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NBS

Publications

NATL INST. OF STAND & TECH

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^''^fAU o* NBS TECHNICAL NOTE 1066

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

'100

5753

NO . 1066

1983
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act ot Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

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a

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THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts

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'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. 'Some divisions within the center are located at Boulder, CO 80303.
;_.„.....:.oil ;fe

Factor of a Reverberating Chamber

Bing-Hope Liu

David C. Chang

Department of Electrical Engineering

University of Colorado

Boulder, Colorado 80309

Mark T. iVIa

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

U.S. Department of Commerce

Boulder, Colorado 80303

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued August 1 983
National Bureau of Standards Technical Note 1066

Natl. Bur. Stand. (U.S.), Tech. Note 1066, 54 pages (August 1983)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1983

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

Price $4.00

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

This report describes theoretical and experimental analyses developed by

staff of the University of Colorado at Boulder in collaboration with the

Electromagnetic Fields Division of the National Bureau of Standards (NBS),

under a contract sponsored by NBS. Professor David C. Chang heads the University team. Dr. Mark T. Ma of NBS serves as the technical contract monitor.

The period covered by this report extends from January 1982 to July 1983.

The work presented in this report represents an initial effort to establish a theoretical basis for the design of a reverberating chamber, and for

the analysis and interpretation of the measurement results made inside the

chamber.

The particular topics addressed herein are to determine (1) the total

number of possible electromagnetic eigenmodes, for a given chamber size and

operating frequency, which may exist inside the chamber for the stirring and

tuning purpose, (2) the mode density within a frequency range, (3) the dependence of mode degeneracy on the dimensions of the chamber, (4) how easy a uniformly homogenous and isotropic field may be generated within a test zone inside the chamber for performing electromagnetic interference/compatibility

tests, and (5) a composite quality factor to represent the chamber as a whole. The theoretical results obtained for each of these topics are con- sidered \/ery useful for the design purpose.

Previous publications under the same effort include:

Tippet, J. C; Chang, D. C. Radiation characteristics of dipole sources located inside a rectangular coaxial transmission line. Nat. Bur. Stand.

(U.S.) NBSIR 75-829; 1976 January.

Tippet, J. C; Chang, D. C. ; Crawford, M. L. An analytical and experimental determination of the cut-off frequencies of higher-order TE modes

in a TEM cell. Nat. Bur. Stand. (U.S.) NBSIR 76-841; 1976 June.

Tippet, J. C; Chang, D. C. Higher-order modes in rectangular coaxial

line with infinitely thin inner conductor. Nat. Bur. Stand. (U.S.) NBSIR

78-873; 1978 March.

Sreenivasiah, I.; 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. Nat. Bur. Stand. (U.S.) NBSIR 79-1606; 1979

May.

Tippet, J. C. ; Chang, D. C. Dispersion and attenuation characteristics of

modes in a TEM cell with a lossy dielectric slab. Nat. Bur. Stand. (U.S.)

NBSIR 79-1615; 1979 August.

Sreenivasiah, I.; Chang, D. C; Ma, M. T. Characterization of electrically small radiating sources by tests inside a transmission line cell.

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

m
Wilson, P. F.; Chang, D. C. ; Ma, M. T. Excitation of a TEM cell by a vertical electric Hertzian dipole. Nat. Bur. Stand. (U.S.) Tech. Note

1037; 1981 March.

Sreenivasiah, I.; Chang, D. C; Ma, M. T. A method of determining the

emission and susceptibility levels of electrically small objects using a TEM cell. Nat. Bur. Stand. (U.S.) Tech. Note 1040; 1981 April.

Wilson, P. F.; Chang, D. C; Ma, M. T. Input impedance of a probe antenna

exciting a TEM cell. Nat. Bur. Stand. (U.S.) Tech. Note 1054; 1982 April.

IV
CONTENTS

Page

1. Introduction 1

2. Number of electromagnetic eigenmodes in a rectangular chamber 4

2.1 Field solution: eigenmodes and eigenfrequencies 5

2.2 Calculation of number of modes and mode density 9

2.3 Design considerations of reverberating chambers 17

2.4 A brief review of mode degeneracy 20

3. A composite Q-factor for lossy rectangular chambers 21

3.1 Previous results of Q's for rectangular cavities 22

3.2 Construction of a composite Q 23

3.3 Discussion of results 28

4. Conclusions 30

5

.

Acknowl edgements 31

6

.

References 31
i
Eigenmodes and the Composite Quality Factor

of a Reverberating Chamber

Bing-Hope Liu', David C. Chang , and Mark T. Ma

Electromagnetic Fields Division

National Bureau of Standards

Boulder, Colorado 80303

The total number N of electromagnetic eigenmodes, with ei genfrequencies not greater than some given value, which can exist

inside a rectangular mode-stirred or mode- tuned reverberating

chamber is important in that it reveals how many modes can be

available at an operating frequency for the "stirring" or tuning

purpose. This is calculated analytically via a lattice-point

counting technique in the k-space (k = wave number), leading to an exact expression for N, which can be split into a smooth component

and a fluctuating part. The former contains, in addition to

Weyl's volume term, an edge term as a second-order correction.

The latter is sensitive to the dimensions of the chamber. Simple

design criteria are then derived in view of the number of available modes and the uniformity of their distribution. To take into

account the ohmic loss in metal walls of the chamber, a composite

Q-factor is also proposed for design purposes. This is achieved

by taking a suitable average of 1/Q-values of all possible modes

within a small frequency interval. Comparison with numerical Qvalues for individual modes shows that the composite Q can be used

as a practical design parameter.

Key words: cavity; composite quality factor; eigenfrequency;

eigenmode; electromagnetic field; mode density; mode number;

reverberating chamber.

1. Introduction

In performing electromagnetic interference (EMI) or compatibility (EMC)

measurements, a uniform plane wave in open space can be used as a reference

environment. Such waves can be obtained, for example, from the far field of a radiating source or by means of a large equiphase radiator; however, it is not

easy to produce high level fields necessary for measurement purposes. Also,

the poor isolation from external interferences and the suffering from the

^Department of Electrical Engineering, University of Colorado, Boulder,

Colorado 80309.
effects of ground reflections at lower operating frequencies (below a few hundred MHz) are inconvenient for performing measurements. Anechoic chambers may

be designed to simulate free-space environments without interferences from

external sources and to produce an approximate plane wave, but the cost is

usually prohibitively high.

Transverse electromagnetic (TEM) cells have been developed recently to

provide a shielded environment for test purposes [1, 13 - 16, 27 - 29]. In

the lower frequency range (which is upper bounded by the cutoff frequency of

the first higher-order mode), and for electrically small equipment under test

(EUT), a plane wave condition can be achieved and successful methods for EMI

measurements have been proposed [24, 27 - 29]. For medium-sized TEM cells,

such as the one at the National Bureau of Standards (NBS) which measures 1.2 m

X 1.2 m X 2.4 m, the limitation on operating frequencies (less than 125 MHz)

could be a drawback. Since the polarization of the field so generated is

fixed, the EUT placed inside a TEM cell has to be physically rotated in order

to make a compatibility assessment. This requirement of EUT rotation could

also be an inconvenient aspect.

There is increasing interest in developing newer EMI/EMC measurement

techniques by utilizing another kind of test field, namely that of a uniformly

homogenous and isotropic field within a local region inside a metal enclosure. The mode-stirred or mode-tuned reverberating chambers have been introduced [3-12, 17, 21, 22, 26] to meet this demand. The isolation from external

interferences being equally good, respectful bandwidths of these reverberating

chambers can also be achieved. As a rule, these chambers are large enough

compared to the wavelengths of the operating frequencies--which are usually in

the microwave range. For example, the NBS reverberating chamber is a steel

welded, rf-tight rectangular room 3.05 m wide, 4.57 m long, and 2.74 m high

with a mode-sti rrer or mode- tuner mounted near the middle of the ceiling.

Conceptually, by rotating the irregularly shaped mode-sti rrer or paddlewheel tuner, the associated boundary conditions are changing either continuously or in steps so that the eigenmodes, which exist simultaneously inside

the shielded metallic chamber, are perturbed accordingly. In this way, a uniformly random field or an average homogenous field can be created in a local
region inside the chamber, and this region can then be used as a test zone. Theoretically and experimentally, there are a number of questions and open

research topics such as general properties, test regions, measurement methods,

designs, interpretations of the measured results, etc., pertaining to a modestirred or mode- tuned reverberating chamber.

To treat electromagnetic field problems associated with reverberating

chambers and to provide basic knowledge for design purposes, two analytical

approaches are possible: the direct approach and the indirect one. The

former involves the direct solution of field problems containing boundaries

with time-varying configurations. A formal solution using this approach is

very difficult to obtain. In the latter approach, suitable linear combinations of eigenmodes of the unperturbed cavity (i.e., without mode stirrer or tuner) are used a priori to approximately satisfy the boundary conditions on the surface of the rotating mode stirrer. The coefficients are generally time

dependent. The main advantage of this approach is the fact that the unperturbed eigenfrequenci es and eigenmodes are usually easier to calculate; therefore, the problem can be reduced to a more familiar one. A necessary condition for the validity of this method is that the total number N of eigenmodes,

with eigenfrequenci es less than or equal to the operating frequency f, be

large enough. Moreover, by knowing N as a function of f, we can also judge

whether there are enough eigenmodes for "stirring" at any given frequency.

Thus, an analytical solution of N(f) is indispensable to the study of rever- berating chambers. This is indeed the approach presented in this report.

In section 2, it will be shown that, starting from a lattice-point counting technique in the k-space (k = wave number), the problem of total number N of possible TE and TM eigenmodes for an unperturbed lossless rectangular

cavity can be treated via Poisson's summation formulas. The degeneracy of

eigenmodes will be suitably taken into account. Besides Weyl 's volume term

[30, 32], the smooth component of N also contains an edge term as a secondorder correction, while the first-order, or surface correction, term will be

seen to vanish. The comparison with exact computer counting reveals that this

result can be used for design purposes in frequency ranges with kaQ > %, where

a^ represents the smallest side of the chamber. The total number of

eigenmodes related to an operating microwave frequency can thus be readily
determined and, simultaneously, a criterion for the applicability of the

indirect approach is also provided.

The ohmic loss of metal walls or, equival ently, the finite Q of a rever- berating chamber will be investigated in section 3. In microwave frequency

ranges, a convenient expression for the composite quality factor Q, as a function of frequency, of the chamber will be introduced. This is achieved by

considering a suitable average of 1/Q-values of all possible eigenmodes in the

k-space. The comparison with exact numerical Q-values of individual modes

will be made with the help of normalized, cumulative distribution curves of

the latter. The new expression for composite Q is useful for the practical

design of reverberating chambers with slightly lossy walls.

Section 4 summarizes our results from a design point of view. For a better understanding of reverberating chambers, further investigations based

on this report are necessary.

2. Number of electromagnetic elgenaodes in a rectangular diamber

The problem of the number of electromagnetic resonant modes existing

inside a cavity was treated by Jeans [19] and Rayleigh [23] as early as 1905

in their studies of thermal radiation of large, rectangular cavities (isothermal black bodies). Weyl [30 - 33] investigated the same problem for more

general cavities by utilizing integral equation techniques. The related problem of mode density was treated by Brownell [4] via a logarithmic Gaussian

smoothing procedure and more recently by Baltes and Kneubuhl [2] via

computational methods. Due to strong and irregular fluctuations of the total

number N of eigenmodes, most results so far available seem to have been

reported either in an asymptotic sense or in terms of smoothed or averaged

expressions.

In this section, a geometric approach will be adopted which, essentially,

consists of a lattice-point counting technique in the k-space (k = wave

number). Complete expressions for both the number N and the mode density

dN/df can be achieved for the case of a rectangular chamber.
2.1 Field solution: eigenmodes and eigenfrequencies

Consider first a closed cavity with lossless metallic walls. The T, IT fields of the free electromagnetic oscillations obey the Maxwell equations

V X r = - jw^i Tf (1)

V X T7 = jwE r (2)

throughout the interior region of the cavity and the boundary conditions

rT X r =

(3)

n . H =

on the enclosing walls, where n is the outward unit normal vector. Equations

(1) and (2) are equivalent to the vector Helmholtz equation

plus the divergence condition

(v^ + k^){^ = (4)

TT

^= . (5)

if the interior region of the cavity is filled with an isotropic, homogenous

material of permittivity e and permeability \i. Here we have assumed (and

suppressed) the time harmonic e*^^^ for all the field quantities and the wave

2 2

number k in (4) is given by k = w |ie .

In order to treat the problem of rectangular cavities with sides a, b, c specifically, we will choose, in an arbitrary way, the z-di recti on to be that

parallel to side c as shown in figure 1. Following standard procedures [18,

p. 129-132], we can formally construct a TE^^' set and a TM^^^ set of eigenmode solutions:
TE modes

mnp

E =

E.. = -^ cos -^ sin -^ sin -^ {6a)

r - mix '""'—

^. m-rtX ^^^ n^y ^,. piiz . SI n -—— cos -r^ SI n -*--—

; a a b c

H =

K, =

H =

1 (k2 . iA cos ^

cos^sin PJ^

a D c

1 -m-Tt p-ji m-Tux jwii a c a

1 i^- -^ cos ——

y jwn b c

b

sin i^ cos £^ :

{6b)

TM„^;^ modes:

mnp

E = J- (k^ - £^) sin

^

sin

^ cos P^

Jtoe

Jwe a C

C

COS I- _ 1 mTi -p-n; ^^^ mirx ^,. nixy ^,. piizE.. = -! —r— COS —r— SI n —rr- SI n !-—-

E = ll!^z£2L sin

^

COS Ilj^ sin P? ; y jwe be a

"^ '^ '

{7a)

H =

H =

n-n; ^,. mux ^^^ n-ny ^^^ p-nz —r-babe SI n -—— cos —r^- cos ^-T—

^

mil cos

a a

mTix ^. nTty sinpTlZ -f^COS ^ (7b)
where m, n, p are nonnegative integers that determine the number of nodes and

anti nodes in the standing wave pattern of the corresponding eigenmode. In

both sets of eigenmode solutions, the k and w are determined by the relation

4nn l^^ = ^L. = i^^f + i^f + i^f ' m, n, p > . (8)

mnp ^ mnp ^a^ ^b^ ^C'' '^

A closer investigation into (6) and (7) reveals that there are only two categories of true eigenmode solutions to the boundary value problem defined by

(3) - (5) as discussed below.

Category I : None of m, n, p is zero (i.e., m > 1, n > 1, p > 1). In this

category, (6) and (7) represent two independent polarizations of field corre- sponding to each set of values m, n, p. Each eignenvalue k^pp is of mutiplicity 2 or, in other words, two independent eigenmodes are associated with each

eigenvalue k^^p. Thus, the degeneracy of mode is 2.

For a given eigenvalue k^^p, the T, TT fields of the associated (degenerated) eigenmodes can be represented by

(E, H)^ = a^(0, E^, E^, H^, H^, H^)^ + b^(E^, E^, E^, 0, H^, H^)* ,

mnp mnp

(9a)

where ( )'' means the column matrix obtained by transposing the row matrix ( ) and where a-^, b-^ are two arbritrary coefficients cor- responding to the degeneracy of 2. By (T, TD , we mean the six components of

the E, H fields under consideration. The six components of (0, E^, E^, H,

:(Z) ^

'mnp

Hj^, Hy)^ of the TeI^^ modes and the six components of (E^, E^^, Ey, 0, H^, Hy)^

of the TM modes are explicitly shown in (6) and (7). The superscript (z)

mnp r ^ r r indicates that the terminology TE or TM has its usual meaning with respect to

z-direction. Equivalently, the T, TT fields of the left-hand side of (9a) can

also be represented by
{%)

or

a^iE^, 0, Ey, H^, H^, H^)^ + b^iE^, E^, E^ , H^ , 0, H^)^

mnp mnp

a^(E , E , 0, H , H , H )^ + b^(E , E , E , H , H , 0)^

3 z' x' ' z' x' y 3 z' x' y' z' x'

mnp mnp

with 32, b2 or 83, b3 being arbitrary coefficients. The expressions for the

field components ^ of TE^^^ TM^^\ JE^^\ TM^^^ modes can be obtained by suitmnp mnp mnp mnp -^ ably permuting the parameters, a, b, c, m, n, p and x, y, z. In Table 1 we

have arbitrarily chosen (9a) to represent this category.

Table 1

Case Restrictions Nonvanishing components Designation Numb^er o^f modes

1 m>l, n>l, p>l

2 m>l, n>l, p>l

Category II : Only one of m, n, p is zero . There are three cases as shown in

Table 2. The names of TmI^^, TE^f , and TEl,f modes are adopted from the mno onp mop

'^ literature [18, p. 190], although the designation of E^, E^, and Ey modes in

the fourth column seems simpler. In each of these three cases, the eigenvalue

kjppp is simple and the degeneracy of mode is 1.

Table 2

Case Restrictions Nonvanishing components Designation Number of modes

3 m>l, n>l, p=0 E^, H^, Hy TM^^^ or E^ modes N3(k)

4 m=0, n>l, p>l Ex, Hy, H^ TE^^^ or E^ modes N4(k)

5 m>l, n=0, p>l Ey, H^, H^ ^Sod ^"^ ^V

"^^^^^ ^5^*^^

^x' ^y ^z' '^x' ^

TM1^] modes

mnp

Ni(k)

p c u u u X ' V ' X ' V ' Z JeIII modes

mnp

N2(k)
In the last column of Table 1 and Table 2 we have, for the purpose of

convenience, also included the items "N]^{k), ^2^'^)' • • • » N^dc), where N-|^(k) means the number of all the TM:^^ modes with eigenvalues k„„r, less than or mnp ^ ninp

equal to some value k. The quantities N2(k), . . . , U^ik) have similar

meaning. Thus, the total number N(k) of electromagnetic resonant modes with

resonant "frequencies" smaller than or equal to some "frequency" k, with all

degeneracies taken into account, is

N(k) = N^(k) + N2(k) + N3(k) + N4(k) + N5(k)

= 2N^(k) + N3(k) + U^{k) + Nglk) (10)

The last step of (10) is due to the fact that N2^(k) = N2(k). In the following

we will calculate N(k) and dN/dk via geometrical considerations in the threedimensional k space. Note that N(k) is invariant with respect to the choice

of z-axis.

2.2 Calculation of number of modes and mode density

From (8) it is apparent that each value k^^ can be represented geometrically by a lattice point P having the coordinates (—, -^, ^] in the threedimensional Euclidian k-space as shown in figure 2. The value k^^- is then

equal to the distance from the origin to the point P. To find the number

Ni(k) of TM eigenmodes with eigenfrequencies k„„„ < k where m > 1, n > 1, p i mnp

-J -t funp r

> 1 (see case 1 of Table 1), we need only count the number of lattice points

with k^pp < k in the first open octant. Thus,

oo N, (k) = y H(k - k ) , (11)

1 ^ mnp

m,n,p>l

where H(x) is the Heaviside unit-step function defined by

'In this report, we will occasionally use tine same word "frequency" to denote

the frequency f, the wave number k = u) /iie , or the normalized wave number

ka. Its meaning should be clear from the context.
Hfv^ - (1 if X > "^^' ^0 otherwise *

By geometrical observations, the expression (11) can be rewritten as follows

ni,n,p=-o° *^

Since

T { I H(k - k ^) + y H(k - k^„^) + y H(k - k^^„)}

2 „ f; -.1 nino „ f;^, onp „ t,^-, mop ^ m,n,>l n,p>l ^ m,p>l *^

m>l n>l p>l

- i H(k - k^„„) . (12)

m>l m=-co

n>l n=-°D

I "'^ - w> 4 I "'^ - ^op> 4 "'^ - ^oo' • P>1 p=-oo

"^

we have

m,n,p=-a'

m,n>l n,p>l *^ m,p>l '^

10
- F n H(k - k„„„) . I H(k - k„„^) ^ I H(k - k )) m=-oo n=-oo p=-oo

"^

+ ^H(k-k„„„). (14)

The quantities N3(k), N^Ck), N^Ck) can be written down immediately. Thus,

Njk) =

J H(k - k^„^) , m,n>l

^4^^) = ^ j^^

H(k - k^^p) . (15)

N.(k) = ^ H(k - k^^^) . ^ , mop m,p>l ^

Substituting (14) and (15) into (10), we arrive at the result

N(k) =^ I H(k - k^^ ) "^ m,n,p=-<. "^"P

-Ul H(^-^moo^^ I '^^-KJ' I ^(^-^op^l

m=-co n=-oo p=-oo

4 H(k - k^^^) , (16)

expressing the total number of resonant modes which can exist inside an unperturbed lossless chamber when the operating frequency is k. By taking the

derivative, we also have the mode density

11
m,n,p=-oo ^

-Til s"^-'<n,oo'^ I «'K-^no>^ I «'^ " ^op'l

[T1=-0D n = -o° P = -aD "^

46(k-k^^^) (17)

where 6{x) is the Dirac's 6-function. For lossless chambers, the mode density

consists of only 6 functions as expected.

Expression (17) can be put into a more convenient form by utilizing the

following three-dimensional and one-dimensional Poisson's summation formulas

for Fourier transforms [20, 25]:

I f(ma. np, Py) = ^ I F(m %, n ^, p % (18)

m,n,p=-oo "^^^ m,n,p=-oo

a p y

I g{ma) = ^ y G(m^) (19) m—oo m—a>

where the Fourier transforms F and G are defined by

F(e, ri, c) = /// f{x,y,z) e^'^^^"'^^"'^^^ dxdydz (20a)

G(c) =

/ g(x) e^^^ dx . (20b)

A straightforward, though somewhat tedious, manipulation can lead to the following Fourier transform pairs:

12
.2 ^ .2 ^ 2, ,, .V _ . _2 sin/r^ + n^ +

^^ f(x,y,z) = 6[q - /x^ + y^ + ?) ^

F(^,n,c) = W lil!ZL_±J_±J_A (21)

L2 , 2 ^ 2 ^

g(x) = 6(q - 1x1)

^

G(e) = 2 cos q ^ . (22)

Thus, the triple sum in (17) can be rewritten, after (18) and (21) are used,

as

I 6(k - k ) -I 6(k - A^)' + [^)' * (Pf)^)

ni,n,p=-°° ^ m,n,p=-co

= 4 abc 4 I s1n 2/a^m . b n -^ c p k

,23)

" "•"•?=- 2/a^m^ + b^n^ . c^p^ k

while the single sums in (17) can be rewritten, with the help of (19) and

(22), as

OO OQ -CO

I S(k - k J -I 6(k - |!^|) = ^ I COS 2amk (24) „ moo „£ ^ a ^

"n; „_ m=-co m--cD m--<»

I 5(k - k^^J =I 6(k - 1^1) = — i

COS 2bnk (25)

^ ono „i: ^ b ^ "n: „_

n=-oo n--<» n--°°

P = -oo ^

p

= -oo p=-oo

Substituting (23), (24), (25), (26) into (17), we get

13
dN ,u^ k^ V

s in iJ^^ + b^n^ + c^p^ k

-ip- = abc -^ ) -^——

' '^ 2/a m +bn +cp k

OO 00 00

^ - {<5^ y COS 2amk + o^ T cos 2bnk + o^ T cos 2cpk} + i 6(k) . m—00 n=-<» p —00

(27)

The last term of the above expression has been formally changed to 2-6(k),

since kgQQ = 0. Both (17) and (27) are complete expressions for the mode

density of a lossless rectangular chamber. By analogy to trigonometrical

Fourier series, we can express (27) as follows:

dN . k a + b + c _, 1 ,,, X _,

. ^ k vi sin 2/a m + b n + c p k ^ = abc -y 2^

+ J

6(k) + abc ^ I

71 TT m,n,p=-oo «/2 2 ^ .2 2 . 2 2 . *^ 2/a m +bn +cp k

CO L ^^ ^

- {2^ 5;' cos 2amk + 2^ J' cos 2bnk ^ ^ V cos 2cpk } (28)

m=-ao n=-o° p=-oo

where the prime (') in the triple sum denotes that the term with m = n = p

=

is to be excluded, while the prime in the single sums denotes that the term

with m = 0, or n = 0, or p

= is not included. The right-hand side of (28)

is thus separated into two constituents: the first three terms may be called

the "smooth component," and the rest the "fluctuating part" in accordance

with the terminology to be adopted for N(k) below.

The total number N(k) of modes can now be calculated by integration,

namely,

N(k) = / -j^dk , (29)

14
with the integration limits so selected that N{0) = as required by physical

reasoning. For k > 0, we have

N{k) =^k^ _±jLbjL£k +^

on

oK^ °° 1 sin 2 r^„^k

4ii; m,n,p=-«> r mnp ^ '^ mnp

'^

1 f V' sin 2 amk , y, sin 2 bnk . v' sin 2 cpki ,-.n,\ ~ '^ ^J m

"

/^ S

"" ^ p ^ ^-^"^

ni=-oo n=-oo p=-oo "^

with

r =/am+bn+cp. (31)

We can separate the right-hand side of (30) into two constituents--a smooth

component and a fluctuating part. The first three terms are the smooth

component N^, i.e.,

N (k) = -^ k^ - ^5-X^-i-^ k + i (k > 0) (32)

o-rt

or, in terms of frequency f,

3

N (f) = % abc ij- - (a + b + c) - + i (f > 0) (33)

S O O \) c V

where u stands for the speed of light in the medium (usually air) inside the

chamber. Note that the first term of (33) coincides with Weyl's asymptotic

expression [30 - 33], which is proportional to the volume abc of the chamber

and the third power of frequency. The second term is the "edge term" which

modifies Weyl's result, especially in lower frequency ranges. The comparison

of (32) or (33) with exact number N by computer-counting (16) reveals that

these formulas are applicable in the frequency range

'There are jt£ static eigenmode solutions to the type of hollow cavity as shown

in figure 1.

15
ka^ > M (34)

where a^ represents the smallest side of the rectangular chamber.

Some typical results are depicted in figures 3, 4, and 5 where the steplike solid curves with label 1 represent the exact values of N, the curves

with label 2 represent our result (33) of the smooth component, and those with

label 3 correspond to Weyl's asymptotic formula. It can be seen mathematically that our smooth-component expression (33) does, on the one hand,

approach Weyl's classical formula as f ^ <° and, on the other hand, gives

better agreement with exact N values in lower frequency ranges (e.g., in

microwave frequency ranges of interest) if the ratios a^ : b^ : c^ of the

rectangular chamber are not too rational (see figures 3 and 4). In this

sense, our smooth component U^{f) (which is deduced via geometrical approach)

is an improvement of Weyl's formula (which was derived by integral equation

technique) for the case of rectangular chambers. Also, our Ng(f) provides a clear insight into the meaning of Weyl's asymptotic formula.

From (28), we can also write down the "smooth component" D^ of the mode

density:

dN

or, in terms of frequency f,

dN dN ., D (f) = _A = _l.d^

^s^^^ df dk df

2 8 ^ abc -^ - ^

"" ^"" ^ + \ 6(f) . (36)

The curves shown in figures 3, 4, and 5 with label 4 correspond to the mode

density (36).

16
The triple sum and the three single sums in (30) represent the fluctuating part N^ of the number of modes N(k). The single sums are purely periodic oscillations with k-periods equal to rc/a, -n/b, or u/c respectively:

^ m

m=-°o

sin 2 amk-=71-2^ i n ^ i , / j \ ka; 0<ka<Ti; (mod n)

r

^"''" I ^^^

= t: - 2 kb ; < kb < 71 (mod 71) (37)

n=-oo

r

^"'" I

^P*^ = 71 - 2 kc ; < kc < 71 (mod 71) p=- r

The triple sum represents an irregular, nonperiodic function whose details

vary depending upon the a, b, c values and the frequency k. This can also be

seen qualitatively from figures 3, 4, and 5.

2.3 Design considerations of reverberating chambers

As already mentioned in the introduction, one of the purposes of investigating the total number N(f) of resonant modes which can exist inside an unperturbed chamber at a given microwave frequency f, is to establish analytical criteria for the validity of the indirect approach to solving the electromagnetic field problem associated with a reverberating chamber. One such

criterion is now supplied by the expression (33), namely.

3

N (f) = % abc ij- - (a + b + c) - + ^ (38)

S o o V c.

which is applicable for k > max {j, ?-, ^} . For example, at frequencies

higher than 200 MHz, the indirect approach can be reasonably applied to the

NBS chamber (2.74 m x 3.05 m x 4.57 m) since there are more than 83 resonant

modes available for eigenmode expansions as predicted by (38) (see curve 2 of

17
figure 3). As another application, we know from (38) that, for frequencies

lower than 110 MHz, the number of resonant modes available in the NBS chamber

is less than 12, so that a uniformly homogenous and isotropic field inside any

local region in the chamber can hardly be produced even with the help of the

rotating mode-sti rrer or tuner.

From a design point of view, we learn from (38) also that, in the microwave range, it is the volume abc of the chamber that essentially determines

the number of modes available. There is no term appearing in (38) which is

related to the inner surface area S = 2(ab + be + ca) of the chamber. The

edge term, i.e., the second term of (38), has only limited influence on Ng(f)

at microwave frequencies because it is of f^-order smaller when compared with

the volume term. In order to have a better stirring or tuning effect and to

achieve a more uniform field, a larger number of modes should be available.

In other words, a chamber of bigger volume will be preferable as is clearly

shown by (38).

Next let us consider another design question such as "what are the best

dimensions for a rectangular chamber, provided that its volume is given and

fixed?" If the volume abc is constant, the inner surface area 2(ab + be + ca)

and the total edge length 4(a + b + c) of the chamber, according to (38), have

little or no effect with respect to the smooth component Ns(f) of the total

number of modes at microwave frequencies or above. However, they do affect

the degeneracy of modes and, hence, the fluctuating part, significantly.

Since the resonant frequencies of eigenmodes are determined by

.2222

>^ = I!I^+I1^+ P^, (39)

-rt a b c

we see that, if the ratios a : b'^ : z^ are rational, there may exist more

than one set of (m, n, p) values giving the same value of k/-n; and the degeneracy of modes arises. Thus, we should design the dimensions of the chamber in

such a way as to keep the ratios a^ : b^ : c^ mutually not too rational in

order to reduce the degeneracy of modes and, hence, to increase the uniformity

in distribution of resonant modes. The cubic chamber (a = b = c) should especially be avoided.

18
To see this design principle, let us compare three rectangular chambers

which have the same volume of 38.191 m^ but different dimensions:

Chamber 1

(NIBS chamber)

a = 2.74 m

b = 3.05 m

c = 4.57 m

Figure 3 and figure 6 (40a)

Chamber 2 (square-base chamber)

a = 2.1755 m

b = 4.1899 m

c = b

Figure 4 and figure 7 (40b)

Chamber 3

(cubic chamber)

a = 3.3676 m

b = a

c = a

Figure 5 and figure 8 (40c)

For the frequency range 40 < f < 220 MHz considered in figures 3, 4, and 5,

the smooth components N5(f) are essentially the same for all three chambers

(see curves with label 2). The fluctuation of exact N-values (solid curves

with label 1) around the smooth components is mild for the NBS chamber (figure

3), relatively strong for the square-base chamber (figure 4), and violent for

the cubic chamber (figure 5), in agreement with our criterion on a^ : b^ : c^

as stated above.

Figures 6, 7, and 8 are further examples to illustrate the same fact,

where the frequency range 140 < f < 230 MHz has been chosen. The three

chambers have 110, 116, or 113 modes, respectively, in this frequency range.

They have about the same number of modes. The numbers AN of modes in each

interval Af = 1 MHz are computer-counted in order to show the distribution of

resonant modes. We can also see that the uniformity in mode distribution is

good for the NBS chamber (figure 6), fair for the square-base chamber (figure

7), and poor for the cubic chamber (figure 8) where no mode exists in a larger

number of frequency sub-bands.

In summary, based on the separation of the total number of eigenmodes

into a smooth component and a fluctuating part, we have established the

19
following two criteria which can be used for the design of rectangular

reverberating chambers for microwave interference measurements:

Criterion A. The volume abc of the chamber should be as large

as possible in order to have large values of ^^{f)

for stirring or tuning purposes. The total edge

length 4{a + b + c) can be set nearly equal to its

minimal value if there is no possible contradiction to criterion B.

(41)

Criterion B. The ratios a : b*^ : c of edge lengths squared

should not be too rational in order to reduce the

fluctuating part N|(f) and, hence, to increase the

uniformity in mode distribution.

(42)

2.4 A brief review of mode degeneracy

As already mentioned, there are two causes of eigenmode degeneracy. The

first cause is due to the vector property of Maxwell's equations or, equivalently, the transverse property of electromagnetic wave motions. As seen in

Table 1, there are two independent eigenmodes corresponding to each threedimensional lattice point (m, n, p) with m > 1, n > 1, p > 1.

The second cause of degeneracy is due to the number-theoretical property

of the eigenfrequency expression (39), where, at most, one of the integers m,

n, p is allowed to be zero. Evidently, there may be more than one set of

integers (m, n, p) corresponding to a given eigenfrequency k, especially when

9 9 9

the ratios a : b^ : c are rational.

To sum up, let Z3(k; m, n, p) be the number of sets of integers (m n, p), m > 1, n > 1, p> 1, satisfying (39); and let Z21^'^' "» P^ denote the number

of sets of integers (0, n, p), m = 0, n>l, p>l, satisfying (39) with

similar denotation of Z22(k; m, p) and Z23(k; m, n). Then the total degeneracy G(k) of modes associated with the eigenfrequency k is given by

20
G(k) = 2 Z3(k; m, n, p) + Z2i(k; n, p) + Zi^i^: m, p) + Z23(k; m, n).

At higher frequencies where k > 1, this can be approximated by

G{k) s 2 Z3(k; m, n, p) .

Since the total number of electromagnetic eigenmodes with eigenfrequencies

less than or equal to some given value k' is

N(k') =

I G(k) , k<k'

k satisfying (39), and since G(k) increases very rapidly with k, a rough

approximation for large k' can be made as follows

N(k') - 2

I z.(k; m, n, p) , k' > 1 , k<k'

^

which means that, at higher frequencies, the total number of electromagnetic

eigenmodes is roughly twice that of scalar (such as acoustic) eigenmodes.

Again, this can be attributed to the transverse property of electromagnetic wave motions in contrast to the longitudinal property of acoustic wave

motions.

Unfortunately, little knowledge concerning the "analytic" properties of

the discrete, number-theoretical functions Z2-j and Z3 is available, so we will

not try to deal with mode degeneracy in more detail. Instead, Criterion B has

been set up in section 2.3 to serve for practical design purposes.

3. A composite Q-factor for lossy rectangular chambers

So far, we have considered only rectangular chambers with perfectly con- ducting metal walls. In practice, however, the steel, aluminum, or other

^In some technical papers, e.g., [6], the mode number and mode density are in error by a factor of, roughly, 2. The expression N == 4ti abc f /(3 u ) is

often used instead of the correct one given by the leading term of our result

(38) for N5(f).

21
metal walls are not perfect so that the electromagnetic fields will penetrate

into walls and cause ohmic losses. For general structures, this phenomenon

may be described with the help of the skin depth and/or the surface resistance. For the case of cavity resonators, the quality factor Q is a more

suitable quantity to account for dielectric and wall losses. In this section

we will be concerned with the calculation of Q-factor of a rectangular chamber

due to its wall losses.

Once the z-di recti on is chosen to be parallel to one of the edges, say

edge c, of the rectangular chamber, the Q-factors of the TE^^^, TE^^^, TE^^^,

TM^P^, and TM^pi modes are already known and explicitly given [18, p. 190].

These, however, are not convenient for practical applications. Since there

are a lot of resonant modes (hundreds, thousands, ten thousand, or more modes)

which can exist and be stirred or tuned inside a reverberating chamber at

microwave frequencies, it is surely more convenient to define a composite Q in

order to describe the ohmic loss behavior of the chamber. In this section, a composite Q will be derived by considering a suitable average of 1/Q in the kspace.

3.1 Previous results of Q's for rectangular cavities

With the coordinate system chosen, as shown in figure 1, such that the zdi recti on is parallel to edge c, the Q due to conductor losses for the various

modes in a rectangular cavity [18 p. 190] as outlined in Tables 1 and 2 are listed below.

/ X -rp n abc k

Te!^^' modes: q'^^ = , !-, ^

(43)

°"P °"P 2R(bc k^ + 2ac k^ + 2ab k^)

(7) TF ^ ^^^ ^r

IVJ-^ modes: q'^„ = , ^j 5- (44)

•"^P '"^P 2R(ac k^ + 2bc k^ + 2ab k^)

22
^) ...... nTE , ^ °"^ ^y ^'

TM^^; modes: Q''' = , 1 5- (46) ""^ "^"° 2R(ab k^ + 2bc kj + 2ac k^)

r X y

TM^^i modes: QTM = " '^' ""^ ^"^ ^

(47

"^"P """P 4R[b(a -f c) kj + a(b + c) k^]

X y

with

k="^ k="^ k=P^ k- /k2 + 1,2

k, - /^FT^ = /(^]'^(^)'-(^)' = k^^p . (48)

Here ti denotes the intrinisic impedance of the medium inside the chamber

(ri = HJz = 120tc ohm:

the metal wall given by

ix] = /|i /e = 120tc ohms for air medium), and R is the surface resistance of

R = -^— (ohms)

where a (siemens/m) is the conductivity of the metal and 6^ the skin depth.

Expressions (43) - (47) are the starting point of our consideration.

3.2 Construction of a composite Q

The central idea of constructing a composite Q is to form a lattice aver- age of 1/Q-values of all resonant modes, taking into account the fact that

each three-dimensional lattice point (m, n, p) with m>l, n>l, p>l

corresponds to two modes, and that the number of these three-dimensional

lattice points is much larger than that of two-dimensional lattice points when

the frequency kp under consideration is high enough. When k is large (see

figure 9), the lattice points (m, n, p) corresponding to (48) inside the

23
elliptical shell k < k^ < k + Ak are densely distributed, so that lattice sums over this shell can be approximately replaced by suitable integrals.

Let us consider the three-dimensional lattice sum of l/oj^pp over the

shell k < kp < k + Ak in the first open octant, namely,

-, k+Ak . h'- J -JW-r :' 1^^ -JW-r -dmdndp. (49)

m,n,p Q^^p(m, n, p) k^=k Q^^p(m, n, p)

After introducing the spherical coordinates k^, e, 4; according to

^ = k^ sine cosci; k^ e (k, k + Ak)

a r r ^ = k^ sine sinci, e e (0, -k/2)

£^ = k^ cose (1; e (0, 7i/2) (50)

and rewriting l/Q^pp in terms of these new coordinates as

TT" " ~^

ir ^^^ "^ ^^^ ^°^^^ ^ ^ sin^(i;)] . (51)

^mnp

^

The integral I^ can be performed, leading to the result

I^ ^ 2Rk

[ab + 1. c(a + b)] Ak . (52)

The three-dimensional lattice sum of l/Q^pp over the shell k < k^ < k + Ak in

the first open octant

1 k+Ak , I2 =

I -j^ = /// -yr dm dn dp (53)

m,n,p ' '^ Q (m, n, p) k =k Q (m, n, p) ^mnp ' ' f" r ^mnp ' ' ^

can be treated analogously. Using the coordinates (50) and the corresponding

expression

24
1 4.R 1 2 2

^mnp

+ 4 c(b + a) cos e(l + ?. I

Z (sin cj; - cos i\>)]] , (54)

we have

I2 = ^

[y (ab + be + ca) + ^ c(a + b)] Ak . (55)

Next, let us turn to the two-dimensional lattice sum of I/qJ^o ^^^^ ^^^ strip k < kp < k + Ak in the first open quadrant of the mn-plane, namely,

, k+Ak ^

' n^.n QTM^{m, n) k^=k Q^^^lm, n)

The introduction of the polar coordinates k^, 4; according to

^ = k^ cosci. k < k^ < k + Ak

^= k^ sin 4. < 4; < J(57)

and the rewriting of l/pJino ^" ^^^ following polar coordinate form

1 2R 1 2 2

"TfT " TTab^ k~ I^*^ "^ ^^^^ cos (]; + a sin c^)] (58)

^mno ^

lead to the result

R

3 tiCti; (ab + be + ca)Ak . (59)

By applying the same technique, we can evaluate the other two-dimensional

lattice sums

25
n,p Q^^p(n, p)

r

ni,P Q^op{m, p)

over one- fourth of the strip k < k^ < k + Ak in the np-plane or mp-plane

respectively. The results are

I. s -^ (ab + be + ca)Ak (62)

Ic = -R- (ab + be + ca)Ak . (63)

r|DTi

In order to calculate the average of 1/Q-values, we now need the total

number of modes AN corresponding to the shell k < k^ < k + Ak in the first

octant. This can be approximated by twice the number of three-dimensional

lattice points (m, n, p) inside that shell. Using the terminology of Tables 1 and 2 of section 2.1, this is equal to'

AN = ANj^ + AN2 + AN3 + AN4 + AN5

= 2 aN^

s 2 /// dm dn dp = ^^

Ak . (64)

k^=k Tu

The average of all the 1/Q-values with z-axis parallel to edge c can be

defined as

'In fact, the approximation AN = (abc/u ) k^Ak used here could have also been

obtained by taking AN s (dN/dk) Ak and retaining only the first term of (28)

for dN/dk.

26
The substitution of (52), (55), (59), (62) - (64) into (65) gives

A> = -i^ 2 |i + 1^ (i + 1 + 1)1 , (66)

Q r\ abck 3 ^ 8k ^a D c^ '

where

S = 2(ab + be + ca) (67)

is the total inner surface area of the chamber. The form of (66) is invariant

with respect to the choice of z-direction, so that it can be taken as a generally valid result, although it is derived under the arbitrary choice of z-axis

along the edge c. Taking the inverse of can be defined for the rectangular chamber;along the edge c. Taking the inverse of <-^> , a composite quality factor Q

^

"" El^ ^a F"" ?^

Noting that

we have the alternative form

Tl abc

2 R S

"^ S 6 ^^^^

^ ^ ^ W ^a b 3^

where V denotes the volume of the chamber. At high frequencies with kaQ > 1,

aQ = min {a, b, c}, our result thus predicts for the reverberating chamber a composite Q-factor which is 1.5 times larger than that of the conventional

V/(S 65) estimate.

As numerical checks of the validity of expression (70), we considered the

NBS chamber at three different frequency "shells" of bandwidth 20 MHz for

each,

180 < f < 200 MHz ( 23 modes) Figure 10

330 < f < 350 MHz ( 87 modes) Figure 11 (71)

480 < f < 500 MHz (178 modes) Figure 12

27
and computed the (normalized) 1/Q-values of resonant modes of various types by

using formulas (43) - (47). Each value of 1/Q was computed three times

according to the three orthogonal directions. For example, 69 1/Q-values were

obtained for the case of 180 < f < 200 MHz. These values are plotted as a normalized cumulative distribution curve depicted in figure 10; their arithmetical mean and standard deviation are also shown. From figures 10, 11, and

12, it is seen that, as the frequency goes higher, the arithmetical mean of

the normalized 1/Q-values moves closer and closer to the theoretical limit of

2/3 = 0.667 as predicted by (70), while the standard deviation decreases from

0.090 to 0.074, showing that more 1/Q-values become concentrated near the

theoretical limit. Hence, the validity of the composite quality factor (70),

which is a relatively simple expression, is checked and can be used for design

purposes.

3.3 Discussion of results

From the derivation leading to (66) and (70), or equival ently, from the

factor

appearing in (66) and (70), we see that the main contribution to the final

result of <7r> or Q comes from the TE^^p and TMj^^ modes (with respect to x-,

y-, or z-direction) that correspond to the three-dimensional lattice points

(m, n, p) with m > 1, n > 1, p > 1. The contribution from the TE^^p, TE^^^p,

and TM^pQ modes (with respect to x-, y-, or z-direction), that correspond to

two-dimensional lattice points (see figures 2 and 9) is of k-order smaller, as the second term of (72) shows.

On the other hand, a clear insight into the individual Q-values of the

various eigenmodes can be obtained by considering the special case of a cubic

cavity with a = b = c. Starting from (43) - (47) and noting (69), we can readily arrive at the interesting, extremely simple result, namely.

i

28
TE,„p modes:

qJ'p

= ^

<"^'

s

TE^op^des: Q^^p = s^ (73b)

TM,noTMdes: QTMo = ^

<"'^'

s

TETMnp"odes: Q^E^

= | (^) (74a)

TMm„pTM°des: QTM^ = | (^) (74b)

where TE or TM means transversal electric or magnetic with respect to any of

the X-, y-, and z-directions which are now equivalent because of the cubic

structure of cavity. From (73) and (74) we see that, for the cubic cavity,

the Q associated with each of the TE^p and TM^^^p modes (corresponding to

V 3V three-dimensional lattice points) has the value ac l » while the Q associated

with each of the TEQpp, TE^^p, and TM^^pQ modes (corresponding to twoX 2V dimensional lattice points) has the value <^

. If the dimension of the

^ ^s

cavity deviates from that of a cube, then the Q-values of the TE^^pp and TMj^^p

3V

modes will be expected to spread out around g^ c » while those of the TEq^^,

s 2V TE^Q , and TM^^^ modes will spread out into a vicinity of

^

. s

Based on these observations, we can now give a plausible interpretation

to the composite quality factor Q for the rectangular chamber with sides a, b,

c. Since <jr> , as gi

z-di recti on, that is,

c. Since <-Fr> , as given by (66), is invariant with respect to the choice of

<^lz,ic = 4^lz,,b = <^lz,a = <k • '"'

we can consider it to be the average of three (identical) quantities obtained

by choosing z-axis parallel to edges c, b, and a, respectively:

Q T I Q Iziic Q Izilb

29

Iziib ' i>\zJ ' ^''^
The quantities V^Lnr ' %^|ziib ' ^U^lziia

^l^^selves are lattice averages of

the individual 1/Q-values which are now distributed either in the vicinities

of

-J (S 6 /V) (corresponding to three-dimensional lattice points) or in the

vicinities around j (S 6-/V) (corresponding to two-dimensional lattice

points), due to deviations from "cubicness." The three cases of deviation

from cubicness, corresponding to three different choices of z-axis, namely,

ziic, ziib, and ziia respectively, will cancel one another to some extent when

the overall average (76) is taken. In other words, the final result

of <-7y> for a rectangular chamber should not differ much from that for a cubic

cavity of the same volume. Indeed, by comparing (73) and (74) with our result

(66), and recalling the fact that the three-dimensional lattice points are dominant in number at higher frequencies, we see that the above interpretation

of <^> and, hence, of Q is satisfactory and can be used to shed some physical

insight into the expressions (70) for Q.

It seems interesting to intuitively account for the factor 2/3 appearing

in the expression (66) for <-^> via arguments similar to those used in [5].

For a randomly varying field, there are 3 degrees of freedom with respect to

the E-polarization, while there are only 2 degrees of freedom concerning the E -polarization of the partial plane waves that add together to form the standing wave patterns in a rectangular chamber.

4. (^inclusions

Among many problems pertaining to the operation and design of modestirred or mode- tuned reverberating chambers, we have solved here two problems

from a design point of view, namely that of the distribution of resonant modes

and that of the composite quality factor of the chamber.

A complete solution is obtained for the total number N(f) of resonant

modes with resonant frequencies less than or equal to f. Due to its stepwise

character, it is conveniently split into a smooth component N^ and a fluctu-

-ating part N^. The former depends at microwave frequencies essentially only

on the volume, while the latter is sensitive to the dimensions of the rectangular chamber. Simple criteria are then formulated concerning the design of

30
the dimensions of a rectangular chamber. Exact expressions for the mode

density dN/df are also obtained.

To take into account the ohmic loss in the metallic walls, a composite

quality factor Q is proposed; it is simple and closed in form. The limiting

value of this unloaded Q differs from the conventional V/(S 65) estimation by

a factor 3/2, where V represents the volume of the chamber, S its inner sur- face area, and 63 the skin depth of the lossy walls of the chamber at the

operating frequency.

Since compromises between low conductor loss (high Q) and broad modal

coverage (low Q) are almost always necessary in the practical design of

reverberating chambers, our results on the distribution of resonant modes and

on the composite quality factor presented in this report will be found convenient and helpful in design applications. We emphasize the fact, however, that

these factors should be used for preliminary design considerations. Under

practical conditions, when equipment is placed inside the chamber for test,

the volume and surface of the chamber and the surface resistance will change,

causing the loaded Q and distribution of resonant modes to be different

depending on the size and shape of the equipment under test and the material

of which the equipment is made.

5. Acknowl edgements

The authors are grateful to Dr. M. Kanda and M. L. Crawford for their

technical discussions during the course of this work. They also wish to thank

C. K. S. Miller for his continued interest and support.

6. References

[1] Adams, J. W. Electromagnetic interference measurement program at the

National Bureau of Standards. USNC/URSI Annual Meeting Digest, p. 33;

1975.

[2] Baltes, H. P.; Kneubuhl , F. K. Thermal radiation in finite cavities.

Helvetica Physica Acta 45^: 481-529; 1972.

[3] Bean, J. L. ; Hall, R. A. Electromagnetic susceptibility measurements

using a mode-stirred chamber. Proc. IEEE International EMC Symposium,

Atlanta: 143-150; 1978 June 20-22.

31
[4] Brownell , F. H. Extended asymptotic eigenvalue distributions for

bounded domains in n-space. J. of Math, and Mech. 6^: 119-166; 1957.

[5] Chang, D. C. ; Prehoda, R. ; Puglielli, V. G. On shielding effectiveness

measurements of metallic enclosures in a coherent and a completely

incoherent environment. Private communication; 1979.

[6] Corona, P. Electromagnetic reverberating enclosures: behavior and

applications. Al ta Frequenza 49^: 154-158; 1980.

[7] Corona, P.; Latmiral, G. Evaluation and use of the reverberating chamber of the Istituto Universitario Naval e. First National Meeting of

Applied Electromagnetism (Italy), Aquila; 1976 June 24-25.

[8] Corona, P.; Latmiral, G. Thermodynamic approach to the study of a variable geometry electromagnetic reverberating room (in Italian). Istituto

Universitario Naval e, Naples, Italy; 1978.

[9] Corona, P.; Latmiral, G; Paolini, E; Piccioli, L. Performance of a reverberating enclosure for power measurements in the microwave range.

Proc. 2nd Symposium and Technical Exhibition on EMC, Montreux, p. 419-

423; 1977 June 28-30.

[10] Corona, P.; De Bonitatibus, A.; Ferrara, G. ; Paolini, E. Electromagnetic enclosures behavior--comparison data analysis. Proc. 3rd Symposium

and Technical Exhibition on EMC, Rotterdam, p. 231-236; 1979 May 1-3.

[11] Corona, P.; Latmiral, G.; Paolini, E. Performance and analysis of a reverberating enclosure with variable geometry. IEEE Trans. EMC-22: 2-

5; 1980.

[12] Corona, P.; Latmiral, G.; Paolini, E.; Piccioli, L. Use of a reverberating enclosure for measurements of radiated power in the

microwave range. IEEE Trans. EMC-18: 54-59; 1976. Also see IEEE

Trans. EMC-J^: 205; 1976 for correctTons.

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

cells. IEEE Trans. EMC-26_: 189-195; 1974.

[14] Crawford, M. L. ; Workman, J. L; Thomas, C. L. Generation of EM susceptibility test fields using a large absorber-loaded TEM cell. IEEE

Trans. IM-26_: 225-230; 1977.

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

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

204-210; 1978.

[16] Crawford, M. L. ; Workman, J. L; Thomas, C. L. Expanding the bandwidth

of TEM cells for EMC measurements. IEEE Trans. EMC-20_: 368-375; 1978.

[17] Donaldson, E. E. ; Free, W. R. ; Robertson, D. W.; Woody, J. A. Field

measurements in an enclosure. Proc. IEEE 66: 464-472; 1978.

32 1
[18] Harrington, R. F. Time-harmonic electromagnetic fields. New York,

NY: McGraw-Hill; 1961.

[19] Jeans, J. H. On the partition of energy between matter and aether.

Phil. Mag. (Ser. 6) 20_: 91-98; 1905.

[20] Jones, D. S. The theory of generalised functions. Cambridge:

Cambridge Univ. Press; 1982.

[21] Lentz, R. R. Use of a reverberating room in microwave oven choke

design. J. of Microwave Power ]A: 29-33; 1979.

[22] Lentz, R. R. ; Anderson, H. C. Reverberating chambers for EMC measurements. Proc. IEEE International EMC Symp., San Diego: 446-451; 1979

October 9-11.

[23] Lord Rayleigh (Strutt, J. W.) The dynamical theory of gases and of

radiation. Nature 72: 54-55; 1905. The constant of radiation as calculated from mol ecuTar data. Nature _72_: 243-244; 1905.

[24] Ma, M. T.; Koepke, G. H. A method to quantify the radiation characteristics of an unknown interference source. Nat. Bur. Stand. (U.S.) Tech.

Note 1059; 1982.

[25] Papoulis, A. The Fourier integral and its applications. New York,

NY: McGraw-Hill; 1962.

[26] Roe, J. M. An improved technological basis for radiated susceptibility

and emission specifications. Proc. International EMC Symposium,

Atlanta: 48-52; 1978 June 20-22.

[27] Sreenivasiah, I.; Chang, D. C. ; Ma, M. T. Characterization of electrically small radiating sources by tests inside a transmission line

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

[28] Sreenivasiah, I.; Chang, D. C; Ma, M. T. A method of determining the

emission and susceptibility levels of electrically small objects using a TEM cell. Nat. Bur. Stand. (U.S.) Tech. Note 1040; 1981.

[29] Sreenivasiah, I.; Chang, D. C. ; Ma, M. T. Emission characteristics of

electrically small radiating sources from tests inside a TEM cell. IEEE

Trans. EMC-23_: 113-121; 1981.

[30] Weyl , H. Das asymptotische verteilungsgesetz der eigenwerte linearer

parti el ler differentialgl eichungen (mit einer anwendung auf die theorie

der hohlraumstrahlung). Math. Annalen_71.: 441-479; 1912.

[31] Weyl, H. Liber die abhangigkeit der eigenschwingungen einer membran von deren begrenzung. J. reine u. angew. Math. 141 : 1-11; 1912.

[32] Weyl, H. IJber das spektrum der hohlraumstrahlung. J. reine u. angew.

Math. 141: 163-181; 1912.

33
[33] Weyl , H. Uber die randwertaufgabe der strahlungstheorie und

asymptotische spektralgesetze. J. reine u. angew. Math. 143 : 177-

202; 1913.

34
Figure 1. A rectangular chamber showing orientation

conventions adopted in this report.

35
ZTt/c

R

PTt/c jQ_

jltiIq.

yrr/b

Figure 2. Lattice points in k-space. Point P represents a general 3-

dimensional lattice point. Also shown are the 2-dimensional

lattice points Qi, Q2, Q3; the 1-dimensional lattice points

R-]^, R2, R3; and the origin.

36
bO o r-\

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43
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45
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46
NBS-n4A (REV. 2-80) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN-1066

2. Performing Organ. Report No 3. Publication Date

August 1983

4. TITLE AND SUBTITLE

Eigenmodes and the Composite Quality Factor of a Reverberating Chamber

5. AUTHOR(S)

B Bing-Hope Liu, David C. Chang, and Mark T. Ma

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

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

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

10. SUPPLEMENTARY NOTES

\2^ Document describes a computer program; SF-185, FIPS Software Summary, is attached.

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

bibliograpliy or literature survey, mention it here) The total number N of electromagnetic eigenmodes, with eigenfrequencies

not greater than some given value, which can exist inside a rectangular

mode-Stirred or mode- tuned reverberating chamber is important in that it

reveals how many modes can be available at an operating frequency for the

"stirring" or tuning purpose. This is calculated analytically via a latticepoint counting technique in the k-space (k = wave number), leading to an exact expression for N, which can be split into a smooth component and a fluctuating part. The former contains, in addition to Weyl's volume term,

an edge term as a second-order correction. The latter is sensitive to the

dimensions of the chamber. Simple design criteria are then derived in view

of the number of available modes and the uniformity of their distribution.

To take into account the ohmic loss in metal walls of the chamber, a composite

Q-factor is also proposed for design purposes. This is achieved by taking a suitable average of 1/Q-values of all possible modes within a small frequency

interval. Comparison with numerical Q-values for individual modes shows

that the composite Q can-be used as a practical design parameter.

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

cavity; composite quality factor; eigenfrequency; eigenmode, electromagnetic

field; mode density; mode number; reverberating chamber.

13. AVAILABILITY

^

Unlimited

I I For Official Distribution. Do Not Release to NTIS

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

\2Z\ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

54

15. Price

$4.00

USCOMM-DC 6043-P80

U.S. GOVERNMENT PRINTING OFFICE: 1983—680-904/442 REGION NO. 8



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