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National Institute of Standards and Technology

N/ST Technical Note 1361

Aperture Excitation of Electrically

Large, Lossy Cavities

David A. Hill

John W. Adams

Mark T. Ma

Arthur R. Ondrejka

Bill F. Riddle

Myron L. Crawford

Robert T. Johnk

//1 361 ^^^^^^^^^^^^^^^^^^H

Aperture Excitation of Electrically

Large, Lossy Cavities

David A. Hill

John W. Adams

Mark T. Ma

Arthur R. Ondrejka

Bill F. Riddle

Myron L. Crawford

Robert T. Johnk

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

September 1993

^W^ OF

U.S. DEPARTMENT OF COMMERCE, Ronald H. Brown, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arati Prabhakar, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1361, 76 pages (September 1993)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1993

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
CONTENTS

Page

Abstract 1

1 . INTRODUCTION 1

2 . CAVITY QUALITY FACTOR 2

2 .

1

Wall Losses 4

2 . 2 Absorption 5 2 . 3 Aperture Leakage 6

2.4 Receiving Antenna 7

3 . APERTURE EXCITATION 8

3 . 1 Apertures of Arbitrary Shape 9

3 . 2 Circular Aperture 11

4 . POWER BALANCE 13

4 . 1 Shielding Effectiveness 13

4 . 2 Time Constant 15

4 . 3 Source inside Cavity 16

5 . MEASUREMENTS 17

5 . 1 Measurement Techniques 17

5.2 Comparison of Measured and Theoretical Results 21

6 . CONCLUSIONS 24

7 . ACKNOWLEDGMENTS 25

8 . REFERENCES 25

APPENDIX A. ABSORPTION BY A HOMOGENEOUS SPHERE 27

APPENDIX B. TRANSMISSION CROSS SECTION OF A SMALL CIRCULAR APERTURE 34

APPENDIX C . RADAR CHARACTERISTICS 35

APPENDIX D_..„ SCALING 36

APPENDIX E. ELECTRICAL PROPERTIES OF SALT WATER 41

APPENDIX F. SOFTWARE TO ANALYZE A RECTANGULAR CAVITY ^1

111

APERTURE EXCITATION OF ELECTRICALLY LARGE, LOSSY CAVITIES

David A. Hill, John W. Adams, Mark T. Ma, Arthur R. Ondrejka,

Bill F. Riddle, Myron L. Crawford, and Robert T. Johnk

Electromagnetic Fields Division

National Institute of Standards and Technology

Boulder, Colorado 80303

We present a theory based on power balance for aperture

excitation of electrically large, lossy cavities. The theory

yields expressions for shielding effectiveness, cavity Q, and

cavity time constant. In shielding effectiveness calculations,

the incident field can be either a single plane wave or a uniformly random field to model reverberation chamber or random

field illumination. The Q theory includes wall loss, absorption

by lossy objects within the cavity, aperture leakage, and power

received by antennas within the cavity. Extensive measurements

of shielding effectiveness, cavity Q, and cavity time constant

were made on a rectangular cavity, and good agreement with

theory was obtained for frequencies from 1 to 18 GHz.

Key words: absorption; aperture; cavity; polarizability ; quality

factor; reverberation chamber; scaling; shielding effectiveness;

time constant; transmission cross section.

1 . INTRODUCTION

In many electromagnetic interference (EMI) problems, the important

electronic systems are located within a metal enclosure with apertures. In

such cases it is important to know the shielding effectiveness (SE) of the

enclosure so that we can relate the interior fields to the external incident

fields. For example, the high intensity radiated fields (HIRFs) suggested

in SAE draft advisory circular AE4R [1] are in the 1000 to 7000 V/m range

for frequencies above 400 MHz, the frequencies used by most radars.

However, the shielding effectiveness of aircraft skins is not well known, so the interior fields that excite the aircraft electronics are not well

characterized.

The purpose of this report is to develop a mathematical model for the

shielding effectiveness of electrically large enclosures that contain
apertures and interior loading. The method that we present uses a power

balance approach and yields a simple, approximate expression for the average

field strength throughout the cavity enclosed by the enclosure. This method

does not yield the fine detail of the interior fields as would be obtained

by a numerical method [2] , but it has the advantages of being applicable to

both cw and pulsed fields and not requiring all the geometrical details of

the enclosure, the apertures, and the loads. The theory is similar to that

developed for reverberation chambers [3-5], where the average power density

is assumed to be uniform throughout the chamber.

The organization of this report is as follows. Section 2 contains a derivation for the cavity quality factor (Q) of an electrically large cavity

where the losses include skin effect in the walls, volume absorption by

lossy objects within the cavity, leakage through apertures, and power

absorbed by receiving antennas. Section 3 covers aperture excitation of the

cavity by an external field. Section 4 contains a power balance derivation

of the shielding effectiveness and the time constant of the cavity. Section

5 presents measurement methods and comparisons of measured and theoretical

values of SE, Q, and time constant. Section 6 presents conclusions and

recommendations for further work. A number of appendices contain

mathematical and software details.

2. CAVITY QUALITY FACTOR

The Q of a cavity is a key quantity in calculating both SE of the

surrounding shield and the time constant of the cavity. We consider an electrically large (multimode) cavity as shown in figure 1. In general the

cavity will have apertures and will contain lossy objects and at least one receiving antenna. The definition for Q is

Q = "U,/P^, (1)

where w is the excitation (angular) frequency, U is the steady state energy

in the cavity, and P is the dissipated power. In our application the
cavity is excited by an exterior field incident on apertures, but the Q

theory does not depend on the excitation method. Thus this theory is also

applicable to reverberation chambers excited by antennas.

We can write the steady state energy U as the product of the energy

density W times the cavity volume V:

U = WV. (2)

s

In eq (2) we assume that W is uniform throughout the cavity volume, and this

uniformity has been shown for the average energy density in reverberation

chambers [4]. In reverberation chamber measurements, an ensemble average is

obtained by rotating a mode stirrer, and in our theory we deal with ensemble

averages that could represent averages over different cavity shapes. It is

useful to write W in terms of the RMS electric field E:

W = eQE^ (3)

where €„ is the permittivity of free space and we have assumed the stored

2 electric and magnetic energies are equal. E should also be interpreted as an ensemble average. It is also useful to write an expression for the power

density S within the cavity:

S^ = eV^Jq = cW, (4)

1/2 1/2 where ry- = (/i /e_) '

, o. = l/(/i_e^) '

, and \i is the magnetic permeability

of free space. S should also be interpreted as an ensemble average.

The dissipated power can by written as the sum of four terms:

P=P +P +P +P (5) d dl ^d2 d3 d4' ^ '
where P,^ is the power dissipated in the cavity walls, P _ is the power

absorbed in loading objects within the cavity, P „ is the power lost through

aperture leakage, and P is the power dissipated in the loads of receiving

antennas. The forms of eqs (1) and (5) suggest that we write an expression

for the inverse of Q:

Q"^ = Q'^ + Q'^ + Q'^ + Q'\ (6)

where Q^ = a.U^/P^^

, Q^ = cU^/P^^ ' ^3 =

'^'^s^^dZ' ^^^ % ='^"'s^^d^- (^>

The smallest of Q^ , Q_ , Q_ , and Q, will be the dominant contributor to Q,

We now consider the individual loss mechanisms.

2.1 Wall Losses

For highly conducting walls the wall loss can be determined from the

skin depth approximation. For reverberation chambers, the wall loss has

been determined both by averaging over the individual cavity modes [6] and

by averaging plane wave losses over all incidence angles and polarizations

[5]. The latter approach is consistent with our ensemble average theory.

In either case, the result for Q^ is

3V 1/2

^1 = 27^' ^^^^^ ^ = (2/a;M„a^) , ^^ = ^^y^^^, (8)

where S is the cavity surface area, 5 is the skin depth, n is the wall

permeability, and o is the wall conductivity.

Complications, such as variations in wall conductivity and permeability

with position, could also be treated using the skin depth approximation, but

the skin depth would then be a function of position. Wall coatings, such as
paint, could also be included, but we would still lump all wall losses into

2.2 Absorption

In general the absorption cross section o of a lossy object depends on

Si the incidence angle and polarization of the incident plane wave [7]. Since

we assume that the fields in an electrically large cavity can be written as

a superposition of plane waves of all incidence angles and polarizations

[8] , the absorbed power can be written as the product of the cavity power

density S and the averaged absorption cross section <a >:

c a

Ph9 = S <a >, (9) dz c a

where < > indicates an incidence angle average over hn steradians and an average over all polarizations

.

The absorption cross section in eq (9) can be that of a single object

or a summation for a number of absorbers. For example, for M absorbers <a >

in eq (9) would be replaced by:

M

<a > = 2 <o .>, (10) a . , ai 1=1

where <o .> is the averaged absorption cross section of the ith absorber.

For the example of an aircraft, the i summation in eq (10) could represent

absorption by people, seats, wiring, etc.

For the simple case of a uniform sphere, the absorption cross section

is independent of the incidence angle and polarization of the incident field

and no averaging is required. The cross section formulation for a uniform

sphere is given in Appendix A.
If we use eqs (2), (4), (7), and (9), we can derive the following

expression for Q-

:

^ 27rV .-,-,. ^2 = I^TT' (11>

where A is the free -space wavelength. The frequency dependence of Q- can be

fairly complicated because of the frequency dependence of <o >. (See

di Appendix A for the frequency dependence of the absorption cross section of a homogeneous sphere .

)

2 . 3 Aperture Leakage

In general the transmission cross section o . of an aperture depends on

the incidence angle and polarization of the incident plane wave [9] . We

again assume that the fields can be written as a superposition of plane

waves of all incidence angles and polarizations , but only the plane waves

that propagate toward the apertures contribute to the transmitted power.

Consequently, we introduce a factor of 1/2 in the product expression for the

leakage power:

^d3 = ^<V/2. (12)

where < > again represents an average over incidence angle and polarization.

The transmission cross section in eq (12) can be that of a single

aperture or a summation for a number of apertures. For N apertures, <o > in

eq (12) would be replaced by

N

<a^> = S <a^^>, (13)

i=l
where <a ..> is the averaged transmission cross section of the ith aperture.

For the example of an aircraft, the N summation in eq (13) could represent

leakage through windows and other apertures of different shapes and sizes.

Specific expressions for transmission cross sections are given in Section 3.

If we use eqs (2), (4), (7), and (13), we can derive the following

expression for Q„

47rV

For electrically large apertures, <o g> is independent of frequency, and Q_

is proportional to frequency. For small or resonant apertures, the

frequency dependence of Q- is more complicated.

2.4 Receiving Antenna

The power P ,, dissipated in the load of the receiving antenna can be

written as the product of the cavity power density and the averaged

effective area of the receiving antenna:

P ,, = S <A >, (15) d4 c e

where the effective area A is also averaged over incidence angle and

polarization. The average effective area <A > of a lossless antenna can be

written as the product of the effective area of an isotropic antenna

(A^/47r), the i of 1/2 [10,11]

2 (A /47r) , the impedance mismatch factor m, and a polarization mismatch factor

<A > = f^. (16)

e on
<A > in eq (16) does not depend on the antenna pattern because it is

averaged over all possible incidence angles and polarizations [8]. The

impedance mismatch factor m is equal to 1 for a matched load and is less

than 1 otherwise.

If we use eqs (2), (4), (7), (15), and (16), we can derive the

following expression for Q ,

Q, - ^. (17)

mA

If there is more than one receiving antenna in the cavity, then <A > in eq

(15) is replaced by a sum of averaged effective areas. For a matched load

(m = 1) , Q, is proportional to frequency cubed. This means that Q, is small

for low frequencies and is the dominant contributor to the total Q in eq

(6). The effect of antenna loading on the Q of reverberation chambers has

been observed experimentally [12]. At high frequencies, Q, becomes large

and contributes little to the total Q.

3. APERTURE EXCITATION

Consider a cw plane wave of power density S. incident on the shield

apertures as shown in figure 1. If the total transmission cross section of

the apertures is o , the power P transmitted into the cavity is

p, = Vi- <"'

(Of course power will also leak out through the apertures, but we lump that

effect under leakage loss P,„ as discussed in Section 2.3.) For the general

di case of N apertures, o can be written as a sum:

I
1=1

where a . is the transmission cross section of the ith aperture. In

ti '^

general, a . and a depend on the frequency, incidence angle, and

polarization of the incident field.

If the shielded cavity is illuminated in a reverberation chamber

(random illumination) , then the transmitted power can be written

P = <a >S./2. (20)

t t i'

The factor 1/2 in eq (20) results from shadowing of the incident field by

the electrically large enclosure and is a good approximation for convex

shields. The average value of the transmission cross section for N

apertures is obtained directly from eq (19)

:

N

<a> = S <a .>. (21)

t , - ti 1=1

The results in (20) and (21) could also be used in applications where the

incidence angle and polarization are unknown.

3.1 Apertures of Arbitrary Shape

Consider a plane wave incident on an aperture in a perfectly conducting

sheet as shown in figure 2. For convenience, we drop the subscript i that

identifies the ith aperture in the shield. Aperture theory has been

developed primarily for apertures in flat, perfectly conducting screens of

infinite extent and zero thickness [9]. Here we assume that the shield is

locally planar and that the shield thickness is small. Aperture theory can

be subdivided into three cases where the aperture dimensions are either

small, comparable, or large compared to the wavelength.
For electrically large apertures, the geometrical optics approximation

yields

a = A cos^^, (22)

where A is the aperture area and d is the incident elevation angle. Thus

o is independent of frequency, polarization, and azimuth of the incident

field. For this case, the average transmission cross section can be

written:

, 27r . 7r/2 . <o > = T^ / d.^^ / d^^ kcosO^ sintf^ = A/2, (23)

where we restrict to angles less than 7r/2 because the field is incident

from only one side of the screen. For electrically small apertures, polarizability theory states that the

transmitted fields are those of induced electric and magnetic dipole moments

[9,13]. This theory yields a transmission cross section that is

proportional to frequency to the fourth power:

a^ = C k^, (24.)

where C depends on incident angle and polarization and aperture size and

shape, but is independent of frequency. The wavenumber k = w/c . The

specific form of C for a circular aperture will be given in the following

section.

In the resonance region, the aperture dimensions are comparable to a wavelength, and the frequency dependence of a depends on the aperture

shape. Numerical methods [9] can be used to compute o for such cases, but

we will not pursue such methods here.

10
3.2 Circular Aperture

The circular aperture is of particular interest because it has an analytical solution and it is easy to work with experimentally. The

geometry for a circular aperture of radius a is shown in figure 3. An exact

solution for the transmission coefficient is available in terms of

spheroidal functions [14] , but we choose to construct a simpler solution in

terms of the simple approximations that are available for electrically large

and small circular apertures.

For electrically large circular apertures, the geometrical optics

approximations in eqs (22) and (23) yield the following expressions for the

transmission cross section and the averaged transmission cross section:

2 i 2 a = Tra cos^ and <o > = na /2

.

(25)

For electrically small circular apertures, polarizability theory [9] can be used to determine the effective electric and magnetic dipole moments

and the resultant transmission cross section. The details are given in

Appendix B. The transmission cross section depends on the polarization and

the elevation angle of the incident field. For the electric field polarized

parallel to the incidence plane (defined by the incident wave vector and the

normal to the aperture) ^ , we write the transmission cross section as a : tpar

64 , 4 6 ,- 1 . 2^i, ,„,, %ar = i7;^^ (^ +^sin ^ ). (26)

For perpendicular polarization, we write the transmission cross section as

tperp

a^ = TT— k a cos 6 . (27) tperp 277r

11
4

Both a and a have the k dependence eiven by eq (24) , and they are tpar tperp ^ '=' y ^ \ / . j

equal for normal incidence (^ = 0) . We assume that an incident random

field will have equal power densities in the parallel and perpendicular

waves. Thus the averaged transmission cross section can be written

^ 7r/2 . <a> = -T r d^^ (a^ + o^ ) sin^^, (28)

t 2 ^ tpar tperp

where we have used the fact that the transmission cross sections are independent of the incident azimuth angle. If we substitute eqs (26) and

(27) into eq (28) and carry out the integration over ^ , we obtain

<V=ifkV. (29)

We do not have a simple expression for the transmission cross section

that is valid in the resonance region, but the circular aperture does not

have strong resonances [15]. Thus we choose to cover the entire frequency

range by using only the electrically small and electrically large

approximations. The crossover wavenumber k where we switch from eq (29) to

eq (25) for the average transmission cross sections is given by equating eqs

(25) and (29):

-K^/l = ^ kV. (30)

yTT C

The solution to eq (30) is

k a = (97r^/32)^/^ = 1.29. (31)

This technique is not valid for long, narrow apertures, which typically have

strong resonances.

12
4. POWER BALANCE

4.1 Shielding Effectiveness

Consider again the geometry in figure 1, where an incident wave is

incident on a shielded cavity with apertures. We wish to determine the

power density S inside the cavity. For steady state conditions, we require

that the power P transmitted through the apertures is equal to the power

P dissipated in the four loss mechanisms considered in Section 2:

^t =^d-

^32)

If we substitute eqs (1), (2), (4), and (18) into (32), we can solve

for the power density S in the cavity:

a AQ

^c = -^

h' (33>

Since we have assumed that the power density S is uniform throughout the

cavity, we can define shielding effectiveness in terras of the ratio of the

incident and cavity power densities:

SE = 10 log^oCS./S^) - 10 logio(;^). dB. (34)

The results in eqs (33) and (34) are consistent with a recent treatment of a related problem [16]. We have defined SE to be a positive number when the

cavity power density is less than the incident power density. Our

definition of SE in (34) depends on the cavity volume and Q in addition to

the transmission cross section o . For the application of measuring SE of

13
materials covering the aperture, it would be more logical to work directly

with a . t The results in eqs (33) and (34) apply to a single incident plane wave

where a depends on the incident direction and polarization. For the case

of uniformly random incidence (as in a reverberation chamber) , we need to

replace a in eqs (33) and (34) by one -half the averaged value <a >/2

.

The Q enhancement of the cavity power density is clear in eqs (33) and

(34), and we can see that a lossy cavity (low Q) has a greater shielding

effectiveness than a high Q cavity. The importance of loss is seen if we consider the special case where the cavity is lossless (P,^ = P,^ = P., = 0) di Q.I d4

except for leakage. In this case Q is given by

47rV

If we substitute eq (35) into eq (33) , we obtain

la

S = S. —^. (36)

c 1 <a„> ^ '

For the case of uniformly random excitation, the transmission cross section

is replaced by one half the averaged cross section. However, the averaged

transmission cross section is equal to the averaged leakage leakage cross section (<a > = <a ^>) , and eq (36) reduces to

S = S. or SE = dB. (37)

c 1

Thus the leakage loss equals the transmitted power, and the cavity has zero shielding. We expect aircraft cavities to have significant loss and fairly

low Q.

14
4.2 Time Constant

Up to this point we have considered only steady-state, single -frequency

excitation. Since radar pulses are important in aircraft applications, we also need to consider transient effects. In general this is a complex

problem that is best handled using Fourier integral techniques. However, we can analyze the special case of a turned-on or turned-off sinusoid in a simpler manner.

We consider first the case of field decay where the source (the

incident power density) is instantaneously turned off. By equating the

change in the cavity energy U to the negative of the dissipated power times

a time increment dt, we obtain the differential equation

dU = - P^ dt. (38) a

We can use eq (1) to replace P in eq (38)

dU = -(wU/Q) dt = - - dt, (39)

T

where the time constant r = Q/w. The initial condition is U = U at t = 0,

s The solution of eq (39) with this initial condition is

U = U e"^/'^, t > 0. (40)

s

The closely related case of a turned-on (step modulated) incident power

density involves the same exponential function and time constant:

U = U (1 - e'^/^), t > 0. (41)

s

The cavity energy density and power density also follow the same exponential

variation with the same time constant, and eqs (40) and (41) agree with [3].

If a radar pulse duration is long compared to r, then the cavity fields will

15
reach their steady- state value. However, if the pulse length is short

compared to t, the fields will not reach their steady-state values before

the incident pulse is turned off. Some common radars and their pulse

characteristics are described in Appendix C.

A high Q (long r) cavity might have poor steady- state SE, but would

require a long time for the cavity fields to reach their steady state

values. Physically, a high Q (long r) means that waves make many bounces

within the cavity before they decay.

4.3 Source inside Cavity

We can use the power balance theory of Section 4.1 to analyze the case

of an internal source, a transmitting antenna located within the cavity.

This configuration is of interest in calculating either the power density

within an aircraft caused by unintentional emitters or the power density

within a reverberation chamber with loading.

If a cw source within the cavity transmits power P , then eq (32) still

holds , but P now represents antenna transmitted power rather than aperture

transmitted power. Using eqs (1), (2), (4), and (32), we can solve for the

power density S within the cavity:

AQP

S = 7-^. (42)

c 27rV

Equation (42) applies to cavities with general loading because Q can include

any of the loss mechanisms discussed in Section 2.

If an impedance -matched, receiving antenna is placed within the cavity,

2 then the received power P is the product of the effective area A /Stt and

the power density S as discussed in Section 2.4. Using eq (42), we obtain

the following expression for the received power:

16
3

P =-^P (43)

167r V

If we solve eq (43) for Q, then we see that the cavity Q can be determined

from a measurement of the power ^ ratio P /P : r^ t

Q = ^

^. (44)

A t

Equation (44) is commonly used for measuring the Q of reverberation chambers

[17].

5 . MEASUREMENTS

5.1 Measurement Techniques

A relatively small, rectangular cavity (which we call the FAA cavity) was constructed and used for measurements to validate the theoretical model.

The theoretical model can be used for any size or shape of electrically

large cavity. The physical cavity was selected to satisfy a number of

practical limitations and yet to allow evaluation of all loss mechanisms in

the theoretical model. Scaling considerations for general, lossy materials

are discussed in Appendix D. The cavity was rectangular, 0.514 m x 0.629 m

X 1.75 m. The cavity was selected to be large enough to provide sufficient

mode density above 1 GHz, but to be light enough to meet weight requirements

of our automated positioner. The three parameters that needed to be

measured were quality factor Q, time constant r, and shielding effectiveness

SE, from 1 to 18 GHz for various loss and/or loading conditions due to

walls, absorption, antenna, and aperture.

Of the various materials considered (galvanized steel, copper, and

aluminum) , aluminum was selected because it has a high electrical

conductivity and is easy to weld. A removable panel was considered as an

17
access door, but was rejected because of the variability of contact

resistance and possible unknown loss in rf gasketing material. Because of

the difficulty in welding and grinding open an end wall each time access was needed to the cavity, a minimum of opening was planned.

Double-ridged waveguide horns were selected because of their wide

frequency range, 1 to 18 GHz. A stirrer and drive motor were installed

inside the chamber to provide mode stirring. The stirrer was also made of

aluminum. The drive motor was enclosed inside a high- isolation aluminum

case so that motor noise would not escape into the cavity.

Q was measured before any aperture was cut in a wall just to ensure

that antennas, cables, and connectors were functioning properly and also to ensure the integrity of the welded seams. A second set of measurements was made with an aperture of radius 0.014 m cut in a side wall. This allowed

coupling energy into and out of the cavity. Later a sphere of radius 0.066 m and containing salt water was put inside the cavity. Still later, a set

of three spheres of radius 0.066 m and containing salt water was put into

the cavity. The salt water was selected as having absorption similar to a human body. The salt concentration was selected as that of ocean water so that we could use the electrical properties of salt water given by Saxton

and Lane [18]. These electrical properties are discussed in Appendix E. Finally, everything was removed and two standard- gain horns were used in

place of the double -ridged waveguide horn antennas. The aperture, of

course, remained. This last step had to be added to try to quantify the

loss of the double-ridged waveguide horn antennas, since there was greater

loss measured than calculated. The standard gain horns have an efficiency

of about 0.98.

Initial measurements of Q relied on eq (44) , previously given by Corona

et al . [17]. Volume and frequency errors introduce very little uncertainty,

but the power ratio P /P requires consistent use of either maximum or

average values. The power transmitted is determined at the same time cable

loss is measured, by shorting the cables to the two antennas together. This removes all stirring effects and maximum and average values are within 0.1

dB, as expected for cw excitation. The received power is coupled through

the resonant cavity modes, and the stirrer introduced a 40 to 50 dB

excursion, including 6 to 10 dB differences between maximum and average.

18
The maximum- average difference must be known if maximum values are to be

used. The most practical way to avoid this is to use average values.

Perhaps later, when the distributions have been adequately characterized,

this requirement can be relaxed.

For measurement of shielding effectiveness where there are two

chambers, two stirrers, and two similar antennas, either maximum or average

values might be used since both receiver powers are coupled through the

resonant, stirred environment. However, there is a difference in type of

distribution and maximum- average ratio of the two chambers, and average

based SE values are a few decibels higher than maximum-based values. The

observed maximum- average ratio is also higher in the outer- to -inner chamber

distribution than in the larger, outer chamber distribution. Therefore, it

is probably safer to use average values

.

The measure of shielding effectiveness of the cavity was determined by

measurements using the setup in figure 4. This was a direct measurement of

two powers. The procedure for measuring the loss within the FAA cavity is

similar, except the cable to the transmit antenna is connected to the

transmit antenna in the FAA cavity. This is a modification to figure 4.

The stirrer must be turning and the frequency sweep must be slow enough to

allow one revolution of the stirrer at each frequency. The average power

loss is the preferred parameter to measure. Cable losses were also measured

and corrections were applied. A power-series, least-squares fit was obtained from the loss -frequency curve to use in eq (44) to obtain a smooth

curve for Q from loss measurements.

The measurement of the time constant required the setup shown in figure

5. This is described previously by Crawford et al. [19]. A burst of rf

energy was provided for a microsecond and the maximum and average decay

responses were recorded for each of 200 stirrer positions. The average

decay for these 200 tuner positions was used to provide a clean decay

response as shown in figure 6. An exponential least- squares fit was applied

to determine the actual time constant. This provides a second measure of Q,

since Q is the product of angular frequency and time constant as indicated

in Section 4.2.

Many factors that affect the measured values; not all are included in

the theoretical model. The method of recording and processing data had to

19
be varied from the procedure used for routine electromagnetic susceptibility

measurements

.

The usual procedure for measuring loss , fields , or shielding

effectiveness in a mode-stirred chamber is to use a 'max hold' feature of

the spectrum analyzer and record the maximum field strength during one or more frequency sweeps. The theoretical model in this work uses average

values, so questions concerning the type of distribution and the relation

between maxima and averages are relevant. Previous measurements [19] for

relatively high-Q, mode -stirred chambers indicated a nominally 8 dB

difference between maximum and average values of a distribution at any

single frequency. The range of this difference was between 6.5 and 9.5 dB.

Despite careful measurements by all of the several methods used, we measured substantially lower values of Q than were predicted by our model.

This indicated measurement errors, substantially higher losses than

predicted, or model error. Investigations revealed a combination of the

first two.

The measurement error was caused mainly by loss in the antennas.

Double -ridged waveguide horn antennas were used initially since they have a large frequency range, 1-18 GHz. Subsequent measurements using standardgain horns with very high efficiency over a more -limited frequency range

indicated about 2 dB less loss per antenna than had been measured using the

double-ridged waveguide horns. This loss may not be the same in other

applications where surface areas of the horn and mounting structures and the

plastic parts are not so strongly excited.

Other additional losses could not be quantified as clearly in all

cases, but each loss was demonstrated. By making reasonable estimates, we were able to show agreement between measured and calculated values within

20 %, which indicates validity of the model.

One of the unaccounted losses was due to lower- than-expected

conductivity of aluminum. We initially used a value near that of pure

aluminum from a reference handbook, but discovered that the metal actually

used was an alloy with about half the conductivity of pure aluminum. An

additional decrease in conductivity was measured at NIST using a parallelplate dielectric resonator technique [20]. This measured value (8.83 x 10

S/m) indicated an additional factor-of-2 decrease in conductivity. Whether

20
this is due to the oxide layer, surface roughness, or other factors has not

1/2 been determined. This total factor of 4 reduces the predicted Q by 4 or

2 (see eq (8))

.

Another substantial unaccounted loss was the wall loss due to

additional surface area of cables, mounting brackets, antennas, stirrer,

stirrer motor case, and bulkhead connectors. This additional surface area

can be approximated, but the loss is difficult to model since the

conductivities and current distributions are not known. We estimated 10 to

20 % additional surface area depending on the complexity and size of the

loading and antennas used. One more unaccounted factor that changes the

calculated Q is the reduction of the volume within the cavity. This was estimated to be 5 to 10 % . Again this depends on the size and complexity of

everything within the cavity.

There is one other minor loss due to absorption of the dielectric cover

on the cables within the chamber. We originally used semi-rigid (solid

metal) coaxial transmission lines, but the mating, precision connectors are very susceptible to loosening due to vibration. Only a slight loosening

causes a large mismatch. We changed to flexible, dielectric-covered cables

and special connectors that were more reliable in this application.

During the absorptive- loading modeling, there were additional losses

due to flexible tubing, corks, glassware, and support structures of foam.

None of these corrections is dominant; different loading cases require

different corrections. All of these loss mechanisms will be even more prevalent in most applications, especially in aircraft compartments.

5.2 Comparison of Measured and Theoretical Results

Samples of measured and theoretical results with four loading

conditions are shown. The theoretical results were obtained from the

computer program described in Appendix F. Previous estimates of field

uncertainties in reverberation chambers excited by sources within the

chamber [4,19] range from ±4 to ±8 dB . Consequently, we consider

agreement between measurements and theory within ± 4 dB to be good and

within ± 8 dB to be acceptable.

21
The results for the (relatively) empty cavity with an aperture of

diameter 0.014 m, but with no corrections applied, are shown in figures 7-9.

Figure 7 shows the several calculated Q values based on the appropriate loss

mechanism, the composite Q, and the measured Qs by both loss and time

constant. Figure 8 shows calculated and measured time constant, while

figure 9 shows measured and calculated SE. The Q values in figure 7 measured by the transmission loss method, eq (44) , and the time constant

method, eq (39), are in good agreement, but are much smaller than the

theoretical values. This is typical of earlier reverberation chamber

measurements [4,19]. The differences between measured and calculated values

of r and SE are also substantial. At first, the reasons for these

differences were not known.

Next, the cavity with the 0.014 m aperture was loaded with one of the

spheres containing salt water. In these and subsequent figures,

conductivity, volume, and surface corrections have been applied. Figure 10

shows Qs due to three of the four loss mechanisms plus the composite Q and

the two measured Qs ; the curves along the bottom are too compressed to read.

Q^ , due to wall loss, has dropped from that in figure 7 because we are using

the measured wall conductivity (8.83 x 10 S/m) . Q„ , due to aperture

leakage, reaches a minumim at the crossover from low to high frequency

behavior as given by eq (31). Figure 11, an expanded view of figure 10,

shows the part of the figure that has the Qs due to the dominant loss

mechanisms plus the composite Q and the two measured Qs . Figure 12 shows

measured and calculated time constants plots. Figure 13 shows measured and

calculated SE plots. There is much better agreement between theoretically

predicted and measured values, but the disagreement indicated the need for

further investigation.

Next we loaded the cavity with three spheres containing salt water.

The results are shown in figures 14-17. Figure 14 shows three of the Q curves ; figure 15 shows the expanded portion that has the Qs with the

dominant loss mechanisms, the composite Q, and the measured Qs . Figure 16

shows the time constant plots, and figure 17 shows the SE plots. Again, the

agreement is good, but not good enough.

22
The last set of figures, 18-21, covers a limited frequency range (12-18

GHz), but reveals the missing loss factor. For these we used a set of

standard- gain, K -band horns. The antennas have a fairly high efficiency,

about 0.98. These results show quite acceptable agreement between predicted

and measured results. This indicates a poorer efficiency for the doubleridge waveguide horns than we had expected. Figure 18 shows the Qs ; figure

19 shows the expanded Q curves. The agreement between theoretical and

measured Q is much better than that typically obtained for reverberation

chambers [4]. Figure 20 shows the time constants, and figure 21 shows the

SE values

.

Shielding effectiveness measurements were also performed by Hatfield

[21] on a rectangular cavity with a circular aperture in the Naval Surface

Warfare Center (NSWC) reverberation chamber. The cavity contains a broadband, double-ridged receiving antenna and a paddle wheel (mode stirrer)

as shown in figure 4. The box is made of the same aluminum alloy

{a = 8.83 X 10 S/m and u = 1) with dimensions: i = 1.213 m, w = 0.603 m,

\] 17 and h = 0.937 m. Two different aperture radii, a = 2.94 cm and 3.51 cm, were used.

Shielding effectiveness' measurements were made from 200 MHz to 18 GHz

for both apertures. A comparison with calculated SE values is shown in

figure 23. The theory is not expected to be valid below 400 MHz because the

box is not electrically large (the mode density is too low). The measured

values show rapid frequency variations which do not appear in the smooth

theory, but the rapid frequency variations are typical of reverberation

chamber measurements [4] . The general agreement between theory and

measurements is good above 400 MHz for both aperture sizes. The smaller

aperture yields greater SE, but the high frequency theoretical SE is low for

both apertures. The reason for the decrease in SE with frequency is the

increase in both cavity Q and transmission cross section with frequency.

Equation (34) shows the dependence of SE on these two quantities.

The measured results presented in this section could be scaled up in

size and down in frequency to more nearly match the dimensions of aircrafts

.

However, the electrical properties of the walls and absorbers would also

23
need to be scaled; the required scaling relationships are discussed in

Appendix D

.

6. CONCLUSIONS

We have presented a theory based on power balance for aperture

excitation of electrically large, lossy cavities. The theory yields

expressions for shielding effectiveness, cavity Q, and cavity time constant,

but does not give the fine structure of the fields (standing waves, etc.)

within the cavity. The advantage of the theory is that it does not require

the detailed geometry of the cavity, absorbers, and apertures, but only

requires a few defining parameters (cavity volume and surface area, wall

conductivity, aperture transmission cross section, and absorption cross section of objects within the cavity).

The theory actually yields ensemble averages for quantities of interest

(SE, Q, and r), and the ensemble can be considered to represent different

cavity geometries . Experimentally the ensemble average is obtained by use

of a mode stirrer within the cavity, and that is how the experimental data

for SE, Q, and r were obtained in Section 5. The agreement of the theory

with the broadband experimental data was good, but a great deal of effort

was required to account for additional losses that lowered measured Q and r and increased measured SE. The theory and measurements of Q and t have

application to improved characterization of reverberation chambers [4,19]

where the losses have been difficult to calculate.

A number of other extensions that would be useful. Measurements for

cavities with multiple apertures would provide additional checks for the

theory and would more nearly model typical aircraft geometries. For

modeling aircraft windows, the effects of window glass (and possibly

coatings) should be included in future calculations. A fifth loss

mechanism, water vapor absorption, could be added to the Q theory. This

loss has been observed in reverberation chamber measurements above 18 GHz

[19]. Measurements with plane wave excitation in an anechoic chamber would

provide additional checks for the angular and polarization features of the

24
aperture theory. Also, measurements on an actual aircraft would be useful

in determining whether additional features need to be added to the model.

Further experimental and theoretical studies of the statistical

properties of fields in large cavities need to be performed. Such studies

would be useful in extending the present theory to include probability

density and distribution functions [22] in addition to average values. Some

theoretical work on the statistical properties of fields in cavities [23,24]

has been done, but a more complete statistical electromagnetic theory is

needed.

7 . ACKNOWLEDGMENTS

This research was supported by the Federal Aviation Administration. We

would like to thank Mike Hatfield, NSWC , for providing shielding

effectiveness data for the NSWC cavity and Richard Geyer, NIST, for

performing the wall conductivity measurement for the FAA cavity.

8 . REFERENCES

[1] Guidance to certification of Aircraft electrical/electronic systems for

operation in the high intensity radiated fields (HIRF) environment.

Society of Automotive Engineers, SAE AE4R Committee Report (July,

1991).

[2] Taflove, A.; Umashankar, K. A hybrid moment method/finite-difference

time-domain approach to electromagnetic coupling and aperture

penetration into complex geometries. IEEE Trans. Ant. Propagat., AP-30:

617-627, 1982.

[3] Richardson, R.E. Mode-stirred chamber calibration factor, relaxation

time, and scaling laws. IEEE Trans. Instrum. Meas .

, IM-34: 573-580;

1985.

[4] Crawford, M.L.; Koepke , G.H. Design, evaluation, and use of a reveberation chamber for performing electromagnetic

susceptibility/vulnerability measurements. Nat. Bur. Stand. (U.S.)

Tech. Note 1092; 1986.

[5] Dunn, J.M. Local, high-frequency analysis of the fields in a modestirred chamber. IEEE Trans. Electromag. Compat. , EMC-32: 53-58; 1990.

25
6] Liu, B.H. ; Chang, D.C.; Ma, M.T. Eigenmodes and the composite quality

factor of a reverberating chamber. Nat. Bur. Stand. (U.S.) Tech. Note

1066; 1983.

7] Van de Hulst, H.C. Light Scattering by Small Particles. New York:

Dover, 1981.

8] Hill, D.A. ; Crawford, M.L.; Kanda, M. ; Wu, D.I. Aperture coupling to a coaxial air line: theory and experiment. IEEE Trans. Electromag.

Compat. : 69-74; 1993.

9] Butler, CM.; Rahmat-Samii , Y. ; Mittra, R. Electromagnetic penetration

through apertures in conducting surfaces. IEEE Trans. Ant.

Propagat. ,AP-26: 82-93; 1978.

10] Tai, C.T. On the definition of effective aperture of antennas. IEEE

Trans. Ant. Propagat., AP-9: 224-225; 1961.

11] Hill, D.A. ; Francis, M.H. Out-of-band response of antenna arrays. IEEE

Trans. Electromag. Compat., EMC-29: 282-288; 1987.

12] Loughry, T.A. Frequency stirring: an alternate approach to mechanical

mode-stirring for the conduct of electromagnetic susceptibility

testing. Phillips Laboratory, Kirtland Air Force Base, NM, PL-TR--91-

1036; 1991.

13] Bethe, H.A. Theory of diffraction by small holes. Phys . Rev., 66: 163-

182; 1944.

14] Meixner, J.; Andrejewski, W. Strenge theorie der beugung ebener

elektromagnetischer wellen an der vollkommen leitenden kreisscheibe und

an der kreisformigen offnung im vollkemmen leitenden eben schirm,

Annalen der Physik, 7: 157-168; 1950.

15] Levine , H. ; Schwinger, J. On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen.

Comm. Pure Appl. Math., 3: 355-391; 1950.

16] Lee, K.S.H.; Yang, F.-C; Trends and bounds in RF coupling to a wire

inside a slotted cavity. IEEE Trans. Electromag. Compat., EMC-34: 154-

160; 1992.

17] Corona, P.; Latmiral , C; Paolini, E. Performance and analysis of a reverberating enclosure with variable geometry. IEEE Trans. Electromag.

Compat., EMC-22: 2-5; 1980.

18] Saxton, J. A.; Lane, J. A. Electrical properties of sea water. Wireless

Engineer, 29: 269-275; 1952.

19] Crawford, M.L.; Ma, M.T. ; Ladbury, J.M.; Riddle, B.F. Measurement and

evaluation of a TEM/reverberating chamber. Nat. Inst. Stand. Tech.

(U.S.) Tech. Note 1342; 1990.

26
[20] Geyer, R.G. , National Institute of Standards and Technology, Boulder,

CO. Private communication.

[21] Hatfield, M. , Naval Surface Warfare Center, Dahlgren, VA. Private

communication

[22] Kostas, J.G.; Boverie, B. Statistical model for a mode-stirred chamber.

IEEE Trans. Electromag. Compat ., EMC-33: 366-370; 1991.

[23] Price, R.H. ; Davis, H.T. ; Bonn, R.H. ; Wenaas , E.P.; Achenbach, R.

Gieri, V.; Thomas, R. ; Alcala, J.; Hanson, J.; Haynes , W. ; McCrea, C;

Montano, C; Peterson, R. ; Trautlein, B. ; Umber, R. Determination of

the statistical distribution of electromagnetic field amplitudes in

complex cavities, JAYCOR Report 88JAL129; 1988.

[24] Lehman, T.H. A statistical theory of electromagnetic fields in complex

cavities. EMP Interaction Note 494; May 1993.

[25] Mie, G. Ann. Physik, 25: 377; 1908.

[26] Harrington, R.F. Time -Harmonic Electromagnetic Fields. New York:

McGraw-Hill; 1961.

[27] Rumsey, V.H. Frequency Independent Antennas. New York: Academic Press;

1966.

[28] Hill, D.A. Electromagnetic scattering by buried objects of low

contrast. IEEE Trans. Geosci. Rem. Sens., GRS-6: 195-203; 1988.

[29] Hasted, J.B. Aqueous Dielectrics. London: Chapman and Hall; 1973, Ch.

6, Electrolytic solutions.

APPENDIX A. ABSORPTION BY A HOMOGENEOUS SPHERE

The absorption loss due to loading objects inside a cavity can be

written as the product of cavity power density and absorption cross section

of the object. To simplify the analysis and facilitate experimental

verifications, we assume that a loading object takes the form of a homogeneous sphere made of sea water to simulate a human.

An incident field with known frequency and intensity propagates toward

the sphere. The problem at hand is then to determine the field penetrating

into the sphere. The classical solution of this problem is due to Mie [25],

based on the formulation of a vector wave equation,

V2e + k^m^E = 0, (Al)

27
in a source -free region satisfying appropriate boundary conditions, where E

is the unknown electric field inside the sphere, k is the free -space wave

number, and m is the refractive index defined as

:^ - ja/icoe^), (A2)

with e as the dielectric constant (permittivity relative to the free-space

value e„) and a the conductivity of the material. In free space, e = 1, a

= 0, and m = 1 . We use m for refractive index to be consistent with common

notation [7], but this m should not be confused with the impedance mismatch

factor in eq (16).

To solve the vector wave equation we have to obtain solutions to the

corresponding scalar wave equation,

V^u + k^m^u = 0, (A3)

in spherical coordinates. Since eq (A3) is a second-order partial

differential equation, we have two independent solutions, Uj and u^ [7].

Let us represent the incident wave outside the sphere where m = 1 as

, _, , ..n 2n + 1 „1 , „v . ^1 V Ui = cos(^ S (-J) . .. P (cos^) J (kr),

^

T - n(n+l) n -^n n=l

(A4)

CO u, = sin<^ 2 (-j) ~TTTT P (cos^) j (kr) , ^ T- X J / n(n+l) n -'n n=l

where (<f> , 6, r) are the spherical coordinates with r = at the sphere

center, P is the associate Legendre function, and j is the spherical

Bessel function.

The scattered wave outside the sphere can then be expressed as

28
00

, T^ J- ^y -vH 2n + 1 _1 . .. , (2),, . u; = cos^ E (-a^)(-j)

^(^^+1^ P^(cos^) h^ ^(kr),

n=l

(A5)

00 u' = sin^ S (-b )(-j)" ^T-TTT P^(cos^) h^^\kr) , 2

T n "^ n(n+l) n n

n=l

(2) where h is the spherical Hankel function, and a and b are the unknown

n n n

coefficients to be determined.

The wave penetrated into the sphere where m is complex may be written

as

uV = cos<i S (mc )(-i) ~—77 P (cos^) i (mkr) , 1 ^ ^ n -^ n(n+l) n -^n n=i

(A6)

oo u^ = sin^ S (md^)(-j)'' J^^;^ P^(cos^) j^(mkr) , n=l

where c and d are another unknown coefficients to be determined, n n

The boundary condition requires that at the sphere surface r = a,

Ui + u( = n'l, and u^ + u^ = u^

(A7)

from which we obtain, after some algebraic simplifications,

a = A/B, b = C/D, c = i/B, and d = i/D, (A8) n n n -" n -^

where

A = $'(y) * (x) - m ^(y) ^'(x),

n n n n

29
(A9)

C = m *'(y) * (X) - * (y) ^>' (x) , n n n n

D = m *^(y) C^(x) - *^(y) e;(x)

,

with x = ka, and y = mka = mx,

In eq (A9) , * and £ = * + i X are the Riccati-Bessel functions [7], and ^ n n n -^ n

$' and ^' are the derivatives with respect to their arguments.

Fortunately, the spherical and Riccati Bessel functions can be written

more specifically when n is an integer, which is the case in our study. The

first few lower orders of them are given below.

Spherical Bessel functions:

Jo(z) = z smz,

Ji(z) = z smz - z cosz.

-3 -1 -2

J2(z) = (3z - z ) sinz - 3z cosz,

-4 -2 -3 -1

jgCz) = (15z - 6z ) sinz - (15z - z ) cosz.

J4(z) = (105z"^ - 45z'^ + z"-'") sinz - (105z'^ - lOz'^) cosz. (AlO)

For still higher orders, they can be deduced from the recursion formula.

30
i (z) = f (z) sinz + (-1) f - (z) cosz, (All) ^n n -n-1

and

f^.]^(z) + fn+1^^^ = ^^" "^ ^^

^'"^ ^n^^^' ^^^^^

with fo=z , fi=z , fj=0, and f 2 = - z

In eqs (AlO) and (All), as z - 0, Jo(z) -* 1, and j (z) -•^ for n ?^ 0,

Riccati-Bessel functions

'J'o(z) = sinz,

T / N -1 • Wj(z) = z sinz - cosz.

-2 -1

$2(2) = (3z - 1) sinz - 3z cosz,

-3 -1 -2

^3(2) = (15z - 6z ) sinz - (15z - 1) cosz.

*4(z) = (105z"^ - 45z'^ + 1) sinz - (105z"^ - lOz'"'") cosz;

Xo(z) = cosz,

Xi(z) = sinz + z cosz,

-1 -2

X2(z) = 3z sinz + (3z - 1) cosz.

31
-2 -3 -1

XaCz) = (15z - 1) sinz + (15z - 6z ) cosz,

and

X4(z) = (105z""^ - lOz""'') sinz + (lOSz"^ - 45z'^ + 1) cosz. (Al3)

The derivatives of the Riccati-Bessel functions required in (A9) are then

$o(z) = cosz,

-2 -1

*i(z) = (-Z + 1) sinz + z cosz,

-3 -1 -2

$2(2) = (-6z + 3z ) sinz + (6z - 1) cosz,

-4 -2 -3 -1

^3(2) = (-45z + 21z - 1) sinz + (45z - 6z ) cosz.

*;(z) = (-420z'^ + 195z'^ - lOz'-*-) sinz + (420z"^ - 55z'^ + 1) cosz;

Xo(z) = - sinz.

-1 -2

Xi(z) = - z sinz + (- z +1) cosz,

-2 -3 -1

X2(z) = (-6z + 1) sinz + (-6z + 3z ) cosz.

-3 -1 -4 -2

XsCz) = (-45z + 6z ) sinz + (-45z + 21z - 1) cosz,

and (A14)

X4(z) = (420z"^ + 55z"^ - 1) sinz + (-420z'^ + 195z" - lOz' ) cosz

32
In eqs (A13) and (A14) , z = x (real) or z = y (complex) . Once the frequency

f (thus co) , sphere radius a, and material constants e and o are specified,

the coefficients a , b , c , and d can be computed from eqs (A8) and (A9)

,

n n n n

'^ t x / and the field distributions inside and outside the sphere determined from

eqs (A4) , (A5)

,

and (A6) . The number of terms required in eqs (A4) to (A6)

depends on ka.

Two other efficiency factors, the total factor rj and the scattered

factor 1] , are needed. They are defined by

n = 2(ka)"^ S (2n + 1) Re (a + b ), (A15)

t - n n

n=l

and

-2 *

r,^ = 2(ka) ^ E (2n + 1) [|a^|2 + |b^|2]. (Al6)

n=l

The efficiency factor rj for absorption is then

"a = "t - ^s- (^1^>

The absorption area of the sphere is obtained from o = (Tra^) n , which in

a a turn is used to compute the power loss due to absorption.

When ka becomes very large, this theory is not convenient because the

summations converge too slowly. In this case, a geometrical optics

approximation [7, Sec. 14.23] can be used to compute a . The computer

program in Appendix F uses this approximation when ka becomes large.

Consequently, the Q curve can show a small discontinuity when the frequency

reaches the crossover value (at ka = 46.0).

33
APPENDIX B. TRANSMISSION CROSS SECTION OF A SMALL CIRCULAR APERTURE

Consider a small circular aperture (ka « 1) in a planar sheet as shown

in figure 3. The transmitted fields can be written as the fields of a tangential magnetic dipole moment p and a normal electric dipole moment p

that can be written as the product of an aperture polarizability times the

appropriate incident field [8,13]:

m m tan e U e n

sc where H is the tangential magnetic field at the center of the shorttan ° °

sc circuited aperture and E is the normal electric field at the center of the n short-circuited aperture. The magnetic and electric polarizabilities , a

and Q , are given by [8,13]

ame= 4a^/3 and a = 2a^/3. (B2)

The dipole moments radiate in the presence of the ground plane (so their

images are included) , and the total transmitted power (radiated into one half-space) is [26]

^^0 2 2 2

We consider the cases of parallel and perpendicular polarizations

separately. For parallel polarization, the short-circuited fields are

Hf^ = 2H. and E^'^ = 2E.sin^^, (B4)

tan 1 n 1

where the incident fields can be related to the incident power density S. by

34
S. = »?_ H? and S. = E?/»7_. (B5)

1 '0 1 1 1"^

From eqs (B1)-(B5), we can write the transmission cross section for parallel

polarization as

o^ = PVS. = r^ k^a^ (1+7 sin^tf^), (B6) tpar V 1 277r 4

which is the result needed in Section 3.2.

For perpendicular polarization, the short-circuited fields are

H^^ = 2H.cos^^ and E^^ = 0. (B7) tan in

From eqs (Bl)-(B3), (B5) , and (B7) , we can write the transmission cross section for perpendicular polarization as

a^ ' 1^ k^a^ cos^tf^, (B8) tperp 277r

which is the other result needed in Section 3.2,

APPENDIX C. RADAR CHARACTERISTICS

Four types of radars that are likely to be encountered in the vicinity

of commercial U.S. airports. A summary of these is given below.

A. FAA ARSR very long range search radar. This radar system operates on fixed frequencies in the 1250-1300 MHz band with a peak power of 4 MW. The

pulse duration is 2-5 /xs with a pulse repetition rate of 300 pulses/s and a rise time of approximately 10% of the pulse width. The radar antenna

completes one revolution every 10 s. This is a moving target indicator

(MTI) radar that uses a pulse stagger sequence in order to avoid blind

speeds

.

35
B. ASR airport search radar. This radar is used for air traffic control in

the immediate vicinity of commercial airports. This system operates in the

2700-2900 MHz band with a peak power of 850 kW. The pulse width is 0.8 us with a pulse repetition rate of 1000 pulses/s and a rise time of 10% of the

pulse width. The antenna rotates every 4.75 s. This is an MTI radar with

no stagger sequence.

C. National Weather Service TDWR radar. This is a Doppler weather radar

that operates in the 5600 MHz band with a peak power of 800 kW. The pulse

width is 3 /xs with a pulse repetition rate of 1000 pulses/s. The antenna

can operate in either a fixed mode (to look at interesting weather) or rotate once every 20 s. There is no staggering in the pulse sequence.

D. ASDE ground radar. This system is used to monitor ground terminal

traffic at many airports (such as parked and taxiing aircraft) . This system

operates in the 22 GHz band with a peak power of 1 kW, with a pulse width of

50-60 ns and a pulse repetition rate of 10 000 pulses/s.

APPENDIX D. SCALING

For applications involving large objects, such as aircraft, laboratory

measurements are more conveniently done on smaller scale models. For the

time-harmonic form of Maxwell's equations, scaling of frequency and length

in nondispersive , lossless media is well known. For the example of

frequency -independent antennas [27], "the entire electrical performance is

frequency- independent if all length dimensions are scaled in inverse

proportion to frequency."

To consider the more general case of lossy media, we begin with the

time -harmonic , source-free form of Maxwell's equations for exp(jcc)t) time

dependence

V X H(r,w) = [jwe(r) + a(r)] E(r,w),

(Dl)

36
V X E(r,w) = -jwM(r) H(r,o;),

where the magnetic permeability n, the permittivity e, and the conductivity

o axe assumed to be independent of frequency, but can be functions of

position r. Suppose that we wish to scale (multiply) lengths by a real

factor 1/s (that can be greater than or less than 1)

r' = r/s or r = s r'

.

(D2)

If s > 1, then the new primed lengths are less than the original lengths.

To examine the scaling possibilities of eq (Dl) , we rewrite the del (V)

operator:

V = u f + u f- + u f- = -(u ^ + G fT + ^ fT)=-^'. (D3) X ax y 5y z dz s X 3x' y dy z dz' s

where u , u , and u are unit vectors that remain unchanged in the primed

coordinate system. If we substitute eq (D3) into eq (Dl) and multiply by s, then we have

V X H(sr') = [ja)se(sr') + sa(sr')] E(sr'),

(D4)

V X E(sr') = -jws/i(sr') H(sr').

The object now is to scale quantities on the right side of eq (D4) to bring

it to the same form given in eq (Dl) . There are two possibilities: (1) we could scale e, a, and /z by s or (2) we could scale w and a by s . The first

possibility is of no value for scaled experiments. The second possibility

is the standard length/frequency scaling for lossless media where a is

either or «. For those cases, scaling of o has no effect on the results.

We choose the second possibility and the following specific scaling:

37
H'(r') = H(sr'), E'(r') - E(sr'), r' = - r,

and (D5)

w' = SO), a'(r') = sa(sr'), e'(r') = e(sr'), /i'(r') = M(sr').

If we substitute eq (D5) into eq (Dl) , we obtain

V xH'(r',w') = [jw'c'(r') +a'(r')] E'(r',w'),

(D6)

V X E'(r',w') = -jw'/i'(r') H(r',w').

Equations (D6) are identical to Maxwell's eqs (Dl) except that all

quantities are primed. So they are equivalent under the scaling

transformations in eq (D5). To summarize, we scale all distances by 1/s

,

frequency by s, and conductivity by s. If we wish to perform a reduced-size

(s > 1) scale model experiment, then we increase frequency by a factor s as expected, but we also need to increase conductivity by a factor s . This

obviously presents a materials problem, but some ways around this problem

are discussed in the remainder of this section.

The required conductivity scaling can be explained in an equivalent

manner. The first eq (Dl) can also be written

V X H(r,a)) = ja^c^(r) E(r,w), (D7)

where

€^(r) = c^(r) - ja(r)/a;. (D8)

Since our frequency scaling requires multiplying w by s , we must also

multiply a by s to keep the imaginary part of the complex dielectric

constant e from changing.

For general cavity applications, if the conductivity is not scaled

according to eq (D5) , the field distributions will change and the resonant

frequencies and Q will also change in an unpredictable manner. However, if

the walls are highly conducting, then the resonant frequencies will not

38
depend on the wall conductivity and will scale as s times the resonant

frequencies of the original cavity.

Consider now the Q of a cavity with highly conducting walls. We

examine the individual Qs separately as in Section 2. The expression for Qis given by eq (8) . For the scaling in eq (D5) , the new primed quantities

are

Q{ = YfjTJT = Q^. where V = —, S' = -^, 5' = ^, w' = so,. (D9)

r s s

and o' = so. If it is not possible to scale wall conductivity, then we have

Q' = Q s'-"-/^, 5' =5 s'"""/^, and a' = a. (DIO)

Thus the Q' of the scaled cavity will decrease if the frequency is scaled up

(s > 1) , lengths are scaled down, and wall conductivity is not scaled.

Consider now Q_ . If the loading objects have high conductivity (as for

-1/2 metal) , then Q' will change by the factor s as in eq (DIO) because the

loading objects have the same loss dependence on frequency and conductivity

as the walls. A different situation arises when the cavity losses are primarily due to objects of low conductivity and low permittivity (nonmetal

objects). In this case the Born approximation [28] states that the field

distribution is not strongly affected by the loading objects. Thus the

resonant frequencies are not significantly changed, and the cavity loss is

proportional to the conductivity a of the loading objects. Then Q. is

inversely proportional to the conductivity of loading objects:

Q2 « ^/o. (DID

If we scale frequency, length, and conductivity, then the new Q' is

39
Q' = Q„ a u)' /a' , where w' = sw and o' = sa

(D12)

If we are not able to scale conductivity, then

Q' = s Q_ , where w' = sw and a' = o

(D13)

Here the change in Q' is in the opposite direction as that in eq (DlO) where

the wall conductivity is unsealed. The actual situation for low

conductivity objects (such as people and nonmetal furniture) is that the

unsealed conductivity is equal to frequency times the imaginary part of the

dielectric constant:

a = w Im(e )(D14)

c

If Im (e ) does not change with frequency, then the proper conductivity

scaling automatically occurs with no change in loading material.

Consider now Q„ and Q. . For aperture leakage losses and antenna

reception in a fixed load impedance, length and frequency scaling are sufficient to maintain the Q and Q, with no change upon scaling.

Thus we have three situations with regard to cavity scaling where

frequency is scaled up (s > 1) , length is inversely scaled, and conductivity

-1/2

is unsealed. For dominant wall losses, the Q' drops by a factor s as given by eq (DlO). For dominant loading losses due to low conductivity, the

Q' increases by a factor s as given by eq (D13) . (This change will be less

or even zero for dielectrics with a nearly constant loss tangent.) For

dominant aperture leakage or antenna reception losses, the Q' is unchanged.

In summary, the resonant frequencies will scale with s if the field

distributions change little. The cavity Q' can be higher or lower than the

original Q if the conductivity is unsealed. However, the magnitude and

direction of change can be predicted if we know the dominant loss mechanism.

40
APPENDIX E. ELECTRICAL PROPERTIES OF SALT WATER

The electrical properties of salt water have been investigated by

Saxton and Lane [18] as a function of frequency, salt concentration, and

temperature. The relative complex dielectric constant e can be

approximated by a combination of Debye relaxation and dc conductivity:

€ - e, (e - Ci )wt

s h . r_g_ s h__r, ,„,

.

^r =:—u -^ ^h - Jf^ -^ ";—n~]' ^^^^

1 + U) t W€_ 1 + W t r r

where w is the angular frequency and €_ is the permittivity of free space.

The remaining parameters that characterize salt water are: static dielectric

e , high-frequency dielectric constant e, , dielectric relaxation time t ,

and dc conductivity a. The real part '^ of the relative dielectric constant varies from e at low

s frequencies to e, at high frequencies. The imaginary part consists of two

terms that represent conduction loss and dielectric relaxation loss. The

dielectric relaxation loss reaches its maximum at w = 1/t . To represent

sea water at room temperature (20°C) , we select the following values for the

-12

parameters [181: e =70.0, e, =4.9, t =9.2x10 s, and a = 4.0 S/m.

s h r The results from eq (El) using these parameters are consistent with more recent investigations of electrolytic solutions [29].

APPENDIX F. SOFTWARE TO ANALYZE A RECTANGULAR CAVITY

We have written a computer program to analyze a rectangular cavity

based on the theory in Sections 2-4. This program was used to provide the

theoretical results for comparison with measured data in Section 5. This

software may not be the ultimate form for application software, but

potential users may find it to be very helpful. We are therefore including

the source code in this appendix, along with instructions on how to use it.

41
The program is written in Fortran 77 and can be compiled to run on whatever computer is available. This includes personal computers with a 386

processor and 387 coprocessor or 486DX processor.

Input data

The compiled program will look for an input data file of whatever name

is selected by the input symbol '<' . For example, run 'DFAA7 < Nil' where

DFAA7 is the executable file and Nil is the input data file.

An example of the input file is shown before the source code. All

input data should be in SI units. The use of 'D' rather than 'E' for

exponents indicates double precision numbers.

On the first two lines, the numbers under A, B, and C are the

dimensions of the rectangular- shaped cavity. The number under WCOND is the

wall conductivity, the number under RMU is the relative permeability of the

walls, the number under NANT is the number of antennas inside the cavity

that couple energy out of the cavity, and the number under NAPRAD is the

number of different sizes of apertures.

On the third line, the column headers APRAD and NAP refer to aperture

radius and number of apertures, respectively. If there are no apertures,

both NAPRAD and all NAP values should be . Up to six different aperture

sizes can be used.

On lines 10 and 11, the number under NHSPH is the number of absorptive

spheres with salt water. Salt water provides a good approximation to the

absorption of people. The number under HSPRAD is the radius of these

spheres. The number under HCNDSP is the conductivity of ocean salt water.

The number under EPSLP is the low- frequency relative dielectric constant of

salt water. The number under EPSHP is the high-frequency dielectric

constant of salt water. The number under TR is the dielectric relaxation

time. See Appendix E. Only the first two numbers should be changed by a program user. On lines 12 and 13, the number of spheres of other material may be

treated. No frequency dependence is allowed. The number under NOSPH is the

number of spheres , and the number under OSPRAD is the radius of these

42
spheres. The number under OCONDSP is the conductivity of the sphere

material, and the number under EPSOP is its relative dielectric constant.

On lines 14 and 15, the number under FREQL is the start frequency, the

number under DF is the frequency increment, and the number under NF is the

number of frequency steps.

Output data

The output data are always put into a file 'CAVO.CSV. The CSV is a comma- separated variable format for use with a spread sheet and for

plotting. This format could be changed to any other desired format.

The first 15 lines repeat the input data and also give the volume and

surface area of the cavity.

The output is in 15 columns with the data within each row identified by

a column header.

These are: frequency, transmission cross -section average, Ql (Q due to

wall loss) , Q2H (Q due to humanoid absorption) , Q20 (Q due to other

absorptive materials), Q3(l) (Q due to first size of aperture), Q3(2) (Q due

to second size of aperture), ... Q3(6) (Q due to sixth size of aperture), Q4

(Q due to antenna coupling), Q (composite of all Qs) , TAU (time constant),

and SE (shielding effectiveness)

.

A spread sheet or plotting program is very helpful in displaying these

data for insights and interpretations.

Input file:

A

.51435D0

APRAD

.014D0

.025D0

.050D0

1.5875D-3

.050D0

.050D0

NHSPH HSPRAD

1 .066D0

NOSPH OSPRAD

.025D0

FREQL DF

.5D9 .25D9

B C WCOND RMU NANT

0.62865D0 0.1750D1 8.83D6 l.DO 2 NAP

1

HCNDSP EPSLP EPSHP TR

4. DO 70. DO 4.9 9.2D-12

OCONDSP EPSOP

2.5D0 4. DO

NF

71

NAPRAD

1

43
Source code:

PROGRAM DFAA7

C THIS PROGRAM IS BASED ON THE WORK OF DAVID HILL AND MARK MA OF

C NIST, BOULDER, CO. THE THEORY WAS DEVELOPED DURING LATE 1992.

C THE PROGRAM WAS WRITTEN JAN. 93 BY DAVID HILL, 303-497-3472.

C IT WAS MODIFIED FEB 93 FOR USE ON A 386 OR 486 PC

C BY ABBIE O'GALLAGHER, 303-497-3266.

C THIS DEVELOPMENT WAS SUPPORTED BY THE FAA TO SUPPORT HIRF TESTING

C REQUIREMENTS, BUT IS BROADLY APPLICABLE TO RESONANT CAVITIES.

C THERE WILL NOW BE 2 SETS OF DATA HAVING TO DO WITH SPHERES. THE ONES

C THAT START WITH H HAVE TO DO WITH HUMANOID (SALT-WATER) SPHERES. NHSPH

C IS THE NUMBER OF HUMANOID SPHERES.

C Q2H IS THE ENERGY LOSS DUE TO HUMANOID SHPERES

.

C QWO WILL BE THE ENERGY LOSS DUE TO OTHER KINDS OF SHERES

.

C NOSPH,OSPRAD,OCNDSP AND EPSOP HAVE TO DO WITH OTHER KINDS OF SPHERES.

INTEGER MAXNAR

PARAMETER ( MAXNAR=6

)

C THE PARAMETER NAPRAD IS THE NUMBER OF DIFFERENT APERTURE RADII

C MAXNAR IS THE (ARBITRARILLY) MAXIMUM VALUE NAPRAD CAN TAKE

DOUBLE PRECISION PI , EPSO , PERMO ,RMU, CL, A, B, C , APRAD(MAXNAR)

,

1 WCOND,HSPRAD,TEMP,

1 HCNDSP , EPSP , V , S , EPSLP , EPSHP , TR , X , DEN

,

1 FREQL,DF,FREQ,FW,DEL,Q1,WL,WK,EPSI,Q2H,SIGABS,

1 SIGAVE (MAXNAR) ,TSIGAV,Q3 (MAXNAR) ,Q4 , TAU.Q, SE, PDRAT, A,

1 OSPRAD.OCNDSP, EPSOP, Q20

INTEGER NAP (MAXNAR) ,NANT,NHSPH,NF, I ,NT,NOSPH ,NAPRAD ,

J

DOUBLE COMPLEX EPSC.REFI

OPEN (3, FILE='CAVO.CSV')

REWIND 3

PI=DACOS(-1.DO)

5 FORMAT (IX)

DO 1 1=1, MAXNAR

NAP(I)=0

1 CONTINUE

EPS0=8.854D-12

PERM0=PI*4.D-7

CL=3.D8

READ(*,5)

READ ( *

, *) A , B , C , WCOND , RMU , NANT , NAPRAD

IF (NAPRAD .GT. MAXNAR) THEN

PRINT*, 'ERROR. MAXNAR SMALLER THAN NAPRAD'

STOP

ENDIF

READ(*

READ(*

READ(*

READ(*

READ(*

READ(*

READ(*

READ(*

5)

*)(APRAD(I) ,NAP(I) ,1=1, MAXNAR)

5)

*) NHSPH, HSPRAD, HCNDSP, EPSLP, EPSHP, TR

5)

*) NOSPH, OSPRAD, OCNDSP, EPSOP

5)

*)FREQL, DF, NF

WRITE (3,10)

44
10 FORMAT (' A,B,C,WALLCOND,RELMU,NAPRAD')

WRITE (3,20) A , B , C , WCOND , RMU , NAPRAD

20 FORMAT (5(E15 . 5 ,','), 110)

WRITE (3,11)

11 FORMAT (' APRAD, NAP')

WRITE (3,31) (APRAD(I) ,NAP(I) ,I=1,MAXNAR)

31 FORMAT (E15.5, '

, ' ,110)

WRITE (3,23)

23 FORMAT (' NANT , NHSPH,HSPHERERAD .HSPHERECOND , SEPSL, SEPSH,RELTIME'

)

WRITE (3,27) NANT,NHSPH,HSPRAD,HCNDSP,EPSLP,EPSHP,TR

27 FORMAT (2 (110 , '

, ' ) ,4(E15 . 5

, ',

' )

, E15 . 5)

WRITE (3,28)

28 FORMAT (' NOSPH, SPHERERAD, SPHERECOND, SPHEREPS , '

)

WRITE (3,29) NOSPH, OSPRAD,OCNDSP,EPSOP

29 FORMAT ( (110 , '

, ' )

, 2 (E15 . 5

, ',

' )

, E15 . 5)

V=A*B*C

C APPLY CORRECTION TO VOLUME FROM BARE CAVITY TO COMPENSATE FOR

C VOLUME OF ENCLOSED STRUCTURES

C V=V*.90D0

S=2 . DO* ( A*B+A*C+B*C

)

C APPLY CORRECTION TO SURFACE OF BARE CAVITY TO COMPENSATE FOR

C SURFACE AREA OF ENCLOSED STRUCTURES

C S=S*1.2D0

WRITE (3,30)

30 FORMAT (' VOL, SURF')

WRITE (3,40) V,S

40 FORMAT (E15 . 5

, '

,

'

, E15 . 5)

WRITE (3,50)

50 FORMAT (' FREQ, SIGAVE,Q1 ,Q2H,Q20,Q3(1) ,Q3(2) ,Q3(3) ,Q3(4) ,Q3(5) ,

1 , 'Q3(6) ,Q4,Q,TAU,SE')

DO 60 1=1, NF

FREQ=FREQL+ ( I - 1

) *DF

FW=2.D0*PI*FREQ

DEL=SQRT(2 . D0/(FW*RMU*PERM0*WCOND)

)

Ql=l . 5D0*V/(S*RMU*DEL)

WL=CL/FREQ

WK=2 . DO*PI/WL

X=FW*TR

DEN=1 . DO+X*X

EPSP=EPSHP+ ( EPSLP - EPSHP) /DEN

EPSI=HCNDSP/(FW*EPSO)+(EPSLP-EPSHP)*X/DEN

EPSC=CMPLX(EPSP, -EPSI)

REFI=SQRT(EPSC)

IF (NHSPH .LE. 0)THEN

Q2H=1.D30

ELSE

CALL ABSORB (HSPRAD , WK , REFI , NT , SIGABS

)

Q2H=2 . DO*PI*V/ (WL*SIGABS*NHSPH)

END IF

IF (NOSPH .LE. 0)THEN

Q2O=l.D30

ELSE

EPSI=OCNDSP/(FW*EPS0)

45
EPSC=CMPLX(EPSOP, -EPSI)

REFI-SQRT(EPSC)

CALL ABSORB ( OS PRAD,WK,REFI, NT, SIGABS)

Q20=2 . D0*PI*V/(WL*SIGABS*NOSPH)

ENDIF

C TSIGAV IS THE SUM OF THE SIGAVES FROM 1 TO NAPRAD

TSIGAV=O.DO

DO 80 J=1,MAXNAR

IF (NAP(J) .LE. 0)THEN

Q3(J)=1.D30

SIGAVE(J)=0.

ELSE

CALL CIRC(WL,APRAD(J),SIGAVE(J))

Q3(J)=2.D0*PI*V/(WL*SIGAVE(J)*NAP(J))

ENDIF

TSIGAV=TSIGAV+SIGAVE(J)*NAP(J)

80 CONTINUE

IF (NANT .LE. 0)THEN

Q4=1.D30

ELSE

Q4=16 . D0*PI*PI*V/(NANT*WL**3)

ENDIF

C Q=l . DO/ (1 . DO/Ql+1 . D0/Q2H+1 . D0/Q2O+1 . D0/Q3+1 . D0/Q4

)

C THE FOLLOWING LOOP REPLACES THE ABOVE ASSIGNMENT STATEMENT

C NOW THAT THERE ARE MULTIPLE Q3S.

C NOTE THE FOLLOWING INCONSISTENCY:

C THE Q3(J) FROM (NAPRAD+1) TO MAXNAR

C ARE PRINTED OUT BELOW BUT THE 1/Q3(J) TERMS ARE NOT

C ADDED INTO Q. THE LATTER SHOULD BE VERY

C SMALL SO IT SHOULD MAKE LITTLE, IF ANY, DIFFERENCE.

TEMP=1 . DO/Ql+1 . D0/Q2H+1 . D0/Q2O+1 . D0/Q4

DO 90 J=l, NAPRAD

TEMP=TEMP+1 . D0/Q3 (J

)

90 CONTINUE

Q=l . DO/TEMP

C

TAU=Q/FW

IF (NAPRAD .LE. 0)THEN

SE=1.E10

ELSE

PDRAT=TSIGAV*WL*Q/( 2 . DO*PI*V)

SE=- 10 . DO*LOG10 (PDRAT)

ENDIF

WRITE (3,70)FREQ,TSIGAV,Q1,Q2H,Q20, (Q3(J) , J=l .MAXNAR) ,Q4,Q,

1 TAU.SE

60 CONTINUE

70 FORMAT (14(E13 .4, '

, ' )

, E13 .4)

STOP

END

SUBROUTINE CIRC(WL,RAD, SIGAVE)

DOUBLE PRECISION WL, RAD, SIGAVE

DOUBLE PRECISION PI.WKA

46
PI-3. 1415926536

PI=DACOS(-1.DO)

WKA=2 . DO*PI*RAD/WL

IF (WKA-1.3D0) 10,20,20

10 SIGAVE=8.D0*(WKA**4)*RAD*RAD/(9.D0*PI)

GO TO 30

20 SIGAVE=PI*RAD*RAD/4.D0

30 RETURN

END

SUBROUTINE ABSORB(A,WK,REFI ,NT, SIGABS)

DOUBLE COMPLEX REFI ,Y, FJY(50) , FH(50) ,RJY,DRJY,RHX, DRHX, ANUM, ADEN,

1 AN,BN,BNUM,BDEN,FJX(50) ,X,X2 ,RJX,DRJX,REFI2 ,RAD,

1 RPER , RPAR

DOUBLE PRECISION A, WK, SIGABS

INTEGER NT

DOUBLE PRECISION PI , AVE, SE, SS ,TE,TS ,QE,QS

INTEGER NTP.NP.N

PI=3. 1415926536

PI=DACOS(-1.DO)

X=WK*A

NT=REAL(X)+3

NT=DBLE(X)+3

IF (NT-49) 20,20,30

30 REFI2=REFI*REFI

RAD=SQRT(REFI2- .5D0)

RPER=( . 707D0-RAD)/( . 707D0+RAD)

RPAR=( . 707D0*REFl2-RAD)/( . 707D0*REFI2+RAD)

AVE=.5D0*(ABS(RPER)**2+ABS(RPAR)**2)

SIGABS=PI*A*A*(1.D0-AVE)

GO TO 40

20 Y=REFI*X

NTP=NT+1

CALL CSPJ(X,NTP,FJX)

CALL CSPJ(Y,NTP,FJY)

CALL CSPH2(X,NTP,FH)

SE=0.D0

SS=0.D0

DO 10 N=1,NT

NP=N+1

RJX=X*FJX(NP)

DRJX=FJX(NP)+X*FJX(N) -NP*FJX(NP)

RJY=Y*FJY(NP)

DRJY=FJY(NP)+Y*FJY(N)-NP*FJY(NP)

RHX=X*FH(NP)

DRHX=FH ( NP ) +X*FH ( N

) - NP*FH ( NP

)

ANUM=DRJY*RJX-REFI*RJY*DRJX

ADEN=DRJY*RHX- REFI *RJY*DRHX

AN=ANUM/ADEN

BNUM=REFI*DRJY*RJX-RJY*DRJX

BDEN=REFI*DRJY*RHX- RJY*DRHX

BN=BNUM/BDEN

TE=(2*N+1)*REAL(AN+BN)

47
TE=(2*N+1)*DBLE(AN+BN)

TS=(2*N+1)*(ABS(AN)**2+ABS(BN)**2)

SE=SE+TE

10 SS=SS+TS

X2=X*X

QE=2.D0*SE/X2

QS=2.D0*SS/X2

SIGABS=PI*A*A*(QE-QS)

40 RETURN

END

SUBROUTINE CSPJ(Z ,NJ , FJ)

INTEGER PJMAX

PARAMETER ( PJMAX=2 50

DOUBLE COMPLEX Z, FJ(50) , PJ (PJMAX) , PC , CARG

INTEGER NJ

DOUBLE PRECISION AO.Al.PI

INTEGER I,JMAX,K,J

PI=3. 1415926536

FJ(1)=SIN(Z)/Z

FJ(2)=SIN(Z)/(Z*Z)-C0S(Z)/Z

I=ABS(Z)

JMAX=I+NJ+10

IF (JMAX-248) 10,20,20

20 DO 30 1=1, NJ

CARG=Z-(I-l)*PI/2.

30 FJ(I)=SIN(CARG)/Z

GOTO 90

10 PJ(JMAX+2)=0.D0

PJ(JMAX+l)=l.D-20

DO 40 J=1,JMAX

K=JMAX+1-J

40 PJ(K)=(2*K+l)*PJ(K+l)/Z-PJ(K+2)

A0=ABS(FJ(1))

A1=ABS(FJ(2))

IF(Al-AO) 50,50,60

50 PC=FJ(1)/PJ(1)

GO TO 70

60 PC=FJ(2)/PJ(2)

70 DO 80 I=l,NJ-2

80 FJ(I+2)=PJ(H-2)*PC

90 RETURN

END

SUBROUTINE CSPH2(Z,NH, FH)

DOUBLE COMPLEX Z , FH ( 50 ) , CJ , FACEX

INTEGER NH

INTEGER N,M,MP,MPP

CJ=CMPLX(0.D0,1.D0)

FACEX=EXP(-CJ*Z)

FH(1)=CJ*FACEX/Z

FH ( 2

) =FACEX* (

- 1 . DO/Z+CJ/ ( Z*Z )

N=NH-2

48
IF(N) 10,10,20

20 DO 30 M=1,N

MP=M+1

MPP=M+2

30 FH(MPP)=(2*M+1)*FH(MP)/Z-FH(M)

10 CONTINUE

RETURN

END

49
Apertures

Figure 1. Aperture excitation of a cavity containing absorbers and a receiving antenna.

50
^ y • • • • ,

& Wall

Figure 2. External field incident on an aperture of arbitrary shape

51
Normal

Circular

Aperture

Figure 3 . External field incident on a circular aperture of radius a

52
NIST Chamber

1 -h

Motor

jr

1 Reference

1 Receive

1 Antenna

DC Si

^_ -JTune

jppiy D—

1

1 FAA Cavity

1 c

1 Receive Ant

nj DC Motor 1

1

1 H H 1 uner . enna / fransmit

Antenna

-Y--

1

1

1 J Transmit

Antpnna

' \

"^ f

j / ^

Switch <

L. —-T: -D—- RF Switcr1 Motor

Controller r^

1 haa

RFAmp

, 488

^ ' Bus

Spectrum

8 Bus Analyzer

Computer

48 Signal

Generator

1

Figure 4, Block diagram of system for measurement of shielding

effectiveness. Block diagram of system for measurement of loss

is similar; reconnect cable from NIST chamber transmit antenna to

FAA cavity transmit antenna.

53
FAA Cavity ^

DC Motor

Tuner

Transmit

Antenna

I Receive V y Antenna

Switch

-E

Motor

Controller

^

488

Bus

Xtal /

Detector

DC Supply

Computer

488 Bus

—/^

fBuikh Bulkhead

JT Xtal Detector

Digitizing

Osilloscope

RFAmp

Signal

Generator

Trigger Out

488 Bus —^

Pulse Out

Pulse

Generator

Figure 5. Block diagram of system for measuring time constant. Output data

files are used to plot time response curves. One example is

shown in figure 6

.

54
7 .

+>3

a.3O

cn

>2

u

50E-5

R ye r a ge

J .-"-^ ^

00E-5

50E-5

/

1

1

I\

/ V

( us )

Figure 6. Time response of FAA cavity loaded with one aperture of radius

0.014 m and one sphere of radius 0.066 m filled with salt water.

This is the average of 200 waveforms from 200 stirrer positions.

An exponential, least- squares fit is applied to the decay time to

obtain the time constant. Frequency = 18 GHz.

55
o

i X

^

Z4UUUU

o 1 nnr^n _ i

if^ ^

0^

1 LiTr Z 1 UUUU

1 Rnnnn i

^

^ rurn k^

1 oUUUU

1 i^AAnn //s^ /^P^

1 DUUUU

1 o nnnn /

f^ /*

1 zUUUU

nr\r\r\n eS n

^ yuuuu

buuuu / / /

/ ^

/

jUUUU

Wffislfef^»w«JWWJ<W

,.».-! 0-

5$$$$«W ;iiiyi>M>Mxi)^ siSuMXD'KP^ S^qXiXJXlALXiX

01

8 10 12 14 16

Frequency (GHz)

-^ Qm:TC

-^5^ 04

-ffl—

-^

Qm:Loss

Figure 7. Five plots of Q for the FAA cavity with aperture of radius 0.014

m, two antennas, no absorptive loading, and no corrections. Ql

is the calculated Q from wall losses. Q4 is the calculated Q

from two antennas. Q3 is calculated for aperture loading, but

the curve is off scale. Q is the composite Q of all calculated

Qs from all loss mechanisms in the model. Qm-TC is Q calculated

from measured time constant values. Qm-Loss is Q calculated from

measured loss values. There is a large discrepancy between

measured and calculated values. See text and later figures for

explanation.

56
CO

(D

2.7e-6

2.4e-5

2.1e-6

.8e-5

1.5e-6

1.2e-6

9e-7

6e-7

3e-7

-^ Calc -^^ Msd-TC

6 8 10 12

Frequency (GHz)

14

Figure 8. Calculated and measured time constant values for the FAA cavity

with aperture of radius 0.014 m, two antennas, no absorptive

loading, and no corrections. Squares indicate calculated and xs indicate measured values. There is a large discrepancy between

the two. See text and later figures for explanation.

57
-a— Calc

->^ Msd, Avg

6 8 10 12

Frequency (GHz)

Figure 9. Calculated and measured values of SE for the FAA cavity with

aperture of radius 0.014 m, two antennas, no absorptive loading,

and no corrections. There is a large discrepancy between the

two. See text and later figures for explanation.

58
o

DOUUUU -

DdUUUU / /

-1- 4yuuuu ?

4z(JU(JU /I OAAAn \ /

Tcnnnn

^ [/ ojUUUU /

o QPtPtnn 1

^

ZC5UUUU

z

o 1

1 nnnn UUUU /

1

1 Annnri 4UUUU /

innnn

f

CTfl=^ ^S=i±l

^^^^^^^^S=S]

/UUUU ^

^^

m^S^^^^$^t^#^^

^

^^J^p^^^*^*^ ifUHlllllTO ijlllTHHMiJt ^M^l^M^;OW*j.j*j»S»

-B6 8 10 12

Frequency (GHz)

14 16

-&-

-ffl- -^

01

Q2H

03(1)

Om:TC

04

Om:Loss

Figure 10. Seven plots of Q for the FAA cavity with aperture of radius 0.014

m, two antennas, one sphere of radius 0.066 m filled with salt

water for absorptive loading. Conductivity (decreased by 4),

volume (10% decrease) , and surface area (20% increase)

corrections are applied. Ql is the calculated Q from wall

losses, Q4 is the calculated Q from two antennas, Q3 is the

aperture loading. Q2H, Q, Qm-TC, and Qm-loss are compressed at

the bottom of the figure. See figure 11 for expanded scale.

59
36000

32000

28000

24000

20000

o 16000

2000

8000

4000

—B— Q1

Q2H —e— Qm:TC

Q4 —ffl— Q

<t> Om:Loss

6 8 10 12

Frequency (GHz)

Figure 11. Six plots of Q for the FAA cavity with aperture of radius 0.014

m, two antennas, one sphere of radius 0.066 m filled with salt

water for absorptive loading. Conductivity (decreased by 4),

volume (10% decrease) , and surface area (20% increase)

corrections are applied. Ql is the calculated Q from wall

losses. Q4 is the calculated Q from two antennas. Q3 is the

aperture loading, off scale. Q2H is the calculated Q for

absorptive loading of the sphere of salt water, Q is the

composite Q of all calculated Qs from all loss mechanisms in the

model, Qm-TC is Q calculated from measured time constant values.

Qm-Loss is Q calculated from measured loss values.

60
in

(D

9e-7

5e-7

7e-7

6e-7

5e-7

4e-7

3e-7

2e-7

1e-7

-^ Calc -^^ Msd

6 8 10 12

Frequency (GHz)

14 16

Figure 12. Calculated and measured time constant values for the FAA cavity

with aperture of radius 0.014 m, two antennas, one sphere of

radius 0.066 m filled with salt water for absorptive loading.

Conductivity (decreased by 4) , volume (10% decrease) , and surface

area (20% increase) corrections are applied. Squares indicate

calculated and xs indicate measured values. There is much better

agreement between the measured and calculated values.

61
6 8 10 12

Frequency (GHz)

Figure 13. Calculated and measured values of SE for the FAA cavity with

aperture of radius 0.014 m, two antennas, one sphere of radius

0,066 m filled with salt water for absorptive loading.

Conductivity (decreased by 4) , volume (10% decrease) , and surface area (20% increase) corrections are applied. There is much

better agreement between the measured and calculated values

.

62
630000

560000

490000

420000

350000

a 280000

210000

40000

70000

Frequency (GHz)

Figure 14. Seven plots of Q for the FAA cavity with aperture of radius 0.014

m, two antennas, three spheres of radius 0.066 m filled with salt

water for absorptive loading. Conductivity (decreased by 4),

volume (10% decrease) , and surface area (20% increase)

corrections are applied. Ql is the calculated Q from wall

losses. Q4 is the calculated Q from two antennas. Q3 is the

calculated Q from aperture loading. Q2H, Q, Qm-TC, and Qm-loss

are compressed at the bottom of the figure. See expanded scale

in figure 15.

63
36000

32000

28000

24000

20000

O 16000

12000

8000

4000

6 8 10 12

Frequency (GHz)

^— Qm:Loss

Figure 15. Six plots of Q for the FAA cavity with aperture of radius 0.014

m, two antennas, three spheres of radius 0.066 m filled with salt

water for absorptive loading. Conductivity (decreased by 4),

volume (10% decrease) , and surface area (20% increase)

corrections are applied. Ql is the calculated Q from wall

losses. Q4 is the calculated Q from two antennas. Q3 is the

calculated Q from aperture loading, off scale. Q2H is the

calculated Q for absorptive loading of the spheres of salt water,

Q is the composite Q of all calculated Qs from all loss

mechanisms in the model, Qm-TC is Q calculated from measured

time constant values. Qm-Loss is Q calculated from measured loss

values. There is good agreement between measured and calculated

values

.

64
w

CD

1.8e-7

1.6e-7

1.4e-7

.2e-7

le-7

5e-^

6e-8

Ae-i

2e-8

-^ ri^^

^^ 1 1 1 1 1 1 1 r prrrrFTT FRFFtFttrrrTi-m- — ^^^^I.J 1 H H t iJ-J-JUzLLL FR-Ht-Ft /

J^f=i=

t^— — >; ^

-B— Calc ^^— Msd

) 8 10 12

Frequency (GHz)

14 16

Figure 16. Calculated and measured time constant values for the FAA cavity

with aperture of radius 0.014 m, two antennas, three spheres of

radius 0.066 m filled with salt water for absorptive loading.

Conductivity (decreased by 4), volume (10% decrease), and surface

area (20% increase) corrections are applied. Squares indicate

calculated and xs indicate measured values. There is good

agreement between measured and calculated values.

65
Calc

Msd

6 8 10 12

Frequency (GHz)

Figure 17. Calculated and measured values of SE for the FAA cavity with

aperture of radius 0.014 m, two antennas, three spheres of radius

0.066 m filled with salt water for absorptive loading.

Conductivity (decreased by 4), volume (10% decrease), and surface area (20% increase) corrections are applied. There is good

agreement between the measured and calculated values.

66
720000

640000

560000

480000

400000

a 320000

240000

160000

80000

::

6 8 10 12

Frequency (GHz)

14 16

—B— Q1

-u- Q3(1)

/S Qm:TC

0 Qm:Loss

04 —ffl— Q

Figure 18. Six plots of Q for the FAA cavity with aperture of radius 0.014

m, two antennas, no absorption. Conductivity (decreased by 4),

volume (5% decrease), and surface area (10% increase) corrections

are applied. Standard gain horns were used for measurements from

12 to 18 GHz. Ql is the calculated Q from wall losses. Q4 is

the calculated Q from two antennas . Q3 is the calculated Q from

aperture loading. Q, Qm-TC, and Qm-loss are compressed at the

bottom of the figure. See figure 19 for expanded scale.

67
a

80000

60000

40000

20000

100000

80000

60000

40000

20000

I

I

I .x^'-

1

_ij-d

^

.^

^^

^

^

^ 1 UfTI \^

w^ , Y-J

_[

V rrtn <i /

^

- i^^

-B- Q6 8 10 12

Frequency (GHz)

16

^— Qm:TC

-*— Qrn:Loss ^^ 04

-ffl— Q

Figure 19. Five plots of Q for the FAA cavity with aperture of radius 0.014

m, two antennas, no absorptive loading. Conductivity (decreased

by 4), volume (5% decrease), and surface area (10% increase)

corrections are applied. Standard gain horns were used for

measurements from 12 to 18 GHz. Ql is the calculated Q from wall

losses. Q4 is the calculated Q from two antennas. Q3 is the

calculated Q from aperture loading, off scale, Q is the

composite Q of all calculated Qs from all loss mechanisms in the

model, Qm-TC is Q calculated from measured time constant values,

Qm-Loss is Q calculated from measured loss values. There is very

good agreement between measured and calculated values; the

differences are within 20%, well within measurement system

uncertainties.

68
en

CD

2.7e-6

2.4e-5

2.1e-6

.8e-6

1.5e-6

.2e-6

9e-7

5e-7

3e-7

/

^-d 1 1 r ^^

^=fq-tj-,

FF^&*^

^^^

/

s ^^

^^^]

»i r

>?-- '^

/

/

-^ Calc

-^<— Msd

6 8 10 12

Frequency (GHz)

14 16

Figure 20. Calculated and measured time constant values for the FAA cavity

with aperture of radius 0.014 ra, two antennas, no absorptive

loading. Conductivity (decreased by 4) , volume (5% decrease)

,

and surface area (10% increase) corrections are applied. Squares

indicate calculated and xs indicate measured values. There is

very good agreement between measured and calculated values; the

differences are within measurement system uncertainties.

69
45

40

35

30

m

if)

25

20

10

1

1

V

) \

\

[

r-i

> TTTTTTS H=F=l=n=FB ra^^iq=f ^ 1 1 1 1 1 FR=FFFFE a=hy±i=Epj 1 1 LL-l-l

^Er- Calc -^^ Msd

6 8 10 12

Frequency (GHz)

14 16

Figure 21. Calculated and measured values of SE for the FAA cavity with

aperture of radius 0.014 m, two antennas, no absorptive loading.

Conductivity (decreased by 4) , volume (5% decrease) , and surface area (10% increase) corrections are applied. There is good

agreement between the measured and calculated values; differences

are within the measurement system uncertainties.

70
Stirrer

Circular

Aperture

Receiving

Antenna

>

h

w

Figure 22. NSWC rectangular cavity with a circular aperture. A mode stirrer

and a receiving antenna are located inside.

71
QQ

LJ

° ° ° = ° Measurement, a = 3.51 cm

o o o o o Measurement, a = 2.94 cm

Theory, a = 3.51 cm

Theory, a = 2.94 cm

FREQUENCY GHz

Figure 23. Comparison of calculated and measured SE for the NSWC rectangular

cavity with a circular aperture.

*U.S. GOVERNMENT PRINTING OFFICE: 1993-774-025/89048

72
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