co.

VNAT'L INST. OF STAND & TECH

A111DS TbfiVflE

\

NBS TECHNICAL NOTE 1064

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

Uncertainties in Extracting

Radiation Parameters for anUnknown Interference SourceBased on Power and Phase

Measurements

NATIONAL BUREAU OF STANDARDS

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Uncertainties in Extracting

Radiation Parameters for anUnknown Interference SourceBased on Power and PhaseMeasurements

N3SRESEARCH

WONce;

QcaOS

\1<3

Mark T. MaGalen H. Koepke

Electromagentic Fields Division

National Engineering LaboratoryNational Bureau of StandardsBoulder, Colorado 80303

&*-**

\e

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued June 1983

National Bureau of Standards Technical Note 1064

Natl. Bur. Stand. (U.S.), Tech. Note 1064, 48 pages (June 1983)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICEWASHINGTON: 1983

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

Price $3.75

(Add 25 percent for other than U.S. mailing)

CONTENTS

Page

1. Introduction 1

2. Derivation of Potential Errors 9

3. A Simulated Example with Assumed Unbiased Measurement Errors 18

4. Another Simulated Example with Assumed Biased Measurement Errors.. 26

5. Sensitivity of Extracted Parameters with Respect to MeasurementAccuracy 32

6. Concluding Remarks 35

7

.

References 37

m

Uncertainties in Extracting Radiation Parameters for an

Unknown Interference Source Based on Power and Phase Measurements

Mark T. Ma and Galen H. Koepke

Electromagnetic Fields DivisionNational Bureau of Standards

Boulder, Colorado 80303

A method for determining the radiation characteristics of a

leaking interference source has been reported in a previous publi-cation [1], in which the unintentional, electrically small leakagesource was modeled by two vectors representing a combination ofequivalent electric and magnetic dipole moments. An experimentalsetup, measurement procedures, and the necessary theoretical basiswere all described therein to explain how the relevant sourceparameters can be extracted from the measurement data of outputpowers and phases taken when the interference source is placedinside a transverse electromagnetic (TEM) cell. Simulated exam-ples were also given to show that the equivalent source parametersof unknown vector dipole moments and, thus, the detailed radiationpattern and the total power radiated by the source in free spacecould be uniquely determined by the method if the measurement datawere not contaminated by noise. This report presents the mathe-matical analysis of the uncertainties in the final, extractedresults when the experimental data are degraded by the backgroundnoise and measurement inaccuracies.

Key words: dipole moments, electrically small source, erroranalysis, interference sources, phase measurements, power measure-ments, radiation pattern, TEM cell, total radiated power, uncer-tainties.

1. Introduction

It was shown in previous reports [1,2] that a general unknown leakage

interference source may be characterized by a composite set of parameters

representing both the equivalent vector electric and magnetic dipole moments

as displayed in figure 1. Necessary theoretical justifications were also

given therein to demonstrate that these parameters of twelve unknowns (six

dipole moment amplitudes and six dipole moment phases) may be determined, with

the experimental setup shown in figure 2 when the unknown source is placed

inside a TEM cell, by taking the output sum and difference powers and the

relative phases between the sum and difference ports for six different source

orientations. The corresponding radiation pattern and the total power radi-

ated by the source in free space can also be computed. Naturally, when the

experimental power and phase readings are not contaminated by noise or other

inaccuracies, the source parameters so extracted and the radiation character-

istics so computed are accurate and unique. In the practical world, however,

the experimental data are always degraded somewhat by the background noise,

equipment limitations, and the reading accuracy. The question is then: how

will these measurement imperfections affect the final results of source param-

eters? This report is intended to answer this question by giving necessary

mathematical derivations for, and performing the analysis of, these uncertain-

ties.

For the convenience of presenting the material, all the equations

required for this report are reproduced as follows with the detailed deriva-

tions and related measurement procedures referred to the previous report [1]:

"4 = ( psl

+ Ps2 " P

s3 - Ps4

+ Ps5

+ Ps6>/<2q

2) U>

4y - (Psl

+ Ps2

+ Ps3

+ Ps4 " P

s5 " Ps6>/<

2<2

> < 2 >

4 " (" Psl " P

s2+ P

s3+ P

s4+ P

s5+ Ps6>/f 2q

2) < 3 >

mexmeycose

el= (P

sl- P

s2 )/(2q2

) (4)

meymez

cosee2

= (Ps3 " p

S4)/(2q2

) (5)

m.,2

ezmex

cosee3 = {P

s5 " Ps6^ (2(^V (6)

eel ~ ^ex " ^ey »

ee2E

^ey " ^ez ee3=

*ez " ^ex > {7)

"4 = < Pdl+ Pd2 - Pd3 " P

d4+ Pd5

+ Pd6>/< 2kV> (8)

"my " < Pdl+ P

d2+ P

d3+ P

d4 " Pd5 " Pd6>/< 2kV) W

mmz = <"Pdl " p

d2+ P

d3+ P

d4+ P

d5+ P

d6>/< 2kV> (10)

"W%y cos0ml = (P

d2 " pdl>/( 2kV> UD

mmymmz cos9m2 = (p d4 " pd3

)/(2k2cl2 > (12)

"WW cosem3 " (P

d6 " pd5>/< 2k2^ < 13 >

eml ~ ^mx " V 9m2

E V " ^mz >em3

E%\z " *mx

* (14)

Here mex , m , and m

ez are the x-, y-, and z-components of the amplitude of

the equivalent vector electric dipole moment; cj;

ex , ^ey>and (\>ez

are the

associated phases of the electric dipole moment; mmx , n^ , and mmz are the x-,

y- 5 and z-components of the amplitude of the equivalent vector magnetic dipole

moment; (j^mx , <\> , and 4>mz are the associated phases of the magnetic dipole

moment; PS1-, i = 1, 2, . . .,6 are the measured sum powers at the required

six different source orientations; Pdi , i = 1, 2, . . ., 6 are the measured

difference powers at the same six source orientations; q is the normalized

amplitude of the vertical electric field which would exist at the center of an

empty TEM cell when it is operated in a receiving mode and is excited by an

input power of one watt; and k is the wave number.

In obtaining (1) through (13), we have assumed that the TEM cell is made

of perfectly conducting material, symmetric, and impedance-matched at 50 Q so

that there exists no horizontal component of the electric field at the center

of a TEM cell. The frequency of the unknown interference source to be

detected by the spectrum analyzer included in the measurement instrumentation

as shown in figure 2 is assumed to allow only propagation of the dominant mode

inside a given TEM cell chosen for this study [2]. The size of the interfer-

ence source to be tested is also required to be small relative to the test

volume of the cell in order to minimize the potential perturbation to the

field distribution inside the cell [3].

Under the above conditions, the amplitudes of all the dipole moment com-

ponents, and the relative phases among the dipole moment components of the

same kind can be extracted based on the power (sum and difference) measure-

ments alone [2]. The relative phases among the dipole moment components of

the mixed kinds are still undetermined. This is the reason why measurements

of relative phases between the sum and difference signals are needed.

Before summarizing the equations for phase information, it is instructive

to examine the dependence of the power pattern, P(0,<t>), and the total power,

P-r, radiated by the source in free space on the source parameters. The

expressions for computing P(e,<{>) and Pj are [1, 2]:

P(9,<t>) = -£-£ [(m2x

+ k2 m2

x ) (cos 9 cos <j> + sin 2 <j>)

+ (m2 + k

2 m2 ) (cos e sin2* + cos *) + (m

2z

+ k2 m2

z ) sin2 e

- 2{mex

mey cosUex - <\>ey ) + k2mmxmmy cos(c|^

x- c|^

y)} sin

2G sin* cos*

- 2{meymez cos(<j,ey

- i\,

ez ) + k2mmymmz cosUmy - <|>mz )} sine cose sin*

- 2{mezmex cos(cpez

- <\>ex ) + k2mmzmmx cos((^

z- c}^

x )} sine cose cos<|>

+ 2k{mexmmy sin(cj,ex

- ipmy ) - meymmx s1n(cpey

- <^x )> cose

+ 2k{meymmz sinUey - <\^z ) - mez

n^y

sin(<pez- <\^y

)} sine cos*

+ 2k{mezmmx sin(cp

ez- cpmx ) - m

exmmz s1nUex - c|,mz )} sine sin*] . (15)

PT = J P(e,*)dQ

T^ (mex

+ <y + 4 + k * « mmx+"my

+ 4» (16)

where (0,<t>) are the spherical coordinates measured from the source center, and

\ is the wavelength.

Note that if one is interested only in determining the total power radi-

ated by the source, which may be the case in practice, all we need to extract

are m^- and m2^-, i = 1, 2, 3 from (1), (2), (3), (8), (9), and (10) based on

the measured powers. It is the radiation pattern for which the relative

phases are required. In particular, for the last three lines in (15), we also

need the relative phases among the mixed components, i|>ex- c^ , <j>e

- <j^x, <j>ey

- 4>mz , <\>

ez- 4> , (j,ez

- <j;mx , and cj;

ex-

<|>mz , which are related to some of the

quantities given in (1) through (14) and some of the measured phases <|>«, i =

1, 2, . . ., 6 between the sum and difference ports as follows:

*ex - V =*1

+ al

" h < 17 >

^ey " *mx=

*ex " % " eel " e

ml »(18)

*ey " V =*3

+ a3 " ?3 < 19 >

^ez " V =^ey " ^mz " e

e2 " em2 >(20)

*ez " ^mx=

^5+ a

5 " ^5 (21)

^ex " *mz=

^ez " *mx " ee3 " e

m3 '(22)

where

. m sine1

. mmv cosem1 - mmwpi

= tan (Jb^' ; (24)mx ml

. m sine9

°3 tan Cr t\ die ) (25)ey ez eZ

., mmw cosem9 - mm,

P3

"^\^ * •

and

1m sine ~

"5= ta" (iZ I mp> cose, J (27)

ez ex e3

. m cose , - mmvp 5 f-'i-Viwr) •

(28)mz m3

Alternatively, eqs (17), (19), and (21) may be replaced respectively by:

^ex " ^my=

^2+ a

2 ' p2 » * 29 )

*ey " ^mz=

^4+ a

4 " ^4 ' ^ 30 ^

and

*ez " *mx=

<>6+ a

6 " ^6 • (31)

where

. -m sine .

p = tan-1

!

mxm

™L W) , (33)

2 "V sineml

and

1-m sine

aA = tan_i

(- ^ -||—) , (34)4 ro_w - m «, cose ~ y

ey ez e2

. -m cose „ - m„ + , -If my mZ mz>, /01- x

P4= tan

( -mmv sine. ){35)

my m2

- -m sine ,as = tan (= ^ ^H—) . (36)6 ^m

ez- m

excose

e3^

- -m cose ,, - ma - -i-„r,-lr mz m3 mx> ,~,

x

h " tanC -m sine. )

' (37)

mz mi

It should be noted that, in principle, either (17) or (29) is required

for computing the mixed phase difference <j;ex- 4^ . They both should probably

be computed for the benefit of checking data consistency. The same note

applies for (19), (30), (21), and (31).

In addition, the possible sign ambiguity for eei

- and emi

- , i = 1, 2, 3

obtained from (4), (5), (6), (11), (12), and (13) may be resolved by the

obvious constraints,

3 3

I e . =, I e . = (38)

i=iei

i=imi

and the following special relationships:

sinem

. = M./N. , sinee

. = m./n. (39)

where

Ml

" 2(mmx - m

my ) mexmey

sin9el

+ (mex " m

ey} (m

mx " mmy } tan(

*l" *2 }

(40)

Nl

=" 2(m

ex " mey

} %mmy

+ 4mexmeyVm

mytan(

*l " V sin9el •

{41)

ml

= 2(mex " m

ey) mmx% sin9

ml+ {m

ex " mey

} (mmx " m

my ) tan(*l

" V(42)

nl

=" 2(V " % ] m

exmey

+ 4mex

meyVmmy

tan(*l " V sin9

ml •(43)

M2

= 2(mL " mmz ) m

eymez

sin9e2

+ (mey " m

ez} (m

my " mmz

} tan(*3 " *4 }

•

(44)

2 2N = - 2(m -m)mm +4mmmm tan(d> - a.) sine , (45)2 ey ez my mz ey ez my mz \3 M e2 '

m2 ' 2(n,

ey" m

ez) %V sine

m2+ (ln

ey " mez>

imly " mL' tan(

*3 " »4> '

(46)

n2

=" 2(m

my " 4' mey

ffl

ez+ 4m

eymezmn,yV tan(

*3 " V sin(W '(47)

M3

= 2(mL " mmx>

mezmex

sinee3

+ (mez " mL' (n4 ' m

mx>tan(

*5 " +6>

(48)

N3 - - 2(>4 - m^) Vm

mx VAlx tan( *5 " »6> sinee3 •

(49)

ffl

3= 2(m

ez - mex> Vm

mxs1n9

m3+ (l4 " m

ex>(mmz " "4' tan ( *5 " *6 ]

and

(50)

2 2n~ = - 2(ni - m) ni ni + 4 ni m m m tan (<j)c - d>c ) sine . (51)3 mz mx ez ex ez ex mz mx y

5y6 m3

In summary, eqs (17) through (28) constitute the complete set of expres-

sions for extracting the necessary relative phases required for computing the

detailed radiation pattern. Equations (29) through (37) serve as alternatives

for checking or averaging purposes. Equations (38) through (51) are necessary

for resolving the sign-ambiguity problem for eei

- and emi

-, i = 1, 2, and 3.

2. Derivation of Potential Errors

Examination of (1) reveals that when all the six sum-power readings are

not contaminated by measurement imperfections, the quantity, mex , so computed

contains no error. On the other hand, when the measurements are not perfect

due to the background noise, equipment imperfections, measurement environment,

or even human errors, the result of mexwill also be in error. To perform a

quantitative analysis of potential errors, we denote the measured values and

the parameters computed therefrom with the primes such as

Psi

= Psi

+ dP$i , i = 1, 2, . . ., 6 (52)

where PS1

- represents the true value, and dP$1

- is the small error contributed

by the measurement system.

For the general case, we also assume that the actual vertical electric

field, q, inside the TEM cell be changed to q' by a perturbation dq due to the

presence of source equipment placed inside the cell, and possible construction

defect and imperfect impedance matching of the cell,

q' = q + dq . (53)

We then have, in accordance with (1),

mexx " < P sl

+ Ps2 " P

s3 - Ps4

+ Ps5

+ ps6^ 2 <l'2

>

a m2x

+ (dPsl

+ dPs2

- dPs3

- dPs4

+ dPs5

+ dPs6

)/(2q'2

)

- {Psl

+ Ps2 " P

s3 " Ps4

+ Ps5

+ Ps6 ) W<? > (54)

or

dmex .

dPsl

+ dPs2 - dP

s3- dP

s4+ dP

s5+ dP

s6 do

^e7~=

2(Psl

+ Ps2 " P

s3 " Ps4

+ Ps5

+ Ps6>

~^'

m;x> , (55)

which gives the percentage error in the source parameter m'excomputed directly

from the measurement data of sum powers P^- involving imperfections dP$1-, i =

X) u j • • • j O •

In (54) and (55), we have neglected the second-order terms such as

(dmex ) , (dq)(dmex ), etc., for the simplifying purpose. This practice is

certainly justified if the first-order errors, dmex , dq, etc., are small.

When the situation warrants, the second-order terms can be included to improve

the error analysis.

Note from the first expression in (54) that m^x

may be negative even

though the true value m2x

is always nonnegative. This can occur only when

dps3

+ dps4

1S relatively large and mex

almost vanishes. Under this condi-

tion, we just assume that dmex/nig X= 1, giving m

ex= 0.

9 9 9 9 9The above analysis also applies for m|

y, m|

z, m^

x , m^ , and mmz

respectively in (2), (3), (8), (9), and (10). For mj^, i = 1, 2, 3, we may

reasonably assume dk = implying no perturbation in frequency. Therefore, we

also have

10

dm«w dP*-i

+ dP oO + dP .-3+ dP

./i " dP .c ~ dP .c Aey si s2 s3 s4 s5 s6 dq, cc \W 27P

-1—T~P—TT1—TT1—^"P 1—

- P' ) 7r >lbb 'm

ey^ lf

slKs2

Ks3

Ks4

Ks5

Ks6' q

dmrt , - dp. - dP e9 + dP - + dP . + dP . + dP c .

ez si s2 s3 s4 s5 s6 dq / c -,v

W 2(- P'—^~P"

1—+T1—+~P~1—+~P~I—+ P' \ cT »

lb/ 'mez

^ l Ksl

Ks2

Ks3

Ks4

Ks5

Ks6

] q

dmmx

dPdl

+ dPd2 - dP

d3 - dPd4

+ dPd5 + dP

d6 dq

™r~ =

2(P' + P* - P' - P' + P' + P' ) " q7"

mx dl d2 d3 d4 d5 d6 M

and

dmmz

;

dPdl

- dPd2

+ dPd3

+ dPd4

+ dPd5 * dP

d6 dq

^T" Z( - pdl- pd2

+Pd3

+Pd4

+Pd5

+Pd6

) "^

(58)

dm dP ., + dP , + dP .- + dP .. - dP ._ - dP ,cmy dl d2 d3 d4 d5 d6 dq, cr.,W~ =

2(p' + p' + P' + P' - P' - P' ) q* (59)my K

dlKd2 d3 d4 d5 d6 H

(60)

The predicted values for mei

- and m^- , i = x, y, z, then become

mei

= m^- - dmei

= m^(l - dm^/m^)

and (61)

"mi=

"4l " dmmi = "^i^ 1 " dm

mi/n

\!ii) '

The above implies that if we have known a priori (or have some basis) to esti-

mate dP$i , dP

dl-, and dq, the percentage errors (dm

ei-/nig.. and dm^/n^) can be

evaluated to predict approximately the true values. Or, at least, these per-

centage errors can be computed as a function of assumed values of dP$1

- , dPdl-,

and dq.

If, however, there is strong belief that dPsi , dP

dl- , and dq are all truly

random, the problem is much simpler. Then, the average values of them, and

thus the average values of dmei

and dmml

- in accordance with (55) through (60)

all vanish. That is,

11

Ws

. = Wd

. = -3q = , (62)

and

mei

=^\n1

=° * (63)

We conclude that

nf". = m . and W. = m . . (64)ei ei mi mi

This implies that, if repeated sets of power measurements are taken, we can

compute the corresponding sets for m^ and m^^ . The average values should

approach the true values when all errors involved in the measurement are

random in nature.

Let us now consider an expression involving the relative phase such as

(4). Using the same notation convention and neglecting the second-order

terms, we obtain

dm dm dP , - dP

where

cose .

ei Ci"'ex

mey

r

si" ' s2

cose;i

(psi

- p; 2

)/2^2

m;x

mv •(66)

which can readily be computed based on the power measurements alone, and the

remaining terms in (65) can also be computed if the values for dP$1

- and dq are

either known or assumed. Taking the inverse cosines of (65) and (66) will

yield eel

and 9^ with the appropriate sign determined by (38) and (39). The

difference, deel

= 9^ - 9el , represents the error incurred in the relative

phase deduced from the power measurements.

Similarly, we have

dm dm dP - dP . .

12

and

dni dm dP c - dP ,. .

cosee3

. cose

'

3 ( 1 ^ + ^* . ?5_ p

,

s6 M), (68)

ej ej mez ex

Ks5

Ks6 q

dmmv *Lw dp ^o ~ dp ^i o^«rn ,

fl = rn^fi' M + .

mx + .

my d2 dl 2dq>, ,

fi<ncose _ coseml [1 + -r-_ + -p-

p, _

p, + -^J , (69)

mx my d2 dl ^

dm dm dP .. - dP ,~ .

cose. ~= cose' (i + "0. + "L - <* ^3 +2dq

}(70)

m2 m2 % mmz

Pd4 "

Pd3 q

dm dm dP .c - dP ... ,

a i r 1 mz . mx do db Zdq, /tincose . = cose. (1 + 7^1— + 7^ p' - p '

_+ -£) {71)

m3 " """rn3 ^ m' m' PT,. - P' q'

mz mx do d5

where

cose;2

(p;3 - p;4

>/< 2«'2

m;y

m^ >(72)

cose;3

(p;5 - p;6

>/< 2«'2

*;zm;*> • (73)

cose;i = (p

d2 - pai>/

(2k^'2mm v •

and

74)

cose;2

(pd4 - Pds'/'*

2"'

2m»f •

(75)

cos9;3

(pd6 - ^b^ 2*

2"'

2m;2

m;x ' •

(76 '

Note from (66) that the situation Icose^l > 1 may arise if dPgl

- dP$2

is relatively large and eel

nearly vanishes. In this case, we just set cose^

a 1 (or e^ = 0). Then we have

13

dP 1- dP„ dm dm « . . ._

d9el

s "el

~=^C p?

1

. pf2

- gi^ - i^ - ^)1/2

• (77)el el p

slps2

mex

mey q

Similarly, we have

d9 - e- g f

"Ps3 - dP

s4dm

ey *ez 24, ,1/2 „.,d9e2

9e2 - ^ ( P;

3- P;

4 ^ ^ "^ '' '

rffl fl- ^7 f

dPs5 - dP

s6dm

ezdm

ex 2^ 1/2, 70 .

dee3 " 9

e3 - ^ t p ' - P' ' HT~ ' HT ?]'

(79 >

so so ez ex ^

dP AO - dP .. dni drn OA . /0dV - V - "hfvrf "i?f -ifif "^ •

,80)d2 dl mx my

dP ., - dP .-, dm dm «. . /0

ha -ft

- /T fd4

.

d3 W .mz 2dqa/2 , ftndGm2 " " 9

m2 " /2t P' - P' ' <T " ^7 " "^j ' (81)

d4 do my mz n

and

^PjG ~ ^^C ^m, dmmv O,4o 1 IOde. - . § . /? (

d6 d5- "« - ™* - m.) 1/z

. (82)m3 m3 P

d6 " Pd5

mmz

mmx q

Thus, eqs (65) through (76) provide the necessary means for estimating

deei

- and demi

- when e^- and e^. computed from the power measurements are not

near zero. Equations (77) through (82) serve the same purpose when eei

- =

and 9^. = 0, with the values for dP$1

- , dPdl

- , and dq known or assumed. On the

other hand, if dPsl-, dP^ and dq are random, we see that He .

a '^RLi = 0>

yielding IT. = e . and el. = 9 . .3ei ei mi mi

Corresponding error expressions for the relative mixed phases, (17)

through (22), can also be derived. Since they involve taking the derivatives

14

of inverse tangent functions, the final forms are quite complicated. For this

reason, they are not presented here. One alternative but simple method of

achieving these is to (i) compute the primed quantities based on the measure-

ment data, (ii) compute the unprimed quantities based on the predicted ampli-

tudes of the component dipole moments and the predicted relative phases among

the same kind, which have already been obtained, and then (iii) take the dif-

ference between the results derived above.

Consider (17) first. Initially, we have the computed value for ^ x- <j^y

derived from the measurements,

*ex " *my = *[+ \ ' Pj * < 83 >

where <j>' represents the actual measured relative phase between the sum and

difference signals for one of the measurement positions of the interference

source, and a1

and p' can be calculated respectively in accordance with (23)

and (24),

. m' sine'

"'. tan (< 'Vcoe') • < 84 >

1 ex ey el

. m' cose1

,- m'

,

o;"--'(-^ sis -

*) •

(85)

1 mx ml

Of course, all the primed quantities in (84) and (85) are based on the

measured data, thus, containing some measurement errors.

The error involved in <j,g X- <\>^y

may be expressed as

d((|;ex " V )=

d<J)l

+ dal " df3

l* (86)

where d<h represents the actual measurement error in the relative phase for

the first source position, do^ = a' - c^ , dp^ = p' -pj_

with a^ and p^ to be

15

computed also from (23) and (24) but with the unprimed quantities deduced in

(61), (65), and (69).

Note that the phase error d<j>^ may be estimated (depending on the type of

equipment involved in the measurement system) or assumed for computational

purpose. The predicted true value for <\>ex- <j> then becomes

^ex " V =^ex " ^my " d{(

^ex " W *{87)

Substituting (87) into (18), we also have <j>ey- (|>mx

.

Similarly, from (19) and (21), we obtain

*ey " C = *3+ a

3 " ^3 (88)

and

^ez " C =4>5

+ a5 - ?5 •

< 89 >

where <|>g and <$>§ are the actual measured phases between the sum and difference

signals for two other measurement positions of the source,

. m' sine'

"3= ^ (ST-tV^%) . (90)

ey ez e2

. m' cose' - m'

my m2

. m' sine'^

•sta"

~\ 7?» < cole; ) •

(92)

ez ex e3

and

16

. m' cose' - m'

h ' tant m - sine - )

• (93)mz m3

The errors involved in these phases are:

d(^ey " W = d(})

3+ da

3 " dh » ( 94 >

d(*ez ' W =d,|)

5+ da

5 " d^5 ' (95)

da3

= a3

- a3

, d£3

= P3- P3 , (96)

and

da5

= a5 " a

5 ' % =^5 " ^5 ' (97 ^

where d$3

and d<t>g represent the phase errors in the measurements, and a3 , p3 ,

ag, P5 may be computed from (25) through (28) with the predicted values for

mei

-, m^ , 9e2,

and em2 (unprimed) deduced from (61), (67), (68), (70), and

(71).

The predicted true values for the mixed phases being considered then

become

^ey " *mz=

^ey " ^mz " d((^ey " W (98)

and

*ez - ^mx = *ez " C " d(<^ez " ^mx ) (99)

The phases deduced above can then be used in (20) and (22) to obtain <j>ez- (j,

and ^ex " *mz*

17

Alternatively, eqs (87) (98), and (99) can also be derived by starting

from (29), (30), and (31) based on the measured phases $£, <^, and <j>

6. Either

the mixed phases obtained here or those obtained in (87), (98), and (99), or

the average values between them may be inserted in (15) to compute the pre-

dicted power pattern, which may be compared with that computed based on the

primed values to reveal the final error in power pattern.

Of course, we may also conclude that, when repeated measurements of P ' •

,

P^. , and <j>| are available and dP$i

, dPdi , and d^ are believed to be random,

dT1

- <\>' = <\> - (\> , 4J1

- <b' = cb -ib (100)vex my

Yex

vmy' Tey mtix

yey mtix ^ '

471

- <b' = d, - <b ,dT - d/ = ib - ib (101)v

eyYmz Yey Ymz '

Tez Tmy Yez Miiyu '

and

<y - V = <\> - <\> » <K - V = <b - <b . (102)ez mx ez mtix ^ex vmz Yex Ymz v

Under this condition, the average power pattern will approach the true power

pattern.

3. A Simulated Example with Assumed Unbiased Measurement Errors

For the purpose of checking the formulation and derivation, an ideal

simulated example was given in the previous report [1], where an unknown

arbitrary interference source was assumed to be characterized by the equiva-

lent electric and magnetic dipole moments with the following values:

mex= 1.4, m

ey=1.8, m

ez= 1.6, (b

ex= 0, ^ey = 80 > *ez = 60,

mmx= °' 8 >

"to= °* 6 ' %z = °- 4 > *mx " "80 > % = -60, *mz = -45 '

(103)

where the normalized amplitude of the electric dipole moment is in meters,

that of the magnetic dipole moment is in meters-squared, and the phase is in

degrees. The value used for the vertical electric field existing inside the

18

particular TEM cell considered therein was q = 11.83 V/m, and the frequency

was 30 MHz. With the source parameters assumed in (103), we computed in [1]

that the sum and difference powers in watts and the relative sum-to-difference

phases in degrees should be:

si

's4

425.107856

27.106038

P s2= 302.626424 ,

Ps5

= 473.027282 ,

S3

's6

784.597583 ,

159.541746 ;

'dl

*d4

2.704333 ,

27.172985 ,

'd2

*d5

52.545279

7.617338

'd3

*d6

1.556813 ,

36.582351 ; (104)

<t> 1= -103.0261

d)/i = 105.5593

<|>2 = -77.0300

<t)5= 48.116

4>3 = -113.5502

<j>6= 91.9132

In the previous report [1], we also demonstrated that if the values in (104)

were the actual measurement results, we would recover uniquely the source

parameters exactly as those given in (103). In this case, of course, there

would be no error involved.

Since this is an example aimed for analyzing the final errors in the

extracted source parameters for given errors in the simulated "measurement

data," we treat the problem here in a deterministic manner. The averaging

technique, by taking repeated measurements, is useful only when we are sure

that the measurement errors are indeed random for the equipment involved in

the measurement system.

Now, let us consider the problem from the practical point of view.

Suppose that the hardware in the measurement system is not capable of taking

perfect readings of the outputs accurate to the fourth or sixth digit after

the decimal point as demanded in (104). Instead, all the readings are only

accurate to the second digit after the decimal point. In addition, suppose

that the combined effect of background noise, equipment imperfections, and

human factors also degrades the readings somewjiat such that the actual

measured powers in watts and phases in degrees are as follows:

19

psl

p s2=

ps3

=

PU =

p s5"

p s6=

pdl

=

p d2=

Pd3=

Pd4

=

Pd5

=

pd6

=

i<t>2

=

*3=

*5

446.36

287.50

745.37

28.46

496.68

151.56

2.57

49.92

1.63

28.53

8.00

34.75

- 92.72

69.33

-124.91

95.00

43.30

(increasing

(decreasing

(decreasing

(increasing

(increasing

(decreasing

(decreasing

(decreasing

(increasing

(increasing

(increasing

(decreasing

(increasing

(increasing

(decreasing

(decreasing

(decreasing

approximately by 5%)

approximately by 5%)

approximately by 5%)

approximately by 5%)

approximately by 5%)

approximately by 5%)

approximately by 5%)

approximately by 5%)

approximately by 5%)

approximately by 5%)

approximately by 5%)

approximately by 5%)

approximately by 10%

approximately by 10%

approximately by 10%

approximately by 10%

approximately by 10%

and

(j>g = 101.10 (increasing approximately by 10%)

(105)

For simplifying the problem, we also assume that there is no perturbation

in the vertical electric field inside the TEM cell. That is, dq = 0, q' = q =

11.83 V/m. We then obtain

m

m

m

ex

2ey,2ez,2

2mym

2.1732 ,

3.0706 ,

2.4588 ,

0.5890 ,

0.3611 ,

0.1848 ,

mex= 1.474 ,

mey

= 1.752 ,

mez

= 1.568 ,

Kx = 0.767 ,

v - 0.601 ,

<z = 0.430 ;

20

dmex/nig

X= 0.050 (meaning an error of +5% in m^

xdeduced above),

dmey/

mey= -°- 026 (meaning an error of -2.6% in m' deduced above),

dmez/nig

Z= -0.019 (meaning an error of -1.9% in m^

zdeduced above),

dmmx/mlJ1x= -0.042 (meaning an error of -4.2% in m

fJlxdeduced above),

dmmy/m'= 0.003 (meaning an error of +0.3% in m^ deduced above),

dmmz/mIJ1z= 0.068 (meaning an error of +6.8% in m^ deduced above);

mex (predicted true value) = m^x (l

- dmex

/nigX )

= 1.400 (practically no error compared with the true value of 1.4),

mey= 1.798 (small error compared with the true value of 1.8),

mez= 1.598 (small error compared with the true value of 1.6), (106)

mmx= 0.799 (small error compared with the true value of 0.8),

n^ = 0.599 (small error compared with the true value of 0.6),

and

mmz= 0.401 (small error compared with the true value of 0.4).

Thus, the method presented does predict very good true values for the

amplitudes of the component dipole moments as long as the arbitrary errors in

power measurements are within 5% of the actual readings. Results with other

percent errors in power readings are presented in figure 3a, b, where the

manner with which the errors occur is assumed to remain the same as that in

(105), and the second-order correction terms are also excluded. From figure

3a, we see that, even with the power measurement error as large as 40% and

with the second-order correction terms neglected, the predicted value for mey

is still about 1.68 vs. 1.80 for the true value, a decrease of only 6.67%.

From (66), (72) through (76), and the constraints given in (38) and (39),

we obtain approximately,

cose^ = 0.219778 , 9^ = -77.30°,

cos9g2

= 0.932363 , 9g2= 21.19°

,

cose^3

= 0.533491 , 9^ = 57.76°,

(107)

21

cose^ = 0.929589 , 9^ = -21.63°,

cose^ = 0.941998 , 9^ = -19.61°,

cosg^ = 0.734008 , 9^3

= 42.78° .

Since none of the above values is near zero, we use (65) and (67) through (71)

to yield

cos9el

= 0.174289 ,9el

= -79.96° (increased by 0.05%) ,

cos9e2

= 0.940726 , 9e2= 19.83° (decreased by 0.85%) ,

cos9e3

= 0.499926 , 9e3

= 60.00° (completely recovered);

(108)

cos9ml

= 0.939814 , 9ml

= -19.98° (increased by 0.1%) ,

cos9m2

= 0.961780 , 9m2

= -15.89° (decreased by 5.93%) ,

cos9m3

= 0.811744 , 9m3

= 35.73° (increased by 2.09%) .

Thus, the final errors in eei

« and 9mi

- in degrees are

d9el

= 9^ - 6el

= 2.66 , d9ml

= 9^ - eml

= -1.65

dee2

= ee2 " e

e2 = l ' 36 dem2 = em2 " em2 = " 3 ' 72 ( 109 >

dee3 = e

e3 " ee3 = " 2 ' 24

*de

m3 = em3 " e

m3 = 7 ' 05'

We, next analyze the mixed phases. From (84), (85), and (83), we have

the following results in degrees,

ol[ = -42.59, $[ = 158.39 , <|^

x- <\^ = 66.30 . (110)

Then, from (23), (24), and (86), we have (all results in degrees):

ax

= -45.94, pj_

= 150.91

giving

dax

= a1

- aj_ = 3.35 , dpj_ = pj -pj_

= 7.48

d((J;ex " W = d<1)l

+ dal " dp

l= 6 * 18

'

22

which, in turn, yields

^ex " V =^ex " 4% " d((

^ex " V }

= 60.12 (increased by 0.2%) , (111)

and

^ey " ^mx=

^ex " % " eel " eml

= 160.06 (increased by 0.04%) .

Similarly, from (90, 91, 88), (92, 93, 89), (25, 26), and (27, 28), we obtain

the following in degrees:

«3

a5

a3

i _

and

Ctc "

10.00 , p3' = 145.98 , *ey " ^mz

= 99.11 ;

27.90 , P5 = -57.10 , ^ez " ^mx= 128.30 ;

9.33 , (3 3= 133.12 ;

27.82 , p 5= -63.68 . (112)

Then,

dao = a\ - «3

~ 0.67 dp3

= p3' - p 3

= 12.86

da5

= a£ - a5

= 0.08, dp5

= p 5- p5

= 6.58 ;

d ^ey " W = d4)3

+ da3 " % = "23 - 55

>

and

d(4>ez ^mx ]

= d<J>5

+ da5

- dp5

= -11.32

yield

*ey " *mz=

^ey " ^mz " d(c^ey " W

= 122.66 (decreased by 1.87%) ,

23

^ez " ^mx=

^ez " ^mx " d((^ez " W

= 139.62 (decreased by 0.27%) , (113)

^ez " ^my=

^ey " ^mz " ee2 " em2

= 118.72 (decreased by 1.07%) ,

and

^ex " ^mz = ^ez " ^mx " ee3 " e

m3

= 43.89 (decreased by 2.47%) .

The above results in (111) and (113) are based on the measured phases <j>'

,

<t>2, 4)5 and the assumed values for d<^, d^, and d^. Alternatively, the

results may also be derived from the measured 4>2 , <j>

4 , <|>g and the assumed

values for d<{>2

, d<j>4 , and d<j>g in accordance with (29) through (37):

a£ = 57.50 , P2'

=" 77,86

' ^ex " ^my= 66,03

a^J= -62.90 , P4

' = - 78.55 , <\,' - 4^z= 110.65

a6

= -57.91 , p£ = -105.10 , <^2- <|^x

= 148.29 ;

a2

= 58.46 , p2= " 78.57

a4= -61.47 , P4 = - 80.47

a6

= -53.47 , p6= -101.76 ;

da2

= a2' - a

2= -0.96 , dp

2= p

2- p2

= 0.71

da4

= a^ - a4

= -1.43 , dp4

= pj - p4= 1.92

da6

= a6 " a

6= "4 * 44

'dp

6=

^6 ' ^6= ~3 * 34 ;

d( ^ex " ^ =d<t>

2+ da

2 " dp2

= 6 * 03

d(<J>ey- c^

mz ) = d<|>4

+ da4

- dj34

= -13.91

d(*ez " V * d*6

+ da6 - <% = 8 ' 09

!

and

^ex " V =*ex " *my " d( ^ex " V }

= 60.00 (completely recovered)

24

^ey mz ^ey " ^mz - d(i>ey- ^mz )

= 124.56 (decreased by 0.35%) (114)

^ez mx=

^ez " ^mx " d((^ez " V

= 140.20 (increased by 0.14%)

The results obtained above by the alternative method may be used either to

compare with those obtained in (111) and (113) to check the consistency of the

measured data, or to be averaged with those in (111) and (113) to minimize the

overall errors in the deduced results, because, in reality, the actual

measurement accuracy is unknown. If the averaged values are desirable, we

have:

((^ex " %{ ^ey - ^mz

((^ez " V

( ^ey " ^mx

(+ez

av

av

av

= 60.06 (vs. the true value of 60.00) ,

= 123.61 (vs. the true value of 125.00) ,

= 139.91 (vs. the true value of 140.00) ,

= ((^ex " Vav " e

el " eml = 160 ' 00

<exact

> >

(115)

Vav =^ey " Vav " e

e2 " 9m2

= 119 ' 67

(vs. the true value of 120.00) ,

and

(^ " i,)^ = <*« " iv)ex 'mz'av wez " vmx'av " ne3 " °m3

(vs. the true value of 45.00) .

= 44.18

The above predicted results together with the predicted relative phases

obtained in (108) and the amplitudes obtained in (106) may be used in (15) to

compute the predicted radiation pattern, which may then be compared with that

computed by the primed values (based on the measured results directly) for any

9 and <|>. This comparison should reveal the error caused by the measurement

inaccuracy assumed in (105), and the contribution by the predicting process

discussed here. Typical graphical results are presented in figure 4, where

the original ideal case without involving measurement errors is also included.

25

The predicted amplitudes of the component dipole moments (given in (106))

alone may be used in (16) to predict the total power radiated in free space by

the interference source being investigated. That is,

PT= 0.4 it

2[m|

x+ m|

y+ m|z + k

2 (n4 + m^ + m*x ) ]

= 32.39 W (vs. the true value of 32.44 W)

.

Should the primed values based on direct measurements be used, we would have

Pj = 32.17 W (decreased by 0.83% compared to the true value).

Thus, from the anlaysis made so far for this particular simulated example, we

may conclude that the final predicted results are not substantially affected

by the measurement errors (e.g., + 5% in power measurements and + 10% in phase

measurements) arbitrarily assumed in (105). Selected results with other com-

binations of percentage errors in power and phase measurements, with the same

manner of error occurrence as assumed in (105), are presented in figure 5.

From these results, one may make a quantitative assessment as to how much per-

centage errors in measurement can be tolerated in order to achieve a certain

accuracy in the final predicted source characteristics.

4. Another Simulated Example with Assumed Biased Measurement Errors

Sometimes, we experience biased errors due to the measurement system. In

this case, the measurement readings are consistently higher than the actual

status. For simulating this situation, we assume that dPsi

= dPdl

- = 20 W and

d<j>,- = 20° for the example considered in the previous section. Thus, the

measured powers in watts and the measured phases in degrees become

P^ = 445.11 , P^2

= 322.63 , P^3

= 804.60 ,

P^4

= 47.11 , P^5

= 493.03 , P's6

= 179.54 ;

P^ = 22.70 , P^2

= 72.55 , P^ = 21.56 ,

P^4

= 47.17 , P^5

= 27.62 , Pjfi

= 56.58 ; (116)

26

${ = -83.03

d>; = 125.56

4>2 = -57.03 ,

<t>5 = 68.11 ,

<J>5= -93.55 ,

4>6= 111.91 .

Note that in the above, the errors in P^, P^, and P^ are quite enormous as

compared to the true values given in (104). Consequently, we should expect

large errors to appear in the final deduced source parameters related to these

difference powers. Here, we once again assume dq = for simplicity. Then,

from (54) through (61), we obtain

mex= 2.1029 nig

X= 1.450 , dmex/mex

= 0.034

mey= 3.3829 m^

y= 1.839 , dmey/nigy

= 0.021

mel= 2.7029 m^

z= 1.644 , dm

ez/mez

= 0.026

mmx= 1.0020 m^ = 1.001 , dmmx/mmx

= 0.181

<y = 0.7220 m^y = 0.850 , dmmy/mmy= 0.251

m'2mmz= 0.5220 m^

z= 0.722 , dm

mz/mmz

= 0.347

mex

(the pred icted \/alue) = mgX(l - dm

ex/m;

x )

= 1.401 (vs. 1 .40, +0.07%) ,

mey

= 1.800 !vs. 1 .80, no error) ,

mez= 1.601 (vs. 1 .60, +0.06%) ,

"\nx= 0.820 [vs. .80, +2.50%) ,

mmy= 0.637 (vs. .60, +6.17%) ,

(117)

and

mmz= 0.471 (vs. 0.40, +17.75%) ,

where, for simplicity, the plus sign in parenthesis denotes increasing while

the minus sign denotes decreasing.

From (66), (72) through (76), and the constraint requirements in (38) and

(39), we also obtain the relative phases in degrees among the same kind of

dipole components directly from the measured data,

27

cose^ = 0.164103 , 9^ = -80.55

cos9g2

= 0.895146 , 9g 2= 26.47

cos9g3

= 0.469845 , 9g3

= 61.98 ;

(118)

cose^ = 0.530216 , 9^ = -57.98

cos9^2

= 0.377654 , 9^2= -67.81

cos9^3

= 0.362634 , 9^3= 68.74 .

Equations (65) and (67) through (71) yield the predicted relative phases in

degrees:

cos9el

= 0.173129 , 9el

= -80.03 (vs. -80.0, -0.04%)

cos9e2

= 0.937218 , 9e2

= 20.41 (vs. 20.0, +2.05%)

9e3

= 60.13 (vs. 60.0, +0.22%) ;

(119)

9ml

= -40.60 (vs. -20.0, -103.00%)

9m2= -52.88 (vs. -15.0, -252.53%)

9m3= 56.35 (vs. 35.0, +61.00%) .

Note that the 9^-s obtained above still deviate substantially from the

constraint requirement in (38) while the 9^s so deduced are very

satisfactory. Note further that only the first-order correction terms are

retained in (69) to (71). This will yield a reasonable predicted result only

when the percentage errors involved are small. Since the actual percent

errors for dm^/m^, i = x, y, z are all relatively large in view of the

measurement errors assumed for P^, P^3

, and P^, it is not surprising to see

that the results for 9mi

- predicted in (119) are what they are. Should the

second-order correction terms also be included in the formulation, the

predicted results for 9ml

- would be more reasonable.

Comparing the results in (118) and (119), we have the errors in degrees

for the relative phases among the same kind of dipole components for the

example considered herein,

cos9e3

— 0.498036 »

COS9ml

= 0.759269 »

cos9m2= 0.603491 »

cosem3

= 0.554105 j

28

deel " e

el " eel = "°- 52

>d9

ml = eml " U ' " 17 ' 38

dee2

= e^ - ee2

= 6.06 , dem2

= e^ - em2= -14.93 (120)

dee3 = e

e3 " ee3 = l ' 85

>de

m3 = em3 " e

m3 = 12 ' 39

Now, we treat the mixed phases. In accordance with (83, 84, 85) and (17,

23, 24), we obtain the following in degrees:

aj = - 46.00 , aj_ = - 45.99 , dax

= - 0.01

p{ = -159.39 , Pj = -178.45 , dpj_ = 19.06

^ex " ^my=

*i+ a

l" P{ = 30 - 36

d((j,ex- 4>my ) = d<j>

1+ da

x- dp

x= 0.93

*ex " V =*ex " ^y " d( *ex " V = 29 * 43

'(121)

The above relative phase can also be deduced from <j>

2. That is,

cxo = 57.67 , <x2

= 58.43 , da2

= - 0.76

p2= - 58.42 , p 2

= - 67.04 , dp2

= 8.62

^ex " Ky =*2

+ a2 " ?2

= 59 ' 06

d(<pex- 4>my ) = d<(>2 + da

2- dp

2= 10.62

*ex " *my = *ex " ^y " d{ *ex " V = 48 * 44 (122)

which may be used to average with that in (121), yielding

(c|>ex - <l>my ) av= 38 - 94 < vs - 60 -°> -35.11%) , (123)

(4-ey" Wav =

{ ^ex ' Wav " eel " eml

= 159.57 (vs. 160.0, -0.27%) . (124)

Similarly, we also have the following in degrees:

ct3 = 12.48 , a3

= 9.60 , da3

= 2.88

p 3' = -153.00 , (3 3

= -170.33 , dp3

= 17.33

29

*ey " *mz=

*3+ a

3 " 6h = 71 ' 93

d( *ey " W = d<t>3 + da3

- dp3

= 5.55

^ey " ^mz=

^ey " *iz " d( ^ey " +nu)= 66.38 ;

a^ = - 63.38 , a^ = - 61.79 , da4

= - 1.59

pj = - 52.96 p4= - 59.30 , dp

4= 6.34

^ey " ^mz=

<t>4 + «4 - P4 = 115.14

d^ey " ^mz^= d<))

4+ da

4" = 12.07

^ey " *mz=^y " *mz " d( *ey " W = 103 ' 07 >

(<|>ey-

<l>mz ) av= 84.73 (vs. 125.0, -32.22%) ,

(125)

(126)

(127)

( *ez " V } av = ((^ey " Wav " ee2 " e

m2= 117.20 (vs. 120.0, -2.33%) ; (128)

ocr - 28.83 ,

p£ = - 47.69 ,

a5

= 27.86 ,

0r = - 54.95 ,

da5

= 0.97

dp5

= 7.26

*ez " *mx = *5+ a

5 " P5 = 144 ' 63

d(<|>ez- <\>m ) = d<t> 5

+ da5

- dp5

= 13.71

*ez " *mx = *ez " «ix " d^ez " W = 130 - 92>

(129)

ag = - 53.05 ,

pg = -118.05 ,

ag = - 53.37 ,

p6= -109.94 ,

dag = 0.32

dpfi

= - 8.11

*ez " Kx =*6

+ a6 " ?6 = 176 ' 91

d( ^ez " W = <% + da6 " <% = 28 ' 43

*ez - *mx=

*ez " C " d( *ez " W = 148 ' 48>

Uez- 4>mx ) av

= 139.70 (vs. 140.0, -0.21%),

(130)

(131)

( *ex " Wav = ((^ ' *my)ez ^mx' "e3 " m3

= 23.22 (vs. 45.0, -48.40%) . (132)

30

The predicted amplitudes of the dipole moment obtained in (117) may be

used in (16) to compute the predicted total power radiated by the source

emitter under study. That is,

PT

= 32.69 W (vs. the true value of 32.44 W, +0.76%) , (133)

which is still very reasonable in view of the large errors incurred in the

magnetic dipole moments. If the primed source parameters are used to compute

the total radiated power, we would get

Pf = 35.82 W (vs. 32.44 W, +10.42%) . (134)

The above result implies that the measured errors, dP • = dPd

- = 20 W and d<t>.

= 20° assumed in this example, will contribute more than 10% of error toward

the derived radiated power if no application of the predicting process as

demonstrated is made. Once the predicting process is used, the overall error

in the deduced total radiated power amounts only to less than 1%.

In addition, the predicted phases among the same kind of dipole

components obtained in (119) and the predicted average mixed phases obtained

in (123, 124, 127, 128, 131, 132), together with the predicted amplitudes just

considered can be used in (15) to evaluate the detailed normalized free-space

radiation pattern. This pattern may then be compared with that computed by

the primed source parameters to reveal the error caused by the measurement

inaccuracy and the correction contributed by the process so presented.

Thus, both the example considered in this section and that given in the

previous section show that the predicting process outlined in this report

(i.e., obtaining the necessary unprimed source parameters from the primed ones

computed directly from the experimental data containing measurement errors) is

indeed useful to extract the source parameters and, therefore, to predict the

radiation characteristics of an unknown inteference source, provided that the

approximate measurement error is known or can be estimated.

31

5. Sensitivity of Extracted Parameters with Respect to Measurement Accuracy

Another useful concept for error analysis is via the sensitivity func-

tion. Mathematically, the sensitivity of a parameter A with repect to a

quantity B may be defined as

CA dA/A „. B 5A , 1or ,

bB

=WB~ = A 9B"

(135)

giving the percentage change in A for a given percentage change in B, or

indicating the sensitivity of dependence of A on B.

Refering to (1), if we let A = mex , and B = PS1-, we can derive an

expression for the percentage error incurred in the deduced source parameter

mex due to a given percentage error in the measured quantity P$1

-

:

c

mex

Psi ^ex _ .

Psi r+ for i = 1,2,5,6 n , fi

,

Psi

=

% * Psi

" "4m

p2xq

2 L f0r 1 " 3 ' 4'

(136)

Similarly, we obtain

m D

c ey _ . si r+ for i = 1,2,3,4 MrnbP .

" * ~2 2 [ - for 1 = 5,6

[unS1 4m

eyq

m P<: ez - 4. si r+ for i = 3,4,5,6 M ,

ft»

si 4mezq2

m Pc mx _ di r+ for i = 1,2,5,6 mq \

P " ± —2—2~2 *- for i = 3 4di 4m

mxk q

32

S"W „ + /d1 (+ for 1 = 1,2,3,4 . .

m P

Smz = + dl {+ for 1 = 3,4,5,6 , .

PJ 2 2 2*- for i=1? 1 141

J

mz ^

P P . 5PT OOO Pc-i

SP

T =p^Lap-L = (20it

Z/\V) -p^- (142)

and

P P

Sp

T= (207i

2/\

2q2

) p^- . (143)Kdi

KT

The corresponding expressions for A = eei

-, 6ml

-, or <p ^- 4, • (i, j =

x,y,z, i * j) and B = P . , Pdl

- , or <|>j are much more involved, and thus not

presented here.

For the simulated example (103) and (104) considered in section 3, we

have

m mftY mOYSD

ex = 0.387 , S pex = 0.276 , S D

ex = -0.715 ,Ksl

Ks2 Ks3

mOY mOY mOYS p

ex = -0.025 , S pex = 0.431 , Sp

ex = 0.145 ;ps4

ps5

ps6

nuw mov/ mQX/S p

ey = 0.234 , Spey = 0.167 , Sp

ey = 0.433 ,Ksl

Ks2Ks3

mftw nipy mpvSp

ey = 0.015 , S pey = -0.261 , S p

ey= -0.088 ;p

s4ps5

Ks6

33

Sp" = -0.297 , Sp" = -0.211 , Spez = 0.547 ,rsl K s2 K s3

mP7 mQVS p

ez= 0.330 , Sp

ez = 0.111 ;

Sp™J= 0.372 , S

pmx = -0.011 ,

Sp™ = 0.054 , Spmx = 0.259 ;

Sp^ = 0.660 , Sp^ = 0.020 ,

SpV = -0.096 , S^ = -0.460 ;

S^J* = -1.486 , Sp1"12 = 0.044 ,

cmez .Sps4

_ 0.019»

cmmx _Spdl

" 0.019 5

P d4-0.192

S

pdl

0.034 »

yA4

0.342 J

Spdl"

-0.076>

Spd4

" 0.768 3

Ksl

0.185 9

Ps4

0.012 >

d2 rd3

!

mz = 0.215 , Spmz

d5 p d6

mm-7 mm ,Spmz = 0.215 , Sp

mz = 1.035 ,

(144)

Sp1 = 0.132 , Sp

1

= 0.341 ,Ks2

ps3

PT PTSp 1 = 0.206 , Sp

' = 0.069 ;r s5 r s6

>

T PT Pt>' = 0.001 , Sp

1

= 0.023 , S p' = 0.001 ,

dlpd2

pd3

*T PT PT,' = 0.012 , Sp 1 = 0.003 , Sp 1 = 0.016 .

d4 pd5 pd6

From the above numbers for this particular example, we may interpret the

results as follows:

1) The accuracy in measuring P s3has the most effect on m

ex. In partic-

ular, the deduced source parameter mex

decreases from the true value

by approximately 0.7% whenever the measured PS 3

is caused to increase

from its true value by 1% due to the measurement error. The accuracy

34

of the measured Ps3

has also relatively large effect on mey and

mez . This observaton is, of course, obvious in view of the fact that

P s3is the largest reading in (104). The implication is that more

care should be exercised for taking the stronger data.

2) The accuracy in measuring Pd2

and Pdg

has more effect on mmi , i =

x,y,z than that in measuring Pdl , P

d3 , Pd4 , and Pd5 .

3) The total radiated power, Py, by the interference source is not yery

sensitive to the accuracy in measuring the sum and difference powers,

because S < 0.4, a = P cn- and PH ,-.a SI ul

6. Concluding Remarks

Based on the previous formulation for quantifying the radiation charac-

teristics of an unknown interference source from power and phase measurements

with the aid of a TEM cell, we have derived in this report the deviation

expressions, due to possible errors in measuring these powers and phases, for

all those source parameters required for computing the radiation pattern and

the total power radiated by the subject source in free space. From these

deviation expressions, we can make corrections to those source parameters

deduced directly from the measured data in order to obtain results closer to

the true values, when the measurement errors are provided either by a

manufacturer's estimation or by assumption. If, on the other hand, the

measurement errors are believed to be random in nature, the derivations con-

tained herein indicate that repeated measurements are required, from which a

set of source parameters may be directly computed. The averages of these

respective computed source parameters should approach the true values.

Two numerical examples based on two sets of measurement errors—one is

arbitrarily assumed and the other is based on a systematic biased error—have

been given to illustrate the general procedure of making error analysis.

Utilization of the sensitivity function to yield a percentage change in one of

the deduced source parameters for a given percentage error in one of the

measurement data has also been applied.

35

In practice, the accuracy in power and phase measurements may be esti-

mated in terms of worst-case bounds such as + x% in power measurements and

+ y% in phase measurements (or in an absolute term such as + x watts and + y

degrees respectively). Under this situation, the formulation outlined in this

report is still applicable by considering all the possible combinations of

power and phase readings to derive an upper and lower bounds for the deduced

source parameters and, consequently, for the radiation characteristics of the

unknown source. When one of the source parameters so computed based on the

measurement data is dominant, more attention can then be paid to the bounds of

this particular parameter, because its accuracy should have more impact on the

accuracy in the final result. For example, if me2

is the largest among the

source parameters based on the direct sum power measurements P'-, i = 1, 2,

. . . ,6, then, in accordance with the first expression given in (54), we

easily obtain the upper and lower bounds for m2x

as follows:

m2J = [P!i(l + x%) + Plpd + x%) - Plo(l - x%)e

' upper -

1 *c b0o

- P^4(l - x£) + P;

5(l + x%) + P^

6(l + x%)] /(2q

z)

(145)

"4l lower= [

p sl^ " *%) + Ps2^ - ^ ' Ps3d + ^- P^

4(l + x%) + P;

5(l - x%) + P^

6(l - x%)] /(2q

2)

where dq = is also assumed.

7 7Based on the same sequence of assumed errors as above, nig and m^

zcan

then be computed. After applying a similar consideration for m^ , the upper

and lower bounds for the total radiated power P-j- may be estimated by (16).

Strictly speaking, the formulation outlined in this report applies only

to the symmetric TEM cell, where the center septum is exactly midway between

the top and bottom outer conductors. For an asymmetric cell designed to pro-

vide more test zone inside either top or bottom half of the cell [4], the

electric fields at the centers of the two halves are no longer identical.

When the vertical component of the electric field in either half, qa , is known

(by theoretical computation or by actual measurement) and the corresponding

36

horizontal component is relatively insignificant--as it should be for a well-

designed cell—the formulation and analysis presented herein can still be

applicable by merely replacing the parameter q in (1) through (13) by qa.

7. References

[1] Ma, M. T.; Koepke, G. H. A new method to quantify the radiation charac-

teristics of an unknown interference source. Nat. Bur. Stand. (U.S.)

Tech. Note 1059; 1982,

[2] Sreenivasiah, I.; Chang, D. C; Ma, M. T. Characteristics of electri-

cally small radiating sources by tests inside a transmission line cell.

Nat. Bur. Stand. (U.S.) Tech. Note 1017; 1980.

[3] Crawford, M. L. Generation of standard EM fields using TEM transmission

cells. IEEE Trans. EMC-16_: 189 - 195; 1974.

[4] Crawford, M. L.; Workman, J. L. Asymmetric versus symmetric TEM cells

for EMI measurements. Proc. IEEE International EMC Symposium, Atlanta:

204 - 210; 1978 June 20 - 22.

37

mi±ex/ ex

mt,ey/ ey %y/ my

Figure 1. Unknown equipment under test (EUT) is made of

equivalent three orthogonal electric and three orthogonal

magnetic dipoles.

38

TEM CELL

n>INSTRUMENTATION

FOR POWER AND

PHASE MEASUREMENT

l±> COMPUTER SYSTEM

Figure 2. Emissions testing measurement system

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43

NBS-114A (rev. 2-ao

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATASHEET (See instructions)

1. PUBLICATION ORREPORT NO.

NBS TN-1064

2. Performing Organ. Report No 3. Publication Date

June 1983

4. TITLE AND SUBTITLE

Uncertainties in Extracting Radiation Parameters for an Unknown Interference SourceBased on Power and Phase Measurements

5. AUTHOR(S)

Mark T. Ma and Galen H. Koepke

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

NATIONAL BUREAU OF STANDARDSDEPARTMENT OF COMMERCEWASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

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

10. SUPPLEMENTARY NOTES

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

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

bibliography or literature survey, mention it here)

A method for determining the radiation characteristics of a leaking interferencesource has been reported in a previous publication [l], in which the unintentional

electrically small leakage source was modeled by two vectors representing a

combination of equivalent electric and magnetic dipole moments. An experimental

setup, measurement procedures, and the necessary theoretical basis were all described

therein to explain how the relevant source parameters can be extracted from the

measurement data of output powers and phases taken when the interference source is

placed inside a transverse electromagnetic (TEM) cell. Simulated examples were

also given to show that the equivalent source parameters of unknown vector dipole

moments and thus the detailed radiation pattern and the total power radiated by

the source in free space could be uniquely determined by the method if the measure-

ment data were not contaminated by noise. This report Dresents the mathematical

analvsis of the uncertainties in the final, extracted results when the experimental

data are degraded by the background noise and measurement inaccuracies.

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

dipole moments; electrically small source; error analysis; interference sources;phase measurements; power measurements; radiation pattern; TEM cell; total radiatedpower; uncertainties.

13. AVAILABILITY

[23 Unlimited

| |

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

[] Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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48

15. Price '

$8.50

•frU.S. GOVERNMENT PRINTING OFFICE: 1983-679 901/346USCOMM-DC 6043-P80

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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.

U.S. Department of CommerceNational Bureau of Standards

Washington, D.C. 20234Official Business

Penalty for Private Use $300

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