ESTABLISHING PROBABILITY OF
FAILURE OF A SYSTEM DUE TO
ELECTROMAGNETIC INTERFERENCE
By
VIGNESH RAJAMANI
Bachelor of Engineering (ECE)
University of Madras, India
2002
Master of Science in Electrical Engineering
Oklahoma State University
Stillwater, OK
2004
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
DOCTOR OF PHILOSOPHY
July 2010
ii
ESTABLISHING PROBABILITY OF
FAILURE OF A SYSTEM DUE TO
ELECTROMAGNETIC INTERFERENCE
Dissertation Approved:
Dr. Charles F. Bunting
Dissertation Adviser
Dr. James C. West
Dr. Rama G. Ramakumar
Dr. Manjunath Kamath
Dr. Mark. E. Payton
Dean of the Graduate College
iii
ACKNOWLEDGMENTS
This thesis has had mentorship from numerous outstanding individuals both from within
the university and outside of it. Many people have been a part of my graduate education,
as friends, teachers, and colleagues. Dr. Charles F. Bunting, first and foremost, has been
all of these. The best advisor and teacher I could have wished for, he is actively involved
in the work of all his students, and clearly always has their best interest in mind. Thank
you for pushing me. On the personal side, he did not hesitate to invite his students to
become an extended part of his family.
I have been indebted in the preparation of this thesis to Dr. James C. West whose
patience and kindness, as well as his academic experience, has been invaluable to me. My
special thanks are to Dr. Rama Ramakumar for his advice in Markov Chain modeling.
Dr. Kamath Manjunath has been instrumental in advising me on the final part of this
thesis and my hearty thanks goes out to him.
I shall always be grateful to Dr. Gustav Freyer for the long telephonic discussions we had
together and the knowledge I gained from him. This work would not have been possible
without the support and encouragement of, Ted Lehman whom I know is wishing me
well from above.
iv
I would like to thank several of my friends for helping with the preparation of this thesis
especially my lab mates. Their honest yet considerate criticisms of this work have helped
much in improving its quality. I cannot end without thanking my wife Subha, the constant
support and encouragement from her which was always there when I needed it.
Lastly, and most importantly, I wish to thank my parents, Rajamani and Bhama. They
bore me, raised me, supported me, taught me, and loved me. To them I dedicate this
thesis.
v
TABLE OF CONTENTS
1.0 INTRODUCTION....................................................................................................... 1
1. 1 OPEN AREA TEST SITE (OATS) ................................................................................ 3
1. 2 FULLY ANECHOIC CHAMBERS ................................................................................... 5
1. 3 SEMI ANECHOIC CHAMBERS ...................................................................................... 7
1. 4 TEM ......................................................................................................................... 8
1. 4. 1 TEM Cell .......................................................................................................... 9
1. 4. 2 TEM Waveguide ............................................................................................... 9
1. 4. 3 GTEM ............................................................................................................... 9
1. 5 REVERBERATION CHAMBERS ................................................................................... 11
1. 6WHY USE REVERBERATION CHAMBER IN THIS STUDY? ........................................... 15
1. 7 ORGANIZATION OF THE THESIS ................................................................................ 16
2.0 ASSESSING THE ELECTROMAGNETIC ENVIRONMENT .......................... 19
2. 1 THEORETICAL ESTIMATION OF K ............................................................................. 21
2. 2 FREQUENCY DOMAIN ............................................................................................... 23
2. 2. 1 Experimental procedure................................................................................. 24
2. 2 TIME DOMAIN .......................................................................................................... 30
vi
2. 2. 1 Calculation of reverb distance from time domain data ................................. 36
2.3 INSERTION LOSS AND ITS IMPLICATION TO REVERBERATION DISTANCE .................... 37
2. 4MEASUREMENTS IN HALLWAY ................................................................................ 44
2. 4. 1 Frequency domain ......................................................................................... 44
2. 4. 2 Time domain................................................................................................... 45
2. 5MEASUREMENTS IN SMALL ROOM ........................................................................... 47
2. 5. 1 Frequency domain ......................................................................................... 47
2. 5. 2 Time domain................................................................................................... 48
2. 6 COMPARISON OF FREQUENCY AND TIME DOMAIN Q MEASUREMENTS FROM
LITERATURE ................................................................................................................... 51
2.7 Q - FREQUENCY AND TIME DOMAIN ......................................................................... 55
2.8 DIFFERENCE BETWEEN FREQUENCY AND TIME DOMAIN Q FACTOR ......................... 67
2.9 CALCULATING ANTENNA TO CHAMBER COUPLING EFFICIENCY ............................. 68
2.10 SUMMARY .............................................................................................................. 70
3.0 EFFECT OF LOADING ON INDEPENDENT SAMPLES ................................. 72
3.1 INDEPENDENT SAMPLES (IS) .................................................................................... 73
3.2 EXPERIMENTAL SETUP ............................................................................................. 76
3. 3 UNIFORMITY INDEPENDENT SAMPLES AND LOADING .............................................. 84
3. 4 ANALYSIS WITH TWO TUNERS ................................................................................. 90
4.0 PREDICTION OF INDUCED CURRENT ON A CABLE OPERATING
INSIDE A CAVITY AND VALIDATION OF MODEL FIDELITY ..................... 101
4. 1 DISTRIBUTION OF POWER CLOSE TO THE WALL ...................................................... 106
4. 1. 1 Measurements .............................................................................................. 106
vii
4. 1. 2 Simulations ................................................................................................... 112
4. 2MODAL STRUCTURE STUDY ................................................................................... 113
4. 3 CURRENT INDUCED ON THE WIRE LOCATED INSIDE THE CAVITY DUE TO AN
EXCITATION OUTSIDE ................................................................................................... 120
4. 3. 1 Simulation .................................................................................................... 120
4. 3. 2 Measurements (Open air) ............................................................................ 121
4. 3. 3 Adding arbitrary impedance to the wire ...................................................... 128
4. 4 CURRENT INDUCED ON THE WIRE LOCATED ON A GROUND PLANE DUE TO AN
EXCITATION OUTSIDE (WITHOUT BOX) ......................................................................... 130
4.5 CHANGE IN ANGLE OF INCIDENCE ........................................................................... 135
4.6 DISTRIBUTIONS FOR THE SHORT CIRCUIT CURRENT CONSIDERING ANGLES OF
INCIDENCE ................................................................................................................... 142
5.0 STATISTICS OF THE INDUCED CURRENT, EMI THREAT MODEL AND
PROBABILTIY OF FAILURE ................................................................................... 145
5.1 STATISTICS OF INDUCED CURRENT (CONDITIONAL PROBABILITY OF THE
OBSERVABLE) .............................................................................................................. 147
5. 2 POISSON PROCESS TO MODEL ELECTROMAGNETIC INTERFERENCE ......................... 153
5. 3 POISSON PROCESS .................................................................................................. 155
5. 4 EMI....................................................................................................................... 156
5.5 PROBABILITY OF THREAT DUE TO PEAK, AVERAGE AND CHANGE OF STATE EMI.... 158
5.6 PROBABILITY OF FAILURE ...................................................................................... 165
5.7MARKOV MODEL FOR EMI .................................................................................... 166
6.0 SUMMARY AND FUTURE WORK .................................................................... 172
viii
6.1 NOVEL ASPECTS OF THIS WORK .............................................................................. 176
6.2 FUTURE WORK ....................................................................................................... 178
REFERENCES .............................................................................................................. 180
APPENDIX 1 ................................................................................................................. 192
APPENDIX 2 ................................................................................................................. 199
ix
LIST OF TABLES
TABLE 2. 1 Q AND K VALUES FOR A REVERBERATION CHAMBER (NO DIRECT COUPLING
BETWEEN THE ANTENNAS) ......................................................................................... 28
TABLE 2. 2 K VALUES FOR A REVERBERATION CHAMBER (DIRECT COUPLING BETWEEN THE
ANTENNAS) ................................................................................................................ 29
TABLE 2. 3MEASURED Q VALUES USING FREQUENCY AND TIME DOMAIN ......................... 55
TABLE 2. 4 Q VARIATION AS A FUNCTION OF BANDWIDTH AROUND THE CENTER FREQUENCY
OF 500MHZ AND 5GHZ ............................................................................................ 59
TABLE 2. 5MEASURED Q VALUES USING TWO DIFFERENT RECEIVE ANTENNAS ................. 63
TABLE 2. 6MEASURED Q VALUES FOR AN EMPTY AND LOADED CHAMBER AT 1 AND 2 GHZ
.................................................................................................................................. 66
TABLE 2. 7 ANTENNA EFFICIENCY CALCULATED FROM TIME DOMAIN MEASUREMENTS .... 69
TABLE 3. 1MEASURED INDEPENDENT SAMPLES FOR 10 REPEATED MEASUREMENTS ......... 77
TABLE 3. 2 CORRELATION VALUES CALCULATED FOR DIFFERENT MEASURED SAMPLES .... 78
TABLE 3. 3 INDEPENDENT SAMPLES AS A FUNCTION OF MEASURED SAMPLES ................... 79
TABLE 3. 4 INDEPENDENT SAMPLES MEASURED AT DIFFERENT FREQUENCIES AND LOADING
CONDITIONS ............................................................................................................... 80
TABLE 3. 5 QUALITY FACTOR MEASURED AT DIFFERENT FREQUENCIES AND LOADING
CONDITIONS ............................................................................................................... 80
x
TABLE 3. 6 INDEPENDENT SAMPLES AS A FUNCTION OF LOAD AND DIFFERENT TUNER
COMBINATIONS; F0 = 1 GHZ ...................................................................................... 93
TABLE 3. 7 INDEPENDENT SAMPLES AS A FUNCTION OF LOAD AND DIFFERENT TUNER
COMBINATIONS; F0 = 2 GHZ ...................................................................................... 93
TABLE 3. 8 INDEPENDENT SAMPLES AS A FUNCTION OF LOAD AND DIFFERENT TUNER
COMBINATIONS; F0 = 3 GHZ ...................................................................................... 93
TABLE 3. 9 INDEPENDENT SAMPLES AS A FUNCTION OF LOAD AND DIFFERENT TUNER
COMBINATIONS; F0 = 4 GHZ ...................................................................................... 94
TABLE 3. 10 INDEPENDENT SAMPLES AS A FUNCTION OF LOAD AND DIFFERENT TUNER
COMBINATIONS; F0 = 5 GHZ ...................................................................................... 94
TABLE 5. 1 CONDITIONAL PROBABILITY OF FAILURE CALCULATED FROM PROBABILITY OF
OBSERVABLE AND PROBABILITY OF EMI THREAT .................................................... 165
xi
LIST OF FIGURES
FIGURE 1. 1 OPEN AREA TEST SITE [64] .............................................................................. 4
FIGURE 1. 2 FULLY ABSORBER LINED CHAMBER OR ANECHOIC CHAMBER [65] ................. 7
FIGURE 1. 3 SEMI ANECHOIC CHAMBER [66] ....................................................................... 8
FIGURE 1. 4 TEM CELL [68] ............................................................................................... 10
FIGURE 1. 5 REVERBERATION CHAMBER ........................................................................... 13
FIGURE 2. 1 ESTIMATION OF K FOR DIFFERENT Q VALUES AT DIFFERENT DISTANCES OF
SEPARATION ............................................................................................................... 22
FIGURE 2. 2 CUMULATIVE DISTRIBUTION PLOT FOR RECEIVED POWER INSIDE A
REVERBERATION CHAMBER ........................................................................................ 26
FIGURE 2. 3 Q MEASUREMENT SETUP ................................................................................. 27
FIGURE 2. 4 K MEASUREMENT SETUP ................................................................................. 28
FIGURE 2. 5 TIME DOMAIN RESPONSE OF RECEIVED POWER (NO DIRECT COUPLING BETWEEN
ANTENNAS) ................................................................................................................ 32
FIGURE 2. 6 TIME DOMAIN RESPONSE OF RECEIVED POWER (NO DIRECT COUPLING BETWEEN
ANTENNAS) ................................................................................................................ 33
FIGURE 2. 7 TIME DOMAIN RESPONSE OF RECEIVED POWER ............................................... 35
FIGURE 2. 8 TIME DOMAIN RESPONSE OF RECEIVED POWER ............................................... 35
FIGURE 2. 9 INSERTION LOSS FOR EMPTY AND LOADED CHAMBER AT 1 GHZ ..................... 39
FIGURE 2. 10 PLOT OF K AND INSERTION LOSS FOR EMPTY CHAMBER AT 1 GHZ ............... 40
xii
FIGURE 2. 11 ILLUSTRATION OF THE SOURCE STIRRING MEASUREMENT POINTS IN A ROOM 42
FIGURE 2. 12 PICTURES OF THE HALLWAY AND THE SMALL ROOM ..................................... 43
FIGURE 2. 13 RECEIVED POWER AS A FUNCTION OF POSITION (SOURCE STIRRING) IN THE
HALLWAY AT 1 GHZ .................................................................................................. 44
FIGURE 2. 14 RECEIVED POWER AS A FUNCTION OF TIME IN THE HALLWAY; SMALLER TIME
WINDOW ..................................................................................................................... 46
FIGURE 2. 15 RECEIVED POWER AS A FUNCTION OF TIME IN THE HALLWAY; LARGER TIME
WINDOW ..................................................................................................................... 46
FIGURE 2. 16 RECEIVED POWER AS A FUNCTION OF POSITION (SOURCE STIRRING) IN THE
SMALL ROOM AT 1 GHZ ............................................................................................. 47
FIGURE 2. 17 RECEIVED POWER AS A FUNCTION OF TIME IN THE SMALL ROOM; SMALLER
TIME WINDOW ............................................................................................................ 49
FIGURE 2. 18 RECEIVED POWER AS A FUNCTION OF TIME IN THE SMALL ROOM; LARGER TIME
WINDOW ..................................................................................................................... 49
FIGURE 2. 19 POWER DECAY PROFILE OF THE HALLWAY AND SMALL ROOM AT 1 GHZ ...... 51
FIGURE 2. 20 AVIONICS BAY QUALITY FACTOR COMPARISONS (FIGURE 4-37.
NSWCDD/TR-97/84 [13]) ........................................................................................ 51
FIGURE 2. 21 PASSENGER CABIN QUALITY FACTOR COMPARISONS (FIGURE 4-38.
NSWCDD/TR-97/84 [13]) ........................................................................................ 52
FIGURE 2. 22 ALUMINUM CAVITY QUALITY FACTOR COMPARISONS (FIGURE 7.13 [14]) ... 53
FIGURE 2. 23MAIN CABIN OF THE HANGAR QUEEN PLANE QUALITY FACTOR COMPARISONS
(FIGURE 11.1 [14]) ..................................................................................................... 53
xiii
FIGURE 2. 24 S21 COMPARISONS OF THE NASA SPF CHAMBER WITH FREQUENCY DOMAIN
MEASUREMENTS FROM NIST AND TIME DOMAIN MEASUREMENTS FROM ITS ............ 54
FIGURE 2. 25 POWER DECAY PROFILES OF SMART 80 CHAMBER AT DIFFERENT OPERATING
FREQUENCIES ............................................................................................................. 57
FIGURE 2. 26 A. POWER DECAY PROFILE OF SMART 80 CHAMBER AT 500MHZ AND
VARYING BANDWIDTHS (80-200MHZ) ..................................................................... 61
FIGURE 2. 27MEASURED S PARAMETERS OF THE SMART 80 CHAMBER AT 1 GHZ WITH LP
ANTENNAS AS TX AND RX .......................................................................................... 65
FIGURE 2. 28 POWER DECAY PROFILE OF AN EMPTY AND LOADED SMART 80 CHAMBER AT
1 AND 2 GHZ .............................................................................................................. 67
FIGURE 3. 1 EXPERIMENTAL SETUP FOR MEASURING INDEPENDENT SAMPLES IN A
REVERBERATION CHAMBER ........................................................................................ 77
FIGURE 3. 2 INDEPENDENT SAMPLES VERSUS NORMALIZED STANDARD DEVIATION [16].... 82
FIGURE 3. 3 INDEPENDENT SAMPLES VERSUS NORMALIZED STANDARD DEVIATION FOR
DIFFERENT NUMBER OF TRIALS FROM MONTE CARLO SIMULATION ........................... 83
FIGURE 3. 4 INDEPENDENT SAMPLES VERSUS NORMALIZED STANDARD DEVIATION FOR
DIFFERENT NUMBER OF TRIALS FROM MONTE CARLO SIMULATION (CLOSER LOOK) .. 84
FIGURE 3. 5 ILLUSTRATION OF Q BANDWIDTH FOR A RECTANGULAR CAVITY .................... 86
FIGURE 3. 6 PICTURE OF THE SECOND TUNER ..................................................................... 92
FIGURE 3. 7 CUMULATIVE DISTRIBUTION PLOT OF SMART 80 AT 200MHZ WITH
DIFFERENT OPERATIONAL TUNERS ............................................................................. 97
FIGURE 3. 8 CUMULATIVE DISTRIBUTION PLOT OF SMART 80 AT 150MHZ WITH
DIFFERENT OPERATIONAL TUNERS ............................................................................. 98
xiv
FIGURE 3. 9 CUMULATIVE DISTRIBUTION PLOT OF SMART 80 AT 100MHZ WITH
DIFFERENT OPERATIONAL TUNERS ............................................................................. 99
FIGURE 3. 10 CUMULATIVE DISTRIBUTION PLOT OF SMART 80 AT 80MHZ WITH
DIFFERENT OPERATIONAL TUNERS ............................................................................. 99
FIGURE 4. 1 EXAMPLE OF CABLE BUNDLES INSIDE AIRCRAFT COCKPIT ............................ 101
FIGURE 4. 2 PICTURE OF A MONOPOLE ON A GROUND PLANE LOCATED AT A CORNER OF THE
REVERBERATION CHAMBER ...................................................................................... 107
FIGURE 4. 3MEASURED S11 OF THE MONOPOLE RESONANT AT 1.4 GHZ ......................... 107
FIGURE 4. 4 QQ PLOT FOR A CHI SQUARE WITH 2 D. O. F DISTRIBUTION ......................... 109
FIGURE 4. 5 QQ PLOT FOR A MEASURED VS. THEORETICAL DISTRIBUTION OF POWER ...... 110
FIGURE 4. 6 QQ PLOT FOR A MEASURED VS. THEORETICAL DISTRIBUTION OF E FIELD ..... 111
FIGURE 4. 7 A PLOT OF MODES INSIDE A REGULAR RECTANGULAR CAVITY (30 X 12 X 30 CM)
................................................................................................................................ 114
FIGURE 4. 8 ILLUSTRATION OF RECTANGULAR CAVITY WITH RECTANGULAR APERTURE
INCIDENT BY A PLANE WAVE .................................................................................... 117
FIGURE 4. 9 SHIELDING EFFECTIVENESS OF THE (30 X 12 X 30) CAVITY WITH (15 X 6)
APERTURE ................................................................................................................ 117
FIGURE 4. 10 SKETCH OF THE LOCATIONS OF THE PROBE ON THE BOTTOM PLATE OF THE
CAVITY ..................................................................................................................... 118
FIGURE 4. 11 IMPACT OF THE WIRE ON THE FIELD COMPUTED INSIDE THE RECTANGULAR
CAVITY ..................................................................................................................... 119
FIGURE 4. 12 SIMULATION SETUP TO COMPUTE THE CURRENT ON THE WIRE LOCATED INSIDE
THE CAVITY .............................................................................................................. 121
xv
FIGURE 4. 13MEASUREMENT SETUP TO COMPUTE THE CURRENT ON THE WIRE LOCATED
INSIDE THE CAVITY .................................................................................................. 122
FIGURE 4. 14MEASURED REFLECTION CO-EFFICIENT OF THE SOURCE ANTENNA ............. 123
FIGURE 4. 15 MEASURED REFLECTION CO-EFFICIENT OF THE WIRE LOCATED INSIDE THE
CAVITY ..................................................................................................................... 123
FIGURE 4. 16 MEASURED TRANSFER FUNCTION BETWEEN THE SOURCE ANTENNA AND ... 124
FIGURE 4. 17 SELF IMPEDANCE (CALCULATED) OF THE WIRE LOCATED INSIDE THE CAVITY
................................................................................................................................ 125
FIGURE 4. 18 CIRCUIT MODEL FOR THE WIRE ANTENNA TO CALCULATE CURRENT ........... 125
FIGURE 4. 19MEASURED AND SIMULATED SHORT CIRCUIT CURRENT ON THE WIRE LOCATED
INSIDE THE CAVITY .................................................................................................. 127
FIGURE 4. 20MEASURED (ISC) AND CALCULATED CURRENT ON THE WIRE TERMINATED BY
ARBITRARY IMPEDANCE LOCATED INSIDE THE CAVITY ............................................ 128
FIGURE 4. 21MEASURED AND SIMULATED CURRENT ON THE WIRE TERMINATED BY A 50
OHM IMPEDANCE LOCATED INSIDE THE CAVITY ....................................................... 129
FIGURE 4. 22MEASURED REFLECTION CO-EFFICIENT OF THE WIRE LOCATED ON A GROUND
PLANE (OPEN AIR MEASUREMENT) .......................................................................... 130
FIGURE 4. 23MEASURED TRANSFER FUNCTION BETWEEN THE SOURCE ANTENNA AND ... 131
FIGURE 4. 24MEASURED AND SIMULATED SHORT CIRCUIT CURRENT ON THE WIRE LOCATED
ON A GROUND PLANE ................................................................................................ 132
FIGURE 4. 25MEASURED AND SIMULATED CURRENT ON THE WIRE TERMINATED BY
ARBITRARY IMPEDANCE LOCATED ON A GROUND PLANE (NOT IN THE CAVITY) ........ 133
xvi
FIGURE 4. 26 MEASURED AND SIMULATED CURRENT ON THE WIRE TERMINATED BY A 50
OHM IMPEDANCE LOCATED ON A GROUND PLANE .................................................... 134
FIGURE 4. 27 SIMULATION SETUP FOR ANGLE OF INCIDENCE SWEEP ALONG PHI AND THETA
AXIS ......................................................................................................................... 136
FIGURE 4. 28MEASURED SHORT CIRCUIT CURRENT FOR SWEEP ALONG PHI ..................... 137
FIGURE 4. 29MEASURED SHORT CIRCUIT CURRENT FOR SWEEP ALONG THETA ............... 137
FIGURE 4. 30 SIMULATED SHORT CIRCUIT CURRENT FOR SWEEP ALONG PHI .................... 138
FIGURE 4. 31 SIMULATED SHORT CIRCUIT CURRENT FOR SWEEP ALONG THETA ............... 138
FIGURE 4. 32MEASURED AND SIMULATED SHORT CIRCUIT CURRENT AT NORMAL ANGLE OF
INCIDENCE ............................................................................................................... 139
FIGURE 4. 33MEASURED SHORT CIRCUIT CURRENT USING MULTIPLE MEASUREMENT
TECHNIQUES ............................................................................................................. 141
FIGURE 4. 34MEASURED SHORT CIRCUIT CURRENT FOR WIRE POSITIONS A, 7 AND 13;
APERTURE 15X6 CM ................................................................................................. 143
FIGURE 4. 35MEASURED SHORT CIRCUIT CURRENT FOR WIRE POSITIONS A, 7 AND 13;
APERTURE 15X1 CM ................................................................................................. 143
FIGURE 5. 1WEIBULL SHAPE PARAMETER FIT FOR THE MEASURED SHORT CIRCUIT CURRENT
AT MULTIPLE FREQUENCIES (15X6 AND A) ............................................................... 149
FIGURE 5. 2WEIBULL SHAPE PARAMETER FIT FOR THE MEASURED SHORT CIRCUIT CURRENT
AT MULTIPLE FREQUENCIES (15X6 AND 13) ............................................................. 151
FIGURE 5. 3WEIBULL SHAPE PARAMETER FIT FOR THE MEASURED SHORT CIRCUIT CURRENT
AT MULTIPLE FREQUENCIES (15X1 AND 7) ............................................................... 152
xvii
FIGURE 5. 4 DISTRIBUTION PLOT OF RAYLEIGH DISTRIBUTION OBTAINED FROM THE MEAN
NORMALIZED MEASURED SHORT CIRCUIT CURRENT ................................................. 153
FIGURE 5. 5 EXAMPLE FOR EMI ARRIVAL AND DURATION ............................................... 154
FIGURE 5. 6 EXAMPLE FOR INFLUENCE OF EMI ON A DIGITAL SIGNAL ............................. 157
FIGURE 5. 7MEAN NORMALIZED MAXIMUM INDUCED CURRENT MEASURED ON THE WIRE
OVER THE 100 DIFFERENT TUNER POSITIONS. APERTURE: 15 X 6 CM.WIRE POSITION
‘A’. ........................................................................................................................... 160
FIGURE 5. 8 STAIRCASE FUNCTION REPRESENTING EMI THREAT FOR THRESHOLD 1.0 VS.
TIME FRAME ............................................................................................................. 161
FIGURE 5. 9 STAIRCASE FUNCTION REPRESENTING EMI THREAT FOR THRESHOLD 2.0 VS.
TIME FRAME ............................................................................................................. 161
FIGURE 5. 10 STAIRCASE FUNCTION REPRESENTING EMI THREAT FOR THRESHOLD 3.0 VS.
TIME FRAME ............................................................................................................. 162
FIGURE 5. 11 STAIRCASE FUNCTION REPRESENTING EMI THREAT FOR THRESHOLD 4.0 VS.
TIME FRAME ............................................................................................................. 162
FIGURE 5. 12 PROBABILITY CHART .................................................................................. 166
FIGURE 5. 13 TWO STAGE MARKOV CHAIN MODEL FOR EMI CAUSING SYSTEM UPSET .... 167
FIGURE 5. 14 THREE STAGE MARKOV CHAIN MODEL FOR EMI CAUSING SYSTEM UPSET . 169
xviii
LIST OF SYMBOLS
V – Volume
D – Directivity
λ – Wavelength and scale parameter
r – Distance between antennas
S21 – Transmission coefficient
2
21
r
t
P
S
P
= - Chamber gain
τ – Chamber time constant
A – Area
c – Speed of light
η - Absorption coefficient
d r - Reverberation distance
S – Surface Area
c T - Wall scattering time
S11 and S22 – Reflection coefficient
ρ - Correlation coefficient
μ - Mean or permeability
σ – Variance
Γ - Gamma function
xix
Sn – Normalized standard deviation
1/e – Time constant
E – Electric Field
< Pr > - Average received power
ε - Permittivity
m, n and p – Mode numbers
a, b and c – Length, Width and Height of a cavity
f0 – Resonant frequency of the first mode
w Z - Self impedance of the wire
w Γ - Reflection coefficient of the wire
0 Z - Characteristic impedance
L Z - Load impedance
ZA - self impedance of the wire
L I - Load current
L V - Load voltage
OC V - Open circuit voltage
SC I - Short circuit current
k – Shape parameter
N(t) - Number of active EM disturbances that occur during the time t
ti - Arrival time ith disturbance
di. - Duration of disturbance
α - Average (mean) number of successes per unit time
xx
p – Conditional probability of failure in the presence of EMI
q and r – Transitional probabilities from EMI absent to present and vice versa
Ts - One step of duration
p1 and p2 – Conditional probabiltity that EMI exists and causes a failure and EMI exists
and does not cause a failure
xxi
LIST OF ACRONYMS
EMI/C – Electromagnetic Interference/Compatability
EME – Electromagnetic Environment
EUT – Equipment Under Test
OATS – Open Area Test Site
AC – Anechoic Chamber
RAM – Radar Absorbent Material
RF – Radio Frequency
RS – Radiated Susceptibility
RC – Reverberation Chamber
FALC – Fully Absorber Lined Chamber
SAC – Semi Anechoic Chamber
TEM – Transverse Electromagnetic Cell
GTEM – Gigahertz Transverse Electromagnetic Cell
LUV – Lowest Usable Volume
Q – Quality Factor
K – Rician Factor
VNA – Vector Network Analyzer
CW – Continuous Wave
LOS – Line of Sight
xxii
IL – Insertion Loss
Tx and Rx – Transmit and Receive antennas
CDF – Cumulative Distribution Function
JFTA – Joint Frequency and Time Analysis
BW – Band Width
LP – Log Periodic Antenna
FD – Frequency domain
TD – Time domain
IS – Independent Samples
VT – Vertical Tuner
HT – Horizontal tuner
ST – Small tuner
HERO – Hazard of Electromagnetic Radiation to Ordnance
D.O.F – Degrees of Freedom
MLFMA – Multi Level Fast Multipole Algorithm
OSU – Oklahoma State University
SE – Shielding Effectiveness
1
Chapter 1
1.0 INTRODUCTION
Electromagnetic interference can be defined as a disturbance, intentional or unintentional,
emitted from an internal or external source that affects an electrical circuit. The
disturbance can be just a nuisance or fatal depending on the severity of the situation. In
order for the electronics to be electromagnetically compatible, the emissions and
susceptibility levels must be established and proper care needs to be taken to shield the
components to prevent damage.
Establishing the threat levels, both emission and susceptibility, are of critical importance.
Most of the electronic components operate inside cavities or enclosures and assessing the
electromagnetic damage and upset of these components under various conditions will
provide insight to minimize such failures during operation. Estimating the failures
becomes an ill defined problem when the details of the equipment, enclosure,
interconnecting cables and surrounding geometrical details are not well known.
Reworking the solution for every change to the system in terms of size, position, etc…
when performing theoretical or numerical analysis, advocates the need for statistical
methods. These problems are challenging to simulate as it requires intricate details which
2
may not be quite consistent from one system to another. For example the cable routing is
not very consistent between the same models of planes. Solving for every problem hence
becomes an impossible task. Statistical methods use all these uncertainties that exist in
such a problem to its advantage. These methods are widely applied in EMI/C problems
due to the very nature of the problem and will be very heavily used in all our discussions.
The final outcome of this work is to establish the probability of failure due to current
coupled onto a cable or a cable bundle that is located close to the wall of a cavity due to
external or internal coupling of EM. The electromagnetic environment of the cavity needs
to be determined to estimate the probability of threat depending on the location of the
cable inside the cavity. Given that the probability of threat exists, then the probability that
the value of the current or field that exceeds a certain threshold must also be determined.
To obtain the threshold probability, the environment in which the EUT operates and the
also the influence of the environment on the observable that is being targeted needs to be
known. Finally, the net probability of failure of a system will be determined from the
individual probabilities.
The important properties of the cavity that need to be known are the shape, size and
volume of the cavity and also the material of the cavity along with what is present inside
the cavity. The cavity could vary anywhere from a compartment in a submarine, aircraft
cockpits, hallways, office spaces, warehouses, etc… to electronic equipments operating
inside rectangular enclosures.
3
Conveniently most of the cavities in which electronics operate are rectangular in nature
or at least the cavities can be approximated as rectangular cavities, hence simplifying
some calculations. These environments can also be recreated in a laboratory setup easily
making research on such problems plausible.
In this study a first order model will be analyzed experimentally to obtain the individual
and net probabilities of observables and EMI failure. Brief introductions of test facilities
that are at the disposal of an EMC engineer to analyze the common EMC/I problems are
given below to enable the reader to understand the different electromagnetic
environments (EME) that could exist and also ways to establish such EME that would
enable testing of equipments in operational conditions. The limitations of each test
facility will also be discussed. The extensive use of reverberation chambers for
measurements and creating different electromagnetic environments for testing the first
order model will be justified. Some of the details that need attention while using
reverberation chambers and using reverberation techniques in other operational
environments are also discussed.
1. 1 Open Area Test Site (OATS)
The Open Area Test Site (OATS) is a 3 and 10 meter emissions test range (Ref: Figure
1.1). An OATS facility must be free from any other electromagnetic disturbance. So the
cables that are used to power the equipments, control lines, signal lines must be isolated
from one another and usually run underground to a control room which will be again a
4
shielded room. The radiated field from the source reaches the equipment under test
(EUT) via direct line of sight and via reflections from the ground plane usually enhanced
by a steel sheet.
Limitations
The EUT must be placed on a turntable and turned to expose all parts of the EUT to the
source field. Most OATS facilities are contaminated with noise from nearby transmission
lines and cell phone towers. In order to perform a statistical study on the EME and also
the effects of EUT to determine the probability of failure of the system, the EUT has to
be tested at all possible angles of incidence and polarizations at every frequency which
might not be practical to perform for all test objects. Hence this test technique will be
used only when necessary.
Figure 1. 1 Open Area Test Site [64]
5
1. 2 Fully Anechoic chambers
An Anechoic chamber is a RF test facility where the electromagnetic wave echoes are
suppressed (Ref: Figure 1.2). Such construction isolates the device under test present
inside the anechoic chamber from any other electromagnetic interference. The EUT
receives EM energy only from the source (usually in the line of sight) and there are
minimum reflections from the walls, floor or roof. Anechoic chambers are used in
antenna radiation pattern measurements, radar cross section measurements and
electromagnetic compatibility measurements.
The lowest operating frequency determines the size of the chamber. The walls are coated
with radar absorbent material (RAM) to minimize reflection. The equipment under test is
placed on a turntable at a specified distance away from the source antenna. The source
antenna is powered by a signal source and an amplifier. When a radiated susceptibility
test is done, the antenna and the EUT on the turntable must be rotated to expose all parts
of the EUT to the RF field. It is required by standards that the EUT must be completely
enclosed in the 3dB bandwidth of the source antenna which becomes impossible when
the EUT is large. Hence only parts of the EUT are exposed to the RF field.
A fully anechoic chamber is preferred for radiated susceptibility (RS) testing if a plane
wave environment (with minimum reflections) is desired. In order to do a complete
radiated susceptibility test, the entire sphere surrounding the device must be sampled to
predict the direction maximum coupling to the device. This test is practically impossible
and not economical to be performed in any situation. Because of the time and cost
6
involved usually the radiated susceptibility test in an anechoic chamber is performed over
a limited number of aspect angles and polarizations which may cause a high risk of
missing (and under-testing) some susceptibilities as many EUT’s are highly directive
(susceptible). The region where the field levels are not significantly different is defined as
the uniformity region. It is necessary that the EUT is enclosed inside the uniformity
region during the test as all parts of the EUT will be equally illuminated.
Limitations
The absorber material used in the construction of the chamber is not efficient in the 30-
200 MHz range. Testing of equipment at microwave frequencies in an anechoic chamber
(AC) may lead to stress levels on electronic components inside the equipment that are
more than 10dB higher than what is achieved in a reverberation chamber (RC) [34],
given the same magnitude of the exciting field and the known directivity of the EUT. The
outcome of a radiated susceptibility test in an AC will strictly depend on the choice of the
direction and polarization of the incident field. In case of an anechoic chamber, the field
uniformity levels are not good (6-12 dB variation) giving rise to uncertainty levels that
are large (not desirable for a susceptibility test). The uncertainty associated with this
testing also limits the capability of arriving at a probabilistic model for failure. A
thorough test will be required to establish the distributions with reasonable accuracy
hence only limited tests will be performed in this study using FALC.
7
Figure 1. 2 Fully Absorber Lined Chamber or Anechoic Chamber [65]
1. 3 Semi Anechoic chambers
A semi anechoic chamber pretty much resembles an anechoic chamber in construction
except for the absence of absorbing material on the floor and usually a metal sheet is
provided as a floor to enhance reflections and simulate ground (Ref: Figure 1.3). The one
advantage of a semi anechoic chamber is that the results can be directly compared to the
results from an open area test site (OATS). Commercial radiated emissions standards
demands testing inside a semi anechoic chamber. The advantages and disadvantages of
using a semi anechoic chamber are similar to an anechoic chamber.
8
Limitations
In a semi anechoic chamber when radiated emissions or susceptibility measurements are
made, reflections can produce unpredictable results. The variable results might be
because of the standing waves (due to reflections). Variation is likely if the position of
the test set up in the chamber is changed. Similar limitations to FALC exist with testing
in semi anechoic chambers with respect to measuring an observable for statistical study.
Figure 1. 3 Semi Anechoic Chamber [66]
1. 4 TEM
TEM is a type of guided wave environment which is a form of plane wave environment.
Because of the uniformity of the test field, TEM cavities are used for probe calibration.
Broadly TEM can be classified into TEM Cell, TEM Waveguide and GTEM cell.
9
1. 4. 1 TEM Cell
A closed measuring device consisting of an inner and an outer conductor in which a
voltage difference creates a transverse electromagnetic (TEM)-mode electromagnetic
field between these conductors (Ref: Figure 1.4). Two-port TEM cells typically have
symmetrical tapered input and output ports, whereas a one-port TEM cell typically has a
tapered input port and a integral, closed non tapered termination in place of the output
port [67].
1. 4. 2 TEM Waveguide
An open or closed transmission line system that uses the TEM mode over the frequency
range of interest is called a TEM waveguide. The TEM mode is defined as an
electromagnetic field in which the electric and magnetic field vectors are orthogonal to
each other and orthogonal to the propagation direction. Common examples are the two-port
TEM cell (Crawford cell), the one-port or wideband-TEM cell (example GTEM),
and the parallel-plate stripline [67].
1. 4. 3 GTEM
A TEM cell that has been altered to extend the usable frequency range is called a
Gigahertz Transverse Electromagnetic Cell. Typically, this is achieved by replacing one
port of a two-port TEM cell with a wideband, non-tapered, hybrid discrete resistor/wave
10
absorber termination [67]. GTEMs are also preferred for some RS testing but they are
usually limited by size.
Limitations
The tapered geometry of TEM cell makes it impossible to use in some radiated
susceptibility tests. It cannot enclose larger test bodies and so the applications are very
limited. The field uniformity also degrades at higher frequencies. To do a 3D test inside
GTEM both the antennas and the EUT must be rotated. TEM cells will not be used in any
of the measurements.
Figure 1. 4 TEM cell [68]
11
1. 5 Reverberation chambers
A reverberation chamber is a large welded enclosure with highly conducting walls.
Reverberation chambers are used to expose the EUT to random electromagnetic fields at
high frequencies (Ref: Figure 1.5). The vulnerability of the equipment is tested under
robust conditions of electromagnetic exposure inside the reverberation chamber. In the
case of an Anechoic or a Semi Anechoic chamber, the field structure can be defined as
plane wave. Because of the presence of absorbing walls, the EUT is exposed to the fields
in only one direction. In order to do a susceptibility test, the EUT must be placed on a
turntable to expose all parts of the EUT to the fields but in the case of a reverberation
chamber, due to highly reflecting walls, the EUT is illuminated by fields from many
aspect angles. Hence the EUT need not be rotated inside the chamber. Reverberation
chamber closely resembles the environment in which the EUT will be operating when it
is placed inside a cavity. The field structure inside the reverberation chamber can be
considered as a superposition of plane waves at random phases [1].
Random fields are created inside the reverberation chamber by varying the boundary
conditions of the chamber walls. This is accomplished usually with the use of a tuner
which is typically both electrically large and occupies an appreciable portion of the cavity
volume. This is referred to as mechanical stirring or mode stirring. The input frequency
can be varied over a band to realize the complexity inside the chamber and this method of
stirring is called frequency stirring. RF energy will be injected into the chamber at one of
the corners and allowed to reflect on the side walls, top and bottom plates several times
12
before reaching the EUT. The revolving tuner also contributes to this phenomenon by
changing the path lengths of the waves. In all through our analysis, phase will be
considered to be evenly distributed (uniform) inside the chamber and no analysis will be
done with the phase information.
The reverberation chamber is an ensemble of a large number of cavities with different
shapes [2]. The well stirred field inside the reverberation chamber also satisfies
Maxwell’s equations. The fields inside a stirred reverberation chamber can be
approximated by a sum of plane waves which highly helps in calculating the responses of
the test objects [1]. Because of the multipath scattering inside reverberation chambers, the
phase information is totally lost [3]. Thus the field inside the well stirred chamber is the
sum of large number of multipath rays with random phases.
Deterministic mode theory is not suitable for predicting the response of the test object
that is present inside a mechanically stirred reverberation chamber because the problem
has to be solved for every change (antenna, EUT position, tuner position etc..) which
becomes impractical and hence statistical theory has to be employed to resolve for the
fields near or on the EUT. The field that is created inside the reverberation chamber is
stochastic in nature, so statistical techniques will be applied to analyze these fields.
The range of possible measured responses, indicated by the uncertainty levels, is
generally smaller in a reverberation chamber test. The EUT is equally illuminated in all
directions hence the effective directivity of the RC is one. The EUT inside the lowest
13
usable volume (LUV) of the RC will still respond depending on its directivity but the
chances of finding more susceptible parts inside an RC is more than in any other test
facility.
Figure 1. 5 Reverberation Chamber
Limitations
For radiated susceptibility testing, the EUT has to be exposed to certain field levels as
specified by a standard [4, 5]. Though frequency stirring is a valid method to expose the
EUT inside a reverberation chamber, the standard calls for mechanical stepping (the tuner
is stopped before taking a measurement) with a minimum of 12 tuner positions at all
frequencies. While this process is efficient, the test time can be significant. Mechanical
stirring (the tuner is rotated continuously) is another means by which a similar field
change could be attained. A more important question may be “How long is the EUT
being exposed to that same/required field level during the continuous rotation of the
tuner?” Rotation of a tuner combined with an effective change in frequency, for which
the equipment must be tested, may be a method to reduce the test time. This will be
14
discussed again after establishing the failure models for different device types. Once the
failure model is developed, an appropriate test method could be chosen to reduce the test
time.
The number of discrete tuner steps increases with decrease in frequency of operation due
to the inability of the tuner to generate statistically independent fields. This increases the
test time considerably. Continuous tuner rotation can reduce the test time provided the
EUT responds quickly to the rapidly changing local fields. To identify the response of the
devices to the EM field, the failure mechanisms of different components and devices that
will be installed in aircrafts/automobiles/consumer electronics must be studied. If most of
the devices respond to a peak field more than an average field, then continuous rotation
of the tuner can be considered.
In a mechanically stepped test, the EUT is exposed to a particular intensity of field for a
period called the dwell time for the EUT. By rotating the tuner at a faster rate, a
cumulative dwell time can be established. The tuner can be rotated at a rate faster than
the decay rate of the modes inside RC which then can retain the same field level for the
period of time needed. Though this seems to be a viable option, more research has to be
done to support this argument. Though this issue is not directly addressed in this work,
the failure models developed will help in answering some of these questions.
15
1. 6 Why use Reverberation Chamber in this study?
The statistical nature of the fields inside the RC represents the field structure that exists in
cavities in which electronics operate. There is also a need for “in-situ testing” i.e.… to
test the equipments during its operation in a particular location and also assessing the
influence of EMI when a new system is installed in the same space. The statistics that
govern the typical RC operation are also well established hence throughout this problem,
most of our tests will be performed in RC but often, comparisons will be made to other
test techniques like OATS and AC for cross checks.
To understand the statistical nature of the problem and how to apply the statistics to very
complex scenarios, this study will focus on analyzing some simple and somewhat ideal
cases to begin with. Such cases involve the determination of field and power distributions
as well as induced currents in the “middle” of sparse or unloaded cavities using simple
aperture models. The study then investigates how the distributions change as a function
of excitation frequency, cavity modes, and more complex aperture models. More
complicated overmoded cavities generated by the variation of frequency, cavity fill etc
will be discussed while closely examining the EM phenomena occurring at/near interior
metallic walls and surfaces representing cables for example.
The complications and complexities that will arise due to the variability of cavity fill,
existence of multiple apertures of various types, EM source characteristics, near-field
proximity effects, and so forth will help answer another critical research question: Are
16
there other effective ways of reducing simulation complexity in addition to applying
statistics? It is anticipated that this research will result in a set of guidelines that may
alleviate the need to unnecessarily generate highly-detailed computational models
especially if there are no enhanced accuracy obtained due to simulation of every minor
change that occurs in the system.
Reverberation chambers will be used in all the measurements performed on the metal box
with apertures. A piece of wire placed inside the metallic box will serve as the equipment
under test and the distributions of current and fields will be calculated via measurements.
From the distribution, the probability that the observable exceeding a certain threshold
can be determined. From the nature of the EME generated, the probability of threat can
be determined. Combining both the probabilities, the net probability of failure of the
system could be determined. Reverberation chambers will be useful in measurements in
this study as they simulate operating conditions of the EUT inside a cavity and as the
EUT is exposed in all directions to the electromagnetic field, the uncertainty is also
reduced. The probability models can provide insight into what type of testing is required
to assure worst case testing with reasonable accuracy.
1. 7 Organization of the thesis
The tasks are broken down into multiple subtasks and a procedure is developed in terms
of subtasks. The final outcome of this research is to find a probability of failure of a
system due to electromagnetic interference while operating in a given electromagnetic
17
environment. In order to calculate the probability, the electromagnetic environment in
which the system operates needs to be understood/captured/analyzed. What kind of data
is required to capture this EME quickly and efficiently? Given the EME, how does the
system react to the EME in which it is operating? Finally, from all these data, the
probability that the system will fail given a threshold has been crossed can be calculated.
The step by step procedure in setting up the problem is enumerated below.
1. How to characterize the electromagnetic environment (EME) of a work space using
standard frequency domain and time domain approaches? What will be advantages and
disadvantages of these methods? What other parameters of the work space can be
determined from the collected data.
2. What are independent samples and why are they important while performing a
measurement and what do they say about EME? What is the effect of loading on
independent samples? Is there a way to increase the number of independent samples by
using a second tuner?
3. How could we apply the lessons learnt from 1 and 2 for a practical problem of current
coupling on to a wire located inside a cavity of known dimensions and what probabilistic
models could be developed to assess the probability of threat for an arbitrary case? These
results will also be supported with simulations where applicable.
4. Can the failure mechanisms due to EMI/C be classified into subsets and failure models
developed for each subset? If a failure model could be established, appropriate test
methods can be chosen to ensure a better test while saving resources (money and time!).
18
These tasks will be explained in separate chapters with each chapter clearly summarizing
the knowledge gained and limitations understood at the end of each chapter.
19
Chapter 2
2.0 ASSESSING THE ELECTROMAGNETIC ENVIRONMENT
Measurement of Q and Reverberation distance
(A Method to Determine the Dominance of Direct or Scattered Path)
When dealing with chambers or enclosed spaces, the quality factor, ‘Q’ which can be
simply defined as the energy storing/dissipating capacity is of importance because it
defines how much energy could be available that could potentially interact with the
operating electronics in that cavity. When the Q is high, the space is reverberant leading
to a significant amount of scattered path energy than the direct component. When the Q is
low, the loss that is associated with the space such as loss through walls, apertures,
seams, etc are larger, resulting in poor reflections. Hence the direct path energy is
dominant over the scattered path energy. Classifying the problem into low, medium and
high Q spaces helps in the determination of the electromagnetic environment in that
space. There needs to be an efficient way to calculate the Q of the space and thereby
determining the dominance of the scattered or direct path energy. This chapter will
explore ways to determine Q of a controllable environment (such as reverberation
chambers) both via frequency and time domain techniques and also examine the
dominance of direct and scattered path via simple measurements. With the knowledge
20
gained from a controllable environment, on the determination of Q and reverberation
distance, the procedures can be extended to simple spaces.
With the advent and abundance of wireless devices and the confined spaces in which they
are installed, the exposure dominance of the direct path and the scattered path needs to be
studied. When there is a line of sight between the transmitter and the receiver, the direct
path seems to dominate and the received power varies as 1/r2.
Aircraft cavities, metal rooms and below deck environments are considered to be highly
reverberant spaces where determination of shielding effectiveness (defined as the ratio of
field strength without the shield to field strength with the shield) is important. The field
measured at any point inside these cavities will be a sum of some direct path (mostly line
of sight) from some transmitter and multipath (due to the reflection of electromagnetic
energy from the metal walls). At some locations, the multipath effects can be higher than
the direct path effects hence the dominance of the direct path compared to the multipath
needs to be explored. In order to qualify the placement of any wireless devices inside
such reverberant cavities the distance at which the multipath effects seem to dominate
needs to be found. In this chapter an attempt has been made to compute the reverberation
distance for such a scenario and the results have been compared and validated in
frequency and time domain.
It was suggested in [6] that the distance between the transmitter and the EUT (in our case
the receiver antenna) can be varied to simulate different Rician environments (K factors).
21
The same idea has been explored in this chapter to obtain different K values and we go
further to calculate the distance at which the dominant and scattered path will carry equal
power.
2. 1 Theoretical estimation of K
The Rician K factor can be defined as the ratio of the direct path component to the
scattered path component [6]. K can be expressed as a function of the chamber and
antenna characteristics. The expressions for K are given below. The detailed calculation
of expressing K in terms of Q is shown in [6].
Direct component power
Scattered component power
K =
2
3
2
K V D
λQ r
= (2.1)
22
100 101 102
100
102
K Vs. Q for varying r
Q (linear) - Log scale
K (linear) - Log scale
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
Figure 2. 1 Estimation of K for different Q values at different distances of separation
From eqn. (2.1), we can see that the K factor can be controlled by varying the directivity
of the antenna, or the volume (hence the Q) of the room and also by varying the distance
between the transmitter and the receiver antenna. To understand the effects, a theoretical
estimation of K for multiple separation distances is calculated using eqn. (2.1) for varying
Q’s (1 to 100 in steps of 10) of the same volume (1.98m3) and frequency of operation (2
GHz) and for antenna directivity, D=1.5 (dipole directivity). The results are plotted in
Figure 2.1. The point where the K equals 1 is of importance as it can be looked on as the
point where the direct path and multipath powers are considered to be equal [7]. From
Figure 2.1, the reverberation distance can be found if the Q of the volume can be
measured. The line for a separation distance of 0.5m, crosses K=1 for a much larger Q
than the line for separation distance of 1.5m. As expected, when the Q is high, even for
23
small separation distances between the antennas, reverberation starts and the multipath
power starts to dominate and when the Q is low, the direct path dominates for longer
distances. The 1/r2 dependence of K can also be seen from the exponential decay trend of
the curves represented as straight lines in Figure 2.1 as it is a Log-Log plot.
The problems with theoretical estimation of the K factor are the unknowns in eqn. (2.1).
The directivity varies as a function of frequency making it harder to have a good
estimation. The variation in the estimation of the Q of the chamber can be as high as 3dB
for different points of measurement [4]. Hence all these factors make it difficult to
calculate K values from eqn. (2.1). The best possible estimation of the K values or the
reverberation distance can be obtained from measurements. Measurements will be
performed in both frequency and time domain to calculate K, Q and eventually the
reverberation distance. Validation of these results will be obtained by comparing one
against the other.
2. 2 Frequency domain
In the frequency domain analysis, the signals are analyzed with respect to frequency and
not as a function of time. By fixing the frequency, the Q of the room (in our case an RC)
is also fixed as Q is also a function of frequency. Thereby for varying distances of
separation between the two antennas, the EME can be determined. The Rician K factor in
terms of scattering parameters (S parameters) following [6] is given by
24
( )2
21
2
21 21
S
K
S S
=
−
(2.2)
K includes ensemble averages as presented in eqn. (2.2) which are denoted by the angled
brackets. The ensemble averages are calculated as the average value over the entire
sweep of the tuner. The numerator is related to the direct component and the
denominator, the mean normalized S21, is related to the multipath component. Thus the
Rician K factor can be measured as the ratio of the direct to the multipath component.
Scatter plots of real and imaginary parts of S21 reveal that the “cluster radius” is
determined by σ R and the distance of the “cluster centroid” is dR from origin [6] (not
shown). When the ratio of the numerator to the denominator is small, then the multipath
dominates and so S21 will be normally distributed.
Because reverberation chamber test methods have been well established and also more
controllable, measurements were performed in a reverberation chamber first and then the
analysis will be performed for other not so predictable environments.
2. 2. 1 Experimental procedure
For a Q measurement to be performed in a reverberation chamber, any direct coupling
between transmit and receive antenna has to be eliminated as direct coupling skews the
statistics of the chamber. This will be true for any spaces that are reverberant. In the Q
measurements that were performed inside the reverberation chamber, the antennas were
25
separated by a large distance and the transmitter is focused on a corner while the receiver
antenna was facing the tuner. The polarization of transmit and receive antenna were
avoided to be the same [4].
VNA in the CW (single frequency or constant wave) mode was used in the measurement
process. The transmitter was connected to port 1 and the receiver was connected to port
2. Calibration was performed until the end of the cables and the antenna efficiency is
assumed to be 100%. S21 or the transfer function of the chamber at that frequency was
measured for different positions of the receiver inside the usable volume of the chamber
at 1 GHz for one complete rotation of the vertical tuner. Maximum number of available
data points (1601) is used in all our measurements.
The dimensions of the chamber are L – 13.2m, W – 6.15m and H – 4.95m. Though there
is a vertical and the horizontal tuner, at 1 GHz the mode density (defined as the number
of modes per bandwidth) is significantly high that any one of the tuners is sufficient to
give good statistics. This is proven in the statistical analysis performed on the data that
was captured during these measurements as shown in Figure 2.2. Autocorrelation
analysis performed on the data that was captured shows that there are nearly 250
independent samples associated with the chamber with the vertical tuner operating at 1
GHz. From the mean received power, Q of the chamber was calculated using [4]
2
3
16 r
t
V P Q
P
π
λ
= × (2.3)
Where, 2
21
r
t
P
S
P
=
26
The Q measurement was repeated with 10 different transmit and receiver antenna
positions to look at the variation of Q and K values as just a function of statistical
sampling.
-20 -15 -10 -5 0 5 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Mean Normalized Received Power (dB)
Cumulative Porbability
Cumulative Distribution
Measured
Theory
Figure 2. 2 Cumulative distribution plot for received power inside a reverberation chamber
To measure the Rician K factor, a direct line of sight component must be present else K
will be close to zero [6]. This was established by locating the antennas facing one another
and the polarization is maintained to be the same (co-polarized). The distance between
the transmitter and the receiver was varied from 0.5 feet to 10 feet in 0.5 feet steps. As
the distance between the transmitter and the receiver was increased, more of the tuner
was exposed to the transmit antenna field (as the tuner is present behind the receiver
antenna) hence the scattered power domination is expected to increase. The ratio of the
received power to transmit power in the form of S21 was captured at every position of the
27
receiver for one complete rotation of the tuner (stirred) using the same VNA
experimental setup discussed earlier. Illustrations of both the measurement setups are
provided in Figures 2.3 and 2.4.
Figure 2. 3 Q measurement setup
Horizontal Tuner
Tx
Rx Vertical
Tuner
28
Figure 2. 4 K measurement setup
The Q and K values were calculated from the measured S parameter data using eqn. (2.3)
and eqn. (2.2) respectively. When there is no direct coupling between the transmitter and
the receiver antenna, K values are reported to be significantly less than 1. Q and K values
measured for ten different receiver antenna positions inside the chamber with no direct
coupling between the transmit and receiver antenna are given in Table 2.1.
Table 2. 1 Q and K values for a reverberation chamber (no direct coupling between the antennas)
Position Q K (from S21)
1 38093 0.01
2 37064 0.01
3 44629 0.00
4 49280 0.10
5 43283 0.01
6 45051 0.11
7 43808 0.00
8 42585 0.04
9 42283 0.02
10 41911 0.01
Horizontal Tuner
Tx Rx Rx
Vertical
Tuner
r = 0.15 m
r = 3.048 m
29
In Table 2.2 K calculated from S21 data measured for different separation distances are
reported (Direct LOS component present for this set of measured data). Referring to
Table 2.2, it can be noted that the K values increase with decreasing separation distance.
The distance at which the K values start to be less than 1 can be reported as the
reverberation distance. As per the definition the reverberation distance is the point at
which the direct path and scattered path power would be equal [7]. For this
chamber/antenna settings and frequency of operation, the reverb distance is estimated to
be about 2.5 feet. The calculated reverberation distance will be verified against the time
domain measurements discussed in the next section. Q values are not reported because
eqn. (2.3) does not hold true when there is a direct coupling between the antennas.
Table 2. 2 K values for a reverberation chamber (direct coupling between the antennas)
r in feet r in m K (from S21)
0.5 0.15 32.46
1 0.30 10.47
1.5 0.46 4.54
2 0.61 1.89
2.5 0.76 0.43
3 0.91 0.40
3.5 1.07 1.32
4 1.22 1.28
4.5 1.37 0.83
5 1.52 0.90
5.5 1.68 0.68
6 1.83 0.38
6.5 1.98 0.41
7 2.13 0.44
7.5 2.29 0.34
8 2.44 0.58
8.5 2.59 0.22
9 2.74 0.16
9.5 2.90 0.06
10 3.05 0.31
30
2. 2 Time domain
In the time domain analysis, the signals are analyzed with respect to time. In the
reverberation chamber, generally the excitation pulse required in time domain
measurement is short compared to the chamber time constant [8]. This was made possible
by using a VNA with time domain option and controlling the bandwidth of the input
signal. The Fourier transform is performed on the signal from the receiver antenna by the
VNA thereby transforming the frequency domain data to time domain. In our
measurements, the bandwidth was chosen to be 200 MHz around 1GHz. Given the size of
the chamber, 200 MHz bandwidth is sufficient in producing a statistically equivalent field
configuration inside the chamber compared to CW operation of the chamber combined
with tuner rotation.
The reverberant environment can be separated into a reverberant and pre reverberant
phase. The pre-reverberant phase transforms into the reverberant phase as the number of
wavefronts that has many polarizations and direction of propagation increases from a
single plane or spherical wavefront [8]. As the pre-reverberant phase transforms into the
reverberant phase, the field starts to decrease exponentially. During the single pulse
excitation, both the phases are separated in time. The pre-reverberant phase is short lived
in a highly reverberant environment and the pre-reverberant phase die out before
transforming to the reverberant phase in a poorly reverberant environment. In the CW
mode of excitation, both the phases coexist. The time domain technique has its
advantages over the frequency domain technique as the contamination of the reverberant
field data by any direct path or pre reverberant field components can be eliminated. The
31
exponential decay of the reverberant field can be used to estimate the Q of that space.
The pre-reverberant phase is very interesting in understanding the mode formation and
mode distribution inside the space at the moment of energy being coupled into the space
which is beyond the scope of this effort hence not studied here.
Figures 2.5 and 2.6 are the time domain response measured between two antennas that do
not have a direct line of sight coupling. The measurements were performed in a
reverberation chamber. The thin lines are measurements at a fixed tuner position, for
multiple tuner positions. Measurements were made at 50 such tuner positions
(equiangular distributed around one rotation of the tuner). The thicker line is the
ensemble average of all the 50 fixed tuner point measurements. All the 50 tuner positions
used in the measurements are considered to be independent. From the reverberation
chamber theory, when the number of measured samples is far less than the total number
of available independent samples (50 << 250 in our case), all the measured samples are
considered to be independent. The measurements were performed for two different time
windows. The time windows represents the time for which the power decay in the
chamber has been observed. The smaller time window is useful in studying the pre-reverberant
phase of the field while the larger time window is useful in studying the
reverberant phase. As maximum number of data points was captured in every
measurement (1601 points), the resolution differs for the time windows. When the time
window is short (~500 ns), the resolution is higher and when the time window is large (~
8us), the resolution is low.
32
Figure 2.5 with the smaller time window shows the time delay of the energy to reach the
receiver antenna and a maximum there after indicating the formation of the first or
dominant mode inside the chamber. The under moded characteristics (direct coupling +
unstirred energy) of the chamber before the establishment of reverberant fields can be
seen in the smaller time window.
Figure 2. 5 Time domain response of received power (no direct coupling between antennas)
0 50 100 150 200 250 300 350 400 450 500
-100
-90
-80
-70
-60
-50
-40
-30
Time (nsec)
Prec (dB)
Time domain Q measurement. 50 Tuner positions.
Average of 50 tuner positions
33
Figure 2. 6 Time domain response of received power (no direct coupling between antennas)
The fixed tuner position measurements at first (~ 35 nsec) seem to follow each other
consistently though not exactly over lapping but later, they deviate from each other
indicating the transformation to a reverberant phase when the wavefronts have different
polarizations and direction of propagation and arriving at the receiver end at different
times. This was an interesting observation which will help us understand the under
moded regime of a chamber and needs further exploration.
In Figure 2.6, with the time window being larger, though the fixed tuner position
measurements are noisy and deviate from each other significantly, suggest an exponential
decay of the field. By fitting a straight line to the ensemble average, the slope of the
exponential curve can be found. From the slope, the chamber time constant or the time
0 1 2 3 4 5 6 7 8
-100
-90
-80
-70
-60
-50
-40
-30
Time (usec)
Prec (dB)
Time domain Q measurement. 50 tuner positions.
Average of 50 tuner positions
34
required for the signal to decay to 37% of the maximum (1/e time) can be estimated. The
slope for this set of measurements was calculated to be 1.4 dB/usec.
The chamber time constant can be calculated as below following [8].
log(1/e)= -0.43429 = 4.3429 dB
Slope = 1.4 dB/usec
Reverberation time = 4.3429
Slope
τ = (2.4)
τ = 3.1 usec
Q of the chamber can be calculated from the chamber time constant, Q =τ × 2π f ,
yielding a Q of 42.9 dB [4].
The same procedure was repeated for calculating the time response of the chamber when
the antennas are facing each other (strong line of sight expected) at different separation
distances. Data was collected for 50 different tuner positions as mentioned before and the
ensemble average was calculated from the fixed tuner position measurements.
Figures 2.7 and 2.8 show the time response for a time window of 500 nsec for separation
distances of 1ft and 10 ft respectively. As the distance between the transmitter and the
receiver increases, there is a larger delay for the energy to reach the receiver end (delay in
direct coupling). The slope of the ensemble curves (for all separation distances) from data
for a time window of 8us were calculated in a similar fashion and the slope were found to
be 1.1 dB/usec. The reverberation time was calculated to be 3.9 usec yielding a Q of 43.9
dB.
35
0 50 100 150 200 250 300 350 400 450 500
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Time (nsec)
Prec (dB)
Time domain K measurement. Distance of seperation b/w Tx and Rx = 1ft
Average of 50 tuner positions
Figure 2. 7 Time domain response of received power
(direct coupling between antennas; separation = 1 ft)
0 50 100 150 200 250 300 350 400 450 500
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Time (nsec)
Prec (dB)
Time domain K measurement. Distance of Seperation b/w Tx and Rx = 10 ft
Average of 50 tuner positions
Figure 2. 8 Time domain response of received power
(direct coupling between antennas; separation = 10 ft)
36
2. 2. 1 Calculation of reverb distance from time domain data
The reverberation time was calculated to be 3.9 usec from the slope decay rate of 1.1
dB/usec. The average absorption coefficient can be found using the reverberation time in
equation below [7].
Absorption coefficient = 4V
c A
η
τ
= (2.5)
Where, V is the volume and A is the surface area. The volume and surface area of the
cavity used in this measurement were calculated to be 401.84 m3 and 353.93 m2
respectively resulting in an absorption coefficient of 0.004. The absorption coefficient
can be used to calculate the reverberation distance [7]:
1 2
Reverberation distance = 1
d 2 r = D Dη A (2.6)
D1 and D2 are the directivities of the two antennas (transmitter and the receiver) used.
Due to the reflective nature of a reverberation chamber and multipath propagation
pattern, the directivities of any device or antenna placed inside the chamber are washed
away. Directivity being an intrinsic property of the device or the antenna in use, is not
lost in a reverberation chamber but the effects of it is less pronounced and hence
considered to be washed away (D1 = D2 = 1). Substituting the directivity values, the
reverberation distance is calculated to be 0.6m or 1.95 ft. the reverberation distance
calculated from the frequency domain approach is around 2ft. The calculation is shown
below.
Inside a reverberation chamber D1 and D2 are 1.
37
Hence 1
d 2 r = η A
0.6 1.95 dr = m = ft
Reverb distance calculated for two horn antennas inside the chamber at 1GHz using time
domain is around 2ft. Though the reverb distance was comparable to frequency domain
calculation of reverberation distance (2.5 ft), the Q that is obtained in time domain was at
least 3dB lower than the frequency domain Q at that frequency. The disagreement raises
the question of which test data to believe? Though time domain data is over a number of
frequencies, the Q variation with respect to frequencies might not be as much as 3dB.
Also the bandwidth over which this test was performed was small hence the difference in
Q should be minimal amongst the frequencies. More measurements were performed in
frequency and time domain at many frequencies and varying bandwidths to answer some
of the questions that were raised here. These measurements and results will be discussed
later.
2.3 Insertion loss and its implication to reverberation distance
Insertion loss in a reverberation chamber is defined as the ratio of maximum power
received to power transmitted. As the maximum power received is a function of number
of measured samples, the mean power received is used which would be a better estimate
(because variation in the mean from run to run is smaller than the variation in maxima
from run to run) to calculate insertion loss.
38
r 2
21
P
t
IL S
P
= = (2.7)
From the measured samples and the analysis of independent samples, the maximum can
be predicted. Received power follows a chi square or exponential distribution in a
multipath environment [4]. From the number of independent samples, an estimate for
maximum can be established for that distribution and the maximum will lie within the
uncertainty (1 or 2 standard deviations). Hence the maximum to mean ratio number can
be added to the mean received power to calculate the insertion loss. Here, mean power
received will be used in all calculations.
The insertion loss measurements were performed for 2 chamber configurations, one with
the unloaded chamber and other with 5 dB of loading. The mean received power will
vary for the two different chamber configurations hence the insertion loss will also vary.
Data were collected using the same setups described earlier. When the antennas are not
pointing at each other, ten different antenna positions are used to find an “average”
insertion loss (this is the insertion loss we expect if we do not have any direct coupling
and the multipath power dominates). The antennas are pointed at each other and the
separation distance is varied in 0.5 feet steps from 0.5 to 10 feet and the insertion loss is
calculated at every step from the ratio mean power received data to the input power
(<S21>).
From Figure 2.9 it can be seen that the insertion loss decreases with increase in the
separation distance. The slope is larger for smaller distances and later it traces close to the
average insertion loss. When the chamber is loaded, the surface area is reduced resulting
39
in fewer reflections and subsequent reduction in Q and thus directs path dominates for a
longer time which is reflected in our measurements. Measurements performed at 300
MHz (not shown here) with log periodic antennas follow the same pattern.
0 1 2 3 4 5 6 7 8 9 10
-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
Distance (ft)
IL (dB)
1GHz. Tx and Rx - Horns
Unloaded Chamber
Average chamber IL - Unloaded Chamber
Loaded Chamber
Average chamber IL - Loaded Chamber
Figure 2. 9 Insertion loss for empty and loaded chamber at 1 GHz
Comparison of the insertion loss of the empty chamber with K measured for varying
distances and empty chamber shows, at the distance where K equals 1 (reverberation
distance), the slope of the insertion loss changes and experiences a more average
behavior after that distance [Ref: Figure 2.10] which again ascertains the fact that the
reverberation distance calculated has more significance. The multipath power dominates
the direct path, reflected by K values less than 1 and the change in insertion loss slope to
follow a more average behavior happens around the same distance.
40
0 2 4 6 8 10
0 1
10
20
30
40
Distance (ft)
K
0.1
0.2
0.3
0.4
0.5
IL
Comparison of K and IL for Unloaded chamber at f = 1GHz
IL
K
K=1; Reverberation distance
Figure 2. 10 Plot of K and Insertion loss for empty chamber at 1 GHz
The quest behind this exercise was to quickly determine the EME of the space with quick
set of measurements which will then enable a test engineer to make a meaningful
estimate about assessing what type of testing must be done on the equipments that will
operate in such environment. Tests were performed in a reverberation chamber in
frequency domain at 1 GHz and the same tests were performed in time domain for
comparison and also to establish the advantage of one method over other if there is any.
Several parameters about the environment were calculated in frequency domain. The two
that were of importance is the Q of the cavity and the reverberation distance, i.e. the
distance at which the direct path and multipath powers are equal. When compared to the
41
results obtained from time domain, the reverberation distances obtained from both were
comparable while the Q was at least 3dB different. The time domain data seems to under
predict the Q compared to frequency domain. This discrepancy needs to be resolved
before concluding the accuracy and efficiency of one method over the other.
The advantage of a time domain method in the estimation of reverberation distance is that
the direct path components can be windowed out the multipath effects can be used in
determining the ‘tau’ hence the absorption coefficient and reverberation distance.
Moving the antennas thereby increasing the separation distance between them like in
frequency domain can be eliminated in time domain resulting in the reduction of test
time.
The analysis of estimation of Q was extended to a small room and a hallway to see the if
difference between the time domain and frequency domain Q persists in the not so
reverberant environments. As the room and the hallway do not have a stirrer to stir the
fields efficiently, source stirring was used in data collection. Source stirring indicates that
the source antenna and the receiver antenna are moved in the space and received power
samples are collected as a function of position of the Tx and the Rx antennas [10 11].
42
Figure 2. 11 Illustration of the source stirring measurement points in a room
50 total measurement points were chosen in the space. Spatial correlation was considered
while choosing every measurement point hence every point is at least half a wavelength
from the previous one [12]. For every Tx antenna position, 10 different Rx antenna
positions were chosen for measurement. This was repeated for 5 different Tx antenna
positions leading to 50 measurement points on the whole as shown in Figure 2.11. The
chosen 50 positions are arbitrary, to do a better analysis, more positions may be needed.
50 positions were chosen mainly depending on the time that was available for data
collection. At any measurement point, direct coupling between the Tx and Rx antennas
was avoided to eliminate the bias in the statistics.
Tx antenna position
Rx antenna position
43
Both time domain and frequency domain measurements were performed in the small
room and the hallway. Pictures of the Hallway and small room are shown in Figure 2.12
along with their dimensions.
Figure 2. 12 Pictures of the hallway and the small room
Hallway –
17.6 x 1.9 x 2.9 (m)
Small room –
5.5 x 2.1 x 2.7 (m)
44
2. 4 Measurements in Hallway
2. 4. 1 Frequency domain
0 10 20 30 40 50
-80
-70
-60
-50
-40
-30
-20
-10
0
Position
Received Power; Ref 0 dBm (dB)
Hallway Source Stirring Data at 1 GHz
Figure 2. 13 Received power as a function of position (source stirring) in the hallway at 1 GHz
The received power as a function of position of the antenna is shown in Figure 2.13. The
stirring ratio which the ratio of the max to the min turns out to be about 30 dB. It has
been suggested that for good reverberation inside a cavity, the stirring ratio should be at
least 20 dB [4]. Hence this space would qualify for the application of RC statistics on the
data collected.
Following the standard procedure of analyzing this data, the Q of the space was found to
be about 18 dB. The cumulative distribution function (CDF) comparison between the
45
measured data and theory is marginal (not shown). The reason for the marginal
agreement is suspected to be the poor reverberant nature of the space due to the
associated losses and insufficient amount of data points.
2. 4. 2 Time domain
For the same antenna positions at which frequency domain data was collected, time
domain measurements were also performed. A bandwidth of 200MHz was chosen around
the center frequency of 1GHz for this measurement. From the 50 different frequency
sweep data, an average of the sweep was calculated and used for analysis of Q and
reverberation distance. Two different time windows were also observed to look at the
early and later effects of cavity buildup.
46
0 50 100 150 200 250 300 350 400 450 500
-160
-140
-120
-100
-80
-60
-40
-20
Time (nsec)
Prec(dB)
Time domain Q measurement. Hallway. 0.5us period.
Average over 50 measurement points
Figure 2. 14 Received power as a function of time in the hallway; smaller time window
0 2 4 6 8 10
-160
-140
-120
-100
-80
-60
-40
-20
Time (us)
Prec(dB)
Time domain Q measurement. Hallway. 12us period
Average over 50 measurements
Figure 2. 15 Received power as a function of time in the hallway; larger time window
47
From Figures 2.14 and 2.15 it can be seen that the losses associated with the space is
large hence beyond 0.5us most of the measured samples are lower than the noise floor.
Fitting a line to the mean curve, the slope was obtained to be 92 dB/usec. Following the
same procedure as before, Q was obtained to be 24.7 dB and reverberation distance = 2.6
m or 8.6 ft. In this set of measurements, time domain Q (decay Q) is higher than the
frequency domain estimate.
2. 5 Measurements in Small room
2. 5. 1 Frequency domain
0 10 20 30 40 50
-80
-70
-60
-50
-40
-30
-20
-10
0
Small room Source Stirring Data at 1 GHz
Position Received Power; Ref 0 dBm (dB)
Figure 2. 16 Received power as a function of position (source stirring) in the small room at 1 GHz
48
The received power as a function of position of the antenna for the small room is shown
Figure 2.16. The stirring ratio which the ratio of the max to the min turns out to be about
30 dB. By the 20 dB criterion explained before, this space would qualify for the
application of RC statistics on the data collected. Following the standard procedure of
analyzing this data, the Q of the space was found to be about 17 dB. The CDF
comparison between the measured data and theory is also marginal (not shown) as the
hallway data previously mentioned.
2. 5. 2 Time domain
For the same antenna positions used in the frequency domain measurements, time domain
measurements were also performed. A bandwidth of 200MHz was chosen around the
center frequency of 1GHz for this measurement. From the 50 different frequency sweep
data, an average of the sweep was calculated and used for analysis of Q and reverberation
distance. Two different time windows were also observed to look at the early and later
effects of cavity buildup.
49
0 50 100 150 200 250 300 350 400 450 500
-140
-120
-100
-80
-60
-40
-20
Time (nsec)
Prec(dB)
Time domain Q measurement. Small room. 0.5us period.
Average over 50 measurement points
Figure 2. 17 Received power as a function of time in the small room; smaller time window
0 2 4 6 8 10
-160
-140
-120
-100
-80
-60
-40
-20
Time (usec)
Prec(dB)
Time domain Q measurement. Small room. 12us period.
Average over 50 measurement points
Figure 2. 18 Received power as a function of time in the small room; larger time window
50
As in the hallway, in the small room losses associated with the space is large hence
beyond 0.5us most of the measured samples are lower than the noise floor as can be seen
in Figures 2.17 and 2.18. Fitting a line to the mean curve, the slope was obtained to be 77
dB/usec. Following the same procedure as before, Q was obtained to be 25.5 dB and
reverberation distance = 1.5m or 4.75 ft. In this set of measurements, time domain again
predicts a higher Q.
Comparing the measurements from hallway and small room, the reverb distance for the
small room is smaller than the hallway. The small room is about 3 times smaller than the
hallway in both volume and surface area so we expect the multipath to take over at a
shorter distance than the hallway and that is reflected in the reverb distance calculation.
A comparison of the time domain measurements for the room and hallway are presented
in Figure 2.19. The decay profile of both the room and the hallway can be clearly
differentiated along with different times to peak power. By comparing the amplitude of
peak received power in the room to that of the hallway, it appears that the Q of the room
is higher than that of the hallway. The time to peak power for the room is also shorter
than that of the hallway which may suggest that the losses associated with the room are
less than the hallway leading to a higher Q value for the room compared to the hallway.
51
0 50 100 150 200 250 300 350 400 450 500
-110
-100
-90
-80
-70
-60
-50
-40
Time (nsec)
Rec. Power (dB)
Room and Hallway power decay profile comparison
Room
Hallway
Both curves are the average
of 50 antenna position
measurements
Figure 2. 19 Power decay profile of the hallway and small room at 1 GHz
2. 6 Comparison of frequency and time domain Q measurements from literature
Figure 2. 20 Avionics bay Quality factor comparisons (Figure 4-37. NSWCDD/TR-97/84 [13])
52
Figures 2.20 and 2.21 are the measured Q values in the avionics bay and the passenger
cabin of an aircraft. As the aircraft cabin is metallic and reflective, its operation can be
considered to be similar to that of a reverberation chamber making it potentially feasible
to use reverberation chamber methods and statistics for analysis.
Figure 2. 21 Passenger cabin Quality factor comparisons (Figure 4-38. NSWCDD/TR-97/84 [13])
Received power measurements were performed using mode stirring while the aircraft
cabins were exited using different excitation sources. Band limited white Gaussian
(BLWGN) noise is used as a source in some cases and constant wave source (CW) in
some. Both steady state Q (frequency domain) and the decay Q (time domain) are
calculated when the chamber was operating in a CW mode and frequency swept mode
respectively [13]. The differences between the two Q values as can be seen in Figure 2.20
and 2.21 are apparent. The time domain measurements seem to predict Q values that are
larger than the frequency domain measurements.
53
Figure 2. 22 Aluminum cavity Quality factor comparisons (Figure 7.13 [14])
Figure 2. 23 Main cabin of the hangar queen plane Quality factor comparisons (Figure 11.1 [14])
54
Figures 2.22 and 2.23 are the measured Q values from an aluminum cavity and main
cabin of the aircraft respectively. Power ratio measurements and decay time
measurements were performed with standard gain horns in the frequency range specified
[14]. The decay time measurements are closer to the theoretical Q values and Hill
suggests that decay time measurements are less affected by antenna efficiency and
impedance mismatch [14].
0 400 800 1200 1600
Frequency (MHz)
-40
-30
-20
-10
0
|s21| (dB)
ITS
NIST
Figure 2. 24 S21 comparisons of the NASA SPF chamber with frequency domain measurements from
NIST and time domain measurements from ITS
Courtesy: Bob Johnk, ITS and Galen Koepke, NIST
In Figure 2.24, more recent measurements performed in the NASA SPF chamber by
National Institute of Standards and Technology (NIST-Boulder) and Institute for
Telecommunication Sciences (ITS) are shown. The time domain measurement by ITS,
used joint frequency and time analysis (JFTA) to estimate 1/e power decay rate to
estimate Q [56]. Frequency domain measurements from NIST are direct S21
measurements made with the use of Vector Network Analyzer (VNA). Antenna
55
mismatch corrections applied to the measured data and S21 is used to infer Q factor as
they are directly proportional. At almost all frequencies there is about a 2 dB difference
between the ITS (time domain) and NIST data (frequency domain) with the time domain
measurements always higher than the frequency domain measurements.
2.7 Q - Frequency and Time domain
In order to investigate the difference in Q that existed in our measurements and the
surprising trend found in measurements made by different organizations in different time
frames and different cavities, more measurements were performed in the SMART 80
chamber at multiple frequencies using the same experimental setup as explained before in
both frequency and time domain. The average Q from CW measurements performed for
multiple frequencies averaged over multiple antenna positions (at each frequency) is
shown in Column 2 of Table 2.3. Q increases with frequency as expected and the average
Q around 1 GHz is calculated to be 44.6 dB.
Table 2. 3 Measured Q values using frequency and time domain
Freq
(MHz)
FD
Q(dB)
TD
Q(dB)
Difference
(dB)
400 37.31 40.29 2.98
500 39.74 41.90 2.16
600 41.47 43.06 1.59
700 41.57 43.52 1.95
800 43.08 44.36 1.28
1000 44.63 45.44 0.81
2000 46.40 47.46 1.06
56
The Q values measured at different frequencies through time domain technique is
tabulated in Table 2.3. As explained earlier, the frequency domain Q values reported at
each frequency are the averages of Q values computed from five receive antenna
positions inside the chamber. The S21 data are corrected for impedance mismatch via the
S11 and S22 data in frequency domain and the antenna efficiency corrections were not
applied. The Q values reported for time domain are calculated from ‘τ’ which in turn is
obtained from the measured slopes of the average power decay curve at the center
frequencies after time gating the initial reflections. The average power decay curves of
the SMART 80 chamber at different frequencies measured in time domain are shown in
Figure 2.25. The curves could be smoothed via averaging to obtain a more smooth decay
curve leading to a better fit of the linear curve but as we are interested, in a quick
measure and gross approximation, averaging is not applied to the measured data. The raw
data was subjected to a linear fit and slope was calculated from that. The different decay
slopes will result in different chamber time constants.
From Table 2.3, the Q values that are calculated using time domain method are seen to be
higher than the CW method under all conditions. These results corroborate the results
presented in [13] and [14]. The interesting questions that arise from this comparison are
• “Why is there a difference between the Q values that are measured using
frequency and time domain?”
• “Which Q value represents the real Q of the environment?”
• “Should the frequency and time domain values be is some way “transformable”?
• “In what way are the measurements inherently different?”
57
• “Do the two different Q values represent two different observables of the cavity?
And if so how?”
Some measurement parameters that influence the time domain measurements will be
discussed in the following sections.
0 1 2 3 4 5 6 7 8
-70
-65
-60
-55
-50
-45
-40
-35
-30
Power decay profile for SMART 80 chamber at varying operating frequencies
(with 200 MHz bandwidth around f0)
Time (usec)
Received power (dB)
400 MHz
500 MHz
600 MHz
700 MHz
800 MHz
1 GHz
2 GHz
Figure 2. 25 Power decay profiles of SMART 80 Chamber at different operating frequencies
a. Choosing a Bandwidth
The wall scattering time defined as average time it takes for the energy to scatter of the
walls [57].
4
c
T V
Sc
=
58
(2.8)
where V is the volume of the chamber, S is the surface area and c, speed of light.
For the SMART 80 chamber used in all the measurements presented here, the wall
scattering time was calculated to be approximately 15 nsec. The bandwidth was chosen
such that the pulse width was less than the wall scattering time by choosing a bandwidth
of at least 150 MHz. The effects of choosing pulse widths larger and smaller than the
wall scattering time will be discussed below.
b. Effects of Smaller Bandwidths (Longer pulse widths)
When the pulse widths are larger than the wall scattering time, the energy decay in the
chamber is affected leading to a slope that was slightly larger than the slope calculated
for pulse widths smaller than the wall scattering time. Measurements have been
performed at a center frequency of 500 MHz and 5 GHz with the bandwidths varying
from 80 MHz to 300 MHz in 20 MHz steps [58]. As can be seen from Table 2.4, the Q
values seem to stabilize above 160 MHz bandwidths around the center frequency. If the
pulse widths are larger than the wall scattering time of the chamber, the Q that is
predicted seems higher (because of larger ‘τ’ values) than the Q’s predicted when the
pulse widths are smaller.
59
Table 2. 4 Q variation as a function of bandwidth around the center frequency of 500 MHz and
5GHz
BW
(MHz)
Pulse
width
(nsec)
Q500
(dB)
Q5000
(dB)
80 25 43.71 51.26
100 20 42.78 51.14
120 16.7 43.02 51.10
140 14.3 42.72 51.18
160 12.5 42.48 50.98
180 11.1 42.37 51.02
200 10 42.26 50.94
220 9.09 42.11 50.82
240 8.33 42.16 50.82
260 7.69 42.06 50.90
280 7.14 42.11 50.67
300 6.67 42.06 50.80
The power delay profiles [69] measured for a center frequency of 500 MHz for varying
bandwidths around the center frequency are shown in Figure 2.26a and 2.26b. The slope
change as the bandwidth increases from 80 MHz can be seen from the graph.
c. Effects on Long Bandwidths: Aliasing time
If the bandwidths chosen are too wide, there could be two disadvantages. One, aliasing
occurs due the FFT operation performed within the VNA. With 1600 measurement
points, and a bandwidth of 200 MHz, the frequency step size is about 0.125 MHz (Δf)
which leads to a 8 μsec alias at 1/Δf. When the bandwidth is extended to1 GHz, aliasing
occurs at 1.6 μsec.
In order to find a slope of the power decay curve, the power decay needs to be observed
for a few μ seconds and the received power should be above the noise floor of the
instrument used. A linear fit to the power decay data will result in a better prediction of
60
‘τ’ values. If the bandwidth chosen is too wide then the observation time is reduced due
to aliasing leading to a poor fit thereby a poor estimation of τ.
Secondly if the chosen bandwidth is too wide, the τ value which is a function of
frequency will be varying significantly across the bandwidth which may lead to a poor
estimation of τ at the intended center frequency. While using a wide bandwidth, the τ
predicted will be a combination of different delay spreads and may not be representative
of the delay spread at the intended center frequency.
It can then be suggested that a bandwidth that is not too wide or too short needs to be
chosen. The lower bound on the frequency bandwidth is provided by the pulse widths
smaller than the characteristic wall scattering time (which is a function of the operational
space) and the upper bound is provided by the non aliasing time and variation of τ across
wider bandwidths.
61
-750 1 2 3 4 5 6 7 8
-70
-65
-60
-55
-50
Power decay profile for SMART 80 chamber at 500 MHz excited by
source of varying bandwidths
Time (usec)
Received power (dB)
80 MHz
120 MHz
140 MHz
160 MHz
180 MHz
200 MHz
Figure 2. 26 a. Power decay profile of SMART 80 Chamber at 500 MHz and varying bandwidths
(80-200 MHz)
-700 1 2 3 4 5 6 7 8
-68
-66
-64
-62
-60
-58
-56
-54
Power decay profile for SMART 80 chambet at 500 MHz excited by
source of varying bandwidths
Time (usec)
Received power (dB)
200 MHz
220 MHz
240 MHz
260 MHz
280 MHz
300 MHz
Figure 2.26 b. Power decay profile of SMART 80 Chamber at 500 MHz and varying bandwidths
(200-300 MHz)
62
d. Effect of Efficiency
When insertion loss measurements are performed in a reverberation chamber, the
physical construction of the antennas contributes to the complexity inside the chamber.
The antenna mismatch and the efficiency of the antenna to couple power into the cavity
(we will call this coupling efficiency) must be considered in the calculation of gain of the
chamber. The efficiency that is defined here is different from the radiation efficiency of
the antenna as there is power reflected back from the chamber to the antenna which is
unique to antennas operating inside high Q cavities. In the frequency domain when
measurements are performed at single frequency, the measured reflection coefficient
consists of energy that is reflected due to the antenna mismatch and energy that is
reflected from the chamber back through the transmit antenna [59]. These two exist
together in a CW measurement and cannot be separated while in a time domain
measurement, these two effects can be separated out.
In a CW measurement, Q is calculated from the power ratio measured with the antennas.
A better estimation of Q of the chamber is possible when the data is corrected for
impedance mismatch and coupling efficiency. In the time domain measurement, Q is
calculated from the slope of the energy decay curve. As no absolute powers are
considered in the time domain calculation, the coupling efficiency and mismatch seems
insignificant. To verify this claim, time domain measurements have been performed at 1
and 2 GHz with a pair of LP antennas used as Tx and Rx and one LP antenna and a piece
of wire used as Tx and Rx respectively.
63
The Q values from the measurements are reported in Table 2.5. The raw received power
in the CW measurements was corrected for impedance mismatch (from reflection
coefficients) and not for initial reflections from the chamber. At 1 GHz the difference
between the Q values (frequency domain) calculated with two different receiver antennas
is ~2dB. The Q values calculated with the wire as the receiver were lower than the Q
calculated with LP as the receiver. This is an expected result as the coupling efficiency of
the wire antenna might be different compared to the coupling efficiency of the LP
antenna. The agreement between the Q values (time domain) obtained using two different
receive antennas at both frequencies is noticeable.
Table 2. 5 Measured Q values using two different receive antennas
Freq
(MHz)
Rx- LP Rx - Wire
FD
Q(dB)
TD
Q(dB)
FD
Q(dB)
TD
Q(dB)
1000 44.63 45.44 42.8 45.61
2000 46.4 47.46 47.08 47.88
The above results shows a possibility of using any available antenna (in-band or out of
band) or easily fabricated from a piece of wire could be used in time domain
measurements for the estimation of quality factor of rooms and cavities provided the
antennas used are able to couple energy into the cavity and the receiver can sense power
levels above the noise floor which is to say that the antennas have enough dynamic range.
e. Effect of impedance mismatch
A typical frequency domain measurement is a steady state measurement implying that the
signal has been present for a time that is usually greater than the cavity fill time. The
64
energy is fully coupled from the transmitter into the cavity and steady state has been
reached inside the cavity before a measurement was made with the receive antenna.
Hence the received power at some position inside the cavity is measured from the steady
state response of the cavity.
The measured reflection coefficient consists of energy that is reflected from the antenna
mismatch and energy that is reflected from the chamber back through the transmitter
antenna. These two effects coexist in frequency domain but could be decoupled in time
domain. Hence while performing a time domain measurement the impedance mismatch
created by the antennas can be neglected.
In Figure 2.27, the S parameters measured with a LP antenna pair at f = 1 GHz are plotted
against time. From the graph the initial reflection from the antennas can be seen both in
S11 and S22 curves. The initial reflections due to the mismatch at the antenna terminal can
be seen at the earlier time frame (1st peak) followed by the reflections from the chamber
(2nd peak). The initial reflections from the chamber can be considered as the time before
the reverberant field is setup inside the cavity (unstirred energy) or the inefficiency of the
tuner to stir the fields. At any case, the initial reflections contribute the coupling
efficiency of the antenna. S21 is a function of the energy coupled into the cavity,
interacting with the walls/tuner and being detected back at the receive antenna. The slope
of measured power decay will lead to a better estimation of the chamber time constant
and a better estimation of the conductivity of the material used in the chamber walls at
65
operational conditions and will not be affected by the initial reflections from the
antennas.
As discussed earlier, the possibility of using any available antennas, even if the antennas
are not matched well at the intended frequencies of operation, in time domain
measurements for an approximate Q measurement is fairly high.
As the time domain measurements measures the τ (from the slope of the energy decay
gated after the initial reflections), and not from power ratio, resulting in minimal
contribution from the antenna impedance mismatch and coupling efficiency, the time
domain methods appears to be efficient in measuring the quality factor compared to
frequency domain methods.
-600 0.2 0.4 0.6 0.8 1.0
-55
-50
-45
-40
-35
-30
-25
-20
-15
Time (usec)
Received power (dB)
S parameters; Tx - LP and Rx - LP
S21
S11
S22
Figure 2. 27 Measured S parameters of the SMART 80 chamber at 1 GHz with LP antennas as Tx
and Rx
Mismatch
Initial reflections
from chamber
66
f. Loading
To simulate loading conditions and to sketch the efficiency of time domain measurements
in detecting the energy decay due to loading, measurements were performed at 1 and 2
GHz using a pair of LP antennas and using absorbing cones as load.
The absorbers were placed at a corner of the chamber away from the antennas. The Q
values calculated using both frequency and time domain measurements at both
frequencies and loading conditions are shown in Table 2.6. The absorbing cones provided
approximately 4 dB of loading at both frequencies. The energy decay curve (average of
50 distinct tuner positions) for both frequencies, empty and loaded conditions are shown
in Figure 2.28. As one would expect when there is a load present in the chamber, energy
absorption by the load would result in less energy available in the chamber and the
energy decay rate would be much faster which is evident from the larger slopes of the
decay curves.
Table 2. 6 Measured Q values for an empty and loaded chamber at 1 and 2 GHz
Freq
(MHz)
Empty chamber Loaded chamber
FD
Q(dB)
TD
Q(dB)
FD
Q(dB)
TD
Q(dB)
1000 44.63 45.44 40.36 41.41
2000 46.4 47.46 42.19 43.53
67
-800 1 2 3 4 5 6 7 8
-75
-70
-65
-60
-55
-50
-45
Power decay profile of SMART 80 chamber at 1 and 2 GHz,
Unloaded and Loaded with excitation pulse width of 5nsec
Time (usec)
Received Power (dB)
1 GHz (Unloaded)
2 GHz (Unloaded)
1 GHz (Loaded ~4dB)
2 GHz (Loaded ~4dB)
Figure 2. 28 Power decay profile of an empty and loaded SMART 80 chamber at 1 and 2 GHz
The versatility of the time domain measurements in capturing the power decay
characteristics of the cavity is clear from the above measurements.
2.8 Difference between Frequency and Time domain Q Factor
The difference in the Q values calculated using the frequency and time domain
techniques are primarily due to the way the antenna losses are handled in each domain.
Though the impedance mismatch that occurs at the antenna terminal can be accounted for
in frequency domain, the coupling efficiency cannot be. The antenna efficiency (treated
as radiation efficiency) values that are used in most standards (LP – 0.75 and DRG – 0.9
[4]) are rather arbitrary and might vary from antenna to antenna. Though the antenna
68
efficiency numbers are a good approximation, they can be manipulated to offer a better
match to theoretical or simulated values.
The time domain measurements are rather not influenced by the impedance mismatch and
efficiency, especially in the measurement of quality factor and decay time constants of
cavities.
2.9 Calculating Antenna to Chamber Coupling Efficiency
If the antenna impedance mismatch can be measured precisely using the VNA and if the
time domain measure of power decay profile of the chamber provides a good estimation
of the chamber characteristics, there exists a possibility to measure the coupling
efficiency precisely under operating conditions.
As the coupling efficiency is the one parameter that is different between the frequency
and time domain measurements, by comparing the S parameter measurements from the
two techniques, the coupling efficiency can be calculated. From the Q calculated using
time domain, the chamber gain can be calculated using eqn. 2.3 and comparing those
values with the frequency domain measurement, the coupling efficiency is calculated.
From the measurements made with this study, coupling efficiency calculated for the Log
Periodic antenna used as a receiver (with another LP antenna used as a transmitter), at
discrete frequencies are tabulated in Table 2.7.
69
Table 2. 7 Antenna efficiency calculated from Time domain measurements
Freq
(MHz)
S21 – FD
(measured)
S21 – TD
(calculated) Eff.%
400 0.036 0.071 50.2
500 0.032 0.053 60.7
600 0.028 0.040 69.3
700 0.018 0.028 64.0
800 0.017 0.023 74.4
1000 0.012 0.015 82.9
2000 0.002 0.004 66.2
A method of calculating coupling efficiency in reverberation chambers is presented
which differs from all previous efforts to measure the antenna efficiency using a
reverberation chamber [60 61 62 63]. The major advantage of this method is that,
efficiency is measured under operational conditions of the antenna. There is no need for a
second or reference antenna (with a known gain) to calculate the efficiency of the antenna
in use. It also eliminates performing measurements inside and outside the chamber to
calculate efficiency.
The coupling efficiency calculated using this measurement procedure again ascertains
that the suggested antenna efficiency of 0.75 for a Log Periodic antenna might not be
always true. The efficiency as expected will vary across frequency as high as 20% as in
this case. Time domain technique has an advantage of neglecting the efficiency of the
antenna to estimate Q.
70
2.10 Summary
The advantage of a time domain method in the estimation of time decay constant of the
chamber is that the direct path components can be windowed out so that only the
multipath effects are considered. The parameters that need attention in performing an
efficient time domain measurement is brought out in this paper. The difference in the
quality factor measured with frequency and time domain has been of interest and has
received some attention in a number of reports, and this work has attempted to address
the possible reasons for the discrepancy. The effects of antennas that are used in these
measurements can be separated in time domain measurements which enable the
measurement of antenna properties while in the frequency domain approach the antenna
affects are inseparable.
The effective wall conductivity and absorption coefficient for rooms/cavities can be
quickly determined from measured time domain data which is more practical as the
effective conductivity is measured in a more realistic operational environment. This
knowledge can be further used in the estimation of max field levels inside the cavity for a
known input power level and reverberation distance.
Coupling efficiency could be calculated under operational conditions of the antenna by
comparing the frequency and time domain measurements. There is no need for multiple
measurements or using a reference antenna to calculate the efficiency of the antenna in
71
operation. Time domain measurements could be highly useful and efficient in the gross
estimation of quality factor of spaces under operating conditions.
The time domain method may be the fastest way to calculate the reverberation distance as
any direct coupling between the antennas if exists can be eliminated using windowing
and also the separation distance between the antennas need not be varied. The insertion
loss calculated using the received power data for both Q and K type of measurements
seem to follow the pattern as expected above and below the reverberation distance. This
provides extra confidence in the calculated reverberation distance. Determination of this
distance could be important while placing wireless systems into enclosed cavities where
an estimate of the maximum field can be quickly established depending on the distance of
separation between the victim and the aggressor.
In the next chapter, the importance of independent samples for arriving at statistical
distributions will be discussed. The influence of loading on independent samples and also
uniformity will be discussed in detail with some experimental data. Some methods to
increase the number of independent samples using a second tuner will be presented along
with discussion about the size and the location of the second tuner.
72
Chapter 3
3.0 EFFECT OF LOADING ON INDEPENDENT SAMPLES
Independence is a key parameter in any statistical analysis. If two events are said to be
independent, then by probability theory it means that the occurrence of one event, makes
it neither more nor less probable for other events to occur. When fitting a probability
distribution for the measured data set (set of random variables), all data points need to be
independent. Use of correlated or dependent data, impacts the attempt to hypothesize a
particular distribution. As the statistics are skewed with correlated data, any distribution
to fit the measured data could be acceptable.
Of more importance is to determine how many independent samples are required to
determine an electric field or power density distribution of a room or cavity. For a cavity,
the number of IS provides information regarding the reflectivity of the cavity and also the
number of statistically independent field configurations that could exist inside the cavity.
In order to characterize a cavity, all the independent samples available should be utilized
to reduce the uncertainty. IS can also be interpreted as the number of measurement points
inside the cavity that should be utilized to produce an estimate of received power or a
prediction of the maximum field with minimum uncertainty.
73
An important question that this work will address is “What happens to the number of
independent samples when the cavity gets populated with either personnel or equipments
or furniture (in an office space)?” Will the established probability distribution for the
empty case still be valid? To explore this, some measurements have been performed
inside a reverberation chamber where the number of independent samples has been
estimated with reasonable accuracy through standard procedures. Then the chamber was
loaded with some absorber materials and the effect of loading on IS is monitored. The
statistical variation in the estimation of IS and the limitations are shown to be present.
Methods to increase the number of independent samples which might help to compensate
for the loading and improve the measurement accuracy are also investigated.
3.1 Independent samples (IS)
For a typical reverberation chamber the number of independent samples can be defined as
the number of statistically independent field configurations that can exist in the chamber
at a particular frequency over one complete rotation of the tuner (or variation over a set of
boundary conditions). The number of independent samples is a function of frequency and
the number of measured samples at each frequency.
The estimation of independent samples is possible using autocorrelation which measures
the relative correlation between a sequence of ‘N’ samples and an offset of the same
sequence. Identifying the number of IS is critical for the estimation of the max fields
74
during the test and the uncertainty associated with the test. The autocorrelation function
that is used to calculate the independence is given below.
1
( )( )
0
2
( )
1
,
,
j
N
xi x x j x i
N
xi x i
i m i m
j
i m N i m
μ μ
ρ
μ
−
Σ − −
= =
⎡ ⎤
⎢ Σ − ⎥
⎢⎣ = ⎥⎦
⎧ − > ⎫
= ⎨ ⎬ ⎩ − + ≤ ⎭
(3.1)
Where xi yj are linear, mean normalized data from an ‘N’ sample sequence. The
distribution of yj is same as xi shifted by a sample offset m. μx is mean of original N
sample sequence. Since the y distribution is same as x except for offset, the means and
standard deviations are equal (μy = μx , σx = σy)
An operationally common practice assumes that ρ < 1/e = 0.367 implies independence.
As the underlying distribution is normal, uncorrelation implies independence. A
statistically more robust criterion is given below [15].
( )
0
1 4
2 2
0
2 1
( ) 2 1
2
2
N
N
N
P d
N ρ
ρ ρ ρ ρ
π
⎛ − ⎞
⎜ ⎟
⎝ ⎠
⎡ ⎛ − ⎞ ⎤ ⎢ Γ⎜ ⎟ ⎥ ≥ = ⎢ ⎝ ⎠ ⎥ −
⎢ Γ⎛ − ⎞ ⎥ ⎢ ⎜ ⎟ ⎥ ⎣ ⎝ ⎠ ⎦
∫ (3.2)
If the probability that N samples of two uncorrelated variables have a correlation
coefficient 0 ρ ≥ ρ . 0 ρ
is the strictness factor and can be chosen based on the amount of
uncorrelation expected. For true uncorrelation, 0 ρ
is chosen to be zero, commonly it is
75
chosen to be 0.367. If PN ( ρ ≥ ρ0 ) ≤ 5%, the correlation is called significant and if it is
less than 1%, the correlation is called highly significant i.e. the samples are highly
correlated. Depending on the procedure that is used to calculate the number of IS there
could be as much as a 40% variability [32]. When the max fields are estimated using the
number of IS, the error margin in the estimation of IS translates to higher uncertainty in
the estimation of test fields hence the over test/under test margin.
In practice, a smaller number of test samples are chosen than the total available IS [4].
Note: The assumption typically made is that all the chosen test samples are independent.
The assumption will only be valid if the number of available IS are far greater than the
test samples. At higher frequencies the total number of IS are large while at lower
frequencies this might not be true. Adding the 40% variability to the estimate of IS
complicates the situation.
When the equipment under test is placed inside the usable volume of the chamber for
testing, the EUT loads the chamber. What is the influence of loading on the number of
IS? If the chamber loading reduces the available number of IS, the impact at the low
frequency side is significant because of the lower number of IS available. In order to
investigate the variation between repeated measurements just as a function of statistical
sampling, the number of measured samples required, criteria used and the effect of
loading on the estimation of independent samples, a series of measurements were
performed as detailed below.
76
3.2 Experimental setup
The measurements were performed in the SMART 80 chamber at frequencies 1-5 GHz
where the mode density of the chamber is high and all the chamber statistics are good
(Ref: Figure 2.2). A pair of dual ridge waveguide horns served as the transmit and receive
antennas while the VNA was used to measure the insertion loss of the chamber. Being a
statistical procedure, there is an uncertainty with the way the fields are sampled. This
uncertainty cannot be neglected. Also the more number of data samples there are, a more
complete statistical description is possible. Hence the number of measured samples is
always chosen to be maximum available which is 1600 samples per complete rotation of
the tuner in our case. To look at the variation in the estimation of IS from a set of
repeated experiments, insertion loss measurements were carried out at 10 different
positions of the receive antenna for an input frequency of 1GHz.
Autocorrelation analysis was performed on the measured data and the number of IS were
estimated from a correlation value of ρ = 0.367. The experimental setup is shown in
Figure 3.1. Table 3.1 contains the estimated number of IS for every repeated
measurement along with the measured and expected max to mean ratios.
77
Figure 3. 1 Experimental setup for measuring independent samples in a reverberation chamber
Table 3. 1 Measured Independent samples for 10 repeated measurements
Measurement
Position IS
Measured
Max/Mean
(dB)
Norm.
SD
Expected
Max/Mean
(dB)
SD
1 230 8.25 0.99 7.68 0.93
2 270 7.69 0.93 7.8 0.91
3 270 8.36 0.98 7.8 0.91
4 270 7.24 0.91 7.8 0.91
5 230 8.03 1.01 7.68 0.93
6 270 8.84 0.98 7.8 0.91
7 270 7.05 0.97 7.8 0.91
8 320 7.1 0.95 7.92 0.89
9 270 7.9 1 7.8 0.91
10 230 8 0.97 7.68 0.93
The variability among the estimation due to the sampling and from Table 3.1 it is
calculated to be 28.1%.
Horizontal Tuner
Tx
Rx Vertical
Tuner
78
Table 3.2 shows the ρ values calculated for two different significant values and different
number of measured samples using eqn. (3.2). Using an extrapolated coefficient value of
0.24 for 1600 measured samples for correlation at a highly significant level (1%); the
number of independent samples for the measured data is estimated to be roughly 200.
Table 3. 2 Correlation values calculated for different measured samples
No. of
Samples P = 5% P = 1%
12 0.576 0.707
20 0.445 0.562
30 0.361 0.463
50 0.278 0.361
100 0.197 0.255
The variation between the IS calculated using eqn. (3.2) and the 1/e criterion is 23.9 %. In
order to characterize the chamber at some frequency of operation, maximum number of
data samples need to be captured. To look at the effect of the number of measured
samples on the estimate of independent samples, the number of measured samples per
rotation of the tuner was increased. 1600 data samples were captured for every 16
degrees rotation of the tuner. The total number of measured samples per rotation was then
increased to 32000 samples. This data was parsed down and IS was estimated from the
available data. Every second sample or third or so forth sample was considered for
autocorrelation analysis and the number of IS was estimated using the 1/e criterion.
79
Table 3. 3 Independent samples as a function of measured samples
Measured Independent
32000 440
16000 440
8000 440
4000 400
2000 400
1000 330
500 250
200 200
100 100
From Table 3.3, it can be seen that at 1 GHz the total number of independent samples that
the chamber can generate is about 440. The maximum available number can only be seen
when the measured samples is at least 8000. Hence to determine the maximum number of
IS at any frequency maximum number of measured samples must be captured.
To evaluate the influence of loading on the estimate of IS, absorbing material was used
inside the chamber to simulate different loading conditions. The number of measured
samples was fixed at 1600 per rotation. At every frequency and every loaded condition,
the receive antenna was placed at 5 fixed positions inside the chamber and the average of
IS from the different measurement positions is the IS (<IS>) for that frequency and
loading condition. By averaging out the IS at every experiment, the only variability that
existed was the sampling and the load. For frequencies 1 - 5 GHz in 1 GHz steps,
insertion loss was measured for a no load condition and load was added in 4 different
steps. The maximum load was at step number 4 (~12 dB; this would be the difference
between empty chamber Q and loaded Q), calculated as the difference between the Q of
the empty chamber and Q of the loaded chamber.
80
Table 3. 4 Independent samples measured at different frequencies and loading conditions
Freq. (GHz) No Load Load 1 Load 2 Load 3 Load 4
1 384 238 213 195 109
2 534 395 368 302 183
3 801 534 464 454 278
4 801 801 587 507 334
5 801 801 747 747 368
As the frequency increased the number of independent samples increased irrespective of
the loading (Ref: Table 3.4). When loading increased, invariably at all frequencies, the
estimated number of IS reduces. This uncertainty is not addressed in current RC
applications.
The quality factor of the chamber also varies with respect to loading and frequency hence
at all frequencies, Q was calculated using insertion loss data in eqn (2.3) (Ref Section
2.2.1 in Chapter 2) and is tabulated in Table 3.5.
Table 3. 5 Quality Factor measured at different frequencies and loading conditions
Freq. (GHz) No Load Load 1 Load 2 Load 3 Load 4
1 46.33 42.5 41.15 40.29 34.69
2 48.25 44.99 43.42 42.54 37.17
3 48.41 45.4 44.28 43.41 38.32
4 49.22 46.11 44.8 44.35 38.69
5 51.6 49.18 47.98 47.11 41.95
As seen in Table 3.5, Q increased with increase in frequency for each loading
configuration. As the load is increased at a particular frequency, Q goes down as
expected. At the maximum load, the loading is about 12 dB at 1GHz and about 10 dB at 5
GHz. Loading the chamber reduces the Q as expected but also influences the independent
81
samples. The estimated IS reduces with loading regardless of frequency. This
phenomenon was also reported in [17 18] but the effects at low frequencies was not
explored. In [19] it has been reported that the number of IS increases with loading. The
minimal increase (10 to 14) is within the 40% variability in just the calculation of IS from
the 1/e criterion as was shown in Table 3.1.
In our effort to validate our claims, the unpublished work of Greg Tait [16] was brought
to bear on this issue. Tait’s procedure utilizes the measurement of maximum power inside
the chamber and calculates the number of IS from a maximum power measurement. From
the multiple maximum measurements, the mean and the standard deviation of the relative
maximums will be calculated and the normalized standard deviation which is the ratio of
the standard deviation of the maximums and the mean of the maximums will be plotted
against the number of IS. The equations used to plot this curve were extracted directly
from Taits work and are given below [16].
2
N
0 2
2 n
2
N
0
w f (w)dw
I(N) 1 S
w f (w)dw
∞
∞
= = +
⎡ ⎤
⎢ ⎥
⎢⎣ ⎥⎦
∫
∫
(3.3)
82
Figure 3. 2 Independent samples versus normalized standard deviation [16]
As the max to mean ratio follows a Chi Square distribution with 2 degrees of freedom
[4], a Monte Carlo analysis was performed by randomly selecting data from a Chi Square
distribution. Figure 3.2 might be a limiting value for a large number of measurements.
Hence the Monte Carlo analysis was performed by varying the number of trials and the
results are given in Figure 3.3.
When the number of trials is reduced to 12 or 5, the deviation from the theoretical curve
and prediction of fewer IS can be seen in Figure 3.3. Also the statistical noise in the
estimation of the normalized standard deviation is large when the number of IS are large
and fewer the measurement trials (12 or 5) making it hard to estimate the actual number
of IS. This implies that the estimation of IS may be strongly dependent on the number of
measured maximums. A closer look of the 12 and 5 measurement trials is shown in
83
Figure 3.4 where the small difference in the calculated normalized standard deviation
resulting in variation of the IS are clearly seen (referred as statistical noise).
A similar approach can be used to estimate the influence of loading on independent
samples but there needs to be a large number of measurements of maximums to estimate
the number of IS with minimal statistical noise.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
100
101
102
103
104
Normalized Standard Deviation
Independent Samples
IS curves from Max values
Number of trials = 5
Number of trials = 12
Infinite number of trials (theoretical curve)
Figure 3. 3 Independent samples versus normalized standard deviation for different number of trials
from Monte Carlo Simulation
84
0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28
100
101
102
103
Normalized Standard Deviation
Independent Samples
IS curves from Max values
Infinite number of trials (theoretical curve)
Number of trials = 12
Number of trials = 5
Statistical
noise
Figure 3. 4 Independent samples versus normalized standard deviation for different number of trials
from Monte Carlo Simulation (closer look)
3. 3 Uniformity Independent samples and Loading
Uniform implies all spatial locations within a working volume are equivalent within an
acceptable uncertainty. Uniformity is usually me