.83, -41.14

28
7. CONCLUSIONS

Experimental procedures described in the report should

enable one to determine the total power radiated in free space by

(and/or the complete free space-radiation pattern of) an

electrically small radiating object that may be characterized by

a system of elementary dipole radiators consisting of three

orthogonal electric dipoles or three orthogonal magnetic dipoles,

each excited with arbitrary phase. The test cell is not

restricted to a specific geometry as long as it can support a TFM

mode and the dimensions are such that no higher-order mode can

propagate at the test frequency. The discussion presented in

section 5 should enable one to extend the measurement procedure

to test cells with mismatched end transitions.

The experimental results presented in section 6 demonstrate

the validity of our test procedure for the case of an electric

dipole source. It is reasonable to assume that the same will be

true for the case of a magnetic dipole source. However, for the

case of a composite source, we need to establish the usefulness

of our method through actual measurements. We plan to do this in

the near future.

P . ACKNOWLFDGMENTS

The authors are very thankful to M. L. Crawford and

J. L. Workman for their assistance during the course of the

experimental work. Also, their permission to use the spherical

dipole radiator and the TEM cell is gratefully acknowledged.

Thanks are also due to Dr. M. Kanda for the technical discussions

we had with him during the course of this study. The continued

and wholehearted support and encouragement from C. K. S. Miller

and F. X. Pies are acknowledged with appreciation.

29
9. REFERENCES

[1] Adams, J. W., Electromagnetic interference measurement

program at the National Bureau of Standards, USNC/URSI,

Commissions I-VIII, Annual Meeting Digest, p. 33

(Oct. 20-23, 1975).

[2] Crawford, PI. L., Generation of standard EM fields using TEM

transmission cells, IEEE Trans, on Electromag. Compat.

EMC-16 , No. 4, 189-195 (Nov. 1974).

[3] Crawford, M. L., Workman, J. L., and Thomas, C. L.

,

Generation of EM susceptibility test fields using a large

absorber-loaded TEM cell, IEEE Trans, on Instrum. and

Measure. IM-26 , No. 3, 225-230 (Sept. 1977).

[4] Crawford, ri. L., and Workman, J. L., Asymmetric versus

symmetric TEM cells for EMI measurements. Conference Rept.

IEEE Inter. Symp. on EMC, 204-210, Atlanta, GA (June 20-22,

197-8) .

[5] Crawford, M. L. , Workman, J. L. , and Thomas, C. L.

,

Expanding the bandwidth of TEM cells for EMC measurements,

IEEE Trans, on Electromag. Compat. EMC-20 , No. 3, 368-375

(Aug. 1978).

[6] Collin, R, E., Field Theory of Guided Waves (New York,

McGraw-Hill Book Company, 1960).

[7] Tippet, J. C, and Chang, D. C. , Radiation characteristics

of electrically small devices in a TEM transmission cell,

IEEE Trans, on Electromag. Compat. EMC-18 , No. 4, 134-140

(Nov. 1976).

[8] Tippet, J. C, and Chang, D. C, Radiation characteristics

of dipole sources located inside a rectangular, coaxial

transmission line, NBSIR 75-829 (Jan. 1976).

30
[9] Ghose, R. N. , Microwave Circuit Theory and Analysis (New

York, McGraw-Hill Book Company, 1963).

[10] Kerns, D. M., and Reatty, R. W, , Basic Theory of Waveguide

Junctions and Introductory Microwave Network Analysis,

pp. 41-48 (New York, Pergamon Press, 1967).

31
APPENDIX I

FAR-FIELD RADIATION PATTERN OF A SYSTEM OF ORTHOGONAL, ELECTRICAND P1AGNETIC-DIP0LE SOURCES EXCITED WITH ARBITRARY PHASES

Consider a system of short, electric- and magnetic-dipole

sources with moments m # m , and m alonq the x.y.z axes and

ex ey ez ^

' ^' m m-^ m

excited with phases il^ , ^ # i|> , respectively. ^

i^^ ey ^ez ^ ^ Let H and 1\^ be the complex moment vectors given by

il' n =

em

X m e ex

m

ex

m

+ y m. e ^ ey

m-^

nil \i) ^ey ^ ^ez

+ z m e m

(Al.l)

The complex vector potential A due to the electric dipole system

is then given by

u -ikr

e 4Tr r

( Al . 2 )

where r is the distance from the dipole source. We use the

relation between the cartesian and the spherical coordinate

system

r > / x sine cos(j) cose COS<j) -sin(j)

y

= sine sin()) cose sine}) COS(j)

z

V ) cose -sine

( Al . 3 )

and write the components of A as

32
A = (M sine cosd) + fl sine sin* + M cose) (u /4Ti)e /r

r ex ^ ey ^ ez o

' (Al.4a)

A^ = (M cose cosd) + M cose sin* - M sine) (u /4Tr)e /r

8 ^ ex ^ ey ^ ez o

' (A1.4b)

A, = (-M sind) + M cos*) (y /47r)e /r

(}) ex ^ ey r / \ hq/ (Al.4c)

where we have defined

M = m exp ( ill; ex ex ex

I \

Using the relations

(A1.4d)

H = (V X A) /y (Al.Sa)

E = —^V (V . A) -iwA (Al.5b)

and retaining only those terms which decay as 1/r, we find that

ik

H- = (-M sind) + M cos*) (-; ) exp (-ikr)

e 'ex ^ ey ^Mirr ^ (A1.6a)

-ik

H = (M cose coscj) + M cose sin(j) - M sine) (;t—) exp (-ikr)

(|) ex ey ez 4Trr

( Al.6b)

^e = ^o% (Al.6c)

33
^ = -?o «6 '^l-«<3'

where

C = / M /e = 1207r . (Al.6e)

o o o

Usinq the duality principle, we could write down the electricand magnetic-field components E ' , E' , H ' , and H' due to an

o (p o (p orthogonal system of magnetic dipoles with moments nij^j^ / "^mv' "^mz'

and phases iJ; , il^ , il; as ^ ^mx ^my ^mz

wy k

E' = (-M sin* + n coscb) ( . °

) exp (-ikr) (A1.7a)

8 ^mx ^ my ^Mirr ^

-a3y k

o

E'= (M cose cosd) + H cose sind) - M sine)(—^j ) exp(-ikr)

(j) ^ mx my ^ mz 4Trr

(Al.7b)

h; = -e;A^ (A1.7C)

h; = E'A^ - (Al.7d)

where we have defined

M = m exp {i^ ) . (A1.8)

mx mx mx

The total fields, due to the combined presence of electric and

magnetic dipoles, are then given by

— iwy .

,

R^ ^ p. + E' = (M + ikM ) (—I

-) e~ , (Al.9a)

e e e ee mc|) 4iTr '

4- ^ ~ ~^^^n -ikr

34
H^ = + eVc , (A1.9C)

6 60

^ where we have defined

M ^ = M cos9 cos* + M cosS sm* - M sine, (Al.lOa)

e6 ex ^ ey ^ ez mm ' m m

and

n , = -M sinc^ + M cos* . Al.lOb)

ed) ex ^ ey ^ m m m^

The time-averaqed , radiated power per unit solid angle, P{Q,^),

is given by

P(6,*) =V2(e; Ef+ E^ E^*}/c^ , (Al.ll)

which, upon using eqs (A1.9a) and (Al.9b), becomes

o 2 2 e6 ed) m6 mcf) or A

+ ik [M , M „- M ^ M + M M ^- Fi ^ M J } m(f) ee ee m(f> ecj) me me e(})

(A1.12)

where X is the wavelength at the operating frequency.

By substituting for M ., M ^ from eq (Al.lO) and noting the

e D e 9

definitions in eqs (A1.4d)"^and ( Al . B ) , we obtain

P(e,(|)) = r—^ [ (m + k m ) (cos 6 cos ^ + sin (jj

)

r \

+ (m

2 + k 22

m ) (cos 2

9 sm2

* + cos

2

A) + (m

2 + k 22.2

m ) sin 1 ey my ^ ^ ez mz

35
2 2 2{m m cos(i|; -i) ) + k m m cos(\J; - ii )} sin sin* cosd) ex ey ^ex ^ey mx my ^ ^mx ^my ^ ^^^<v

2 2{m m cos(ii; -ih ) + k m m cosfij; -t1) )\ sine cos9 sin*

ey ez ^ ^ey ^ez my mz ^ ^my ^mz ^ ^>.>i^

2 2{m m cos(il; - tI) ) + k m m cos(\j; - ij; )} sine cos0 cos(

ez ex ^ez ^ex mz mx ^ ^mz ^mx

+ 2kjm m sinfij; - ip ) - m m sin(\l; - ^ )] cose ' ex my ^ex ^my ey mx ^ey ^mx^

+ 2k{m m sinfil; - i) ) - m m sinfiii - ij; )} sine cos* ^ ey mz ^ey ^mz ez my ^ez ^my ^ ^

+ 2kjm m sin(il; - i) ) - m m sin(4; - ii; )} sine sinil .

^ ez mx ^ez ^mx ex mz ^ex ^mz ^ ^^ (A1.13)

The total radiated power PTM is given by

P^ = /P(e,(),)d^

4071^ r2^ 2^ 2^,2,2^ 2^ 2 ^^^-l^) —r— {m + m +m +k(m +m +m)} 2 ex ey ez mx my mz

where we have made use of the fact that c = 120Tr.

o

36
APPENDIX II

MEASUREMENT WITH A TEST CELL THAT CANNOT SUPPORT TEM MODE

If the waveguide section of the test cell cannot support TEM

mode, the electric field corresponding to the fundamental mode

may also have a z-component, and, as a result, it is not possible

to decompose the electric and magnetic dipoles using the sum and

difference technique discussed in sections 3 and 4. However, if

the source is entirely made of electric or magnetic dipoles, it

is still possible to obtain the x,y, z-components of the dipole by

the following procedure:

Terminate one port of the cell with a matched load and

measure output powers p^, i = 1,2... ,6, at the other port, for

the orientations of the test object given by (x'-»-x, y'->-y)/

(x'-»-x, y'->'-y), (y'->-x, z'-»-y), (y'-»-x, z'-»--y), (z'->-x, x'->-y),

and (z'^x, x'->-y), respectively. Then, from eqs (2.5) and

(2.6), we have

p M^+ q^M2+ ^^^^^3+ 2pqM,2+ 2qrM23+ 2rpM2,= P, (A2.1a)

p n^+ q n^+ r M^- 2pqMj^2'^ 2qrM23- 2rpM3,= P^ (A2.1b)

p M2+ g^fl3+ r^M^+ 2pqM23+ 2qrM3,+ 2rpM,2= P3 (A2.1c)

p^M2+ ^^^3+ r^M^- 2pqM23+ 2qr^^^- 2rpMj^2= ^4 (A2.1d)

p^M^+ q^M^+ r^M2+ 2pqM2,+ 2qrM,2+ 2rpM23= P^ (A2.1e)

37
p^H^H- q^M^+ r^M^- 2pqM3^+ IqrH^^- 2rpM23= Pg , (A2.1f)

where we have used the following notation

(a) for an electric dipole source.

M. = m

ex

(A2.2a)

p = e ; a = e ; r = e ^ ox oy oz

M, „ = m n cos (i) - i) ). etc 12 ex ey ^ex ^ey '

(A2.2b)

(A2.2c)

(b) for a magnetic dipole source.

1 o mx

(A2.3a)

p = h ; q = h ; r = h ^ ox ^ oy oz

2 2

M, „ = 0) y iTi m cos(iJ; -4) ), etc 12 o mx my ^mx ^my '

(A2.3b)

(A2.3C)

The solution to eqs (A2.1a) — (A2.1f) is given by

/ > M

M

M

12

23

1

4 pa"

q

2

31

\ y

q

-qr

-qr

2

P, - P,

P^ - P

q p^ - P.

y *x y

(A2.4)

38
f 4

P - 2 2q J^ 4 r

2 2

- p q

4

q - 2 2

- r p

2

"2

1

'2

4

q

2 2

- r p

4

P - 2 2

- q r

4 r

2 2

- p q

4 r2 2

- p q

4

q - 2 2

- r p

4

P - 2 2

- q r

B

B

B

(A2.5)

where

3 3

Aj^ = q + r , 6,6,6 - 2 2 2

A2 = P +q +r -3pqr ,

B2 = V2 (P3 + P4) - 2qr M3^ ,

(A2.6a)

(A2.6b)

(A2.7a)

(A2.7b)

and

= V2 (P5 + Pg) - 2qr M^. (A2.7C)

The normalized components of modal fields e ox ' ^oy' ^oz' ^ ox

h^ , and h^ must either be known theoretically or be measured

using a calibrated probe.

39
APPENDIX III

EQUIVALENT CIRCUIT OF AN ARBITRARY, LOSSLESS, RECIPROCAL TWO-PORT

NETWORK

The scattering matrix of a lossless, reciprocal junction has

the following symmetry and unitary properties [9]

:

S = S

T

(A3. la)

S S" = I , (A3. lb)

where I is the unit matrix and all the ports of the junction are

assumed to have the same characteristic impedance.

Using the above relations, the scattering matrix of a 2-port

lossless, reciprocal network may be written as [10]

S = 11

i^

' ' - ^1^*1 ^ 12

i^

' ' - ^11^11 ^ 12

i^

^11 1 ^

22

(A3. 2a)

with T^ , being given in terms of ij;,, and \j;_^ by

^2 =^/2 (^n + 4^22) " % -/2 ' (A3. 2b)

where N^ is an odd integer and ^b is the phase of S^^.

We show that such a tvi/o-port may be represented by the

equivalent circuit shown in figure 8, which consists of two

sections of lossless, transmission lines coupled by a transformer. To do this, we first obtain the cascading

40
matrix R of this equivalent circuit as the product of the

cascading matrices of the individual sections as follows:

P =

-ikJi

ikJl

z'

2n (n^ + 1) (1 - n^)

(1 - n^) (n^ + 1)

-ikii.

ikii,

/

2^

2n

(1 + n^) e

^ (1 - n^) e

ik{Si,-i ) ik(il +£ ) (l-n'^)e

-^^

{l + n^)e ^ ^ (A3. 3)

The scattering matrix S is then given by

S = 22

12

1

-r 21

r

2 -''""'l

(1 - n ) e

1 + n 2ne-ik{z^+i^)

2ne

ikii^ + Z^)

i2k£

(n -De (A3. 4)

Comparing eq (A3.2a,b) with eq (A3. 4), we obtain the following

values for i,, £„, and n for equivalency:

2kSi = \j^^ + (2N - 1 )Tr (A3. 5a)

41
-2kJi2 "^ ^^22 "^ ^^''^

n^ = (1 + |S^^|)/(1 - IS^^I)

(A3. 5b)

(A3. 5c)

where M and N are arbitrary integers subject to the constraint,

2(M+N) -1=Nq, which is imposed by eq (A3. 2b).

42
APPENDIX IV

EXPRESSIONS FOR THE COMPLEX FIELD AMPLITUDES a AND b

AT THE TWO PORTS OF A TEH CELL

DUE TO A DIPOLE RADIATOR INSIDE THE CELL

In this appendix we consider a TEM cell with mismatched

transitions which may be represented by the equivalent circuit

shown in figure 9, where the equivalent circuits of the

individual transitions are enclosed by dotted rectangles. Each

of the transitions is represented by two sections of transmission

lines coupled by an ideal transformer. As shown in Appendix III,

any arbitrary, lossless, reciprocal two-port junction can be

represented by such an equivalent circuit. In figure 9 the

complex amplitudes a and b, as given by eq (2.ina), depend upon

the strength and the nature of the dipole radiating system and

are real if the radiator consists of a single electric dipole.

The complex amplitudes of the incident and reflected fields at

the input ends of the transformers are indicated by a^, b-j^, a',

and b' , as shown. Noting that the scattering matrix of an ideal

transformer, with an input to output turns ratio of l:n, is given

by

S =

1 + n'

^

1 - n'

2n

2n

-(1 - n-

(A4.1)

we obtain the following relations among a, b, a, , and b

b

S,. exp(-ikA ) 12a 2a

b b

where

(A4.2a)

S = 2n /(n + 1) , ''g g g

(A4.2b)

43
and we made use of the fact that the output ports of the cell are

terminated by matched loads. Noting further that

b'

= S

iig ( A4 . 3 )

we can write the following equations for a-j^ and b-, in terms of a and b by physical inspection,

-ik(.'^-z) -ik(.'^+.'^)

a^= a e + S^ib'^l^ (A4.4a)

where

-ik(£' +z) -ik(Jl' +£' ) b = be -^° + S,, a,e ^^ ^^ , (A4.4b)

J. J. JL a J.

S^^ = (1 - n^)/(l + n^) (A4.4c)

and

l\ = I + V2 L . la la ^ b b

(A4.4d)

Eliminating a^ and b^ from eqs (A4.4a) and (A4.4b) with the aid

of eq (A4.2a), we can write a and b in terms of a and b as

'^'

ik(ii' +1 -z)

1 la 2a

e 12a

'lib -^"^'^Ib-Sb^^'

- e

S

12b

1 ^"^'^Ib+^Sb^^'

e 12b

b

V ;

(A4.5a)

44
which upon inversion gives

\ ik(£'^+£2b+z)

e

a 12b

11a la 2a

-:::; e

'12a

^llb -i^^^ib-^2b-'^)

s ^ ^12b

1 i^(na-^^2a-^)

12a

^b^

(A4.5b)

where

1 r la lb 2a 2b

A = ;;— {e

12a 12b

^lla^llb

2a 2b la lb i e } .

(A4.5c)

For the case of an electric dipole radiator, a and b are equal

and the ratio of the complex amplitudes at the positive port with

the radiator at z = +g and z = -g is given from eq (A4.5b) as

a =

ik(«' +g)

~ e + S e a(z=+q) _

^

''lib ^ -ik(£^^+g)

/ \ ik(JiT. -g) a(z=-g)

e ^^ + S e ^^^^

-ik(£|j^-g)

.i2kg ^ -i2k£'^

^llb^ -i2kg

^llb^

(A4.6)

Using this relation, we can write

a e

r i2kg ^ j^ i2kg , a e ^ - 1 r

-i2kJl' = [cos(2kg) + S^^j^ e ]/i sin(2kg) ,

45
and using eqs (A4.4c) and (A4.4d), we can obtain the solution for

nj^ and ii , , in the following form:

1 - n^ -i2kJl,. a e^^'^^ + 1 . e

^ = {i{~ ^ot; ) sin(2kq) - cos (2kg)} e^^^" . 1,2 ~ i2kg , 1 + n,b ae^-1 r (^4.7)

In a similar manner, we start with the ratio b = b(-g)/b{+g) and

obtain

1 - n^ -i2kJl, b e "^ + 1 ., 1 e ^^

= {i{^ jj^ ) sin(2kg) - cos ( 2kg) }e^^^.

1 + n

a be^-1 r

( A4 . 8 )

46
APPENDIX V.

THE RELATION BETWEEN THE CASCADING PARAMETERS OF A TEM CELL

AND THE EQUIVALENT CIRCUIT PARAMETERS, i and S, , OF THE INDIVIDUAL TRANSITIONS^

A TEM cell with mismatched transitions may be represented by

the equivalent circuit shown in figure 9, which is essentially

made of two dissimilar networks, of the type shown in figure 8,

cascaded in reverse order. Hence, the cascading matrix of the

entire cell is given by [see eq (A3. 3)]

R = R^ F^ (AS. la)

where

R, =

and

R =

b 2n.

a 2n

, 2, -^'^'^VSb'

(1+n^) e b

, 2, i'^<^2b-'lb'

-d-n ) e D

, 2, -^'^'Sa*^2a'

(1+n ) e

a

,, 2, ^'^'na-^2a'

(1-n ) e

a

-(1-n ) e b

, 2, i'^'Sb^^2b'

(1+n^) e b

ik(£ -£' ) >

2 2a la

(1-n ) e

a

(1+n ) e la 2a

a

(A5.1b)

(A5.1C)

We note that R, is not obtained from R by simple interchange

a of the subscripts a and b. This is because of the reverse order

in which the parameters ^i^f ^2b' ^^^ "^b P^^^sent themselves

in the equivalent circuit as opposed to the parameters ^ia'

Jl2af si^d ^g^^ Taking the product R = R, R , we obtain

47
"u = IhV [(l+"a>(l+n^) exp {-mL' + i^^)]

-(l-n^)(l-n^) exp { ik( L'-Jl^^ ) } 1 / (A5.2a)

R 22 = 4fnr [(l+n2)(H.n2) exp {ik(L'+« )] a b

-(l-n^)(l-n^) exp { -ik (L'-Jl^g ) } 1 , (A5.2b)

and

R

12 = «21 = 4^ [(l-n2)(l+n2) exp {-ikCL'-l^^ )

}

a b

- (l+n^)(l-n^) exp ( ik( L'+Jl2^ ) } ] , (A5.2c)

where

^" = ha * hb^ ^ '

X*^j "~ ^ *-\ ^ ^-^i_ / 2d 2a 2b

^2s ^2a ^2b

(A5.2d)

(A5.2e)

(A5.2f

)

Denoting

"ll

= 2n^n^ R^(R^^) = n^n^^tR^^ + R^j) (A5.3a)

^n = 2n^n^ I^(R^^) = n^Hj^CRj^ - R^^) / i (A5.3b)

"l2 = ^n^nj, R^(R^2' = "a"b"'l2 * '*2l' "^^-^'^^

^2 = 2n^n^ I^(R^2' = "a"b"'l2 " "21' ^ ^ ' '^5. 3d)

48
we can establish the following matrix equations in a straightforward manner:

/^ 2 2

-(1+n n ) sin kL'

a b

2 2

-(n +n, ) cos kL'

a b

2 2

(n +n ) cos kL

a b

2 2

-( l+n n, ) sin kL'

a b

r

J

sin(kJl ) 2s

u

11

cos(k£ ) 2s V ^1

(A5.4)

2 2 2 2 (1-n n, ) sin(kL') -(n -n, ) cos(kL') ^ ab ab

2 2 -(n -n. ) cos ( kL' ) a b -d-n^n^) sin(kL')

sin(kJi2^ )

cos (k^pj )

> < > "l2

/

'hi

(A5.5)

which have the solution:

sin(kJl ) 2s

cos(kil ) 2s

2 2 2 2 -d+n n ) sin(kL') -(n +n^) cos(kL')

a b

si 2 2

(n +n ) cos(kL'

)

a b

a b

2 2

-(l+n n, ) sin(kL' ) a b

U.11

(A5.6a)

sin(kJl2(^)

cos(kJl„ ,

)

2 2 2 2 -(1-n n ' ) sin(kL') (n -n " ) cos(kL') ab ab

2 2 2 2

(n^ab-n, ) cos(kL') ^ (1-n n, ) sin(kL') ab

J

TI 12

^12

(A5.6b)

where

22 2 2

2 22 2

A sab = (n +n, ) cos kL' + (l+n n, ) sm kL' ,

' a b ' (A5.6c)

49
2

A, = -(1- n^n^) sin^ kL' - (n^-n^) cos^ kL' . (A5.6d) d a b a b

The above eqs (A5.6a) to (A5.6d) uniquely determine the values of

i ^ and ^o^f which in turn will give the parameters

£_ and Ji^u

.

2a 2b

I

50
o

ffi C

fO

4-1 OE

10

S-i DfO

0)

•H

51
E"-«— J//" ^E

C

________

J

^ ___________= ^^^_

I z =0

Figure 2. An arbitrary current filament in a uniform cylindrical

guide

52
^^ 1 1 X +IX

wo

•H

4J • W w

•H (U u u

Q) ^

+J 3 U

fC

m

U

fd a; jC -h u

Cl. c •H

-D •H

-u

u

fD

H

•H -U T3 <D

fC

C

5^ CP

(T3 cp e c

•H ^ c

•H \fc T3 ^ C

0) fC +J

0) o

t:; •H

!^ ^ 4J o O

<4-l CD 1—

a. d' a

jj r—

<D ^ W fOe

+J

w

w

(U U-l E^ua

•H

53
z, z'

^

X

Figure 4a, One orientation of the equipment under test (EUT) with

respect to x-, y-, z-axes of the cell.

54
z,x'

Figure 4b. Another orientation of the EUT with respect to the

cell axes.

55
z, r

»• X

Figure 4c. A third orientation of the ETIT with respect to the

cell axes.

56
E <V O O . oj CN >

<v

O i-H • QJ

&: c E- CM• I—

iH (0HX

(0o o o

•H 3 >

<D 4J C U

<u

H 5^ a O

<D •H

TOI

^1

TO (D TO e • e cc ^ >,CC

CJ TO 2

if)

57
5" EHP-CL Absorber

8" EHP-CL Absorber

Figure 6. Absorber loading in the TEM cell shown in figure 5.

58
E

>

^

14

o|2

O

UJ

(/)

Q

-I

UJ

8-

^

6

Q

Ld

M

Theoretical distribution corresponding to unloaded

cell.

rieasured distribution in the loaded cell (dotted part

of this curve is extrapolated )

Fgure 7

99.2 cm

K-IZOcmH

30

y cm

Normalized field strength distribution in the trm cell

shown in figure 5.

59
\-iz-M

I

Figure 8. Scattering matrix representation of a cell with

mismatched transitions.

60
a:o

I-o

UJ

fLUQQ

UJXo' -o Q I- < <

UJXo

UJ

-I q:

-pK

Eh

fOoco

•H

4J

(C 4Jcm

0)^ • a wco

+J 4J -H -H D CO u c H Sj U 4J

-PC> e

<TiU3

•H

Cm

61
NBS-114A (REV. 9-78)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS TN-1017

4. TITLE AND SUBTITLE

Characterization of Electrically Small Radiating

Sources by Tests Inside a Transmission Line Cell

5. Publication Date February 1980

ization Code

7. AUTHOR(S)

Ippalapalli Sreenivasiah, David C. Chang, and Mark T, Ma

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

^Btolt^it m.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS rsfreef. City, state. ziP)

Same as I tern 9

13. Type of Report & Period Covereii

14. Sponsoring Afjen^y Code

15. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, F I PS Software Summary, is attached.

16. ABSTRACT (A 200-word or teas factual summary ol moat significant information. If document includes a significant bibliography or

literature survey, mention it here.) An electrically small radiating source of arbitrary nature may be modeled by an equivalent dipole system consisting of three orthogonal electric dipoles and three

orthogonal magnetic dipoles, each excited with arbitrary phase. An experimental

procedure for determining the emission characteristics of such an equivalent dipole

system by tests inside a single-mode transmission line cell is described in this

report, followed by some experimental results.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Electrical dipole; electrically small radiators, magnetic dipole, TEM cell.

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Penalty for Private Use. S300

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE

COM-215

SPECIAL FOURTH-CLASS RATE BOOK