A CONFORMAL TAGUCHI OPTIMIZED E-PATCH ANTENNA
By
C. M. GARDNER

A THESIS
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
MASTER OF SCIENCE
ELECTRICAL ENGINEERING
2010

ABSTRACT
A CONFORMAL TAGUCHI OPTIMIZED E-PATCH ANTENNA
By
Chad Michael Gardner

The performance of a cavity-backed E-patch antenna placed conformal to a cylindrical conducting surface is explored numerically using the ﬁnite element boundary
integral (FE-BI) method. From using rigorous full-wave simulations based on the
ﬁnite element method, the eﬀects of curvature are assessed for such antenna characteristics as bandwidth and impedance. The necessary changes to the patch geometry
to maintain an acceptable impedance match and a desired bandwidth for rectangular cavity-backed E-patch antennas are determined. Cylindrically conformal cavitybacked E-patch antennas are simulated and perform well with many diﬀerent cylinder
radii. A experimental cylindrical conformal cavity-backed E-patch antenna was built
to verify simulations and preformed well at the L1 and L2 frequencies. Radiation patterns of both experimental and simulated cylindrically conformal cavity-back E-patch
antennas are analyzed and perform well.
Taguchi’s method of optimization is used to optimize |S11 | of rectangular cavitybacked E-patch antennas. Using Matlab and rigorous full-wave simulations based
on the ﬁnite element method in combination, the eﬀects of having diﬀerent ﬁtness
functions, level diﬀerences, and input parameters in Taguchi’s method of optimization
are explored and determined.

To my family and friends.

iii

ACKNOWLEDGMENTS

Dr. Edward Rothwell, Dr. Leo Kempel, Dr. Shanker Balasurbamium, Dr. Prem
Chahal, Air-force Research Labs, Michael Varney, Gordon Shetler, Melissa Sisk, Jose
Hejase, Raoul Ouedraogo, and the entire EM research group at Michigan State University.

iv

PREFACE

O Ye, who in some pretty little boat,
Eager to listen, have been following
Behind my ship, that singing sails along,
Turn back to look again upon your shores;
Do not put out to sea, lest peradventure,
In losing me, you might yourselves be lost.
The sea I sail has never yet been passed;
Minerva breathes, and pilots me Apollo,
And Muses nine point out to me the Bears
Ye other few who have the neck uplifted
Betimes to th’ bread of Angels upon which
One Liveth here and grows not sated by it
Well may you launch upon the deep salt-sea
Your vessel, Keeping still my wake before you
Upon the water that grows smooth again.
The divine comedy, In Pardiso Canto II Verses 1-15. Colonial Press, 1895.

v

Table of Contents
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures

x

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 E-patch Background . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Background on Taguchi’s Method of Optimization . . . . . . . . . . .
2 Conformal E-patch Antenna . . . . . . . . . . . . . .
2.1 Inﬁnite ground plane simulations . . . . . . . . . . . .
2.2 K-brick simulations . . . . . . . . . . . . . . . . . . . .
2.3 HFSS ﬂat bottom cavity simulations . . . . . . . . . .
2.4 HFSS cylindrically conformal cavity simulations . . . .

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65

4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Rectangular cavity-backed E-patch antenna . . . . . . . . . . . . . .
4.2 Cylindrical conformal cavity-backed E-patch antenna . . . . . . . . .

76
76
98

3 Taguchi’s Method of Optimization for E-patches
3.1 Orthangonal Arrays . . . . . . . . . . . . . . . . .
3.2 Taguchi’s method and optimization example . . .
3.3 Taguchi-K-brick-Matlab results . . . . . . . . . .

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Appendix A: Setting up Sonnet for E-patch antenna simulations
Appendix B: Matlab Code for a two variable function
Appendix C: Start up guide to K-brick

. . 129

. . . . . . . . 147

. . . . . . . . . . . . . . . . . 154

Appendix D: Matlab Code for GUI . . . . . . . . . . . . . . . . . . . . 161
Appendix E: Matlab Code for Taguchi-K-brick-Epatch . . . . . . . . 180
Appendix F: Matlab Code for Taguchi-K-brick-Rectangular Patch . 196
Appendix G: Matlab code for Rosenbrock Taguchi optimization
vi

. . 210

Appendix H: Matlab code for gain measurements at MSU engineering
building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Bibliography

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

vii

List of Tables
1.1

Dimensions for an E-patch example given in [1]. . . . . . . . . . . . .

6

1.2

An Orthogonal Array OA(9,4,3,2) . . . . . . . . . . . . . . . . . . . .

9

2.1

Geometry values for the second E-patch example given in [1]. . . . . .

13

2.2

Geometry values for the third E-patch example given in [1].

. . . . .

14

2.3

Geometry values for the 40% enlarged E-patch dimensions. . . . . . .

15

2.4

Geometry values for E-patch dimensions generated from equations
(1.3) and (2.1) - (2.3). . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Geometry values for E-patch dimensions generated from equations
(1.3) and [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Dimensions of a cavity-backed rectangular patch antenna. Values were
generated from a Taguchi optimization. . . . . . . . . . . . . . . . . .

22

2.7

Dimensions of a cavity-backed E-patch antenna. . . . . . . . . . . . .

24

2.8

Dimensions of a cavity-backed E-patch antenna optimized using the
GUI Matlab K-brick interface. . . . . . . . . . . . . . . . . . . . . . .

25

Curvature deﬂection from using (2.10). . . . . . . . . . . . . . . . . .

30

2.10 HFSS settings for the cylindrically conformal cavity-backed E-patch
antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.11 Dimensions and probe placement of a cylindrically conformal cavitybacked E-patch antenna. . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.12 Dimensions and probe placement of a cylindrically conformal cavitybacked E-patch antenna. . . . . . . . . . . . . . . . . . . . . . . . .

35

2.13 Dimensions and probe placement of a cylindrically conformal cavitybacked E-patch antenna. . . . . . . . . . . . . . . . . . . . . . . . .

36

2.5
2.6

2.9

viii

2.14 HFSS settings for the cylindrically conformal cavity-backed E-patch
antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

2.15 Dimensions and probe placement of a cylindrically conformal cavitybacked E-patch antenna. . . . . . . . . . . . . . . . . . . . . . . . .

39

2.16 Dimensions and probe placement of a cylindrically conformal cavitybacked E-patch antenna. . . . . . . . . . . . . . . . . . . . . . . . .

40

3.1

An Orthogonal Array OA(18,7,3,2) . . . . . . . . . . . . . . . . . . .

54

3.2

An Orthogonal Array OA(9,4,3,2) . . . . . . . . . . . . . . . . . . . .

58

3.3

An Orthogonal Array OA(9,2,3,2) . . . . . . . . . . . . . . . . . . . .

59

3.4

The Orthogonal Array OA(9,2,3,2) and level values in the ﬁrst iteration
of Equation (3.3) optimization. . . . . . . . . . . . . . . . . . . . . .

60

The Orthogonal Array OA(9,2,3,2), level values, S/N ratio, and experiment outcome in the ﬁrst iteration of Equation (3.3) optimization. .

61

3.6

Response table for the optimization of (3.3). . . . . . . . . . . . . . .

61

3.7

An Orthogonal Array OA(27,13,3,2) . . . . . . . . . . . . . . . . . . .

66

3.8

An Orthogonal Array OA(7,13,3,2) . . . . . . . . . . . . . . . . . . .

67

3.9

The input parameter ranges for E-Patch Taguchi optimization. Fitness
function is (3.14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

3.10 The input parameter ranges for E-patch Taguchi optimization. Fitness
function is (3.15). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

3.11 Dimensions of a cavity-backed E-patch antenna optimized by TaguchiK-brick code. Input parameter ranges used to generate the geometry
values are given in Table 3.9. . . . . . . . . . . . . . . . . . . . . . . .

72

3.12 Dimensions of a cavity-backed E-patch antenna optimized by TaguchiK-brick code. Input parameter ranges used to generate the geometry
values are given in Table 3.10. . . . . . . . . . . . . . . . . . . . . . .

72

3.5

4.1
4.2

Dimensions of the experimental rectangular cavity-backed E-patch antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

HFSS settings for the rectangular cavity-backed E-patch antenna. . .

79

ix

4.3

Dimensions of a experimental cylindrical conformal cavity-backed Epatch antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4

HFSS settings for the cylindrically conformal cavity-backed E-patch
antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.5

Dimensions of a experimental cylindrical conformal cavity-backed Epatch antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

x

List of Figures
1.1

E-patch topology from [1]. . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Field conﬁgurations (modes) of a E-patch antenna. . . . . . . . . . .

4

1.3

(a) An LC circuit representing the transmission line model for the lower
resonance frequency of an E-patch antenna. (b) An LC circuit representing the transmission line model for the higher resonant frequency
of an E-patch antenna. . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.4

|S11 | vs frequency plot of a E-patch antenna found using Sonnet. . .

2.1

Sonnet simulation frequency response of the second E-patch in [1] (L =
70 mm, W = 45 mm). Dimensions provided in Table 2.1. . . . . . . .

12

Sonnet simulation frequency response of the third E-patch in [1] (L =
70 mm, W = 50 mm). Dimensions provided in Table 2.2. . . . . . . .

14

Sonnet simulation frequency response of a ﬁrst designed L-band Epatch (L = 98 mm, W = 75 mm). Dimensions provided in Table
2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Sonnet simulation frequency response of the second designed L-band
E-patch (L = 107.14 mm, W = 91.1731 mm). Dimensions provided in
Table 2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Sonnet simulation frequency response of the third designed L-band Epatch (L = 142 mm, W = 90 mm). Dimensions provided in Table
2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

A typical rectangular cavity-backed E-patch antenna analyzed in Kbrick. Depicted cavity dimensions are not to scale and typically h
Cx , h
Cy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.7

Top and side view of an E-patch in a cavity. . . . . . . . . . . . . . .

21

2.8

Top and side view of a rectangular patch backed by a rectangular cavity. 23

2.2
2.3

2.4

2.5

2.6

xi

7

2.9

K-brick simulation frequency response of the cavity-backed rectangular
patch. Table 2.6 contains the dimension values. . . . . . . . . . . . .

25

2.10 K-brick simulation frequency response of the E-patch antenna with
geometry values given in Table 2.7. . . . . . . . . . . . . . . . . . . .

26

2.11 Matlab GUI interface for varying the geometry of an E-patch antenna.
For interpretation of the references to color in this and all other ﬁgures,
the reader is referred to the electronic version of this thesis. . . . . . .

26

2.12 K-brick simulation frequency response of an E-patch antenna with geometry values given in Table 2.8. . . . . . . . . . . . . . . . . . . . .

27

2.13 Curvature diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.14 A conceptual depiction of an cavity-backed E-patch conformed to a
cylinder. Depicted cavity and E-patch dimensions are not to scale. . .

32

2.15 HFSS screen capture of an cylindrically conformal cavity backed Epatch antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.16 Simulated frequency response of an cylindrically conformal cavity-backed
E-patch antenna (ρ = 100 cm). Dimensions provided in Table 2.11. . 34
2.17 Simulated frequency response of an cylindrically conformal cavity-backed
E-patch antenna(ρ = 70 cm). Dimensions provided in Table 2.12. . . . 35
2.18 Simulated frequency response of an cylindrically conformal cavity-backed
E-patch antenna (ρ = 50 cm). Dimensions provided in Table 2.13. . . 36
2.19 Simulated frequency responses of cylindrically conformal and rectangular cavity-backed E-patch antennas. Dimensions provided in Tables
2.11 - 2.13 and 3.12. . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.20 An angled-view conceptual depiction of the E-patch antenna conformed
to the surface of a cylinder with a radius ρ. . . . . . . . . . . . . . . .

41

2.21 A 3-view depiction of the E-patch antenna conformed to the surface of
cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2.22 An screen capture of the cylindrically conformal cavity-backed E-patch
antenna in HFSS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

2.23 An closer view screen capture of the cylindrically conformal cavitybacked E-patch antenna in HFSS. . . . . . . . . . . . . . . . . . . . .

44

xii

2.24 Simulated frequency response of a cylindrically conformal cavity-backed
E-patch antenna (ρ = 100 cm). Dimensions provided in Table 2.11. .

45

2.25 Simulated frequency response of a cylindrically conformal cavity-backed
E-patch antenna(ρ = 70 cm). Dimensions provided in Table 2.12. . . .

46

2.26 Simulated frequency response of a cylindrically conformal cavity-backed
E-patch antenna (ρ = 50 cm). Dimensions provided in Table 2.13. . .

47

2.27 Simulated frequency response of a cylindrically conformal cavity-backed
E-patch antenna (ρ = 25 cm). Dimensions provided in Table 2.15. . .

48

2.28 Simulated frequency response of a cylindrically conformal cavity-backed
E-patch antenna (ρ = 15.4 cm). Dimensions provided in Table 2.16. .

49

2.29 Simulated frequency responses of cylindrically conformal and rectangular cavity-backed E-patch antennas. Dimensions provided in Tables
2.11 - 2.13, 2.15, 2.16 and 3.12. . . . . . . . . . . . . . . . . . . . . .

50

3.1

Three-dimensional solution surface of (3.3). . . . . . . . . . . . . . . .

55

3.2

Flowchart of Taguchi’s optimization method from [3]. . . . . . . . . .

56

3.3

X value of (3.3) versus iteration number in Taguchi’s Method. . . . .

62

3.4

Y value of (3.3) versus iteration number in Taguchi’s Method. . . . .

63

3.5

Value of (3.3) versus iteration number in Taguchi’s Method. . . . . .

64

3.6

Simulated |S11 | of an E-patch antenna versus iteration number in
Taguchi’s Method. Input parameter ranges are given in Table 3.9. . .

70

Simulated |S11 | of an E-patch antenna versus iteration number in
Taguchi’s Method. Input parameter ranges are given in Table 3.10. .

71

Simulated |S11 | of an E-patch antenna optimized in K-brick.. Geometry values are given in Table 3.11. . . . . . . . . . . . . . . . . . . .

73

Simulated |S11 | of an E-patch antenna optimized in K-brick. Geometry
values are given in Table 3.12. . . . . . . . . . . . . . . . . . . . . . .

74

3.10 Simulated |S11 | of an E-patch antenna optimized in HFSS. Geometry
values given in Table 3.12. . . . . . . . . . . . . . . . . . . . . . . . .

75

3.7
3.8
3.9

4.1

A screen capture of the rectangular cavity-backed E-patch antenna in
HFSS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiii

80

4.2

A front view of the rectangular cavity-backed E-patch experimental
antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

A side view of the rectangular cavity-backed E-patch experimental antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

A back view of the rectangular cavity-backed E-patch experimental
antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

A picture of the copper tape covering up the thickness of the cardboard
box top. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.6

A close up picture of the feed point of the experimental cavity. . . . .

85

4.7

Top and side view depiction of an experimental E-patch in a cavity. .

86

4.8

Coordinate system for the radiation patterns. . . . . . . . . . . . . .

88

4.9

A picture of the Uchida 3500 Super Hot-wire Foam-Cutter Crafting
Tool used to cut the cavity to the proper dimensions. . . . . . . . . .

88

4.10 |S11 | versus frequency measured on a Hewlett Packard 8753D network
analyzer for experimental rectangular cavity-backed E-patch antenna.
Dimensions provided in Table 4.1. . . . . . . . . . . . . . . . . . . .

89

4.11 |S11 | versus frequency of a simulated rectangular cavity-backed Epatch antenna. Dimensions provided in Table 4.1. . . . . . . . . . . .

90

4.3
4.4
4.5

4.12 |S11 | versus frequency for experimental and simulated rectangular cavitybacked E-patch antenna. Dimensions provided in Table 4.1. . . . . . 91
4.13 HFSS gain radiation pattern of the simulated cavity-backed E-patch
antenna, co-polarization. . . . . . . . . . . . . . . . . . . . . . . . . .

92

4.14 HFSS gain radiation pattern of the simulated cavity-backed E-patch
antenna, cross-polarization. . . . . . . . . . . . . . . . . . . . . . . .

93

4.15 Gain radiation pattern of the experimental rectangular cavity-backed
E-patch antenna, cross-polarization. . . . . . . . . . . . . . . . . . . .

94

4.16 Gain radiation pattern of the experimental rectangular cavity-backed
E-patch antenna, co-polarization. . . . . . . . . . . . . . . . . . . . .

95

4.17 Gain radiation patterns of the experimental and simulated rectangular
cavity-backed E-patch antenna, co-polarization. . . . . . . . . . . . .

96

xiv

4.18 Gain radiation patterns of the experimental and simulated rectangular
cavity-backed E-patch antenna, cross-polarization. . . . . . . . . . . .

97

4.19 A angled-view conceptual depiction of the E-patch antenna conformed
to the surface of a cylinder with a radius ρ. . . . . . . . . . . . . . . . 104
4.20 A 3-view depiction of the E-patch antenna conformed to the surface of
cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.21 A front view of the cylindrically conformal cavity-backed E-patch experimental antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.22 A top view of the cylindrically conformal cavity-backed E-patch experimental antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.23 A close up picture of the front cavity E-patch experimental antenna. . 108
4.24 A close up picture of the feed point of the experimental cylindrically
conformal cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.25 A closer up picture of the feed point of the experimental cylindrically
conformal cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.26 A picture of the cylinder cardboard tube used to construct the cylindrically conformal cavity-backed E-patch antenna. . . . . . . . . . . . 110
4.27 A screen capture of the cylindrically conformal cavity-backed E-patch
antenna in HFSS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.28 A closer view screen capture of the cylindrically conformal cavitybacked E-patch antenna in HFSS. . . . . . . . . . . . . . . . . . . . . 112
4.29 |S11 | versus frequency measured on a Hewlett Packard 8753D network analyzer for experimental cylindrically conformal cavity-backed
E-patch antenna. Dimensions provided in Table 4.1. . . . . . . . . . 113
4.30 |S11 | versus frequency of a simulated cylindrically conformal cavitybacked E-patch antenna. Dimensions provided in Table 4.3. . . . . . 114
4.31 |S11 | versus frequency for experimental and simulated cylindrically
conformal cavity-backed E-patch antennas. Dimensions provided in
Table 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.32 |S11 | versus frequency for experimental cylindrically conformal and
rectangular cavity-backed E-patch antennas. Dimensions provided in
Table 4.3 and Table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . 116

xv

4.33 |S11 | versus frequency for a cylindrically conformal cavity-backed Epatch antenna. Dimensions provided in Table 4.5. . . . . . . . . . . . 117
4.34 Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna, X-Y plane and co-polarized.
Dimensions provided in Table 4.3. . . . . . . . . . . . . . . . . . . . . 118
4.35 Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna, X-Y plane cross-polarized.
Dimensions provided in Table 4.3. . . . . . . . . . . . . . . . . . . . . 119
4.36 Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna, Z-X plane and co-polarized.
Dimensions provided in Table 4.3. . . . . . . . . . . . . . . . . . . . . 120
4.37 Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna, Z-X plane cross-polarized.
Dimensions provided in Table 4.3. . . . . . . . . . . . . . . . . . . . . 121
4.38 Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna, Z-Y plane and co-polarized.
Dimensions provided in Table 4.3. . . . . . . . . . . . . . . . . . . . . 122
4.39 Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna,Z-Y plane cross-polarized.
Dimensions provided in Table 4.3. . . . . . . . . . . . . . . . . . . . . 123
A.1 New Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.2 Circuit Unit box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.3 Circuit box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.4 Dielectric ﬁrst layer Dielectric selection . . . . . . . . . . . . . . . . . 133
A.5 Dielectric second layer parameter selection . . . . . . . . . . . . . . . 134
A.6 Completion of Dielectric selection . . . . . . . . . . . . . . . . . . . . 135
A.7 Analysis Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.8 Adding a metal rectangle . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.9 Rectangle dimension selection . . . . . . . . . . . . . . . . . . . . . . 138
A.10 Rectangle placement . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

xvi

A.11 Adding a Via . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
A.12 Via circle parameter selection . . . . . . . . . . . . . . . . . . . . . . 141
A.13 Via placement on a rectangle

. . . . . . . . . . . . . . . . . . . . . . 142

A.14 3-D view of a patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.15 Save as screen shot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.16 Viewed Response Box . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.17 Finished E-patch topology . . . . . . . . . . . . . . . . . . . . . . . . 146
F.1 |S11 | as a function of Taguchi iteration number for rectangular patches
optimizing at 1.5 and 1.2 GHz. . . . . . . . . . . . . . . . . . . . . . 209

xvii

Chapter 1
Introduction
1.1

E-patch Background

E-patch antennas were introduced in [1] as a novel way to increase the bandwidth∗
of conventional rectangular patch antennas. Unlike some wider bandwidth patch
antennas and their conﬁgurations† , the E-patch is a single uniform patch that is easily
constructed. Figure 1.1 depicts the topology of the antenna. The patch dimensions
are denoted by L, W, h and the antenna is normal to the patch surface fed by a coaxial
probe at position (W − Xf , Yf ). To increase the antenna bandwidth, two parallel
slots are incorporated and positioned in vertical symmetry with respect to the feed
point. Due to the patch resembling the letter “E”, the name E-patch antenna is used.
To maintain vertical symmetry about the top and bottom of the patch, Yf must be
1 L. The placement of the probe at X can aﬀect the properties of the antenna and
f
2
is an essential parameter; however, it is not the focus of discussion for this thesis.
The choice of feeding the antenna at (W − Xf , Yf ), was chosen primarily to excite
Mode 2 of the E-patch antenna (discussed by the authors in [1]). Mode 1 and Mode
2 excitation is indicated using the black dotted lines in Figure 1.2. The operation of
∗
†

Bandwidth is deﬁned as the frequency span between -10 dB values of |S11 |.
Some additional patch antennas are discussed by researchers in [4], [5] and [6].

1

Dielectric extends
past the patch to infinity.

Y

Ls

Ws
.

L

Xf

slot
Ps

h

coaxial
probe feed

Y
f
dielectric

X

W

Ground Plane

Top View

Side View

Figure 1.1: E-patch topology from [1].

2

the antenna using Mode 2 excitation is explained later. The slot length Ls , slot width
Ws , slot placement Ps , and dielectric height h are crucial to control the bandwidth of
the antenna. The slots of the E-patch antenna allow it to resonant at two frequencies,
and the bandwidth primarily is determined by the separation of those two frequencies.
Figure 1.3 is from [1] and shows two parallel circuits containing capacitors and
inductors. The parallel circuits of Figure 1.3 can be used to represent the transmission line resonant circuit model for E-patch antennas. The resonance frequency of a
parallel LC circuit is
f=

1
√
2π LC

(1.1)

and can explain the existence of the two resonance frequencies. Assuming mode two
excitation, currents ﬂow from the probe to the top and bottom edges of the patch.
Due to the slots, some currents travel a longer distance to reach the extreme edges
of the patch. This added distance that the current travels is modeled as a added
inductance ∆Ls in series with Ls . This added inductance forces the antenna to
resonate at a lower frequency. Current paths which aren’t detoured by the slots do
not have the additional inductance ∆Ls , and thus the antenna resonates at a higher
frequency because the inductance is lower in (1.1).
Controlling the inductance of the slots determined by Ps , Ws , and Ls speciﬁes
where the two resonant frequencies of the antenna will be. Refer to Figure 1.4 for a
sample simulated E-patch frequency response. The dimensions for corresponding Epatch are given in [1] and are in Table 1.1. Figure 1.4 was simulated with the Method
of Moments (MoM) simulator, Sonnet. The two resonance points of the antenna are
clearly visible at 2.3 GHz and 2.82 GHz.
Another parameter that aﬀects the frequency response of the E-patch antenna is
the dielectric height, h. The resonance frequency for a cavity-model of a rectangular

3

Probe
placement

Mode 1
(Fudamental mode)

slot

.

Ground plane
Probe
placement

slot

.

Mode 2

Ground Plane
Figure 1.2: Field conﬁgurations (modes) of a E-patch antenna.

4

L

∆ Ls

C
L

C
Figure 1.3: (a) An LC circuit representing the transmission line model for the lower
resonance frequency of an E-patch antenna. (b) An LC circuit representing the
transmission line model for the higher resonant frequency of an E-patch antenna.

5

patch antenna is

(fr )mnp =

1
√

2π µ

mπ 2
nπ 2
pπ 2
+
+
.
h
L
W

(1.2)

The values p, n, and m are the mode numbers for the cavity in the x, y, and z
directions respectively. The dielectric height of the antennas explored in this thesis
have insigniﬁcant contribution to equation (1.2) frequency of resonance since it is
suﬃciently less then a wavelength so that the dominant mode has m = 0; however, the
dielectric h has a signiﬁcant aﬀect on the bandwidth of the antenna. The authors in [7]
explored how the bandwidth of a rectangular patch can increase up to 20% by varying
the height of patch antenna lying on top of a dielectric slab. The dielectric material
used for exploration in this thesis is air. Air substrates have the advantage that
the dielectric height is simple to vary, and as such is a viable solution to increase the
bandwidth of E-patch antennas. In addition, the bandwidth of a low-order microstrip
when εreﬀ is given by (2.1). Hence, εr = 1 will yield
patch mode varies as √ε1
reﬀ
the largest bandwidth.
Table 1.1: Dimensions for an E-patch example given in [1].
E-patch Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h

Value in mm
70
45
7
35
35
4
9
11

Other papers, such as [8], [9], [10], and [11], manipulate the E-patch topology
to obtain better antenna properties such as bandwidth, radiation pattern and Sparameters. In particular, [11] gives empirical formulas for optimal (broadband op6

Figure 1.4: |S11 | vs frequency plot of a E-patch antenna found using Sonnet.
eration) E-patch dimensions when a inﬁnite ground plane is implemented. These
empirical formulas are

Ls = .39564λ0 , Ws = .026336λ0 , Ps = .051134λ0
h = .054557λ0 , Xf = .025488λ0, Yf = .25λ0 .

(1.3)

The symbol λ0 refers to the center wavelength. The center wavelength is deﬁned as
the wavelength in-between the two resonate wavelengths of the E-patch antenna. For
example, the center wavelength between 169.49 mm and 265.49 mm is 217.49 mm. In
[11] the authors reported that equations (1.3) yielded approximately a 40% increase
in bandwidth compared to a rectangular patch, an improvement of about 11% over
that given in [1]. For many E-patch antenna designs the equations (1.3) provide a
simple method for improving the frequency response of the antenna.
In [12] a Particle Swarm Optimization (PSO) was used to further increase the
7

bandwidth of the E-patch from 40% to 55%. In [9] the authors used capacitive
input probes to improve the input impedance of the antenna. The researchers in [13]
implemented a Genetic Algorithm (GA) to optimize the radiation pattern of the Epatch antenna. Reference [10] documents methods to reduce the size of the E-patch
while maintaing working functionality at the two frequencies of interest. None of the
authors in [1], [4], [5], [11], [14], [12], [9], [13], and [10] explored how the antenna would
behave on a cylindrical curved surface, nor did they address how a cavity-backed Epatch antenna would function. Airborne applications could beneﬁt from exploring the
functionality and feasibility of a cavity-backed and cylindrically-conformal E-patch.
A goal of this thesis is to present simulated and experimental data of the antenna
properties of a cavity-backed and cylindrically-conformal E-patch antenna.

1.2

Background on Taguchi’s Method of Optimization

Taguchi’s Method of optimization was developed by Dr. Genchi Taguchi as a way
of using statistics to design and improve quality in manufactured goods [15]. It is
a fractional factorial approach to optimization. Instead of exhausting all possible
combinations of parameters, a smaller number of the parameter combinations is used
to sample the entire exhaustive set. This fraction of possibilities achieves a comparable
outcome to the full factorial approach. In order to use Taguchi’s Method the concept
of Orthogonal Arrays (OAs) needs to be understood. OAs were introduced in the
1940s in a series of seminal papers [16], [17], and [18]. OAs provide a convenient
and orderly way to utilize the fractional factorial approach to optimization. An
Orthogonal Array is deﬁned in Deﬁnition 1 [19].
Deﬁnition 1. Let S be a set of s symbols or levels. A matrix A, commonly called an
array, of N rows and k columns with entries from S is said to be an orthogonal array
8

with s levels and strength t (0≤t≤k) if in every N x t subarray of A, each t-tuple
based on S appears exactly the same number of times as any other t-tuple.
OAs are typically represented by the notation OA(N, k, s, t). In Table 1.2 is an
example of an OA with nine rows N, four columns k, three levels s, and a strength t
of two. Each parameter in this OA is selected from three numbers, e.g. s = (0,1,2).
Thus Table 1.2 is a 3-level OA. For example, If one forms a sub-array of any two
(strength t = 2) columns, one can see 9 possible combinations of rows: (0,0), (0,1),
(0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2). Each of these nine combinations appears
exactly the same times as in a row. If Table 1.2 were used in a optimization problem,
the four columns would represent four parameters that need optimization. Each level
number (0,1,2) corresponds to a variation in one of the parameters that is being
optimized. Every row corresponds to a combination of varying level values that aﬀect
optimization. For example, the second row indicates that columns two and three take
on the value of level two; column one takes on the value of level one, and column four
takes on the value of level three. If one were to count all the possible combinations of
Table 1.2: An Orthogonal Array OA(9,4,3,2)
0
0
0
1
1
1
2
2
2

0
1
2
0
1
2
0
1
2

0
1
2
1
2
0
2
0
1

0
2
1
1
0
2
2
1
0

a problem with four parameters that had three diﬀerent possibilities each, it would be
34 or 81 diﬀerent combinations. Using Taguchi’s method and OAs, only nine diﬀerent
experiments are carried out. Due to this being a fraction of the total combinations,
Taguchi’s Method has much to oﬀer in terms of reducing the time and cost required
9

to run simulations and obtain results. Chapter 3 discusses Taguchi’s method and use
of OAs in more detail.
In [20] the authors discuss application of Taguchi’s Method to electromagenetics
(EM). The results and methods of [20] are expanded on in [3] and [21]. The researchers
in [20] and [21] address how to use Taguchi’s method for optimizing radiation patterns,
ﬁlter designs, and antenna factors in various EM problems. They do not discuss
explicitly the application to patch antennas. However, the researchers do provide
ample groundwork to create a Taguchi based optimizer for optimizing |S11 | of an
antenna. Another goal of this thesis is to investigate Taguchi’s Method and apply it
to optimizing the E-patch antenna in both cavity-backed and cylindrically-conformal
implementations.

10

Chapter 2
Conformal E-patch Antenna
2.1

Inﬁnite ground plane simulations

As discussed in Chapter 1, other authors have looked into improving the E-patch
antenna parameters by using a variety of diﬀerent methods. Most of these researchers
used inﬁnite ground plane simulations for the E-patch. In order to better understand
how the E-patch antenna can be simulated, it would prove useful to try to replicate
some results in [1] and use the formulas in [11] to create a E-patch.
The MoM simulator Sonnet was used as a comparison simulation tool for the
simulations appearing in [1]. It is important to note that Sonnet uses a inﬁnite
grounded slab for simulating antenna properties. Refer to Appendix A for step-bystep directions of how to use Sonnet for E-patch antenna simulations. One will note
that the cell size is 1 mm by 1 mm in the Sonnet setup for this work. When Sonnet
incorporates E-patch antenna parameters, the dimensions of the metallic patches are
thus rounded to the nearest mm. Figure 2.1 shows the Sonnet simulation of the
second E-patch in [1]. The dimensions for the antenna used to generate Figure 2.1
are given in Table 2.1. In [1] the authors found that an antenna constructed using
the values in Table 2.1 resonated in their simulation at 2.12 GHz and 2.66 GHz. It is

11

Figure 2.1: Sonnet simulation frequency response of the second E-patch in [1] (L = 70
mm, W = 45 mm). Dimensions provided in Table 2.1.

clear in Figure 2.1 that the results are slightly diﬀerent than in [1]. In Figure 2.1 the
resonant points are 2.30 GHz and 2.81 GHz; this is a diﬀerence of a bit less than 200
MHz from the results in [1]. Figure 2.2 shows the Sonnet simulation of the third Epatch in [1]. The dimensions for the antenna used to generate Figure 2.2 are in Table
2.2. In [1] the authors found that the antenna constructed using Table 2.2 values
resonated in their simulation at 1.8 GHz and 2.3 GHz. Figure 2.2 shows the Sonnet
simulated resonant points are approximately 2 GHz and 2.6 GHz, a diﬀerence again
of about 200-300 MHz. With a 200 MHz shift in frequency noted, one can extrapolate
how an E-patch antenna designed in Sonnet would performed compared to E-patches
simulated in [1]. The 200 MHz shift in frequency maybe due to a diﬀerent resolution
cell size between Sonnet and the MoM simulator in [1].
The L1 (1.57542 GHz) and L2 (1.2276 GHz) frequencies are used to communicate
12

Table 2.1: Geometry values for the second E-patch example given in [1].
E-patch Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h

Value in mm
70
45
7
35
35
4
9
11

Global Positioning System (GPS) signals. This thesis investigates the performance of
the E-patch antenna designed to operate at the frequencies at and between L1 and L2.
The authors in [1] primarily investigated the performance of an E-patch in the S-band
range of frequencies. An L-band E-patch antenna is designed starting from [1] with
the antenna dimensions modiﬁed so that the antenna resonates at L1 and L2. Refer
to Table 2.2 and Table 2.3. Table 2.3 values were generated by scaling according to
the ﬁrst resonant frequency taking Table 2.2 values and multiplying them by the ratio
1.8
1.227 ≈ 1.4. Since given the same substrate and electrical thickness, the resonant
frequencies are primarily determined by the relevant geometric dimensions. Figure
2.3 shows a Sonnet simulation of a designed L-band E-patch using the values of Table
2.3.
It is clear that |S11 | in Figure 2.3 is not the desired -10 dB or less at the L1 and
L2 frequencies. Instead of manipulating the dimensions of the antenna by trial and
error, one could use the empirical equations (1.3) in hopes of improving |S11 |. If one
uses 214 mm (1.4 GHz frequency) as λo , the E-patch dimensions become those in
Table 2.4. The L and W values were determined using [22]

εreﬀ =

h −1/2
εr + 1 εr − 1
+
1 + 12
2
2
W

13

(2.1)

Figure 2.2: Sonnet simulation frequency response of the third E-patch in [1] (L = 70
mm, W = 50 mm). Dimensions provided in Table 2.2.

Table 2.2: Geometry values for the third E-patch example given in [1].
E-patch Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h

14

Value in mm
70
50
6
35
40
6
10
15

Figure 2.3: Sonnet simulation frequency response of a ﬁrst designed L-band E-patch
(L = 98 mm, W = 75 mm). Dimensions provided in Table 2.3.

Table 2.3: Geometry values for the 40% enlarged E-patch dimensions.
E-patch Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h

15

Value in mm
98
75
8.4
49
56
8.4
14
21

W + 0.264
h
∆L = 0.412h
εreﬀ − 0.258 W + .8
h
εreﬀ + .3

W =
L=

νo
2fr

2
εr + 1

1

− 2∆L
√
2fr εreﬀ µo o
√

(2.2)

(2.3)

where r is the relative permittivity of the dielectric, µo is the permeability of free
space, o is permittivity of free space, h is the height of the dielectric, fr is the
center frequency between the resonance points of the antenna, and νo is the speed of
light in a vacuum. Equations (2.1) - (2.3) are the typical equations used to design a
patch antenna with a microstrip line feed. Figure 2.4 shows the Sonnet simulation
of the E-patch designed using equations (1.3) and (2.1) - (2.3). The dimensions for
the antenna used to generate Figure 2.4 are given in Table 2.4∗ . Figure 2.5 shows
the Sonnet simulation of the E-patch designed using equations 1.3 and the online
calculator found at [2]. The dimensions of the antenna used to generate Figure 2.5
are given in Table 2.5. The L and W values in Table 2.5 were chosen from [2]. The
probe placement for E-patch antennas is typically near the edge of the patch. To
provide a better input impedance match, the online calculator in [2] can estimate the
radiation edge impedance of a rectangular patch antenna. It was approximated that
a 50 Ω radiation edge impedance would yield a better input impedance match to a
E-patch antenna (assuming 50 Ω input impedance).
∗

In Sonnet the values from Table 2.4 were rounded to the nearest mm except h, which was
rounded down to the nearest mm.

16

Table 2.4: Geometry values for E-patch dimensions generated from equations (1.3)
and (2.1) - (2.3).
E-patch Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h

Value in mm
107.14
91.1731
5.4617
53.57
84.7800
5.6434
10.9573
11.6908

Figure 2.4: Sonnet simulation frequency response of the second designed L-band Epatch (L = 107.14 mm, W = 91.1731 mm). Dimensions provided in Table 2.4.

17

Figure 2.5: Sonnet simulation frequency response of the third designed L-band Epatch (L = 142 mm, W = 90 mm). Dimensions provided in Table 2.5.

Table 2.5: Geometry values for E-patch dimensions generated from equations (1.3)
and [2].
E-patch Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h

18

Value in mm
142
90
5
71
85
6
11
11

2.2

K-brick simulations

K-brick is a ﬁnite-element boundary-integral (FE-BI) program developed by Professor
Leo Kempel at Michigan State University. K-brick simulates antenna characteristics
(radiation pattern, |S11 |, TM modes, TE modes, etc.) for rectangular-cavity-backed
patch antennas. K-brick serves (in this thesis) as a comparison program to HFSS,
a general purpose ﬁnite element method simulation program (discussed in section
2.3). K-brick is a compiled Fortran program run within a Unix operating system.
Geometry and operating parameters are speciﬁed using an input text ﬁle (.txt) which
must be edited each time the program is executed. Appendix D contains a sample
input text ﬁle for K-brick and a start-up guide. Figure 2.6 is a depiction of a simulated
rectangular cavity-backed E-patch antenna in K-brick. Dimensions Cx, and Cy are
the length and width of the cavity and h is the depth of the cavity. Figure 2.7 shows
the top view of the cavity backed E-patch antenna with associated dimension labels† .
As discussed in [23], the procedure of formulating the FE-BI begins with a weak form
of the vector wave equation. Quoting from [23],
“The FE-BI formulation begins with the weak form of the vector wave
equation followed by speciﬁcation of appropriate vector shape functions
and dyadic Greens function. The resulting FE-BI equations are then used
to solve for the total electric ﬁelds within the cavity and on the aperture.”
In [23], the authors discussed only circular cavity-backed patch antennas. K-brick was
written after the publication; however, it is functionally similar to programs developed by Prof Jian-Ming Jin and others. K-brick adheres to the same principles used
and discussed in [23], namely the formulation of the FE-BI equations in rectangular
coordinates.
†

It should be noted that K-brick uses the center of the cavity when referencing the probe placement. In other words, the probe is placed irrespective of the cell size of the cavity. This is discussed
in Appendix D.

19

E-patch antenna flush
with top of the cavity

Grounded cavity

Cx

Cy
h
Coaxial probe feed
Figure 2.6: A typical rectangular cavity-backed E-patch antenna analyzed in K-brick.
Depicted cavity dimensions are not to scale and typically h
Cx , h
Cy .

20

y
PEC

dielectric

Ls

Ws

h

slot

Xf

L

.

Cy x

P
s

probe
feed

Yf

PEC

W
Side View

Cavity

Cx

Top View
Figure 2.7: Top and side view of an E-patch in a cavity.

21

As a starting example for simulating an E-patch antenna in a cavity, a simple
rectangular patch antenna is simulated in K-brick. Figure 2.8 shows the top view
topology of a rectangular patch with associated dimension labels backed by a rectangular. Table 2.6 contains the geometry values for the simulated rectangular cavitybacked patch. Figure 2.9 depicts the K-brick simulated frequency response of the
cavity-backed rectangular patch antenna in a cavity. The response of the antenna
show in Figure 2.9 is what is excepted of a single rectangular cavity-backed patch
antenna.
In section 2.1 a simulated E-patch with the lowest |S11 | at the L1 and L2 frequencies was found in a Sonnet Simulation (no Taguchi method was used); refer back
to Figure 2.5 and Table 2.5. Figure 2.10 shows the frequency response of a antenna
with Table 2.7 values simulated in K-brick. Comparing Figure 2.10 and Figure 2.5 the
simulated frequency response was not the same. Figure 2.10 shows that the simulated
|S11 | is not -10 dB and below for L1 or L2 frequencies. The large diﬀerence in the
|S11 | of the Sonnet simulation and K-brick simulation is due to the presence of the
cavity in the K-brick simulation.
Table 2.6: Dimensions of a cavity-backed rectangular patch antenna. Values were
generated from a Taguchi optimization.
Dimension
L
W
Xf
Yf
Cy
Cx
H

Value in mm
60
89
2.1
30
200
200
8.6

In the pursuit of a value of |S11 | of -10 dB or below at the L1 and L2 frequencies,
a Matlab graphical user interface (GUI) was written to allow the user to specify a
range of E-patch dimensions to vary. Matlab generates an input ﬁle (K-brick .txt
22

y

coaxial
probe feed
PEC

h

Xf

.

L

Cy x

dielectric

Yf

PEC

W
Side View

Cavity
Cx

Top View
Figure 2.8: Top and side view of a rectangular patch backed by a rectangular cavity.

23

Table 2.7: Dimensions of a cavity-backed E-patch antenna.
Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy

Value in mm
142
90
5
71
85
6
11
11
200
200

ﬁle) for each set of E-patch dimensions, executes K-brick, and places the results in
seven variables h, Ls , Ws , L, Xf , W , and Ps ; Yf is always placed along the vertical
symmetry plane of the antenna. At the end of all of the programs run, the desired
output parameters (typically return loss) are written to an output text ﬁle. Figure
2.11 shows a screen capture of the GUI. Any or all of these variables may be varied
by specifying starting and ending values, and a step size. K-brick is executed for all
combinations of the parameters that are varied. Appendix F has the Matlab code
associated with this program. After a series of parameter runs, a simulated E-patch
antenna frequency response was found that has a -10 dB for |S11 | at the L1 and L2
frequencies. The simulated frequency response of that E-patch is shown in Figure
2.12 and the antenna dimensions are provided in Table 2.8. However, this result is
still not optimal, as the |S11 | between the frequencies 1.227 and 1.575 raises above
-10 dB. In section 3.3 a Taguchi based optimizer is used in conjunction with K-brick
to further improve the simulated frequency response of the cavity-backed E-patch
antenna.

24

Figure 2.9: K-brick simulation frequency response of the cavity-backed rectangular
patch. Table 2.6 contains the dimension values.

Table 2.8: Dimensions of a cavity-backed E-patch antenna optimized using the GUI
Matlab K-brick interface.
Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy

Value in mm
101
88
6
50.5
75
5
9
13
200
200

25

Figure 2.10: K-brick simulation frequency response of the E-patch antenna with geometry values given in Table 2.7.

Figure 2.11: Matlab GUI interface for varying the geometry of an E-patch antenna.
For interpretation of the references to color in this and all other ﬁgures, the reader is
referred to the electronic version of this thesis.

26

Figure 2.12: K-brick simulation frequency response of an E-patch antenna with geometry values given in Table 2.8.

27

2.3

HFSS ﬂat bottom cavity simulations

To begin the study of conforming the E-patch antenna to a cylinder, it would prove
useful to approximate how the curvature varies according to diﬀerent cylinder radii.
Consider Figure 2.13, which shows a simple top down view of a cylinder with a triangle
inscribed inside of the circle. From Figure 2.13 the deﬂection d can be estimated.
Using Pythagoras’ theorem,
w 2
a2 + ( ) = (a + d)2 .
2

(2.4)

Then expanding out the squared terms gives
w2
= a2 + 2ad + d2 .
a2 +
4

(2.5)

Collecting terms and putting them on the left side of the equal sign gives
w2
= 0.
4

(2.6)

4a2 + w 2
.
2

d2 + 2ad −

(2.7)

Using the quadratic formula then gives

d = −a +

Simplifying and bringing an a outside the square root produces

d = −a + a 1 +

28

w2
.
4a2

(2.8)

W
W/2

d

a

a

Figure 2.13: Curvature diagram.

Now, assuming that
√
sion of 1 + x ≈ 1 +

w 2
1 and using the ﬁrst two terms in the binomial expan2a
x , (2.8) becomes,
2
d ≈ −a + a 1 +

Simplifying (2.9) then gives

1 w 2
.
2 2a

1 w2
d≈
.
8 a

(2.9)

(2.10)

Table 2.9 assumes that the cavity dimensions (Cx , Cy , and h) do not change and the
radius a is changing. From Table 2.9 one can note how the curvature of a rectangular
cavity-backed antenna depends on cylinder radius for a ﬂat bottom cavity.
29

Table 2.9: Curvature deﬂection from using (2.10).
Radius of Cylinder a in cm
2000
200
100
70
50
25

Deﬂection d in cm
.025
.25
.5
.714
1
2

High Frequency Structure Simulator (HFSS) was used to simulate the cylindrically
conformal cavity-backed E-patch antenna. HFSS provides E and H-ﬁelds, currents,
S-parameters and near and far radiated ﬁeld simulation results. Table 2.10 contains
the settings used in the HFSS simulations. Using the standard cylindrical coordinate
y
system where ρ = x2 + y 2 , φ = tan−1 x and z = z (x, y, and z are variables in
the rectangular coordinate system), one can develop a topology for the cylindrically
conformal cavity-backed E-patch antenna. Figure 2.14 shows a conceptual depiction
of a cavity-backed E-patch conformed to a cylinder. Figure 2.15 is a screen capture
of an cavity-backed E-patch conformed to a cylinder’s surface in HFSS. In order to
analyze how the curvature of the cylinder changes the antenna’s performance, as
compared to the planar cavity-backed case, the E-patch dimensions need to stay
consistent between both cases. Therefore, the width dimensions of the E-patch (W
and Ls ) will be speciﬁed using φ, at a distance ρ from the z axis. Further, the arc
length along the circumference of the cylinder is

g = ρφ.

(2.11)

The variable g in (2.11) is the E-patch width dimension W or Ls . The length dimensions of the E-patch (L, Ws , and Ps ) will be along the z axis. To keep the cavity
in the same conﬁguration as the rectangular cavity-backed case, the dielectric h will

30

be speciﬁed along ρ, Cx will be speciﬁed by a φ and Cy will be speciﬁed along the
z. The probe feed (Xf , Yf ) will be placed consistent with that of the E-patch. In
other words Xf will be an arc length adhering to (2.11) where g = Xf and Yf will
be speciﬁed along the z axis. In all HFSS simulations, the height of the cylinder is
ﬁxed at 100 cm and the cavity dimensions, Cx and Cy , are both 200 mm.
Figure 2.16 shows the simulated frequency response of the cylindrically conformal
(100 cm radius) cavity-backed E-patch antenna. The E-patch antenna in Figure 2.16
has the same dimension values as the rectangular cavity case in Table 3.12. At a
radius of 100 cm, d in (2.10) is 0.5 cm. Figure 2.17 shows the simulated frequency
response of the cylindrically conformal (70 cm radius) cavity-backed E-patch antenna.
The E-patch antenna in Figure 2.17 has the same dimension values as the rectangular
cavity case in Table 3.12. At a radius of 70 cm, d in (2.10) is .714 cm. In Figure
2.18 the simulated frequency response of the cylindrically conformal (50 cm radius)
cavity-backed E-patch antenna is shown. The E-patch antenna in Figure 2.18 has the
same dimension values as the rectangular cavity case in Table 3.12. At a radius of 50
cm, d in (2.10) is 1 cm.
Table 2.10: HFSS settings for the cylindrically conformal cavity-backed E-patch antenna.
HFSS setting
Frequency of solving
Adaptive passes
Maximum Delta S
Sweep Type
Frequency Set up
Initial mesh options
Solution Options
Excitation
Port Renormalization
Deembed setting

HFSS setting value(s)
1.6 GHz
7
.01
Fast
linear step 1 to 2 GHz step size .02
do lambda reﬁnement .3
First order
Waveport at the coaxial probe feed
50 Ω
checked -35 mm

Figure 2.19 contains all frequency response data in Figures 2.16-2.18 and Figure
31

z

z
y
x

x

y

y

z
x
Figure 2.14: A conceptual depiction of an cavity-backed E-patch conformed to a
cylinder. Depicted cavity and E-patch dimensions are not to scale.

32

Figure 2.15: HFSS screen capture of an cylindrically conformal cavity backed E-patch
antenna.

3.10. Using Figure 2.19 to compare frequency responses of diﬀerent E-patch antennas
on diﬀerent radii of cylinders, a trend is noted for the ﬂat bottom cavity. As the
radius ρ is increased, |S11 | decreases for the lower resonant frequency and increases
for the higher resonance frequency. In addition the higher resonance frequency has
shifted upwards in frequency. These diﬀerences in |S11 | versus frequency might be
due to the mode two resonance point being altered by curvature of the cylinder. With
the exception of the 50 cm radius, at the L1 and L2 frequencies |S11 | is less then -10
dB. For the 50 cm radius cylinder case, |S11 | at the L2 frequency is approximately
-9 dB, and is less than -10 dB at the L1 frequency.

2.4

HFSS cylindrically conformal cavity simulations

Refer to Figure 2.20 and Figure 2.21 for depictions of the E-patch antenna and associated cavity conformed to the surface of a cylinder. In [24] the authors discussed
conforming a rectangular patch antenna to the surface of a cylinder. The authors in
33

Table 2.11: Dimensions and probe placement of a cylindrically conformal cavitybacked E-patch antenna.
Dimensions
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy
ρ

Value in mm
96
83
37.5
48
65
7
13
15.6
200
200
1000

Figure 2.16: Simulated frequency response of an cylindrically conformal cavity-backed
E-patch antenna (ρ = 100 cm). Dimensions provided in Table 2.11.

34

Table 2.12: Dimensions and probe placement of a cylindrically conformal cavitybacked E-patch antenna.
Dimensions
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy
ρ

Value in mm
96
83
37.5
48
65
7
13
15.6
200
200
700

Figure 2.17: Simulated frequency response of an cylindrically conformal cavity-backed
E-patch antenna(ρ = 70 cm). Dimensions provided in Table 2.12.

35

Table 2.13: Dimensions and probe placement of a cylindrically conformal cavitybacked E-patch antenna.
Dimensions
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy
ρ

Value in mm
96
83
37.5
48
65
7
13
15.6
200
200
500

Figure 2.18: Simulated frequency response of an cylindrically conformal cavity-backed
E-patch antenna (ρ = 50 cm). Dimensions provided in Table 2.13.

36

Figure 2.19: Simulated frequency responses of cylindrically conformal and rectangular
cavity-backed E-patch antennas. Dimensions provided in Tables 2.11 - 2.13 and 3.12.

37

[24] did not discuss cavity-backed scenarios and assumed a ﬁnite dielectric beneath
the surface of the patch with a free space air dielectric surrounding the patch and
corresponding grounded cylinder. However, the authors in [24]did provide data for
insight into the eﬀect of curvature on patch antennas. The eﬀect of conforming a
patch antenna to a cylinder was to produce a broader radiation pattern and larger
bandwidth for the T M01 mode. As discussed in section 2.3, HFSS provides E and Hﬁelds, currents, S-parameters and near and far radiated ﬁeld simulation results. HFSS
is used to provide |S11 | for the cylindrically conformal cavity-backed E-patch antenna
for the conformed cavity case. Figures 2.22 and 2.23 show some screen captures of
the simulations from HFSS.
The HFSS settings for the simulations involving the cylindrically conformal cavitybacked E-patch antenna are listed in Table 2.14. All simulations were done in a driven
set-up.
Table 2.14: HFSS settings for the cylindrically conformal cavity-backed E-patch antenna.
HFSS setting
Frequency of solving
Adaptive passes
Maximum Delta S
Sweep Type
Frequency Set up
Initial mesh options
Solution Options
Excitation
Port Renormalization
Deembed setting

HFSS setting value(s)
1.6 GHz
7
.01
Fast
linear step 1 to 2 GHz step size .02
do lambda reﬁnement .3
First order
Waveport at the coaxial probe feed
50 Ω
checked -35 mm

Figure 2.24 shows the simulated frequency response of the cylindrically conformal
(100 cm radius) cavity-backed E-patch antenna. The E-patch antenna in Figure 2.24
has the same dimension values as the rectangular cavity case in Table 3.12. Figure
2.25 shows the simulated frequency response of the cylindrically conformal (70 cm
38

radius) cavity-backed E-patch antenna. The E-patch antenna in Figure 2.25 has
the same dimension values as the rectangular cavity case in Table 3.12. In Figure
2.26 the simulated frequency response of the cylindrically conformal (50 cm radius)
cavity-backed E-patch antenna is shown. The E-patch antenna in Figure 2.26 has
the same dimension values as the rectangular cavity case in Table 3.12. In Figure
2.27 the simulated frequency response of the cylindrically conformal (25 cm radius)
cavity-backed E-patch antenna is shown. The E-patch antenna in Figure 2.27 has
the same dimension values as the rectangular cavity case in Table 3.12. In Figure
2.28 the simulated frequency response of the cylindrically conformal (15.4 cm radius)
cavity-backed E-patch antenna is shown. The E-patch antenna in Figure 2.28 has the
same dimension values as the rectangular cavity case in Table 3.12.
Table 2.15: Dimensions and probe placement of a cylindrically conformal cavitybacked E-patch antenna.
Dimensions
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy
ρ

Value in mm
96
83
37.5
48
65
7
13
15.6
200
200
250

Figure 2.29 contains all frequency response data in Figures 2.24-2.28, and 4.11.
Using Figure 2.29 to compare frequency responses of E-patch antennas on diﬀerent
radii of cylinders, a trend is noted. As the radius ρ is decreased, |S11 | stays approximately the same for the lower resonant frequency and decreases for the higher
resonance frequency. In addition the higher resonance frequency has shifted lower in
39

Table 2.16: Dimensions and probe placement of a cylindrically conformal cavitybacked E-patch antenna.
Dimensions
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy
ρ

Value in mm
96
83
37.5
48
65
7
13
15.6
200
200
154

frequency as the radius decreases. Also, bandwidth enlarges as the radius decreases.
Notably, the combined ﬁrst and second resonance bandwidth for the 100 cm case is
245 MHz (-10 dB or lower at 1245 MHz to 1330 MHz and 1590 MHz to 1750 MHz)
while the bandwidth for the 15.4 cm case is 320 MHz (-10 dB or lower at 1250 MHz
to 1340 MHz and 1520 MHz to 1750 MHz). At 50, 25, and 15.4 cm radius, the L2
frequencies |S11 | is less then -10 dB. The L1 frequency stays approximately -9 dB for
all the cases. These diﬀerences in |S11 | versus frequency might be due to the mode
two resonance point being altered by curvature of the cylinder.

40

Coaxial Probe
Xf

x
Cy
Yf

PEC

h

Cx
ρ

θ

z

f

ф
y

Figure 2.20: An angled-view conceptual depiction of the E-patch antenna conformed
to the surface of a cylinder with a radius ρ.

41

x

z
x
y
y

Z-axis into page
x

z
y
Figure 2.21: A 3-view depiction of the E-patch antenna conformed to the surface of
cylinder.

42

Figure 2.22: An screen capture of the cylindrically conformal cavity-backed E-patch
antenna in HFSS.

43

Figure 2.23: An closer view screen capture of the cylindrically conformal cavitybacked E-patch antenna in HFSS.

44

Figure 2.24: Simulated frequency response of a cylindrically conformal cavity-backed
E-patch antenna (ρ = 100 cm). Dimensions provided in Table 2.11.

45

Figure 2.25: Simulated frequency response of a cylindrically conformal cavity-backed
E-patch antenna(ρ = 70 cm). Dimensions provided in Table 2.12.

46

Figure 2.26: Simulated frequency response of a cylindrically conformal cavity-backed
E-patch antenna (ρ = 50 cm). Dimensions provided in Table 2.13.

47

Figure 2.27: Simulated frequency response of a cylindrically conformal cavity-backed
E-patch antenna (ρ = 25 cm). Dimensions provided in Table 2.15.

48

Figure 2.28: Simulated frequency response of a cylindrically conformal cavity-backed
E-patch antenna (ρ = 15.4 cm). Dimensions provided in Table 2.16.

49

Figure 2.29: Simulated frequency responses of cylindrically conformal and rectangular
cavity-backed E-patch antennas. Dimensions provided in Tables 2.11 - 2.13, 2.15, 2.16
and 3.12.

50

Chapter 3
Taguchi’s Method of Optimization for
E-patches
3.1

Orthangonal Arrays

As discussed in section 1.2, Taguchi’s method involves using OAs to perform optimization. When using OAs for optimization problems two questions arise: Does an
OA exist, and if so how can it be constructed? To determine the existence of an OA
one must, for given values of k, s, and t, determine the minimum number of rows N
so that an OA(N, k, s, t) exists. One would like to minimize N, as it reduces the
number of experiments carried out. Rao’s inequalities are established as a mathematical answer to the existence question. To minimize N, the parameters of OA(N, k, s,
t) must satisfy the following Rao inequalities [14],
u

k
(s − 1)i , if t = 2u, u > 0
i

(3.1)

k
k−1
(s − 1)i +
(s − 1)u+1 , if t = 2u + 1, u ≥ 0.
u
i

(3.2)

N≥
u
N≥

i=0

i=0

51

More improvements on the Rao’s inequalities can be found in [14] for diﬀerent t
strengths and s levels. Numerous techniques to construct OAs are discussed in [14]
and some of these techniques are based on Galois ﬁelds and error-correcting codes.
Galois ﬁelds are ﬁnite sets of elements where one can add, subtract, multiply and
divide within the set. The virtue of the Galois ﬁelds is that it is possible to multiply
and divide vectors with a speciﬁed number of components. Error-correcting codes
are designed to correct errors in the transmission of data over noisy communication
channels. These error-correcting codes can also be used to construct OAs. One may
consult [14] for more information on error-correcting codes, Galois ﬁelds, and their
use in construction of OAs.
An online database, [25], of OAs has been established by the authors in [14]. OAs
with diﬀerent parameters, levels, and strengths have been developed and archived in
this online database, which can also be found in books related to OAs or Taguchi’s
Method. The online database contains the OAs used in this thesis. The authors in
[20] discuss explicitly how to use Matlab code to produce OAs that adhere to the
Rao inequalities. However, the authors in [20] only discuss constructing OAs with an
arbitrary odd-level (s = 3, 5, 7, ...) and two-strength (t = 2).
OAs have many important properties that are associated with them. The ability
of OAs to lessen the number of experiments in optimization problems is discussed
in Section 1.2. Another property of OAs is that all possible combinations of the
parameters of up to the strength t occur equally; this ensures a balanced and fair
comparison of all interactions of levels (s). In Table 3.1 if one picks any level value
(s = (0, 1, 2)) in a column, that value occurs exactly the same number of times as the
other level values. For example, the level values zero, one, and two all occur six times
in a column. This means that all possible level values are tested fairly within the OA.
Further, if one looks at a combination of any two columns (strength t = 2) as a row,
one sees exactly the same number of combinations. For example, two combinations

52

of (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2) can be noted as a row in any
two columns. To consider more interactions of parameters, one could increase the
strength t, but then more rows (experiments) would need to be included to maintain
the properties of the OA. The OAs in this thesis use a strength of two, as this seems
suﬃcient for the problems considered.
Another practical property of OAs is discussed by the authors in [20] and is quoted
here:
“...

property of OAs is that any N × k subarray of an existing

OA(N, k , s, t ) is still an OA with a notation OA(N, k , s, t ), where t =
min(k , t). In other words, if one or more columns are deleted from an
OA, the resulting array is still an OA but with a smaller number of parameters. For example, if we delete the last two columns in [Table 3.1],
we can obtain an [OA(18, 5, 3, 2)]. This property is especially useful when
selecting an OA from an existing OA database. If an OA with a certain
number of columns (k ) cannon be found in a database, one can choose
an OA with a larger number of columns (k > k ) and manually delete the
redundant (k − k ) columns to obtain the desired OA."
This property inﬂuences the choice of an OA for Taguchi optimization and is used
later in section 3.2.

3.2

Taguchi’s method and optimization example

Taguchi’s method can be understood using an example optimization problem. In [26]
the authors discuss ﬁnding the global minimum of the function

f (x, y) = x sin(4x) + 1.1y sin(2y), 0 ≤ x ≤ 10, 0 ≤ y ≤ 10.

53

(3.3)

Table 3.1: An Orthogonal Array OA(18,7,3,2)
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2

0
1
2
0
1
2
1
2
0
2
0
1
1
2
0
2
0
1

0
1
2
1
2
0
0
1
2
2
0
1
2
0
1
1
2
0

0
1
2
2
0
1
2
0
1
0
1
2
1
2
0
1
2
0

0
1
2
1
2
0
2
0
1
1
2
0
0
1
2
2
0
1

0
1
2
2
0
1
1
2
0
1
2
0
2
0
1
0
1
2

0
0
0
0
0
0
1
1
1
1
1
1
2
2
2
2
2
2

This function has a global minimum of -18.5547 located at x = 9.0390, y = 8.6682.
The diﬃculty in ﬁnding a global minimum for this function is that there are many
local minima close to the global minimum. Figure 3.1 is a plot for the solution
surface of (3.3). To ﬁnd the global minimum, one may establish a procedure of
operation for Taguchi’s method. Consider Figure 3.2, which shows a ﬂow chart for
Taguchi’s Method of optimization from [3]. As shown there, selecting an OA for the
optimization problem is the ﬁrst step. To select the proper OA, ﬁrst determine the
number of variables that needs to be optimized. For the optimization of (3.3), two
variables are needed because x and y are both varying; thus k = 2. A strength of
t = 2 is chosen by default because there are only two variables in the optimization. A
suitable OA can be constructed from Table 3.2 (from the online database [25]) with
the last two columns deleted. As discussed in Section 3.1 the deletion of columns
is an acceptable practice as it does not invalidate the properties of an OA. The OA

54

Figure 3.1: Three-dimensional solution surface of (3.3).

constructed from Table 3.2 becomes the OA in Table 3.3.
The next step is to select a ﬁtness function. A ﬁtness function is selected based
on its ability to reach the optimization goal. The selection of the ﬁtness function may
be the most important factor in Taguchi’s method. If one chooses a wrong ﬁtness
function, then an optimal solution may not be reached. The ﬁtness function for (3.3)
should take on its lowest value when (3.3) is at the global minimum. The designed
ﬁtness function for (3.3) is,

F itness = exp (x sin(4x) + 1.1y sin(2y)).

(3.4)

The role of the exponential function in the ﬁtness of (3.4) will be explained later in
this section. All one needs to know at this point is that a smaller ﬁtness means a
result closer to the optimum.
A range of numbers needs to be selected for the variables (input parameters) in

55

Problem Initialization:
Select OA
Determine ﬁtness function

c
E

Design input parameters using
OA

c

Conduct experiments and build a
response table

Reduce the
optimization
range

c

T

Identify optimal level values and
conduct conﬁrmation experiment

c
 d
  d
 
d
d
No  
Termination d
d criteria met?  
d
 
d
 
d  
d  Yes
' c

$

&

%

End

Figure 3.2: Flowchart of Taguchi’s optimization method from [3].

56

the ﬁtness function. When an OA is used in Taguchi’s Method, the level values
correspond to numbers in the input parameter range. The input range of (3.3) is
between 0 and 10 for both x and y variables. To determine which input numbers to
be used as level values in each iteration, a level diﬀerence (LD) needs to be calculated.
The level diﬀerence is the amount of separation between level 2 and level 1 (also level
2 and level 3). The equation,

LD 1 =

max-min
10 − 0
=
= 2.5
number of levels + 1
3+1

(3.5)

is used to calculate the ﬁrst LD of the x variable of (3.3). Max and min are the
maximum and minimum values of the input range, in this case 10 and 0 respectively.
If the input range was instead 20 and -10 for the maximum and minimum values,
then the value of LD would be 7.5. As Taguchi’s Method is implemented, the LD
changes depending on the iteration (discussed later in this section). For the ﬁrst
iteration of (3.3) optimization, the middle of the input range was selected and thus
level 2 becomes 5. Table 3.4 is an OA with input values corresponding to the level
values of the ﬁrst iteration of (3.3) optimization. Level 1 corresponds to 2.5 (5 − 2.5)
and Level 3 becomes 7.5 (5 + 2.5).
Now that the OA level values have input parameters associated with them, experiments can be conducted and a response table can be built. Conducting experiments
means computing the ﬁtness function with the level values for a given row in the OA.
For example, the ﬁrst experiment outcome is,

F itness = exp (2.5 sin(4 ∗ 2.5) + 1.1 ∗ 2.5 sin(2 ∗ 2.5)) = .0184.

(3.6)

The seventh experiment outcome is,

F itness = exp (7.5 sin(4 ∗ 7.5) + 1.1 ∗ 2.5 sin(2 ∗ 2.5)) = .000043303.
57

(3.7)

Table 3.5 shows the OA with the associated experiment outcome (ﬁtness). Once
all experiments have been conducted and ﬁtnesses determined, the ﬁtness values are
converted to a signal-to-noise (S/N) ratio η. The S/N is determined by the following,

η = −20 log10 (Fitness).

(3.8)

As (3.4) decreases, (3.8) results in a large S/N ratio. The role of the exponential
function in (3.4) now becomes evident. If (3.3) were used as the ﬁtness function,
then the exponential function compensates for the logarithm in the S/N making the
S/N proportional to the function value. The scaling is such that less than unity gives
positive S/N values. A ﬁtness function of

F itness = 20 + (x sin(4x) + 1.1y sin(2y)).

(3.9)

could also be used as a ﬁtness function for this optimization. However, it relies on
knowing where the minimum point already is. The ﬁtness function (3.9) yields the
same results presented in this section. The S/N ratio for the ﬁrst iteration (ﬁtness
function (3.3)) is included in Table 3.5.
Table 3.2: An Orthogonal Array OA(9,4,3,2)
0
0
0
1
1
1
2
2
2

0
1
2
0
1
2
0
1
2

0
1
2
1
2
0
2
0
1

0
2
1
1
0
2
2
1
0

When the S/N ratios are obtained, then a response table is built. A response

58

Table 3.3: An Orthogonal Array OA(9,2,3,2)
0
0
0
1
1
1
2
2
2

0
1
2
0
1
2
0
1
2

table is populated with the averaged S/N ratios for each parameter n and each level
m using the following,
η (m, n) =
¯

s
N

ηi .

(3.10)

Here N is the experiment (row) number, s is the total level value (3 in this case) and
i is the iteration number. For example, the average S/N ratio for the 2nd column (y),
1st iteration, 3rd level is:

η (3, 2) =
¯

3
9

1
η1 = [(−34.7854) + (−86.2474) + (17.7658)] = −34.4224(dB).
3
(3.11)

The rest of the average S/N ratios are found in the response Table 3.6. After the
response table has been built, the largest S/N ratio for each parameter is identiﬁed.
Italicized in Table 3.6 are the largest S/N ratios. A conﬁrmation experiment is performed using the combination of the optimal levels identiﬁed in the response table,
in this case, level 3 for column 1 (x = 7.5) and level 2 for column 2 (y = 5.0). As
mentioned in Section 1.2, Taguchi’s method is a fractional factorial approach. Therefore, the optimal combination may not be included in Table 3.3. The conﬁrmation
experiment is therefore not a redundant experiment and the ﬁtness value obtained
from the conﬁrmation experiment is regarded as the ﬁtness values of the current i
iteration. If the results of the current iteration do not meet the optimization criteria
59

Table 3.4: The Orthogonal Array OA(9,2,3,2) and level values in the ﬁrst iteration
of Equation (3.3) optimization.
2.5
2.5
2.5
5.0
5.0
5.0
7.5
7.5
7.5

2.5
5.0
7.5
2.5
5.0
7.5
2.5
5.0
7.5

or termination criteria (discussed later), then the process is repeated in the next iteration. The optimal level values of the current iteration are used as level 2 values
in the next iteration. For example, after the ﬁrst iteration of (3.3) optimization, 7.5
would be the new level 2 value for the 1st element (x variable) and 5.0 would be the
level 2 value for the 2nd element (y variable). To reduce the optimization range for a
converged result, the LD is multiplied by a reduction rate to obtain a new LD for the
new iteration. An example reduction rate is rr i , where i is the iteration. The larger
rr is, the slower the convergence rate. Typically, rr is chosen to be between .9 and
.5. For (3.3) optimization, the rr is chosen to be .9. After the ﬁrst iteration in (3.3)
optimization process, LD 2 would be
LD 2 = LD 1 ∗ rri = 2.5 ∗ (.9)1 = 2.25.

(3.12)

If LD i is large enough, and the level 2 of the next generation is close to the upper or
lower input parameter bounds, then a danger exists that the corresponding level 1 or
level 3 values may lay outside the input parameter bounds. A simple solution is to
use an if-then statement at the start of the iteration. If the level value is below the
minimum input parameter, then that level is set to the minimal value of the input
parameter. Similarly, if the level value is above the maximum input parameter, then
60

that level is set to the maximum value of the input parameter.
Table 3.5: The Orthogonal Array OA(9,2,3,2), level values, S/N ratio, and experiment
outcome in the ﬁrst iteration of Equation (3.3) optimization.
Experiment (row)
1
2
3
4
5
6
7
8
9

Elements
2.5
2.5
2.5
5.0
5.0
5.0
7.5
7.5
7.5

(columns)
2.5
5.0
7.5
2.5
5.0
7.5
2.5
5.0
7.5

Fitness
S/N ratio
.0184
34.7183
.0129
37.8025
54.8621
-34.7854
6.8736
-16.7437
4.8192
-13.6595
20529
-86.2474
.000043303 87.2696
.000030361 90.3537
.1293
17.7658

Table 3.6: Response table for the optimization of (3.3).
Level (row)
1
2
3

Averaged S/N ratio (columns)
12.5784
35.0814
-38.8835
38.1655
65.1297
-34.4224

A termination criteria in the optimization procedure may be established by using
the following condition
LD i
< converged value.
LD 1

(3.13)

Normally the converged value is set between 0.0001 and 0.01 depending on the problem. The iterative optimization process is stopped if the optimization criterion is met
or if (3.13) is satisﬁed.
Appendix C shows the Matlab code used in the optimization of (3.3) and can be
used as a guide for coding other optimization problems. Figure 3.5 shows the value
of (3.3) versus the iteration number of Taguchi’s method. Figure 3.3 and Figure
3.4 show the x and y variables versus the iteration number of Taguchi’s method.
The optimization goal (global minimum of (3.3)) was not met. The global minimum
61

Figure 3.3: X value of (3.3) versus iteration number in Taguchi’s Method.

of -18.5547 located at x = 9.0390, y = 8.6682 was not found after 68 iterations of
Taguchi’s method. The reason for not ﬁnding the global minimum is that LD 1 wasn’t
large enough to contain for the necessary x and y variables. If LD 1 is enlarged by
50% then the input optimization range is wider and contains the location of the global
minimum. This global minimum is found after 71 iterations.
Techniques of decreasing the number of iterations to more quickly achieve the optimization result, such as increasing the level number, changing boundary treatments,
and using a Gaussian reduce function, are discussed in [26]. The Rosenbrock function
was also optimized using Taguchi’s method. It is included in Appedix .

62

Figure 3.4: Y value of (3.3) versus iteration number in Taguchi’s Method.

63

Figure 3.5: Value of (3.3) versus iteration number in Taguchi’s Method.

64

3.3

Taguchi-K-brick-Matlab results

In section 3.2 a Taguchi based optimizer was described that locates a global minimum
in a two variable function. Techniques discussed in section 3.1 and 3.2 provide introductory information (OA selection, ﬁtness selection etc.) needed to create a Taguchi
based optimizer for |S11 | of a cavity-backed E-patch antenna.
Following the procedure shown in Figure 3.2 for developing a Taguchi optimizer,
one of the ﬁrst steps is determining a suitable OA. First, consider the number of
parameters of a cavity-backed E-patch antenna to be optimized. For simplicity, the
cavity’s width Cx (x-direction) and length Cy (y-direction) are ﬁxed at 20 cm x 20
cm. This is done so that the cell size in K-brick is 1mm x 1mm x h (and thus Kbrick’s calculation time is lessened considerably). The probe placement Yf (in the
y-direction) is always at the vertical symmetric middle of antenna (since the desired
mode is symmetric about the y-direction of the probe placement); this eliminates the
need for Yf to be a parameter of optimization. The rest of the E-patch dimensions,
h, Ls , Ws , L, Xf ∗ , W , and Ps are the same dimensions as shown in Figure 2.7.
The restriction of the dimensions of the cavity to 20 cm x 20 cm, and Yf to the
vertical symmetric middle of antenna, yields seven diﬀerent dimensional parameters
that aﬀect the geometry of the E-patch, and thus the value of |S11 |.
An OA of seven columns is needed to include the eﬀects of the seven parameters
that determine |S11 |. An OA (27,13,3,2) (selected from the online database [25]),
modiﬁed by deleting the last six columns, yields a suitable OA. As discussed in section
3.1, the deletion of columns or variables in an OA is an acceptable practice. Table 3.7
is the OA (27,13,3,2) and Table 3.8 is the OA (27,7,3,2). An OA (27,7,3,2) ﬁts all the
criteria for this optimization in that it contains seven columns or variables, each of
the variables has a middle, higher, and lower value, (three levels) and it satisﬁes the
∗
Probe placement Xf is diﬀerent then the previous E-patch designs in that that the probe
placement is given from the center of the cavity instead of placed at (W − Xf ).

65

equations 3.1 and 3.2. As with all the Taguchi optimizations in this thesis, a strength
of two is suﬃcient for the problems considered.
Table 3.7: An Orthogonal Array OA(27,13,3,2)
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2

0
0
0
1
1
1
2
2
2
0
0
0
1
1
1
2
2
2
0
0
0
1
1
1
2
2
2

0
1
2
0
1
2
0
1
2
1
2
0
1
2
0
1
2
0
2
0
1
2
0
1
2
0
1

0
1
2
1
2
0
2
0
1
0
1
2
1
2
0
2
0
1
0
1
2
1
2
0
2
0
1

0
1
2
1
2
0
2
0
1
1
2
0
2
0
1
0
1
2
2
0
1
0
1
2
1
2
0

0
2
1
1
0
2
2
1
0
1
0
2
2
1
0
0
2
1
2
1
0
0
2
1
1
0
2

0
2
1
2
1
0
1
0
2
1
0
2
0
2
1
2
1
0
2
1
0
1
0
2
0
2
1

0
0
0
2
2
2
1
1
1
2
2
2
1
1
1
0
0
0
1
1
1
0
0
0
2
2
2

0
1
2
0
1
2
0
1
2
2
0
1
2
0
1
2
0
1
1
2
0
1
2
0
1
2
0

0
2
1
1
0
2
2
1
0
0
2
1
1
0
2
2
1
0
0
2
1
1
0
2
2
1
0

0
1
2
2
0
1
1
2
0
1
2
0
0
1
2
2
0
1
2
0
1
1
2
0
0
1
2

0
0
0
1
1
1
2
2
2
2
2
2
0
0
0
1
1
1
1
1
1
2
2
2
0
0
0

0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2

The next step in designing a Taguchi based K-brick optimizer is the selection of
a proper ﬁtness function. Consider the equation

F itness = |(db1 + 40)| + |(db2 + 40)|.

(3.14)

The variables db1 and db2 are the simulated |S11 | in dB produced by K-brick at the
66

Table 3.8: An Orthogonal Array OA(7,13,3,2)
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2

0
0
0
1
1
1
2
2
2
0
0
0
1
1
1
2
2
2
0
0
0
1
1
1
2
2
2

0
1
2
0
1
2
0
1
2
1
2
0
1
2
0
1
2
0
2
0
1
2
0
1
2
0
1

0
1
2
1
2
0
2
0
1
0
1
2
1
2
0
2
0
1
0
1
2
1
2
0
2
0
1

67

0
1
2
1
2
0
2
0
1
1
2
0
2
0
1
0
1
2
2
0
1
0
1
2
1
2
0

0
2
1
1
0
2
2
1
0
1
0
2
2
1
0
0
2
1
2
1
0
0
2
1
1
0
2

0
2
1
2
1
0
1
0
2
1
0
2
0
2
1
2
1
0
2
1
0
1
0
2
0
2
1

L1 and L2 frequencies, respectively. For optimization one would want |S11 | to be a
minimum at the L1 and L2 frequencies. The absolute value ensures equal weight in
the ﬁtness calculation. For example, if db1 was -50 and db2 was -3, this would not
be optimal. Indeed, the ﬁtness would be 47, whereas a db1 and db2 value of both -40
would yield 0. Obviously, the antenna is optimized when the ﬁtness function is 0.
The value of 40 was chosen as a suﬃcient number to add to an optimal dB response
of |S11 | at the L1 and L2 frequencies. Another ﬁtness function considered was
F itness = |(db1 + 20)| + |(db2 + 20)|.

(3.15)

Diﬀerent but similar ﬁtness functions yield diﬀerent results as discussed later in this
section.
In Table 3.9 is the input parameter selection of seven variables used in the optimization of (3.14). In Table 3.10 is the input parameter selection of the seven
variables used in the optimization of (3.15). With the OA, ﬁtness function, and input
Table 3.9: The input parameter ranges for E-Patch Taguchi optimization. Fitness
function is (3.14).
E-patch dimension variable Min Value in mm Max Value in mm
W
60
90
L
85
110
Ls
40
89
Ws
1
10
Ps
5
25
h
8
20
Xf
0
70

parameters chosen, the Taguchi optimization can proceed. However, special attention
needs to be given to undesirable E-patch geometries that may occur during optimization. For example, if Ls was 75 mm and W was 65 mm then the slots would cut
through the patch and would yield three separate patch antennas. To exclude these
68

Table 3.10: The input parameter ranges for E-patch Taguchi optimization. Fitness
function is (3.15).
E-patch dimension variable Min Value in mm Max Value in mm
W
60
90
L
85
110
Ls
40
89
Ws
1
7
Ps
8
15
h
8
16
Xf
0
70

geometries a simple if-then statement is used in the Matlab code. Another parameter
to monitor is the Xf placement. If the probe is placed oﬀ the antenna, then the
impedance result would not be practically useful. Once again during optimization, a
simple if-then statement is used to check this condition.
The reduction rate (rr) used for all cases was .9. Note, however, that this value
of reduction rate poses a problem with K-brick input. With the cell size restricted
to 1 mm by 1 mm by h, only integer mm multiples of the dimensions parameters
will be accepted into K-brick (otherwise K-brick crashes). To make sure K-brick will
not crash, after the reduction rate has been applied to the current iteration of input
parameter values, the E-patch dimensions are rounded to the nearest mm† . Refer to
Appendix F for the Matlab code used in the K-brick Taguchi optimization. Due to the
rounding of E-patch parameters during Taguchi optimization, there is potential for
wasting computation time after the iteration number has grown signiﬁcantly. Wasting
of computation time by the rounding of input parameters is not fully addressed in
this thesis and remains a future improvement to the Matlab code.
The |S11 | at the L1 and L2 frequencies versus the numbered iteration in Taguchi’s
method determing using the ﬁtness function (3.14) is shown in Figure 3.6. Figure 3.7
†

E-patch geometry values h and Xf can be integer values as they are independent of the cell
size of the cavity and therefore not rounded in the matlab code.

69

Figure 3.6: Simulated |S11 | of an E-patch antenna versus iteration number in
Taguchi’s Method. Input parameter ranges are given in Table 3.9.

shows |S11 | at the L1 and L2 frequencies versus the iteration number in Taguchi’s
method with a ﬁtness function of (3.15). Comparing Figure 3.6 and Figure 3.7, it is
seen that |S11 | is diﬀerent at all iteration numbers. The diﬀerence in |S11 | is primarily
because of the diﬀerent ﬁtness functions used. Comparing Figure 3.6 and Figure 3.7,
|S11 | seems to be greatly diﬀerent between L1 and L2 when the ﬁtness function (3.14)
is used. This may imply that |S11 | results from K-brick between 0 and -20 dB are
more frequent than between 0 and -40 dB. In the ﬁtness function optimizations using
(3.15) and (3.14) the input range of parameters were slightly diﬀerent, and could have
also aﬀected the values of |S11 | for a simulated E-patch in K-brick. However, there is
no conclusive evidence that the diﬀerent input parameter ranges are a major factor
in the diﬀerent results shown in Figures 3.6 and 3.7. Another factor that may have
eﬀected both optimizations was the if-then statements used to correct for undesirable
E-patch geometries. These if-then statements, however, executed under the same

70

Figure 3.7: Simulated |S11 | of an E-patch antenna versus iteration number in
Taguchi’s Method. Input parameter ranges are given in Table 3.10.

conditions for both optimizations possibly unaﬀecting the end optimization value.
The smallest value of |S11 | in Figures 3.6 and 3.7 (iterations 26 and 21 respectively) have E-patch dimensions given in Tables 3.11 and 3.12 respectively. Figures 3.8
and 3.9 show |S11 | versus frequency for E-patch antennas optimized in the TaguchiK-brick program. In both Figures 3.8 and 3.9, a good result of -10 dB and below for
the L1 and L2 frequencies is noted.
To provide a comparison simulation, HFSS was used to simulate the antenna with
geometry values given in Table 3.12. Figure 3.10 shows |S11 | versus frequency found
by HFSS. From Figure 3.10 it is clear that the resonance points are the nearly the
same. However, |S11 | is about approximately 6 dB higher. This could be due to
diﬀerences of probe modeling in HFSS and K-brick.
Refer to Appendix G for a matlab K-brick-Taguchi code that optimizes a simple
cavity-backed rectangular patch antenna in K-brick.

71

Table 3.11: Dimensions of a cavity-backed E-patch antenna optimized by Taguchi-Kbrick code. Input parameter ranges used to generate the geometry values are given
in Table 3.9.
E-patch Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy

Value in mm
96
81
31.7
48
71
8
11
15
200
200

Table 3.12: Dimensions of a cavity-backed E-patch antenna optimized by Taguchi-Kbrick code. Input parameter ranges used to generate the geometry values are given
in Table 3.10.
E-patch Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy

72

Value in mm
96
83
37.5
48
65
7
13
15.6
200
200

Figure 3.8: Simulated |S11 | of an E-patch antenna optimized in K-brick.. Geometry
values are given in Table 3.11.

73

Figure 3.9: Simulated |S11 | of an E-patch antenna optimized in K-brick. Geometry
values are given in Table 3.12.

74

Figure 3.10: Simulated |S11 | of an E-patch antenna optimized in HFSS. Geometry
values given in Table 3.12.

75

Chapter 4
Experimental Results
4.1

Rectangular cavity-backed E-patch antenna

An experimental rectangular cavity-backed E-patch antenna was built to validate
simulations done in HFSS. Figures 4.2 through 4.6 show detailed pictures of the
experimental antenna. Figure 4.7 shows a top and side view depiction of the Epatch antenna with dimension variables. The ground plane, as shown in Figure 4.2,
was constructed out of a corrugated rectangular cardboard box. The cardboard box
length, width, and depth was 54 cm by 69 cm by 11.2 cm. A hole was cut in the
center of the rectangular box in order to place the dielectric cavity. The hole’s length
and width dimensions are the same as those of the dielectric cavities (20 cm by 20
cm). The cavity and experimental antenna’s dimensions are provided in Table 4.1.
In order to have a good conductor for the ground plane in the experiment, aluminum
tape was adhered to the surface of the rectangular cardboard box. The hole cut out for
containment of the cavity has a small thickness in which copper tape was wrapped
around; refer to Figure 4.5 for a picture of this thickness covered with aluminum
tape. The dielectric cavity material was a rectangular Styrofoam piece cut to the
dimensions of the cavity given in Table 4.1. To cut the Styrofoam to the proper

76

dimensions, a Styrofoam cutter was used; Figure 4.9 shows the Styrofoam cutting
tool. Styrofoam’s µ and

material properties are nearly identical to a free-space air

dielectric and for experimental construction purposes Styrofoam is more practical.
The Styrofoam cavity’s outer and bottom walls had copper tape adhered to them
to provide a good ground plane conductor. The E-patch was constructed with four
pieces of copper tape. The copper tape used to assemble the E-patch was cut to size
using a standard exact-o knife. The E-patch dimensions are provided in Table 4.1.
The probe feed, as shown in Figure 4.6, is a female SMA connector and is placed at
the proper position (Xf , Yf ) as depicted in Figure 4.7. The non-threaded end of the
female SMA connector has a wire soldered to it. The lead was cut to the dielectric
height h. The non-threaded end of the female SMA connector (lead soldered and cut
to h) pierces the copper tap adhered to the bottom surface of the cavity and through
the copper tape conductor of the E-patch antenna. Once the lead end was through
the dielectric cavity and the E-patch, it was soldered to the copper tape (E-patch
copper tape).
Table 4.1: Dimensions of the experimental rectangular cavity-backed E-patch antenna.
E-patch and cavity dimension
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy

Value in mm
96
83
37.5
48
65
7
13
15.6
200
200

The experiment to measure |S11 | versus frequency was conducted using a Hewlett
Packard 8753D network analyzer. The power setting was 10 dBm, averaging set
77

to 4, IF Bandwidth set to 3700 KHz, the start and stop frequencies set to 1 and
2 GHz. The Hewlett Packard 8753D was calibrated for |S11 | using the 3.5 mm
SMA connector Agilent test calibration set 8753D. The experimental antenna was
orientated perpendicular to the ground and rested on a 3 foot block of Styrofoam.
Radiation gain measurements of the E-patch antenna were conducted at 1300
MHz and 1640 MHz in the anechoic chamber located at the engineering building
at Michigan State University. Unfortunately, the measurement setup used is not
calibrated for gain measurements and a reference antenna with a known gain should
be used to determine the true gain of the E-patch antenna. This is a fairly simple
procedure. For instance, if the gain of the reference antenna is known to be Gref and
the uncalibrated measurement produces a gain value Gref _un and if the uncalibrated
measurement of the E-patch antenna gain is measured to be Ge_un then, the true
gain of the E-patch antenna Ge is determined using

Ge = Gref

Ge _un
.
Gref _un

(4.1)

In our case, the reference antenna used is a broad band pyramidal horn∗ and its
gain is evaluated using a two identical antenna gain measurement approach. In this
approach, the ratio of the power received (Pr ) to the power transmitted (Pt ) between
two identical antennas placed a distance R away from each other is used in conjunction
with the Friis equation given in (2).
Pr
=
Pt

λ 2
G0t G0r
4πR

(4.2)

From equation (4.2), λ is the operating wavelength, G0t and G0r are the gains of the
transmitting and receiving antenna respectively. Since the two antennas are identical,
∗

The
antenna
speciﬁcations
H-1734%20Data%20Sheet.pdf

are

at

78

http://www.cobham.com/media/101070/

G0t =G0r and equation (2) reduces to equation (3).
4πR 2 Pr
= Gref
λ
Pt

(4.3)

In our setup, both transmitting and receiving antennas are connected to a network
analyzer, and the ratio Pr is the transmission coeﬃcient (|S21 |2 ) measured by the
Pt
network analyzer. Note that the network analyzer needs to be calibrated at the end
of the cables connecting the two antennas in order to remove losses from the cables.
Figure 4.1 is a screen capture of the HFSS simulation used to compare with the
experimental rectangular cavity-backed E-patch antenna. The HFSS settings are
listed in Table 4.2. Figure 4.11 is the frequency response data for the simulation of
the experimental antenna.
Table 4.2: HFSS settings for the rectangular cavity-backed E-patch antenna.
HFSS setting
Frequency of solving
Adaptive passes
Maximum Delta S
Sweep Type
Frequency Set up
Initial mesh options
Solution Options
Excitation
Port Renormalization
Deembed setting

HFSS setting value(s)
1.4 GHz
10
.01
Fast
linear step 1 to 2 GHz step size .01
do lambda reﬁnement .15
First order iterative solver .001
Waveport at the coaxial probe feed
50 Ω
checked -15.6 mm

To understand gain measurements in HFSS realized gain needs to be deﬁned.
Realized gain in HFSS is

realized gain = 4π

U
Pincident

.

(4.4)

where U is the radiation intensity in watts per steradian in the direction speciﬁed,
79

Figure 4.1: A screen capture of the rectangular cavity-backed E-patch antenna in
HFSS.

80

69 cm
Ground plane
20 cm

54 cm

Cavity

20 cm

Figure 4.2: A front view of the rectangular cavity-backed E-patch experimental antenna.

and Pincident is power associated with the conjugate match of the accepted power
Paccepted . In HFSS Paccepted is deﬁned as,
Paccepted = Re
A

E × H ∗ .dS

(4.5)

Where Re is the real part of a complex number, A is the union of all port boundaries
in the model (including wave and lumped Ports, but excluding Floquet ports), E is
the electric ﬁeld and H ∗ is the conjugate of the magnetic ﬁeld and dS is the local
port-boundary unit normal directed out of the 3D HFSS model.
Figure 4.10 shows |S11 | versus frequency measured on an Hewlett Packard 8753D
network analyzer. Comparing Figure 4.10 with Figure 4.11 in Figure 4.12, |S11 | is
near in agreement with simulation and experiment. The bandwidth of the simulation

81

11.2 cm

Figure 4.3: A side view of the rectangular cavity-backed E-patch experimental antenna.

82

Copper tape is adhered to the bottom
Female SMA connector
and side walls of the cavity.

Figure 4.4: A back view of the rectangular cavity-backed E-patch experimental antenna.

83

The thickness of the rectangular box
is covered with copper tape
to maintain continuity of the top ground plane.

Figure 4.5: A picture of the copper tape covering up the thickness of the cardboard
box top.

84

Female SMA connector

Figure 4.6: A close up picture of the feed point of the experimental cavity.

85

y
PEC

dielectric

Ls

Ws

h

slot

Xf

L

.

Cy x

P
s

probe feed

Yf

PEC

W
Side View

Cavity

Cx

Top View
Figure 4.7: Top and side view depiction of an experimental E-patch in a cavity.

86

was approximately 243 MHz while the bandwidth of experimental antenna was 330
MHz. The return loss diﬀerence at both points of resonance (1.3 and 1.64 GHz)
and bandwidth diﬀerences may be attributed to diﬀerences between the constructed
antenna and the HFSS model.
Refer to Figure 4.8 for a graphical depiction of the coordinate system used to
describe the radiation patterns in HFSS and for the experimental antenna. For the
Z-X plane case, φ = 0◦ and θ is varying from -180 to 180 degrees. -180 degrees
to 0 degrees refers to negative x values and 0 to 180 degrees refers to positive x
values. For the Z-Y plane case, φ = 90◦ and θ is varying from -180 to 180 degrees.
-180 degrees to 0 degrees refers to negative y values and 0 to 180 degrees refers to
positive y values. Figure 4.13 and Figure 4.14 are HFSS simulated radiation patterns
of the experimental antenna. The realized gain for the experiment is approximately
10 dB in the forward broadside direction of the antenna, indicating that the antenna
transmitting capabilities in that direction are good. Figure 4.15 and 4.16 are the
radiation gain measurements of the experimental antenna, and the gain was found
using (4.1). The gain at 0 degrees in Figure 4.16 is about 10 dB and about 7 dB in
Figure 4.15. Comparing Figure 4.13 and Figure 4.16 in Figure 4.17, the simulated
results match reasonably well with the experiment. Diﬀerences could be attributed to
human construction of the experimental antenna. Comparing Figure 4.14 and Figure
4.15 in Figure 4.18, the simulated results did not match well with the experiment.
Diﬀerences could be attributed to human construction of the experimental antenna
and HFSS’s radiation pattern accuracy.

87

z

θ

x

ф

y
Figure 4.8: Coordinate system for the radiation patterns.

Figure 4.9: A picture of the Uchida 3500 Super Hot-wire Foam-Cutter Crafting Tool
used to cut the cavity to the proper dimensions.

88

Figure 4.10: |S11 | versus frequency measured on a Hewlett Packard 8753D network
analyzer for experimental rectangular cavity-backed E-patch antenna. Dimensions
provided in Table 4.1.

89

Figure 4.11: |S11 | versus frequency of a simulated rectangular cavity-backed E-patch
antenna. Dimensions provided in Table 4.1.

90

Figure 4.12: |S11 | versus frequency for experimental and simulated rectangular cavitybacked E-patch antenna. Dimensions provided in Table 4.1.

91

Figure 4.13: HFSS gain radiation pattern of the simulated cavity-backed E-patch
antenna, co-polarization.

92

Figure 4.14: HFSS gain radiation pattern of the simulated cavity-backed E-patch
antenna, cross-polarization.

93

Figure 4.15: Gain radiation pattern of the experimental rectangular cavity-backed
E-patch antenna, cross-polarization.

94

Figure 4.16: Gain radiation pattern of the experimental rectangular cavity-backed
E-patch antenna, co-polarization.

95

Figure 4.17: Gain radiation patterns of the experimental and simulated rectangular
cavity-backed E-patch antenna, co-polarization.

96

20

Figure 4.18: Gain radiation patterns of the experimental and simulated rectangular
cavity-backed E-patch antenna, cross-polarization.

97

4.2

Cylindrical conformal cavity-backed E-patch antenna

A experimental cylindrical conformal cavity-backed E-patch antenna was built to
compare and verify simulation results in HFSS. Comparing with results from the
experimental rectangular cavity-backed E-patch antenna presented in section 4.1, the
eﬀects of curvature can be investigated.
Using the standard cylindrical coordinate system where ρ =
x2 + y 2 , φ =
y
tan−1 x and z = z (x, y, and z are variables in the rectangular coordinate system),
one can develop a topology for the cylindrically conformal cavity-backed E-patch
antenna. Figure 4.19 and Figure 4.20 show conceptual depictions of a experimental
cavity-backed E-patch conformed to a cylinder. In order to analyze how the curvature
of the cylinder changes the antenna’s performance, as compared to the planar cavitybacked case, the E-patch dimensions need to stay consistent between both cases.
Therefore, the width dimensions of the E-patch (W and Ls ) will be speciﬁed using
φ, at a distance ρ from the z axis. Further, the arc length along the circumference of
the cylinder is
g = ρφ.

(4.6)

The variable g in (4.6) is the E-patch width dimension W or Ls . The length dimensions of the E-patch (L, Ws , and Ps ) will be along the z axis. To keep the cavity in
the same conﬁguration as the rectangular cavity-backed case, the dielectric height h
will be speciﬁed along ρ, Cx will be speciﬁed by a φ and Cy will be speciﬁed along
the z. The probe feed (Xf , Yf ) will be placed consistent with that of the E-patch.
In other words Xf will be an arc length according to (4.6) where g = Xf and Yf will
be speciﬁed along the z axis.
Figures 4.21 through 4.25 show detailed pictures of the experimental antenna. As
discussed earlier, Figures 4.19 and 4.20 show depictions of the cylindrically conformal
98

cavity-backed E-patch antenna. The ground plane, as shown in Figure 4.21, was
constructed out of a cardboard cylinder. The cardboard cylinder’s height along the zaxis was 121.92 cm and radius ρ was 15.24 cm. Further E-patch and cavity dimensions
for the experiment are provided for Table 4.3. Figure 4.26 shows the cardboard
cylinder as it was purchased from Home Depot. A hole was cut in the center of
the cardboard cylinder in order to place the dielectric cavity. The hole’s length and
height dimensions are the same as those of the dielectric cavity’s (Cx by Cy ). In
order to have a good conductor for the ground plane, aluminum tape was adhered
to the surface of the cardboard cylinder. The hole cut out for containment of the
cavity has a small thickness in which aluminum tape was wrapped around; this was
done to maintain continuity of the top ground plane to the outer and bottom walls
of the dielectric cavity. The dielectric cavity material is made out of two pieces
of Styrofoam as shown in Figure 4.23. To cut the Styrofoam pieces to the proper
dimensions, a Styrofoam cutter was used; Figure 4.9 shows the Styrofoam cutting
tool. The Styrofoam’s µ and

material properties are nearly identical to an free-

space air dielectric and for experimental construction purposes it is more practical to
use Styrofoam than air. The Styrofoam cavity’s outer and bottom walls had copper
tape adhered to them to provide a good ground plane conductor. The E-patch was
constructed with four pieces of copper tape. The copper tape used to assemble the
E-patch was cut to size using a standard exact-o knife. The probe feed, as shown in
Figure 4.24, is a female SMA connector and is placed at the position (Xf , Yf ) as
depicted in Figure 4.19. The non-threaded end of the female SMA conductor has a
wire lead soldered to it. The lead was cut to the dielectric height h. The non-threaded
end of the female SMA connector (wire lead soldered and cut to the length h) pierces
the copper tap adhered to the bottom surface of the cavity and is placed in between
both pieces of the Styrofoam cavity through the copper tape conductor of the E-patch
antenna. Once the lead end was through the dielectric cavity and the E-patch, it was

99

soldered to the copper tape (E-patch copper tape).
Table 4.3: Dimensions of a experimental cylindrical conformal cavity-backed E-patch
antenna.
E-patch and Cavity Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy
f
ρ

Value in mm
96
83
37.5
48
65
7
13
15.6
200
200
509.6
154

The experiment for the |S11 | versus frequency was conducted using a Hewlett
Packard 8753D network analyzer for measurement. The Hewlett Packard 8753D is
located in the Engineering Research Center, at MSU. The power setting was 10 dbm,
the averaging set to 4, the IF Bandwidth set to 3700 KHz, and the start and stop
frequencies were set to 1 and 2 GHz. The Hewlett Packard 8753D was calibrated
for |S11 | using the Female SMA connector Agilent test calibration set 8753D. The
experimental antenna was orientated perpendicular to the ground and rested on a 3
foot block of Styrofoam.
Figure 4.27 and Figure 4.28 are screen captures of the HFSS simulation used
to compare with the experimental cylindrically conformal cavity-backed E-patch antenna. The HFSS settings are listed in Table 4.4. Figure 4.30 is the frequency response
data for the simulation of the experimental cylindrically conformal cavity-backed Epatch antenna.
As discussed in the previous section, the radiation reference gain for the measured

100

Table 4.4: HFSS settings for the cylindrically conformal cavity-backed E-patch antenna.
HFSS setting
Frequency of solving
Adaptive passes
Maximum Delta S
Sweep Type
Frequency Set up
Initial mesh options
Solution Options
Excitation
Port Renormalization
Deembed setting

HFSS setting value(s)
1.6 GHz
20
.005
Fast
linear step 1 to 2 GHz step size .02
do lambda reﬁnement .3
First order
Waveport at the coaxial probe feed
50 Ω
checked -18 mm

experimental E-patch antenna frequencies is determined using equations (4.3) - (4.1)† .
Appendix H contains the matlab ﬁle used to convert the gain measurements in the
anechoic chamber at Michigan State University engineering building to true gain
measurements.
Figure 4.29 is |S11 | versus frequency measured on a Hewlett Packard 8753D network analyzer. Comparing Figure 4.29 with Figure 4.30 in Figure 4.31, |S11 | versus frequency is similar. In Figure 4.31 there are two distinct points of resonance.
However, the points of resonance are in diﬀerent positions and |S11 | is lower in the
simulated case. The bandwidth of the simulated antenna is 320 MHz (-10 dB or
lower at 1250 MHz to 1340 MHz and 1520 MHz to 1750 MHz). The bandwidth of
the experimental antenna is 321 MHz (-10 dB or lower at 1254 MHz to 1310 MHz
and 1520 MHz to 1785 MHz). The diﬀerences in |S11 | versus frequency in Figure
4.31 may be due to construction techniques and an inability of HFSS to simulate the
E-patch antenna exactly.
In order to achieve -10 dB and below for |S11 | at the L1 and L2 frequencies,
another experimental antenna was built. Figure 4.33 is the |S11 | versus frequency for
†

The frequencies of interested tested in the cylindrical case were 1.3 GHz and 1.64 GHz.

101

a experimental cylindrically conformal cavity-backed E-patch antenna. Dimensions
are provided in Table 4.5. The bandwidth of the experimental antenna in Figure 4.33
is 273 MHz (-10 dB or lower at 1195 MHz to 1253 MHz and 1500 MHz to 1715 MHz).
At the L1 and L2 frequencies |S11 | was measured as -12 dB and -20 dB respectively.
Table 4.5: Dimensions of a experimental cylindrical conformal cavity-backed E-patch
antenna.
E-patch and Cavity Dimension
L
W
Xf
Yf
Ls
Ws
Ps
h
Cx
Cy
f
ρ

Value in mm
96
83
37.5
48
67.5
7
13
15.6
200
200
509.6
154

Figure 4.32 shows the experimental |S11 | versus frequency of the cylindrically
conformal and rectangular cavity-backed E-patch antennas. The bandwidth of the
experimental rectangular cavity-backed patch antenna is 330 MHz (-10 dB or lower
at 1270 MHz to 1370 MHz and 1550 MHz to 1780 MHz). The bandwidth of the
experimental cylindrically conformal cavity-backed E-patch antenna is 321 MHz (10 dB or lower at 1254 MHz to 1310 MHz and 1520 MHz to 1785 MHz). The
eﬀect of curvature of the experimental antennas indicates a minor loss in bandwidth
and a slightly lower shift in frequencies of the main points of resonance. However,
the simulation of the experiments shown in Figures 4.31 and 4.12 seem to be good
results verifying HFSS operation. The eﬀect of curvature may have changed mode
two’s operation of the antenna and contributed to the loss in bandwidth and shift
in resonant frequency. Comparing the simulation and experiment bandwidths, the
102

diﬀerence may be due to construction techniques of the experimental antennas and
an inability of HFSS to simulate the E-patch antenna exactly.
Refer to the blue arrows in Figure 4.19 for the φ and θ values used to describe
the radiation patterns in HFSS and for the experimental antenna. For the Z-X plane
case, φ = 0◦ and θ is varying from -180 to 180 degrees. -180 degrees to 0 degrees
refers to negative x values and 0 to 180 degrees refers to positive x values. For the
Z-Y plane case, φ = 90◦ and θ is varying from -180 to 180 degrees. -180 degrees to
0 degrees refers to negative y values and 0 to 180 degrees refers to positive y values.
For the X-Y plane case, θ = 90◦ and φ is varying from -180 to 180 degrees.
Figures 4.34 - 4.39 are the radiation gain measurements of the experimental and
simulated antenna. The realized gain in the simulation is approximately 10 dB in the
forward broadside direction of the antenna indicating that the antenna transmitting
and receiving capabilities in that direction are good. Comparing the simulated and
experiment radiation patterns in Figures 4.34, 4.36, and 4.39 the simulated results
are near in agreement with the experiment. The gain at 0 degrees in Figure 4.36 and
Figure 4.34 is about 6.7 dB and about 4.5 dB in Figure 4.35 and Figure 4.37. Small
diﬀerences could be attributed to construction issues of the experimental antenna.
Comparing Figure 4.35, 4.37, and 4.38 the simulated results were not in agreement
with the experiment. Diﬀerences could be attributed to human construction of the
experimental antenna and how the antenna was mounted (non-air dielectric platform)
for the radiation pattern measurement.
Comparing the experimental rectangular and cylindrically conformal cavity-backed
E-patch antenna radiation patterns (Figures 4.15 and 4.16 and Figures 4.34, 4.35, 4.36
and 4.37), it can be noted that the radiation pattern of the cylindrically conformal
case is broader then that of the rectangular case. The broadening of the radiation
pattern might be due to creeping wave eﬀects and subsequent diﬀraction occurring
around the cylinder.

103

Female SMA connector
Xf

x

Cy
Aluminium tape is adhered to the
surface of the outer walls
of the cylinder.

Cx

ρ

Yf

h

θ

z

f

ф
y

Figure 4.19: A angled-view conceptual depiction of the E-patch antenna conformed
to the surface of a cylinder with a radius ρ.

104

x

z
x
y
y

Z-axis into page
x

z
y
Figure 4.20: A 3-view depiction of the E-patch antenna conformed to the surface of
cylinder.

105

Ground plane

50.96 cm
121.92 cm

Cavity
20 cm

20 cm

Figure 4.21: A front view of the cylindrically conformal cavity-backed E-patch experimental antenna.

106

Cavity side and bottom walls have
copper tape adhered to them.

Female SMA connector

Figure 4.22: A top view of the cylindrically conformal cavity-backed E-patch experimental antenna.

107

Female SMA connector

10 cm

10 cm
Cavity consists of two separate
piecies of Styrofoam, and the separation
is denoted by the dashed line.

Figure 4.23: A close up picture of the front cavity E-patch experimental antenna.

108

Female SMA connector soldered
to the copper tape.

Figure 4.24: A close up picture of the feed point of the experimental cylindrically
conformal cavity.

109

Female SMA connector
sleeve soldered to the copper
groundplane.

Figure 4.25: A closer up picture of the feed point of the experimental cylindrically
conformal cavity.

Figure 4.26: A picture of the cylinder cardboard tube used to construct the cylindrically conformal cavity-backed E-patch antenna.

110

Figure 4.27: A screen capture of the cylindrically conformal cavity-backed E-patch
antenna in HFSS.

111

Figure 4.28: A closer view screen capture of the cylindrically conformal cavity-backed
E-patch antenna in HFSS.

112

Figure 4.29: |S11 | versus frequency measured on a Hewlett Packard 8753D network
analyzer for experimental cylindrically conformal cavity-backed E-patch antenna. Dimensions provided in Table 4.1.

113

Figure 4.30: |S11 | versus frequency of a simulated cylindrically conformal cavitybacked E-patch antenna. Dimensions provided in Table 4.3.

114

Figure 4.31: |S11 | versus frequency for experimental and simulated cylindrically conformal cavity-backed E-patch antennas. Dimensions provided in Table 4.3.

115

Figure 4.32: |S11 | versus frequency for experimental cylindrically conformal and rectangular cavity-backed E-patch antennas. Dimensions provided in Table 4.3 and Table
4.1.

116

Figure 4.33: |S11 | versus frequency for a cylindrically conformal cavity-backed Epatch antenna. Dimensions provided in Table 4.5.

117

Figure 4.34: Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna, X-Y plane and co-polarized. Dimensions
provided in Table 4.3.

118

Figure 4.35: Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna, X-Y plane cross-polarized. Dimensions
provided in Table 4.3.

119

Figure 4.36: Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna, Z-X plane and co-polarized. Dimensions
provided in Table 4.3.

120

Figure 4.37: Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna, Z-X plane cross-polarized. Dimensions
provided in Table 4.3.

121

Figure 4.38: Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna, Z-Y plane and co-polarized. Dimensions
provided in Table 4.3.

122

Figure 4.39: Gain radiation pattern of the experimental and simulated cylindrically
conformal cavity-backed E-patch antenna,Z-Y plane cross-polarized. Dimensions provided in Table 4.3.

123

Chapter 5
Conclusion
Inﬁnite ground plane simulations for E-patch antennas were thoroughly investigated
at the L1 and L2 frequencies. The simulated |S11 | for the E-patch antenna at the L1
and L2 frequencies performed well when the design equations (1.3) were used. The -10
dB bandwidth of the E-patch antenna in the inﬁnite ground plane simulation extended
between the L1 and L2 frequencies. However, when the same E-patch antenna, whose
dimensions were determined from equations (1.3), was simulated with a rectangular
cavity backing, it did not perform well.
In the pursuit of a value of |S11 | of -10 dB or below at the L1 and L2 frequencies
in the rectangular cavity-backed case, a Matlab graphical user interface (GUI) was
written to allow the user to specify a range of E-patch dimensions to vary. Using
K-brick and the Matlab GUI, acceptable values for |S11 | at L1 and L2 frequencies
were determined.
A cylindrically conformal cavity-backed E-patch antenna with a ﬂat bottom cavity
was simulated for multiple radii of cylinders. As the radius ρ of the cylinder was
increased, |S11 | decreased for the L1 frequency and increased for the L2 frequency.
In addition the higher resonance frequency shifted upwards beyond the L2 frequency.
With the exception of the 50 cm radius case, at the L1 and L2 frequencies |S11 |

124

was less then -10 dB. For the 50 cm radius cylinder case, |S11 | was approximately -9
dB at the L2 frequency, and was less than -10 dB at the L1 frequency. For the ﬂat
bottom cavity case of the conformal E-patch antenna, it is unclear if the operation
of the cylindrically conformal cavity-backed E-patch antenna would be eﬀective and
reliable at the L1 and L2 frequencies.
A cylindrically conformal cavity-backed E-patch antenna with a conformal cavity
was simulated for multiple radii of cylinders. As the radius ρ decreased, |S11 | stayed
approximately the same for the lower resonance frequency and decreased for the
higher resonance frequency. In addition the higher resonance frequency shifted lower
in frequency as the radius decreased. The bandwidth does enlarge as the radius
decreases. Notably, the bandwidth for the 100 cm simulated cylindrically conformal
cavity-backed E-patch antenna case is 245 MHz (-10 dB or lower at 1245 MHz to
1330 MHz and 1590 MHz to 1750 MHz) while the bandwidth for the 15.4 cm case is
320 MHz (-10 dB or lower at 1250 MHz to 1340 MHz and 1520 MHz to 1750 MHz).
At 50, 25, and 15.4 cm radius, |S11 | at the L2 frequencies is less then -10 dB. The
L1 frequency stays approximately -9 dB for all the cases. These diﬀerences in |S11 |
versus frequency might be due to the mode two resonance point being altered by
curvature of the cylinder.
Two experimental E-patch antennas were built to verify simulations done in HFSS.
The rectangular cavity-backed E-patch antenna simulation and experiment matched
well. |S11 | at frequencies of interest for the experimental rectangular cavity-backed
E-patch antenna was -10 dB less than the HFSS simulation at the two resonant
frequencies. The bandwidth of the experimental rectangular cavity-backed patch
antenna was 330 MHz (-10 dB or lower at 1270 MHz to 1370 MHz and 1550 MHz
to 1780 MHz). The bandwidth of the experimental cylindrically conformal cavitybacked E-patch antenna was 321 MHz (-10 dB or lower at 1254 MHz to 1310 MHz
and 1520 MHz to 1785 MHz). The eﬀect on curvature of the experimental antennas

125

indicates a loss in bandwidth and a shift in the main points of resonance to lower
frequency. The curvature may have eﬀected mode two’s operation and contributed to
the loss in bandwidth and shift in resonant frequency. Comparing the simulation and
experiment bandwidths, the diﬀerence may be due to construction techniques of the
experimental antennas and the lack of HFSS ability to simulate the E-patch antenna
exactly in both cases.
An experimental antenna was built to achieve a -10 dB or lower value for |S11 |
at the L1 and L2 frequencies; that antenna’s |S11 | at the L1 and L2 frequency was
acceptable with -12 dB at 1227 MHz and -20 dB at 1575 MHz.
The radiation pattern of the two experimental antennas was also investigated and
simulated. The experimental radiation pattern of the cylindrically conformal antenna
was broader than that of the rectangular case. The broadening of the radiation
pattern might be due to creeping wave eﬀects and subsequent diﬀraction occurring
around the cylinder. In addition the broadside gain of the experimental cylindrically
conformal cavity-backed E-patch antenna was found to be 8 dB, matching well with
the simulated gain from HFSS of approximately 10 dB.
Future investigations into the cylindrically conformal cavity-backed E-patch antenna should include better techniques for construction and better modeling for the
feed in HFSS. An implementation of Taguchi’s method of optimization might improve
|S11 | at the L1 and L2 frequencies and the overall bandwidth of the cylindrically conformal cavity-backed E-patch antenna.
Taguchi’s method of optimization for E-patches has been proven to be an effective method for optimizing |S11 | for rectangular cavity-backed E-patch antennas.
Through the use of simple ﬁtness functions and orthogonal arrays, |S11 | at the two
main resonance frequencies of the E-patch antenna can be minimized. Computation time and ease of coding make Taguchi’s method of optimization an attractive
alternative to genetic algorithms and particle swarm optimizations.

126

Further research using Taguchi’s method for E-patch antennas should be done
to improve the overall bandwidth between both points of resonance of the antenna.
Increasing the bandwidth of the antenna may be accomplished with modiﬁcation to
the ﬁtness functions presented in earlier chapters.

127

APPENDICES

128

Appendix A: Setting up Sonnet for
E-patch antenna simulations
1. Refer to Figure A.1. Create a new Geometry.
2. Refer to Figure A.2. Left click on circuit tab in the upper left corner of the new
geometry layout. Highlight “units" and left click. Once the Unit dialog box has
appeared, highlight the length box and scroll until mm is highlighted. Left click
when mm is highlighted and then left click on the “Ok" button.
3. Refer to Figure A.3. Locate the circuit tab again and highlight “box" and left
click it to bring up the Box dialog box. To initialize the geometries of Figure
2.1 and Figure 2.2 initialization, change the “Cell" size to 1 for the X and Y
directions. Change the box size to 150 for the X and Y directions. Change the
number of cells to 150 for the X and Y directions. To initialize the geometries
Figure 2.3 and Figure 2.4 initialization, change the “Cell" size to 1 for the X and
Y directions. Change the box size to 200 for the X and Y directions. Change
the number of cells to 200 for the X and Y directions. Change the Top metal
to “Free Space" in both cases. Left click on the“apply” button, then left click
on the “Ok" button. For L-band E-patch antenna designs, one should specify
that the box be at least 1λ x 1λ in the X and Y directions respectively. Here λ
is the largest wavelength of L-band. Note that 200 mm is not large enough to
cover the 1λ x 1λ; however, 200 mm yields good approximations for |S11 |.
4. Refer to Figure A.4. Locate the circuit tab again and highlight “dielectric layers"
and left click. The dielectric layer box should now appear. Left click the ﬁrst
dielectric layer so that it is highlighted. Then left click the “edit" button and
change the material name to “Air". Change the thickness of the dielectric to
the designs h height. In this case, the value is 10 mm.
5. Refer to Figure A.5. Left click the second dielectric layer so that it is highlighted.
This is the layer above the metallic patches. Then left click the “edit" button
and change the material name to “Air". Change the thickness of the dielectric
to the height of the antenna. In this case, the value is 10 mm.
6. Refer to Figure A.6. Left click on the “Ok" button on the dielectric layer box.
The dielectric box should now appear as below.

129

7. Refer to Figure A.7. Left click on the Analysis tab in the upper left corner.
Highlight and left click setup. A setup box will appear. For the geometry of
Figure 2.1, enter 2 and 3 GHz for the start and stop frequencies, respectively.
For the geometry of Figure 2.2, enter 1.5 and 3 GHz for the start and stop
frequencies, respectively. For the geometry of Figure 2.3 and Figure 2.4 enter 1
and 2 GHz for the start and stop frequencies, respectively. Check the computer
density box if you wish to view surface ﬁelds and radiation patterns. Leave the
Adaptive Sweep drop down box as is. Left click on “Ok" when ﬁnished.
8. Refer to Figure A.8. Left click on the tools tab in the upper left corner. Highlight “add metallization". Then highlight “add rectangle".
9. Refer to Figure A.9. The rectangle attributes should appear on your screen.
For the width and and length, type in the dimensions of the patch. Left click
on “Ok" when ﬁnished. For the Sonnet simulations, four patches were used then
merged when all were placed in the box.
10. Refer to Figure A.10. Place rectangle.
11. Refer to Figure A.11. Left click on the tools tab once again and highlight “via",
then left click on “down to ground". With “via" still highlighted, highlight and
left click on “circular".
12. Refer to Figure A.12. Another warning dialog box may appear. Left click on
“ok" if it does. The circular attributes dialog box should now appear. Change
the diameter to 1.2 mm this is the approximate diameter of a coaxial probe.
The circle should be 10 sided. Left click “Ok" when the changes have been
made.
13. Refer to Figure A.13. Place the via at the designated Xf and Yf point by left
clicking.
14. Refer to Figure A.14. Left click on the tools tab once again and highlight and
left click on “add port". The pointer now has a boxed “n" around it. Left click
on the via just made. A warning dialog box will pop up. Left click “ok" to close
it. To view the patch geometry in 3-D, left click the 3-D box as shown in the
screen shot below.
15. Refer to Figure A.15. When done viewing the 3-D simulation, left click on the
“x" in the upper right corner. Now left click on the ﬁle tab in the upper left
corner and save the geometry ﬁle.
16. Refer to Figure A.16. Left click on the project tab in the upper left corner and
highlight and click the analyze tab. A view response dialog box should now
appear and show your |S11 | and other parameters. Click on the view response
box to see the |S11 | vs Frequency plot.
17. Refer to Figure A.17. Below the ﬁnal E-patch geometry.
130

Figure A.1: New Geometry

Figure A.2: Circuit Unit box

131

Figure A.3: Circuit box

132

Figure A.4: Dielectric ﬁrst layer Dielectric selection

133

Figure A.5: Dielectric second layer parameter selection

134

Figure A.6: Completion of Dielectric selection

135

Figure A.7: Analysis Setup

136

Figure A.8: Adding a metal rectangle

137

Figure A.9: Rectangle dimension selection

138

Figure A.10: Rectangle placement

139

Figure A.11: Adding a Via

140

Figure A.12: Via circle parameter selection

141

Figure A.13: Via placement on a rectangle

142

Figure A.14: 3-D view of a patch

143

Figure A.15: Save as screen shot

144

Figure A.16: Viewed Response Box

145

Figure A.17: Finished E-patch topology

146

Appendix B: Matlab Code for a two
variable function
close all
clear all
[filename pathname] = uigetfile('*.txt','Choose the OA file from
online or generated code MUST BE ACII Formated!');
OA=dlmread([pathname filename]);
SizeofOA=size(OA);
Element = 2;% input number of Elements to be tested
if SizeofOA(2) == Element
else
newColummnsize=SizeofOA(2)−Element;
for n=1:newColummnsize % reduces the columm size for expriment
OA(:,SizeofOA(2))=[];
SizeofOA=size(OA);
end
end

% Level Design User specifies level
Level=0:2; %Input by user assuming 3 levels
lengthflevel=length(Level);
Max1=10; % Input by User
Min1=0; % Input by User
Max2=10; % Input by User
Min2=0; % Input by User
rr=.9; % Reduction rate used by User
i=0;
diffrentlevels=0:1; % Because not all levels will have the same
range. Specified here is how many are different
Ld1= (Max1 − Min1)/(lengthflevel+1); % first specifed level differnce
first interation

147

Ldi1=Ld1*rr^i;
Ld2= (Max2 − Min2)/(lengthflevel+1); % first specifed level differnce
first interation
Ldi2=Ld2*rr^i;
while (Ldi1/Ld1)>.0001;
clear LevelOA2 LevelOA3 LevelOA1
Level=0:2;
clear n b g c

Ld1= (Max1 − Min1)/(lengthflevel+1); % first specifed level differnce
first interation
Ldi1=(1.5*Ld1)*rr^i;
Ld2= (Max2 − Min2)/(lengthflevel+1); % first specifed level differnce
first interation
Ldi2=(1.5*Ld2)*rr^i;

for n=1:SizeofOA(2)

if i==0
switch n
case 1
Level(2)= (Max1−Min1)/2 + Min1;% Middle of range
Level(1)=Level(2)−Ldi1;
Level(3)=Level(2)+Ldi1;
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 2
Level(2)=(Max2−Min2)/2 + Min2; %Middle of range
Level(1)=Level(2)−Ldi2;
Level(3)=Level(2)+Ldi2;
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
end
else
switch n
case 1
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi1;

148

Level(3)=Level(2)+Ldi1;
if Level(1)≤Min1
Level(1)=Min1;
else
end
if Level(3)≥Max1
Level(3)=Max1;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 2
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi2;
Level(3)=Level(2)+Ldi2;
if Level(1)≤Min2
Level(1)=Min2;
else
end
if Level(3)≥Max2
Level(3)=Max2;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
end

end

end

countoflevel=0;

clear LevelOA

for n=1:SizeofOA(1) % Filling up a Level based OA
for b=1:SizeofOA(2)

if OA(n,b)==0
LevelOA(n,b)=LevelOA1(b);

149

elseif OA(n,b)==1
LevelOA(n,b)=LevelOA2(b);
else
LevelOA(n,b)=LevelOA3(b);
end
end

end
clear FitnessF
for n=1:SizeofOA(1)

%Fitness=Fitness+(LevelOA(n,g))^2−10*cos(2*pi*(LevelOA(n,g)))+10;
%Fitness=1−abs(sin(pi*(LevelOA(n,1)−3))/(pi*(LevelOA(n,1)−3)))*abs
(sin(pi*(LevelOA(n,2)−3))/(pi*(LevelOA(n,2)−3)));
Fitness=exp((LevelOA(n,1)*sin(4*LevelOA(n,1))+1.1*LevelOA(n,2)*sin
(2*LevelOA(n,2))));
FitnessF(n,1)=Fitness;
end
clear StoN
StoN=−20*log10((FitnessF)); % Signal to Noise Ratio
clear FinalResponse
%Build Response table

for n=1:SizeofOA(2)
response=0;
response2=0;
response3=0;
countoflevel=0;
for h=1:SizeofOA(1)
if OA(h,n)==0
response=response+StoN(h);
countoflevel=countoflevel+1;
elseif OA(h,n)==1
response2=response2+StoN(h);
elseif OA(h,n)==2
response3=response3+StoN(h);
end
end

150

for c=1:lengthflevel
if c==1
FinalResponse(c,n)=response/countoflevel;
elseif c==2
FinalResponse(c,n)=response2/countoflevel;
elseif c==3
FinalResponse(c,n)=response3/countoflevel;
end
end
end
clear Optparameters

clear OptMat
Optparameters=max(FinalResponse);
for n=1:SizeofOA(2) % Matrix use to single out the good s/n's
for c=1:lengthflevel
if Optparameters(n)==FinalResponse(c,n)
OptMat(c,n)=FinalResponse(c,n);
else
OptMat(c,n)=pi;
end
end
end
clear OAr
for n=1:SizeofOA(2) % matrix used to specify final levels that are
optimal
for c=1:lengthflevel
if OptMat(c,n)==pi;
else
switch c
case 1
OAr(n)=LevelOA1(n);
case 2
OAr(n)=LevelOA2(n);
case 3
OAr(n)=LevelOA3(n);
end
end
end

151

end
% The design for this Fitness Function Raguchi for use only for
Taguchi

%FitnessG=1−abs(sin(pi*(OAr(1)−3))/(pi*(OAr(1)−3)))*abs(sin(pi*(OAr
(2)−3))/(pi*(OAr(2)−3)));
FitnessG=(OAr(1)*sin(4*OAr(1))+1.1*OAr(2)*sin(2*OAr(2)));
meck(i+1)=(FitnessG);
%
if FitnessG >.9
%
break
%
else
i=i+1;
Store(i+1)=OAr(1);
Store2(i+1)=OAr(2);
end

x=[0:.01:10];
y=[0:.01:10];
ta=length(x);

for bah=1:ta
for cah=1:ta
meckfitness(cah,bah)=x(bah)*sin(4*x(bah))+1.1*y(cah)*sin(2*y(
cah));
end
end

figure;
mesh(x,y,meckfitness)
xlabel('x values')
ylabel('y values')
title('global minimun')

meck(i+1)=FitnessG;

152

plotness=1:length(meck);
figure,plot(plotness,meck);
h = legend('Final Fitness');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 27 expriements')
ylabel('Fitness')
title('Fitness vs Runs')

figure,plot(plotness,Store);
h = legend('x values');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 9 expriements')
ylabel('x values')
title('x values')
figure,plot(plotness,Store2);
h = legend('y vaules');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 9 expriements')
ylabel('y values')
title('y values')

153

Appendix C: Start up guide to
K-brick
Contained in this appendix are a start up guide to K-brick and a sample input text
ﬁle. Comments indicated by ^^^ should note appear in an actual input ﬁle.
K-brick uses the FE-BI equations to solve for the electric and magnetic ﬁelds of
a cavity-backed patch antenna. To do this, the cavity is represented using a ﬁnite
number of cells. The dimensions and number of cells are determined by the user.
K-brick starts the cell topology in the lower left (LL) corner of the cavity. In the
input ﬁle below, the length, width and depth of the cavity are 20 cm x 20 cm x 1.5
cm respectively. The corresponding input line in the text ﬁle is 200 200 1. The input
20
20
line 200 200 1 means that each cell is 200 = .1 cm = 1 mm by 200 = .1 cm = 1 mm
by h (h is the dielectric height, ((Layer 1 thickness (cm))of the cavity ). Smaller
cell sizes increase computation time considerably. For most simulations 1 mm x 1 mm
x h cells will yield good results. The layer concept is discussed later in this appendix.
As discussed earlier, K-brick begins constructing the cavity out of the cells starting
in LL corner. Due to cells being used there isn’t a 0 cell, so the cell numbering starts
at 1. This also means that the input into K-brick for cells has to be a whole number.
This is important for when metallic patches are added in the input ﬁle. Inﬁnitesimally
thin rectangular patches are created and placed by cell number within the cavity. In
the example text ﬁle below, four rectangular patches are added to the top layer of

154

the cavity. The numbers for placement of these patches are cell positions, and are
not in x, y, z coordinates. For example, the cell size in the text ﬁle below is 1 mm
by 1 mm by 1.5 cm. The patches are each placed by ﬁrst specifying where they start
to enlarge. Patches enlarge from left to right and down to up. So the ﬁrst patch’s
LL corner in the text ﬁle below is at the 60th cell from the left and 52nd from the
bottom of the cavity; at that cell point the patch enlarges by 10 cells to the right and
96 cells to the top of the cavity. This is repeated 4 times until the proper topology
of the patch is created. The probe is not placed using cell length or position. The
probe is placed relative to the exact middle of the cavity instead of the LL corner of
the cavity like metallic patches, material blocks, or metal bricks are. This means that
probe placement and h dielectric height can be a real integer. In the example below
the probe is placed 3.17 cm to the right, and 0 cm from the middle (center) of the
cavity.
K-brick also has the ability to layer the dielectric in the cavity. Typically this is
done if metal bricks are added to the simulation in the input ﬁle. Metal bricks are
perfect electric conductors that are composed of cells (same cell size as initialized in
the input ﬁle). Metal blocks are constructed from the bottom of the cavity to one
layer below where the patch antenna resides. Metal blocks are added so that the
probe can rest on top of the metal blocks. The probe is placed on top of the metal
blocks to yield results closer to those of a typical coaxial vertical fed probe. However,
when metal blocks are added in this way, a considerable capacitance is added and this
yields a poor |S11 | calculated by the probe. Because of the added capacitance of the
metal bricks, all the simulations done in K-brick did not have metal bricks included.
At each point of resonance in the E-patch simulation, a sign change was noted in
the imaginary portion of the impedance calculated at the probe, indicating a good
simulated result. If one were to use metal bricks, they would be added exactly the
same way a patch is, i.e by cell position and number.

155

The rest of the input ﬁle is self-explanatory except the convergence tolerance,
minimum iterations, and maximum iterations, which deal how close the numerical
solutions in the FE-BI equations need to get to before K-brick terminates operation
and outputs simulated antenna parameters. The convergence tolerances for the problems considered in this thesis are 0.01 5 60000, which seem to be suﬃcient.

1
^^^ 1 = save geo file, 2=read it, 0=do nothing
patch.geo
^^^ filename for geometry
20.0 20.0
^^^ (x,y) cavity dimensions in cm
200 200 1
^^^ x,y,z dimension cells
0
^^^ 0 = cavity, 1=slot
1.50
^^^ Layer 1 thickness (cm)
4
^^^ Number of patches
60 52 1
^^^ Lower left corner of patch + layer
10 96
^^^ Patch dimension in edges
60 52 1
^^^ Lower left corner of patch + layer
81 33
^^^ Patch dimension in edges
60 93 1
^^^ Lower left corner of patch + layer
81 14
^^^ Patch dimension in edges
60 115 1
^^^ Lower left corner of patch + layer
81 33
^^^ Patch dimension in edges
0
^^^ Number of slots
0
156

^^^ Number of shorting pins
0
^^^ Number of metal blocks
0
^^^ Number of material blocks
1
^^^ 1 = uniform material fill
0
^^^ 0 = non-dispersive, 1 = dispersive
1.0 -0.00
^^^ Complex epsilon (real, imaginary)
1.0 -0.00
^^^ Complex mu (real, imaginary)
0.01 5 60000
^^^ convergence tolerance, min. iterations, max. iterations
0
^^^ 0 = BiCG, 1 = QMR
0
^^^ 1 = monitor iterations
0
^^^ 0=initial guess, 1=automatic guess
1.227 1.6 .348
^^^ Start, Stop, INCR frequency [GHz]
0 0 1
^^^ Start, Stop, Incr phi angles [deg]
-90.0 90.0 1.0
^^^ Start, Stop, Incr theta angles [deg]
0
^^^ 0 = do not compute far-zone, 1=compute far-zone fields
patchX_test
^^^ Six (6) character file prefix
0
^^^ 0 = driven, 1=S-paramters, 2=bistatic
E_patch_chad_impedance_Opt.zin
^^^ Input impedance file
50.0 0.0
^^^ Reference impedance
1
^^^ Number of feeds
3
^^^ direction of feed (1=x, 2=y, 3=z)
3.17 0.00
^^^ (x,y) location of feed in cm
1
^^^ Layer of feed
157

1.0
^^^
0
^^^
0
^^^
0
^^^

0.0
Magnitude and phase of current
Number of loads
0 = do NOT save interior fields, 1= save surface fields
Number of field probe

Below are the steps taken to get K-brick running on a Unix machine at Michigan
State University.

1. Obtain fortran code from Professor Kempel.
2. Create a folder in the Unix station to house input, output, and fortran ﬁles.
3. Once the ﬁles have been successfully uploaded, type make in the command line
(you should still be in the same directory where the fortran ﬁles are located).
This builds and complies the fortran program K-brick. If the program fails to
compiles, double check the .mak ﬁle. Make sure it uses the proper compiler for
the unix station you are using, i.e pgf90, f90, or f77 compilers.
4. Once K-brick has successfully compiled, obtain or create a input text ﬁle such
as that given earlier in the appendix.
5. K-brick can output many simulated antenna parameters such as radiation pattern, impedance measured at the probe, and magnitude and phase of the electric and magnetic ﬁelds. Generally you would want have two diﬀerent types
of input ﬁles, one for a frequency sweep of the impedance calculated at the
probe, and a second one for the radiation pattern, electric ﬁeld, and magnetic
ﬁeld at a particular frequency. If you did a frequency sweep that had 100
frequencies, the there would be 100 diﬀerent radiation patterns and 100 different electric and magnetic magnitude and phase ﬁeld ﬁles. K-brick would
158

take too long to compute your simulated antenna parameters. Once you have
determined what you would like for an output, modify your input ﬁle at the
appropriate line. For impedance frequency sweeps change the Start, Stop,
INCR frequency [GHz] to the range you require. For a test case use 1 2 .01.
This will make K-brick simulate the antenna at 100 frequencies between 1 and
2 GHz. Also change Start, Stop, Incr phi angles [deg] to 0 0 1 and
Start, Stop, Incr theta angles [deg] to -90 90 1. This will prevent Kbrick from computing any E-ﬁelds, H-ﬁelds, or radiation pattern ﬁles because
the angle range is zero. To get the electric ﬁeld (magnitude and phase), magnetic ﬁeld(magnitude and phase), and radiation pattern ﬁles, do the opposite.
Fix the Start, Stop, INCR frequency [GHz] to sweep one frequency (e.g 2,
2, and 1) and vary the Start, Stop, Incr phi angles [deg] and Start,
Stop, Incr theta angles [deg].
6. After the input ﬁle has been modiﬁed, now is the time to run K-brick. The Unix
command nohup will make K-brick run even if you log out of the Unix machine.
It is a useful command when K-brick takes a long time to run. A typical
run command for K-brick looks like this: nohup k_brick<inputfile.txt >
out.txt &. This will make K-brick run in the back round of the server and
will allow exit of the server while the program still runs. out.txt is an output
ﬁle containing basic run logs of the inputs from the input ﬁle. It also contains
impedance sweep information.
7. As mentioned earlier K-brick outputs many antenna parameters. For a frequency sweep that contains impedance information at the probe refer to the
.zin ﬁle. The 0 above the line, ^^^0 = do not compute far-zone, 1=compute
far-zone fields is what is changed to calculate the radiation pattern. For the
magnitude and phase of the electric and magnetic ﬁelds the 0 above the line,

159

^^^0 = do NOT save interior fields, 1= save surface fields is what is
changed. Then, specify the component of the ﬁelds desired.

160

Appendix D: Matlab Imaging Code
for GUI

function varargout = gui(varargin)
% GUI M−file for gui.fig
%
GUI, by itself, creates a new GUI or raises the existing
%
singleton*.
%
%
H = GUI returns the handle to a new GUI or the handle to
%
the existing singleton*.
%
%
GUI('CALLBACK',hObject,eventData,handles,...) calls the local
%
function named CALLBACK in GUI.M with the given input
arguments.
%
%
GUI('Property','Value',...) creates a new GUI or raises the
%
existing singleton*. Starting from the left, property value
pairs are
%
applied to the GUI before gui_OpeningFunction gets called. An
%
unrecognized property name or invalid value makes property
application
%
stop. All inputs are passed to gui_OpeningFcn via varargin.
%
%
*See GUI Options on GUIDE's Tools menu. Choose "GUI allows
only one
%
instance to run (singleton)".
%
% See also: GUIDE, GUIDATA, GUIHANDLES
% Edit the above text to modify the response to help gui
% Last Modified by GUIDE v2.5 28−Oct−2009 01:13:25
% Begin initialization code − DO NOT EDIT
gui_Singleton = 1;
gui_State = struct('gui_Name',
mfilename, ...
'gui_Singleton', gui_Singleton, ...
'gui_OpeningFcn', @gui_OpeningFcn, ...

161

'gui_OutputFcn', @gui_OutputFcn, ...
'gui_LayoutFcn', [] , ...
'gui_Callback',
[]);
if nargin && ischar(varargin{1})
gui_State.gui_Callback = str2func(varargin{1});
end
if nargout
[varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
gui_mainfcn(gui_State, varargin{:});
end
% End initialization code − DO NOT EDIT
% −−− Executes just before gui is made visible.
function gui_OpeningFcn(hObject, eventdata, handles, varargin)
% This function has no output args, see OutputFcn.
% hObject
handle to figure
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% varargin
command line arguments to gui (see VARARGIN)
% Choose default command line output for gui
handles.output = hObject;
% Update handles structure
guidata(hObject, handles);
% UIWAIT makes gui wait for user response (see UIRESUME)
% uiwait(handles.figure1);

% −−− Outputs from this function are returned to the command line.
function varargout = gui_OutputFcn(hObject, eventdata, handles)
% varargout cell array for returning output args (see VARARGOUT);
% hObject
handle to figure
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Get default command line output from handles structure
varargout{1} = handles.output;

function edit1_Callback(hObject, eventdata, handles)
% hObject
handle to edit1 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit1 as text

162

%

str2double(get(hObject,'String')) returns contents of edit1
as a double

% −−− Executes during object creation, after setting all properties.
function edit1_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit1 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit2_Callback(hObject, eventdata, handles)
% hObject
handle to edit2 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit2 as text
%
str2double(get(hObject,'String')) returns contents of edit2
as a double

% −−− Executes during object creation, after setting all properties.
function edit2_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit2 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

% −−− Executes on selection change in popupmenu1.
function popupmenu1_Callback(hObject, eventdata, handles)
% hObject
handle to popupmenu1 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: contents = get(hObject,'String') returns popupmenu1 contents
as cell array

163

%

contents{get(hObject,'Value')} returns selected item from
popupmenu1
if isequal(get(hObject,'Value'),1)
set(handles.text2,'Visible','off');
set(handles.edit2,'Visible','off');
else
set(handles.text2,'Visible','on');
set(handles.edit2,'Visible','on');
end

% −−− Executes during object creation, after setting all properties.
function popupmenu1_CreateFcn(hObject, eventdata, handles)
% hObject
handle to popupmenu1 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: popupmenu controls usually have a white background on
Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit3_Callback(hObject, eventdata, handles)
% hObject
handle to edit3 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit3 as text
%
str2double(get(hObject,'String')) returns contents of edit3
as a double

% −−− Executes during object creation, after setting all properties.
function edit3_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit3 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

164

% −−− Executes on button press in pushbutton2.
function pushbutton2_Callback(hObject, eventdata, handles)
% hObject
handle to pushbutton2 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)

% −−− Executes on button press in pushbutton1.
function pushbutton1_Callback(hObject, eventdata, handles)
% hObject
handle to pushbutton1 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
global results;
sprintf('%10s%10s%10s%10s%10s%10s%10s%10s%10s%10s','1.227 GHz','1.575
GHz','Width','Length ','SlotWidth ','SlotLength','Ps','Probe X',
'Probe Y','Height')
if isequal(get(handles.popupmenu1,'Value'),1)
lmin=str2num(get(handles.edit1,'String'));
%
lmin=round(str2num(lmin{1}));
lmax=lmin;
else
lmin=str2num(get(handles.edit1,'String'));
%
lmin=round(str2num(lmin{1}));
lmax=str2num(get(handles.edit2,'String'));
%
lmax=round(str2num(lmax{1}));
end
if isequal(get(handles.popupmenu2,'Value'),1)
wmin=str2num(get(handles.edit3,'String'));
wmax=wmin;
else
wmin=str2num(get(handles.edit3,'String'));
wmax=str2num(get(handles.edit4,'String'));
end
if isequal(get(handles.popupmenu3,'Value'),1)
slmin=str2num(get(handles.edit5,'String'));
slmax=slmin;
else
slmin=str2num(get(handles.edit5,'String'));
slmax=str2num(get(handles.edit6,'String'));
end
if isequal(get(handles.popupmenu4,'Value'),1)
swmin=str2num(get(handles.edit7,'String'));
swmax=swmin;
else
swmin=str2num(get(handles.edit7,'String'));
swmax=str2num(get(handles.edit8,'String'));
end
if isequal(get(handles.popupmenu5,'Value'),1)
psmin=str2num(get(handles.edit9,'String'));
psmax=psmin;
else
psmin=str2num(get(handles.edit9,'String'));
psmax=str2num(get(handles.edit10,'String'));

165

end
if isequal(get(handles.popupmenu6,'Value'),1)
xmin=str2num(get(handles.edit11,'String'));
xmax=xmin;
xstep=.05;
else
xmin=str2num(get(handles.edit11,'String'));
xmax=str2num(get(handles.edit12,'String'));
xstep=str2num(get(handles.edit15,'String'));
end
if isequal(get(handles.popupmenu7,'Value'),1)
hmin=str2num(get(handles.edit13,'String'));
hmax=hmin;
hstep=.05;
else
hmin=str2num(get(handles.edit13,'String'));
hmax=str2num(get(handles.edit14,'String'));
hstep=str2num(get(handles.edit16,'String'));
end

num=1;
% E−patch length
for l=lmin:lmax
% E−patch width (must be even)
for w0=wmin:2:wmax
% slot length
for sl=slmin:slmax
% slot width (must be even)
for sw=swmin:swmax
% Ps (distance from line of symmetry to middle of slot)
for ps=psmin:psmax
% Probe x placement (cm)
for probex=xmin:xstep:xmax
% Dielectric Height
for h=hmin:hstep:hmax
% Define all patches
if not(mod(sw,2)==mod(w0,2))
w=w0+1;
else
w=w0;
end

x1=round((200−l)/2);
l1=l−sl;
y1=round((200−w)/2);
w1=w;
x2=round((200−l)/2);
l2=l;
y2=round((200−w)/2);

166

w2=round(w/2−sw/2−ps);
x3=round((200−l)/2);
l3=l;
y3=round(100+sw/2−ps);
w3=2*ps−sw;
x4=round((200−l)/2);
l4=l;
y4=round(100+ps+sw/2);
w4=round(w/2−sw/2−ps);
sw1=y3−y2−w2;
sw2=y4−y3−w3;
probey=mod(w3,2)/20

%
%
%
%
%
%
%
%

figure;
rectangle('position',[0 0 200 200]);
hold on;
fill([x1 x1 x1+l1 x1+l1],[y1 y1+w1 y1+w1 y1],'r');
fill([x2 x2 x2+l2 x2+l2],[y2 y2+w2 y2+w2 y2],'r');
fill([x3 x3 x3+l3 x3+l3],[y3 y3+w3 y3+w3 y3],'r');
fill([x4 x4 x4+l4 x4+l4],[y4 y4+w4 y4+w4 y4],'r');
plot(probex*10+100,probey*10+100,'−−rs','MarkerFaceColor','g','
MarkerSize',5);

%

% output kbrick input file
fid=fopen(['text.txt'],'w');
fprintf(fid,'1\n^^^ 1 = save geo file, 2=read it, 0=do nothing\
npatch.geo\n^^^ filename for geometry\n20.0 20.0\n^^^ (x,y)
cavity dimensions in cm\n200 200 1\n^^^ x,y,z dimension cells\
n0\n^^^ 0 = cavity, 1=slot\n');
fprintf(fid,'%.2f\n^^^ Layer 1 thickness (cm)\n',h);
fprintf(fid,'4\n^^^ Number of patches\n');
fprintf(fid,'%d %d 1\n^^^ Lower left corner of patch + layer\n',[
x1; y1]);
fprintf(fid,'%d %d\n^^^ Patch dimension in edges\n',[l1; w1]);
fprintf(fid,'%d %d 1\n^^^ Lower left corner of patch + layer\n',[
x2; y2]);
fprintf(fid,'%d %d\n^^^ Patch dimension in edges\n',[l2; w2]);
fprintf(fid,'%d %d 1\n^^^ Lower left corner of patch + layer\n',[
x3; y3]);
fprintf(fid,'%d %d\n^^^ Patch dimension in edges\n',[l3; w3]);
fprintf(fid,'%d %d 1\n^^^ Lower left corner of patch + layer\n',[
x4; y4]);
fprintf(fid,'%d %d\n^^^ Patch dimension in edges\n',[l4; w4]);
fprintf(fid,'0\n^^^ Number of slots\n0\n^^^ Number of shorting
pins\n0\n^^^ Number of metal blocks\n0\n^^^ Number of material
blocks\n1\n^^^ 1 = uniform material fill\n0\n^^^ 0 = non−
dispersive, 1 = dispersive\n1.0 −0.00\n^^^ Complex epsilon (
real, imaginary)\n1.0 −0.00\n^^^ Complex mu (real, imaginary)\
n');

167

fprintf(fid,'0.01 5 60000\n^^^ convergence tolerance, min.
iterations, max. iterations\n0\n^^^ 0 = BiCG, 1 = QMR\n0\n^^^
1 = monitor iterations\n0\n^^^ 0=initial guess, 1=automatic
guess\n1.227 1.6 .348\n^^^ Start, Stop, INCR frequency [GHz]\
n0 0 1\n^^^ Start, Stop, Incr phi angles [deg]\n−90.0 90.0 1.0
\n^^^ Start, Stop, Incr theta angles [deg]\n0\n^^^ 0 = do not
compute far−zone, 1=compute far−zone fields\npatchX_test\n^^^
Six (6) character file prefix\n0\n^^^ 0 = driven, 1=S−
paramters, 2=bistatic\nE_patch_chad_impedance_Opt.zin\n^^^
Input impedance file\n');
fprintf(fid,'50.0 0.0\n^^^ Reference impedance\n1\n^^^ Number of
feeds\n3\n^^^ direction of feed (1=x, 2=y, 3=z)\n');
fprintf(fid,'%.2f %.2f\n^^^ (x,y) location of feed in cm\n',[
probex; probey]);
fprintf(fid,'1\n^^^ Layer of feed\n1.0 0.0\n^^^ Magnitude and
phase of current\n0\n^^^ Number of loads\n0\n^^^ 0 = do NOT
save interior fields, 1= save surface fields\n0\n^^^ Number of
field probe\n');
fclose(fid);
%run kbrick
!kbrick<text.txt >out.txt
%input db's from kbrick output file
fid=fopen('out.txt','rt');
while feof(fid) == 0
clear tline;
tline = fgetl(fid);
if length(tline)>6
if tline(1:7)==' 1.2270'
db1=str2num(tline(length(tline)−5:length(tline)));
end
if tline(1:7)==' 1.5750'
db2=str2num(tline(length(tline)−5:length(tline)));
end
end
end
fclose(fid);

%save results
results(num,1)=db1;
results(num,2)=db2;
results(num,3)=w;
results(num,4)=l;
results(num,5)=sw;
results(num,6)=sl;
results(num,7)=ps;
results(num,8)=probex;
results(num,9)=probey;
results(num,10)=h;
disp(results(num,:))
num=num+1;
% !rm out.txt

168

end
end
end
end
end
end
end
fid=fopen(['results.txt'],'w');
fprintf(fid,'%10s%10s%10s%10s%10s%10s%10s%10s%10s%10s\n','1.227 GHz',
'1.575 GHz','Width','Length ','SlotWidth ','SlotLength','Ps','
Probe X','Probe Y','Height');
for i=1:num−1
fprintf(fid,'%10g%10g%10g%10g%10g%10g%10g%10g%10g%10g\n',results(i,:)
);
end
fclose(fid);

% −−− Executes on button press in pushbutton4.
function pushbutton4_Callback(hObject, eventdata, handles)
% hObject
handle to pushbutton4 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)

function edit4_Callback(hObject, eventdata, handles)
% hObject
handle to edit4 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit4 as text
%
str2double(get(hObject,'String')) returns contents of edit4
as a double

% −−− Executes during object creation, after setting all properties.
function edit4_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit4 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

169

% −−− Executes on selection change in popupmenu2.
function popupmenu2_Callback(hObject, eventdata, handles)
% hObject
handle to popupmenu2 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: contents = get(hObject,'String') returns popupmenu2 contents
as cell array
%
contents{get(hObject,'Value')} returns selected item from
popupmenu2
if isequal(get(hObject,'Value'),1)
set(handles.text4,'Visible','off');
set(handles.edit4,'Visible','off');
else
set(handles.text4,'Visible','on');
set(handles.edit4,'Visible','on');
end
% −−− Executes during object creation, after setting all properties.
function popupmenu2_CreateFcn(hObject, eventdata, handles)
% hObject
handle to popupmenu2 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: popupmenu controls usually have a white background on
Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit5_Callback(hObject, eventdata, handles)
% hObject
handle to edit5 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit5 as text
%
str2double(get(hObject,'String')) returns contents of edit5
as a double

% −−− Executes during object creation, after setting all properties.
function edit5_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit5 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called

170

% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit6_Callback(hObject, eventdata, handles)
% hObject
handle to edit6 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit6 as text
%
str2double(get(hObject,'String')) returns contents of edit6
as a double

% −−− Executes during object creation, after setting all properties.
function edit6_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit6 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

% −−− Executes on selection change in popupmenu3.
function popupmenu3_Callback(hObject, eventdata, handles)
% hObject
handle to popupmenu3 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: contents = get(hObject,'String') returns popupmenu3 contents
as cell array
%
contents{get(hObject,'Value')} returns selected item from
popupmenu3
if isequal(get(hObject,'Value'),1)
set(handles.text6,'Visible','off');
set(handles.edit6,'Visible','off');
else
set(handles.text6,'Visible','on');
set(handles.edit6,'Visible','on');
end
% −−− Executes during object creation, after setting all properties.
function popupmenu3_CreateFcn(hObject, eventdata, handles)

171

% hObject
% eventdata
% handles
called

handle to popupmenu3 (see GCBO)
reserved − to be defined in a future version of MATLAB
empty − handles not created until after all CreateFcns

% Hint: popupmenu controls usually have a white background on
Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit7_Callback(hObject, eventdata, handles)
% hObject
handle to edit7 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit7 as text
%
str2double(get(hObject,'String')) returns contents of edit7
as a double

% −−− Executes during object creation, after setting all properties.
function edit7_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit7 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit8_Callback(hObject, eventdata, handles)
% hObject
handle to edit8 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit8 as text
%
str2double(get(hObject,'String')) returns contents of edit8
as a double

% −−− Executes during object creation, after setting all properties.
function edit8_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit8 (see GCBO)

172

% eventdata
% handles
called

reserved − to be defined in a future version of MATLAB
empty − handles not created until after all CreateFcns

% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

% −−− Executes on selection change in popupmenu4.
function popupmenu4_Callback(hObject, eventdata, handles)
% hObject
handle to popupmenu4 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: contents = get(hObject,'String') returns popupmenu4 contents
as cell array
%
contents{get(hObject,'Value')} returns selected item from
popupmenu4
if isequal(get(hObject,'Value'),1)
set(handles.text8,'Visible','off');
set(handles.edit8,'Visible','off');
else
set(handles.text8,'Visible','on');
set(handles.edit8,'Visible','on');
end
% −−− Executes during object creation, after setting all properties.
function popupmenu4_CreateFcn(hObject, eventdata, handles)
% hObject
handle to popupmenu4 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: popupmenu controls usually have a white background on
Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit9_Callback(hObject, eventdata, handles)
% hObject
handle to edit9 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit9 as text

173

%

str2double(get(hObject,'String')) returns contents of edit9
as a double

% −−− Executes during object creation, after setting all properties.
function edit9_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit9 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit10_Callback(hObject, eventdata, handles)
% hObject
handle to edit10 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit10 as text
%
str2double(get(hObject,'String')) returns contents of edit10
as a double

% −−− Executes during object creation, after setting all properties.
function edit10_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit10 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

% −−− Executes on selection change in popupmenu5.
function popupmenu5_Callback(hObject, eventdata, handles)
% hObject
handle to popupmenu5 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: contents = get(hObject,'String') returns popupmenu5 contents
as cell array

174

%

contents{get(hObject,'Value')} returns selected item from
popupmenu5
if isequal(get(hObject,'Value'),1)
set(handles.text10,'Visible','off');
set(handles.edit10,'Visible','off');
else
set(handles.text10,'Visible','on');
set(handles.edit10,'Visible','on');
end
% −−− Executes during object creation, after setting all properties.
function popupmenu5_CreateFcn(hObject, eventdata, handles)
% hObject
handle to popupmenu5 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: popupmenu controls usually have a white background on
Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit11_Callback(hObject, eventdata, handles)
% hObject
handle to edit11 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit11 as text
%
str2double(get(hObject,'String')) returns contents of edit11
as a double

% −−− Executes during object creation, after setting all properties.
function edit11_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit11 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit12_Callback(hObject, eventdata, handles)

175

% hObject
% eventdata
% handles

handle to edit12 (see GCBO)
reserved − to be defined in a future version of MATLAB
structure with handles and user data (see GUIDATA)

% Hints: get(hObject,'String') returns contents of edit12 as text
%
str2double(get(hObject,'String')) returns contents of edit12
as a double

% −−− Executes during object creation, after setting all properties.
function edit12_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit12 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

% −−− Executes on selection change in popupmenu6.
function popupmenu6_Callback(hObject, eventdata, handles)
% hObject
handle to popupmenu6 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: contents = get(hObject,'String') returns popupmenu6 contents
as cell array
%
contents{get(hObject,'Value')} returns selected item from
popupmenu6
if isequal(get(hObject,'Value'),1)
set(handles.text12,'Visible','off');
set(handles.edit12,'Visible','off');
set(handles.text15,'Visible','off');
set(handles.edit15,'Visible','off');
else
set(handles.text12,'Visible','on');
set(handles.edit12,'Visible','on');
set(handles.text15,'Visible','on');
set(handles.edit15,'Visible','on');
end
% −−− Executes during object creation, after setting all properties.
function popupmenu6_CreateFcn(hObject, eventdata, handles)
% hObject
handle to popupmenu6 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called

176

% Hint: popupmenu controls usually have a white background on
Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit13_Callback(hObject, eventdata, handles)
% hObject
handle to edit13 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit13 as text
%
str2double(get(hObject,'String')) returns contents of edit13
as a double

% −−− Executes during object creation, after setting all properties.
function edit13_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit13 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit14_Callback(hObject, eventdata, handles)
% hObject
handle to edit14 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit14 as text
%
str2double(get(hObject,'String')) returns contents of edit14
as a double

% −−− Executes during object creation, after setting all properties.
function edit14_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit14 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.

177

%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

% −−− Executes on selection change in popupmenu7.
function popupmenu7_Callback(hObject, eventdata, handles)
% hObject
handle to popupmenu7 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: contents = get(hObject,'String') returns popupmenu7 contents
as cell array
%
contents{get(hObject,'Value')} returns selected item from
popupmenu7
if isequal(get(hObject,'Value'),1)
set(handles.text14,'Visible','off');
set(handles.edit14,'Visible','off');
set(handles.text16,'Visible','off');
set(handles.edit16,'Visible','off');
else
set(handles.text14,'Visible','on');
set(handles.edit14,'Visible','on');
set(handles.text16,'Visible','on');
set(handles.edit16,'Visible','on');
end
% −−− Executes during object creation, after setting all properties.
function popupmenu7_CreateFcn(hObject, eventdata, handles)
% hObject
handle to popupmenu7 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: popupmenu controls usually have a white background on
Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit15_Callback(hObject, eventdata, handles)
% hObject
handle to edit15 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit15 as text

178

%

str2double(get(hObject,'String')) returns contents of edit15
as a double

% −−− Executes during object creation, after setting all properties.
function edit15_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit15 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

function edit16_Callback(hObject, eventdata, handles)
% hObject
handle to edit16 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
structure with handles and user data (see GUIDATA)
% Hints: get(hObject,'String') returns contents of edit16 as text
%
str2double(get(hObject,'String')) returns contents of edit16
as a double

% −−− Executes during object creation, after setting all properties.
function edit16_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit16 (see GCBO)
% eventdata reserved − to be defined in a future version of MATLAB
% handles
empty − handles not created until after all CreateFcns
called
% Hint: edit controls usually have a white background on Windows.
%
See ISPC and COMPUTER.
if ispc && isequal(get(hObject,'BackgroundColor'), get(0,'
defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end

179

Appendix E: Matlab Imaging Code
for Taguchi-K-brick-Epatch

close all
clear all
format short
db1=0;
db2=0;
% 1 Epatch Width
% 2 Epatch Lenght
% 3 Slot Length
% 4 Slot Width
% 5 Ps
% 6 Dielectric Height
% 7 Probe placement x direction
[filename pathname] = uigetfile('*.txt','Choose the OA file from
online or generated code MUST BE ACII Formated!');
OA=dlmread([pathname filename]);
SizeofOA=size(OA);
Element = 7;% input number of Elements to be tested
if SizeofOA(2) == Element
else
newColummnsize=SizeofOA(2)−Element;
for n=1:newColummnsize % reduces the columm size for expriment
OA(:,SizeofOA(2))=[];
SizeofOA=size(OA);
end
end
A=1; % factor for db1

180

B=1; % factor for db2
% Level Design User specifies level
Level=0:2; %Input by user assuming 3 levels
lengthflevel=length(Level);
Max1=110; % Input by User patch width in mm
Min1=60; % Input by User patch width in mm
Max2=110; % Input by User patch lenght in mm
Min2=60; % Input by User patch lenght in mm
Max3=95; % Input by User slot lenght in mm
Min3=50; % Input by User slot lenght in mm
Max4=10; % Input by User slot width in mm
Min4=1; % Input by User slot width in mm
Max5=25; % Input by User Ps in mm
Min5=5; % Input by User Ps in mm
Max6=1.5; % Input by User dielectric hight in cm
Min6=.1; % Input by User dielctric height in cm
Max7=7; % Input by User probe placment x in cm
Min7=0; % Input by User probe placement x in cm
rr=.9; % Reduction rate used by User
i=0;
diffrentlevels=0:1; % Because not all levels will have the same
range. Specified here is how many are different
LDtime=1.5;
Ld1= LDtime*(Max1 − Min1)/(lengthflevel+1); % first specifed level
differnce first interation

Ld2= LDtime*(Max2 − Min2)/(lengthflevel+1); % first specifed level
differnce first interation

Ld3= LDtime*(Max3 − Min3)/(lengthflevel+1); % first specifed level
differnce first interation

Ld4= LDtime*(Max4 − Min4)/(lengthflevel+1); % first specifed level
differnce first interation

Ld5=LDtime*(Max5 − Min5)/(lengthflevel+1); % first specifed level
differnce first interation

Ld6= LDtime*(Max6 − Min6)/(lengthflevel+1); % first specifed level
differnce first interation

181

Ld7=LDtime*(Max7 − Min7)/(lengthflevel+1); % first specifed level
differnce first interation

Ldi1=round(Ld1*rr^i);
Ldi2=round(Ld2*rr^i);
Ldi3=round(Ld3*rr^i);
Ldi4=round(Ld4*rr^i);
Ldi5=round(Ld5*rr^i);
Ldi6=(Ld6*rr^i);
Ldi7=(Ld7*rr^i);

while (Ldi1/Ld1)>.0001;
clear Level LevelOA2 LevelOA3 LevelOA1 db1 db2
Level=0:2;
clear n b g c

Ldi1=round(Ld1*rr^i);
Ldi2=round(Ld2*rr^i);
Ldi3=round(Ld3*rr^i);
Ldi4=round(Ld4*rr^i);
Ldi5=round(Ld5*rr^i);
Ldi6=(Ld6*rr^i);
Ldi7=(Ld7*rr^i);

for n=1:SizeofOA(2)

182

if i==0
switch n
case 1
Level(2)= (Max1−Min1)/2 + Min1;% Middle of range
Level(1)=Level(2)−Ldi1;
Level(3)=Level(2)+Ldi1;
case 2
Level(2)= (Max2−Min2)/2 + Min2;% Middle of range
Level(1)=Level(2)−Ldi2;
Level(3)=Level(2)+Ldi2;
case 3
Level(2)= (Max3−Min3)/2 + Min3;% Middle of range
Level(1)=Level(2)−Ldi3;
Level(3)=Level(2)+Ldi3;
case 4
Level(2)= (Max4−Min4)/2 + Min4;% Middle of range
Level(1)=Level(2)−Ldi4;
Level(3)=Level(2)+Ldi4;
case 5
Level(2)= (Max5−Min5)/2 + Min5;% Middle of range
Level(1)=Level(2)−Ldi5;
Level(3)=Level(2)+Ldi5;
case 6
Level(2)= (Max6−Min6)/2 + Min6;% Middle of range
Level(1)=Level(2)−Ldi6;
Level(3)=Level(2)+Ldi6;
case 7
Level(2)= (Max7−Min7)/2 + Min7;% Middle of range
Level(1)=Level(2)−Ldi7;
Level(3)=Level(2)+Ldi7;
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
else
switch n
case 1
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi1;
Level(3)=Level(2)+Ldi1;
if Level(1)≤Min1
Level(1)=Min1;
else
end
if Level(3)≥Max1
Level(3)=Max1;
else
end

183

LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 2
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi2;
Level(3)=Level(2)+Ldi2;
if Level(1)≤Min2
Level(1)=Min2;
else
end
if Level(3)≥Max2
Level(3)=Max2;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 3
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi3;
Level(3)=Level(2)+Ldi3;
if Level(1)≤Min3
Level(1)=Min3;
else
end
if Level(3)≥Max3
Level(3)=Max3;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 4
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi4;
Level(3)=Level(2)+Ldi4;
if Level(1)≤Min4
Level(1)=Min4;
else
end
if Level(3)≥Max4
Level(3)=Max4;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 5

184

Level(2)= OAr(n);
Level(1)=Level(2)−Ldi5;
Level(3)=Level(2)+Ldi5;
if Level(1)≤Min5
Level(1)=Min5;
else
end
if Level(3)≥Max5
Level(3)=Max5;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 6
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi6;
Level(3)=Level(2)+Ldi6;
if Level(1)≤Min6
Level(1)=Min6;
else
end
if Level(3)≥Max6
Level(3)=Max6;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 7
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi7;
Level(3)=Level(2)+Ldi7;
if Level(1)≤Min7
Level(1)=Min7;
else
end
if Level(3)≥Max7
Level(3)=Max7;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);

end

185

end

end

countoflevel=0;

clear LevelOA

for n=1:SizeofOA(1) % Filling up a Level based OA
for b=1:SizeofOA(2)

if OA(n,b)==0
LevelOA(n,b)=LevelOA1(b);
elseif OA(n,b)==1
LevelOA(n,b)=LevelOA2(b);
else
LevelOA(n,b)=LevelOA3(b);
end

if b==7
if LevelOA(n,1)≤LevelOA(n,3)
LevelOA(n,3)=Min1−5;
else
end
if (LevelOA(n,1)/20)≤LevelOA(n,7)
LevelOA(n,7)=(LevelOA(n,1)/20)−.2;
else
end
else
end

end
LevelOATrack(n,b,(i+1))=LevelOA(n,b);

end

186

clear FitnessF
clear F1 F2 F3
% 1 Epatch Width
% 2 Epatch Lenght
% 3 Slot Length
% 4 Slot Width
% 5 Ps
% 6 Dielectric Height
% 7 Probe placement x direction
FirstT=clock;
for n=1:SizeofOA(1)
tic;
if n==1
else
!rm out.txt
end

if not(mod(LevelOA(n,4),2)==mod(LevelOA(n,2),2))
w=LevelOA(n,2)+1;
else
w=LevelOA(n,2);
end

l=LevelOA(n,1);

sl=LevelOA(n,3);
sw=LevelOA(n,4);
ps=LevelOA(n,5);
h=LevelOA(n,6);
probex=LevelOA(n,7);
x1=round((200−l)/2); % probe placement x direction piece 1
l1=(l−sl); % slot length of the first piece
y1=round((200−w)/2); % probde placement y direction piece 1
w1=(w); % width of E−patch
x2=round((200−l)/2); % probe placement x direction piece 2
l2=(l); % lenght of E−patch
y2=round((200−w)/2); % probe placement y dircetion piece 2
w2=round(w/2−sw/2−ps); %with of piece 2 based on how wide slot needs
to be
x3=round((200−l)/2); % probe placement x direction piece 3
l3=(l); % lenght of piece 3

187

y3=round(100+sw/2−ps); % probe placement y direction of piece 3 based
on PS
w3=(2*ps−sw); % width of piece 3 based on PS
x4=round((200−l)/2); % probe placement piece 4 x
l4=(l); % lenght of piece 4
y4=round(100+ps+sw/2); % probe placem piece 4 in y
w4=round(w/2−sw/2−ps); % width of piece 4
probey=mod(w3,2)/20; % probe placement in y direction always in the
symetric center

fid=fopen(['text.txt'],'w');
fprintf(fid,'1\n^^^ 1 = save geo file, 2=read it, 0=do nothing\
npatch.geo\n^^^ filename for geometry\n20.0 20.0\n^^^ (x,y)
cavity dimensions in cm\n200 200 1\n^^^ x,y,z dimension cells\
n0\n^^^ 0 = cavity, 1=slot\n');
fprintf(fid,'%.2f\n^^^ Layer 1 thickness (cm)\n',h);
fprintf(fid,'4\n^^^ Number of patches\n');
fprintf(fid,'%.0f %.0f 1\n^^^ Lower left corner of patch + layer\
n',[x1; y1]);
fprintf(fid,'%.0f %.0f \n^^^ Patch dimension in edges\n',[l1; w1
]);
fprintf(fid,'%.0f %.0f 1\n^^^ Lower left corner of patch + layer\
n',[x2; y2]);
fprintf(fid,'%.0f %.0f \n^^^ Patch dimension in edges\n',[l2 ; w2
]);
fprintf(fid,'%.0f %.0f 1\n^^^ Lower left corner of patch + layer\
n',[x3; y3]);
fprintf(fid,'%.0f %.0f \n^^^ Patch dimension in edges\n',[l3; w3
]);
fprintf(fid,'%.0f %.0f 1\n^^^ Lower left corner of patch + layer\
n',[x4; y4]);
fprintf(fid,'%.0f %.0f \n^^^ Patch dimension in edges\n',[l4; w4
]);
fprintf(fid,'0\n^^^ Number of slots\n0\n^^^ Number of shorting
pins\n0\n^^^ Number of metal blocks\n0\n^^^ Number of material
blocks\n1\n^^^ 1 = uniform material fill\n0\n^^^ 0 = non−
dispersive, 1 = dispersive\n1.0 −0.00\n^^^ Complex epsilon (
real, imaginary)\n1.0 −0.00\n^^^ Complex mu (real, imaginary)\
n');
fprintf(fid,'0.01 5 60000\n^^^ convergence tolerance, min.
iterations, max. iterations\n0\n^^^ 0 = BiCG, 1 = QMR\n0\n^^^
1 = monitor iterations\n0\n^^^ 0=initial guess, 1=automatic
guess\n1.227 1.6 .348\n^^^ Start, Stop, INCR frequency [GHz]\
n0 0 1\n^^^ Start, Stop, Incr phi angles [deg]\n−90.0 90.0 1.0
\n^^^ Start, Stop, Incr theta angles [deg]\n0\n^^^ 0 = do not
compute far−zone, 1=compute far−zone fields\npatchX_test\n^^^
Six (6) character file prefix\n0\n^^^ 0 = driven, 1=S−
paramters, 2=bistatic\nE_patch_chad_impedance_Opt.zin\n^^^
Input impedance file\n');
fprintf(fid,'50.0 0.0\n^^^ Reference impedance\n1\n^^^ Number of
feeds\n3\n^^^ direction of feed (1=x, 2=y, 3=z)\n');

188

fprintf(fid,'%.2f %.2f\n^^^ (x,y) location of feed in cm\n',[
probex; probey]);
fprintf(fid,'1\n^^^ Layer of feed\n1.0 0.0\n^^^ Magnitude and
phase of current\n0\n^^^ Number of loads\n0\n^^^ 0 = do NOT
save interior fields, 1= save surface fields\n0\n^^^ Number of
field probe\n');
fclose(fid);
%run kbrick
!k_brick <text.txt > out.txt
%input db's from kbrick output file
fid=fopen('out.txt','rt');
while feof(fid) == 0
clear tline;
tline = fgetl(fid);
if length(tline)>6
if tline(1:7)==' 1.2270'
db1=str2num(tline(length(tline)−5:length(tline)));
end
if tline(1:7)==' 1.5750'
db2=str2num(tline(length(tline)−5:length(tline)));
end
end
end
fclose(fid);
% build fitness function

%

FitnessF(1,n) =int((db1+15),x,1.177,1.277)+int((db2+15),x,1.525,1
.625);
FitnessF(1,n)=abs(A*(db1+40))+abs((db2+40));
disp('FITNESS for Experiment');disp(n);disp('Iteration');disp(i)
disp(FitnessF(1,n));

end
Time2=clock;
TimeF=Time2−FirstT;
disp('Time for one whole set of 27 expriments');disp(TimeF)
disp(' LevelOA for Iteration');disp((i+1))
disp(LevelOATrack(:,:,(i+1)));
clear StoN
StoN=−20*log10((FitnessF)); % Signal to Noise Ratio
clear FinalResponse
%Build Response table

for n=1:SizeofOA(2)

189

response=0;
response2=0;
response3=0;
countoflevel=0;
for h=1:SizeofOA(1)
if OA(h,n)==0
response=response+StoN(h);
countoflevel=countoflevel+1;
elseif OA(h,n)==1
response2=response2+StoN(h);
elseif OA(h,n)==2
response3=response3+StoN(h);
end
end
for c=1:lengthflevel
if c==1
FinalResponse(c,n)=response/countoflevel;
elseif c==2
FinalResponse(c,n)=response2/countoflevel;
elseif c==3
FinalResponse(c,n)=response3/countoflevel;
end
end
end
clear Optparameters
disp('Final Response table for iteration');disp(i)
disp(FinalResponse)
clear OptMat
Optparameters=max(FinalResponse);
for n=1:SizeofOA(2) % Matrix use to single out the good s/n's
for c=1:lengthflevel
if Optparameters(n)==FinalResponse(c,n)
OptMat(c,n)=FinalResponse(c,n);
else
OptMat(c,n)=pi;
end
end

190

end
clear OAr
for n=1:SizeofOA(2) % matrix used to specify final levels that are
optimal
for c=1:lengthflevel
if OptMat(c,n)==pi;
else
switch c
case 1
OAr(n)=LevelOA1(n);
case 2
OAr(n)=LevelOA2(n);
case 3
OAr(n)=LevelOA3(n);
end
end
end
end
% The design for this Fitness Function for use only for Taguchi

!rm text.txt
!rm out.txt
if OAr(1)≤OAr(3) % redundent check of slotlenght and patch width
OAr(3)=Min1−5;
else
end
if (OAr(1)/20)≤OAr(7)%make sure probe is on patch
OAr(7)=(OAr(1)/20)−.2;
else
end
if not(mod(OAr(4),2)==mod(OAr(2),2))
w=OAr(2)+1;
else
w=OAr(2);
end

l=OAr(1);
sl=OAr(3);
sw=OAr(4);

191

ps=OAr(5);
h=OAr(6);
probex=OAr(7);
x1=round((200−l)/2); % probe placement x direction piece 1
l1=(l−sl); % slot length width of the first piece
y1=round((200−w)/2); % probde placement y direction piece 1
w1=(w); % width of E−patch
x2=round((200−l)/2); % probe placement x direction piece 2
l2=(l); % lenght of E−patch
y2=round((200−w)/2); % probe placement y dircetion piece 2
w2=round(w/2−sw/2−ps); %with of piece 2 based on how wide slot needs
to be
x3=round((200−l)/2); % probe placement x direction piece 3
l3=(l); % lenght of piece 3
y3=round(100+sw/2−ps); % probe placement y direction of piece 3 based
on PS
w3=(2*ps−sw); % width of piece 3 based on PS
x4=round((200−l)/2); % probe placement piece 4 x
l4=(l); % lenght of piece 4
y4=round(100+ps+sw/2); % probe placem piece 4 in y
w4=round(w/2−sw/2−ps); % width of piece 4
probey=mod(w3,2)/20; % probe placement in y direction always in the
symetric center

fid=fopen(['text.txt'],'w');
fprintf(fid,'1\n^^^ 1 = save geo file, 2=read it, 0=do nothing\
npatch.geo\n^^^ filename for geometry\n20.0 20.0\n^^^ (x,y)
cavity dimensions in cm\n200 200 1\n^^^ x,y,z dimension cells\
n0\n^^^ 0 = cavity, 1=slot\n');
fprintf(fid,'%.2f\n^^^ Layer 1 thickness (cm)\n',h);
fprintf(fid,'4\n^^^ Number of patches\n');
fprintf(fid,'%.0f %.0f 1\n^^^ Lower left corner of patch + layer\
n',[x1; y1]);
fprintf(fid,'%.0f %.0f \n^^^ Patch dimension in edges\n',[l1; w1
]);
fprintf(fid,'%.0f %.0f 1\n^^^ Lower left corner of patch + layer\
n',[x2; y2]);
fprintf(fid,'%.0f %.0f \n^^^ Patch dimension in edges\n',[l2 ; w2
]);
fprintf(fid,'%.0f %.0f 1\n^^^ Lower left corner of patch + layer\
n',[x3; y3]);
fprintf(fid,'%.0f %.0f \n^^^ Patch dimension in edges\n',[l3; w3
]);
fprintf(fid,'%.0f %.0f 1\n^^^ Lower left corner of patch + layer\
n',[x4; y4]);
fprintf(fid,'%.0f %.0f \n^^^ Patch dimension in edges\n',[l4; w4
]);
fprintf(fid,'0\n^^^ Number of slots\n0\n^^^ Number of shorting
pins\n0\n^^^ Number of metal blocks\n0\n^^^ Number of material
blocks\n1\n^^^ 1 = uniform material fill\n0\n^^^ 0 = non−
dispersive, 1 = dispersive\n1.0 −0.00\n^^^ Complex epsilon (

192

real, imaginary)\n1.0 −0.00\n^^^ Complex mu (real, imaginary)\
n');
fprintf(fid,'0.01 5 60000\n^^^ convergence tolerance, min.
iterations, max. iterations\n0\n^^^ 0 = BiCG, 1 = QMR\n0\n^^^
1 = monitor iterations\n0\n^^^ 0=initial guess, 1=automatic
guess\n1.227 1.6 .348\n^^^ Start, Stop, INCR frequency [GHz]\
n0 0 1\n^^^ Start, Stop, Incr phi angles [deg]\n−90.0 90.0 1.0
\n^^^ Start, Stop, Incr theta angles [deg]\n0\n^^^ 0 = do not
compute far−zone, 1=compute far−zone fields\npatchX_test\n^^^
Six (6) character file prefix\n0\n^^^ 0 = driven, 1=S−
paramters, 2=bistatic\nE_patch_chad_impedance_Opt.zin\n^^^
Input impedance file\n');
fprintf(fid,'50.0 0.0\n^^^ Reference impedance\n1\n^^^ Number of
feeds\n3\n^^^ direction of feed (1=x, 2=y, 3=z)\n');
fprintf(fid,'%.2f %.2f\n^^^ (x,y) location of feed in cm\n',[
probex; probey]);
fprintf(fid,'1\n^^^ Layer of feed\n1.0 0.0\n^^^ Magnitude and
phase of current\n0\n^^^ Number of loads\n0\n^^^ 0 = do NOT
save interior fields, 1= save surface fields\n0\n^^^ Number of
field probe\n');
fclose(fid);
%run kbrick
!k_brick <text.txt >out.txt
%input db's from kbrick output file
fid=fopen('out.txt','rt');
while feof(fid) == 0
clear tline;
tline = fgetl(fid);
if length(tline)>6
if tline(1:7)==' 1.2270'
db1=str2num(tline(length(tline)−5:length(tline)));
end
if tline(1:7)==' 1.5750'
db2=str2num(tline(length(tline)−5:length(tline)));
end
end
end
fclose(fid);
% build fitness function
% FitnessG=int((db1+15),x,1.177,1.277)+int((db2+15),x,1.525,1.625);
FitnessG=abs(A*(db1+40))+abs((db2+40));
disp('This is the Fitness of the "bestfit" function for iteration');
disp(i);
disp(FitnessG);
meck(i+1)=FitnessG;
for n=1:Element
FinalDs(i+1,n)=OAr(n);
end

193

disp('Values for E−patch for iteration');disp(i);
disp(OAr);
!rm out.txt
i=i+1;
Store_db1(i+1)=(db1)
Store2_db2(i+1)=(db2)
%Write and update a output file
fid=fopen(['Taguchi_Epatch_heightrange_5.txt'],'w+');
fprintf(fid,'Max1
Min1
Max2
Min2
Max3
Min3
Max4
Min4
Max5
Min5
Max6
Min6
Max7
Min7 \n')
fprintf(fid,' %.2f
%.2f
%.2f
%.2f
%.2f
%
.2f
%.2f
%.2f
%.2f
%.2f
%.2f
%.2f
%.2f
%.2f
', [ Max1; Min1; Max2; Min2; Max3; Min3;
Max4; Min4; Max5; Min5; Max6; Min6; Max7; Min7])
fprintf(fid,'\n Iteration
Epatch Width
Epatch Length
Sloth
Lenght
Slot Width
Ps
Dielectric Height
Probe X 1
.227 1.575')
for j=1:i

for n=1:Element
if n==1
fprintf(fid,'\n
%.2f
%.2f
', [j; FinalDs(j,n)]);
else
fprintf(fid,'
%.2f
',FinalDs(j,n));
end
if n==Element
fprintf(fid,' %.2f %.2f ' , [Store_db1(j+1); Store2_db2(j+1)
]);
else
end
end
end
fclose(fid);
end

meck(i+1)=FitnessG;
plotness=1:length(meck);
figure,plot(plotness,meck);
h = legend('Final Fitness');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 27 expriements')
ylabel('Fitness')
title('Fitness vs Runs')

194

figure,plot(plotness,Store_db1);
h = legend('db1');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 27 expriements')
ylabel('db1 values')
title('db1 values')
figure,plot(plotness,Store2_db2);
h = legend('db2');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 27 expriements')
ylabel('db2 values')
title('db2 values')

195

Appendix F: Matlab Imaging Code
for Taguchi-K-brick-Rectangular
Patch
This attached code is for a Taguchi optimizer for a rectangular-cavity-backed patch
antenna, simulated in K-brick. Figure F.1 shows the iteration of Taguchi-K-brick
optimization vs |S11 | in dB of the simulated patch antenna. For both frequencies a
return loss of at least 35 dB was found. After 27 iterations |S11 | was -36.23 dB for
1.5 GHz and -37.27 dB after 24 iteration for 1.2 GHz.

close all
clear all
format short
db1=0;
db2=0;
% 1 Epatch Width
% 2 Epatch Lenght
% 3 Slot Length
% 4 Slot Width
% 5 Ps
% 6 Dielectric Height
% 7 Probe placement x direction
[filename pathname] = uigetfile('*.txt','Choose the OA file from
online or generated code MUST BE ACII Formated!');
OA=dlmread([pathname filename]);
SizeofOA=size(OA);
Element = 4;% input number of Elements to be tested
if SizeofOA(2) == Element
else
newColummnsize=SizeofOA(2)−Element;

196

for n=1:newColummnsize % reduces the columm size for expriment
OA(:,SizeofOA(2))=[];
SizeofOA=size(OA);
end
end
A=1; % factor for db1
B=1; % factor for db2
% Level Design User specifies level
Level=0:2; %Input by user assuming 3 levels
lengthflevel=length(Level);
Max1=130; % Input by User patch width in mm
Min1=60; % Input by User patch width in mm
Max2=130; % Input by User patch lenght in mm
Min2=60; % Input by User patch lenght in mm
Max3=6.5; % Input by User probe in x in cm
Min3=0; % Input by User probe in x in cm
Max4=1.5; % Input by User dielectric h in cm
Min4=.5; % Input by User tric hdielec in cm

rr=.9; % Reduction rate used by User
i=0;
diffrentlevels=0:1; % Because not all levels will have the same
range. Specified here is how many are different
LDtime=1.5;
Ld1= LDtime*(Max1 − Min1)/(lengthflevel+1); % first specifed level
differnce first interation

Ld2= LDtime*(Max2 − Min2)/(lengthflevel+1); % first specifed level
differnce first interation

Ld3= LDtime*(Max3 − Min3)/(lengthflevel+1); % first specifed level
differnce first interation

Ld4=LDtime*(Max4 − Min4)/(lengthflevel+1); % first specifed level
differnce first interation

Ldi1=round(Ld1*rr^i);
Ldi2=round(Ld2*rr^i);

197

Ldi3=(Ld3*rr^i);
Ldi4=(Ld4*rr^i);

while (Ldi1/Ld1)>.0001;
clear Level LevelOA2 LevelOA3 LevelOA1 db1 db2
Level=0:2;
clear n b g c

Ldi1=round(Ld1*rr^i);
Ldi2=round(Ld2*rr^i);
Ldi3=(Ld3*rr^i);
Ldi4=(Ld4*rr^i);

for n=1:SizeofOA(2)

if i==0
switch n
case 1
Level(2)= (Max1−Min1)/2 + Min1;% Middle of range
Level(1)=Level(2)−Ldi1;
Level(3)=Level(2)+Ldi1;
case 2
Level(2)= (Max2−Min2)/2 + Min2;% Middle of range
Level(1)=Level(2)−Ldi2;
Level(3)=Level(2)+Ldi2;
case 3
Level(2)= (Max3−Min3)/2 + Min3;% Middle of range
Level(1)=Level(2)−Ldi3;
Level(3)=Level(2)+Ldi3;
case 4
Level(2)= (Max4−Min4)/2 + Min4;% Middle of range
Level(1)=Level(2)−Ldi4;
Level(3)=Level(2)+Ldi4;
case 5

198

end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);

else
switch n
case 1
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi1;
Level(3)=Level(2)+Ldi1;
if Level(1)≤Min1
Level(1)=Min1;
else
end
if Level(3)≥Max1
Level(3)=Max1;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 2
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi2;
Level(3)=Level(2)+Ldi2;
if Level(1)≤Min2
Level(1)=Min2;
else
end
if Level(3)≥Max2
Level(3)=Max2;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 3
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi3;
Level(3)=Level(2)+Ldi3;
if Level(1)≤Min3
Level(1)=Min3;
else
end

199

if Level(3)≥Max3
Level(3)=Max3;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 4
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi4;
Level(3)=Level(2)+Ldi4;
if Level(1)≤Min4
Level(1)=Min4;
else
end
if Level(3)≥Max4
Level(3)=Max4;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
end

end
end

countoflevel=0;

clear LevelOA

for n=1:SizeofOA(1) % Filling up a Level based OA
for b=1:SizeofOA(2)

if OA(n,b)==0
LevelOA(n,b)=LevelOA1(b);
elseif OA(n,b)==1
LevelOA(n,b)=LevelOA2(b);
else
LevelOA(n,b)=LevelOA3(b);
end

200

if b==4 % check for probe is on patch
if (LevelOA(n,1)/20)≤LevelOA(n,3)
LevelOA(n,3)=(LevelOA(n,1)/20)−.2;
else
end
end
LevelOATrack(n,b,(i+1))=LevelOA(n,b);

end

end

clear FitnessF
clear F1 F2 F3
% 1 Epatch Width
% 2 Epatch Lenght
% 3 Slot Length
% 4 Slot Width
% 5 Ps
% 6 Dielectric Height
% 7 Probe placement x direction
FirstT=clock;
for n=1:SizeofOA(1)
tic;
if n==1
else
!rm out.txt
end

w=LevelOA(n,2);
l=LevelOA(n,1);

h=LevelOA(n,4);
probex=LevelOA(n,3);
BlocksX=floor(probex*10+99);
BlocksY=99;
x1=round((200−l)/2); % probe placement x direction piece 1

201

l1=round(l); % patch width of the first piece
y1=round((200−w)/2); % probde placement y direction piece 1
w1=round(w); % lenght of patch

probey=mod(w,2)/20; % probe placement in y direction always in the
symetric center
fid=fopen(['text_normal.txt'],'w');
fprintf(fid,'1\n^^^ 1 = save geo file, 2=read it, 0=do nothing\
npatch.geo\n^^^ filename for geometry\n20.0 20.0\n^^^ (x,y)
cavity dimensions in cm\n200 200 1\n^^^ x,y,z dimension cells\
n0\n^^^ 0 = cavity, 1=slot\n');
fprintf(fid,'%.3f\n^^^ Layer 1 thickness (cm)\n',h);
fprintf(fid,'1\n^^^ Number of patches\n');
fprintf(fid,'%.0f %.0f 1\n^^^ Lower left corner of patch + layer\
n',[x1; y1]);
fprintf(fid,'%.0f %.0f \n^^^ Patch dimension in edges\n',[l1; w1
]);
fprintf(fid,'0\n^^^ Number of slots\n0\n^^^ Number of shorting
pins\n0\n^^^ Number of metal blocks\n0\n^^^ Number of material
blocks\n1\n^^^ 1 = uniform material fill\n0\n^^^ 0 = non−
dispersive, 1 = dispersive\n1.0 −0.00\n^^^ Complex epsilon (
real, imaginary)\n1.0 −0.00\n^^^ Complex mu (real, imaginary)\
n');
fprintf(fid,'0.01 5 60000\n^^^ convergence tolerance, min.
iterations, max. iterations\n0\n^^^ 0 = BiCG, 1 = QMR\n0\n^^^
1 = monitor iterations\n0\n^^^ 0=initial guess, 1=automatic
guess\n1.2 1.6 .448\n^^^ Start, Stop, INCR frequency [GHz]\n0
0 1\n^^^ Start, Stop, Incr phi angles [deg]\n−90.0 90.0 1.0\n
^^^ Start, Stop, Incr theta angles [deg]\n0\n^^^ 0 = do not
compute far−zone, 1=compute far−zone fields\npatchX_test\n^^^
Six (6) character file prefix\n0\n^^^ 0 = driven, 1=S−
paramters, 2=bistatic\nE_patch_chad_impedance_Opt.zin\n^^^
Input impedance file\n');
fprintf(fid,'50.0 0.0\n^^^ Reference impedance\n1\n^^^ Number of
feeds\n3\n^^^ direction of feed (1=x, 2=y, 3=z)\n');
fprintf(fid,'%.3f %.3f\n^^^ (x,y) location of feed in cm\n',[
probex; probey]);
fprintf(fid,'1\n^^^ Layer of feed\n1.0 0.0\n^^^ Magnitude and
phase of current\n0\n^^^ Number of loads\n0\n^^^ 0 = do NOT
save interior fields, 1= save surface fields\n0\n^^^ Number of
field probe\n');
fclose(fid);
!k_brick <text_normal.txt >out.txt
fid=fopen('out.txt','rt');
while feof(fid) == 0
clear tline;
tline = fgetl(fid);
if length(tline)>6
if tline(1:7)==' 1.2000'
db1=str2num(tline(length(tline)−5:length(tline)));
end

202

end
end
fclose(fid);
% build fitness function

%

FitnessF(1,n) =int((db1+15),x,1.177,1.277)+int((db2+15),x,1.525,1
.625);
FitnessF(1,n)=abs(A*(db1+50));
disp('FITNESS for Experiment');disp(n);disp('Iteration');disp(i)
disp(FitnessF(1,n));

end
Time2=clock;
TimeF=Time2−FirstT;
disp('Time for one whole set of 27 expriments');disp(TimeF)
disp(' LevelOA for Iteration');disp((i+1))
disp(LevelOATrack(:,:,(i+1)));
clear StoN
StoN=−20*log10((FitnessF)); % Signal to Noise Ratio
clear FinalResponse
%Build Response table

for n=1:SizeofOA(2)
response=0;
response2=0;
response3=0;
countoflevel=0;
for h=1:SizeofOA(1)
if OA(h,n)==0
response=response+StoN(h);
countoflevel=countoflevel+1;
elseif OA(h,n)==1
response2=response2+StoN(h);
elseif OA(h,n)==2
response3=response3+StoN(h);
end
end
for c=1:lengthflevel

203

if c==1
FinalResponse(c,n)=response/countoflevel;
elseif c==2
FinalResponse(c,n)=response2/countoflevel;
elseif c==3
FinalResponse(c,n)=response3/countoflevel;
end
end
end
clear Optparameters
disp('Final Response table for iteration');disp(i)
disp(FinalResponse)
clear OptMat
Optparameters=max(FinalResponse);
for n=1:SizeofOA(2) % Matrix use to single out the good s/n's
for c=1:lengthflevel
if Optparameters(n)==FinalResponse(c,n)
OptMat(c,n)=FinalResponse(c,n);
else
OptMat(c,n)=pi;
end
end
end
clear OAr
for n=1:SizeofOA(2) % matrix used to specify final levels that are
optimal
for c=1:lengthflevel
if OptMat(c,n)==pi;
else
switch c
case 1
OAr(n)=LevelOA1(n);
case 2
OAr(n)=LevelOA2(n);
case 3
OAr(n)=LevelOA3(n);
end
end
end
end

204

% The design for this Fitness Function for use only for Taguchi

!rm text.txt
!rm out.txt

w=OAr(2);
l=OAr(1);

if (OAr(1)/20)≤OAr(3)
OAr(3)=(OAr(1)/20)−.2;
else
end

h=OAr(4);
probex=OAr(3);
BlocksX=floor(probex*10+99);
BlocksY=99;
x1=round((200−l)/2); % probe placement x direction piece 1
l1=round(l); % patch width of the first piece
y1=round((200−w)/2); % probde placement y direction piece 1
w1=round(w); % lenght width of E−patch

probey=mod(w,2)/20; % probe placement in y direction always in the
symetric center

fid=fopen(['text_normal.txt'],'w');
fprintf(fid,'1\n^^^ 1 = save geo file, 2=read it, 0=do nothing\
npatch.geo\n^^^ filename for geometry\n20.0 20.0\n^^^ (x,y)
cavity dimensions in cm\n200 200 1\n^^^ x,y,z dimension cells\
n0\n^^^ 0 = cavity, 1=slot\n');
fprintf(fid,'%.3f\n^^^ Layer 1 thickness (cm)\n',h);
fprintf(fid,'1\n^^^ Number of patches\n');
fprintf(fid,'%.0f %.0f 1\n^^^ Lower left corner of patch + layer\
n',[x1; y1]);
fprintf(fid,'%.0f %.0f \n^^^ Patch dimension in edges\n',[l1; w1
]);
fprintf(fid,'0\n^^^ Number of slots\n0\n^^^ Number of shorting
pins\n0\n^^^ Number of metal blocks\n0\n^^^ Number of material
blocks\n1\n^^^ 1 = uniform material fill\n0\n^^^ 0 = non−
dispersive, 1 = dispersive\n1.0 −0.00\n^^^ Complex epsilon (
real, imaginary)\n1.0 −0.00\n^^^ Complex mu (real, imaginary)\
n');

205

fprintf(fid,'0.01 5 60000\n^^^ convergence tolerance, min.
iterations, max. iterations\n0\n^^^ 0 = BiCG, 1 = QMR\n0\n^^^
1 = monitor iterations\n0\n^^^ 0=initial guess, 1=automatic
guess\n1.2 1.6 .448\n^^^ Start, Stop, INCR frequency [GHz]\n0
0 1\n^^^ Start, Stop, Incr phi angles [deg]\n−90.0 90.0 1.0\n
^^^ Start, Stop, Incr theta angles [deg]\n0\n^^^ 0 = do not
compute far−zone, 1=compute far−zone fields\npatchX_test\n^^^
Six (6) character file prefix\n0\n^^^ 0 = driven, 1=S−
paramters, 2=bistatic\nE_patch_chad_impedance_Opt.zin\n^^^
Input impedance file\n');
fprintf(fid,'50.0 0.0\n^^^ Reference impedance\n1\n^^^ Number of
feeds\n3\n^^^ direction of feed (1=x, 2=y, 3=z)\n');
fprintf(fid,'%.3f %.3f\n^^^ (x,y) location of feed in cm\n',[
probex; probey]);
fprintf(fid,'1\n^^^ Layer of feed\n1.0 0.0\n^^^ Magnitude and
phase of current\n0\n^^^ Number of loads\n0\n^^^ 0 = do NOT
save interior fields, 1= save surface fields\n0\n^^^ Number of
field probe\n');
fclose(fid);
!k_brick <text_normal.txt >out.txt

%run kbrick

%input db's from kbrick output file
fid=fopen('out.txt','rt');
while feof(fid) == 0
clear tline;
tline = fgetl(fid);
if length(tline)>6
if tline(1:7)==' 1.2000'
db1=str2num(tline(length(tline)−5:length(tline)));
end
end
end
fclose(fid);

FitnessG=abs(A*(db1+50));
disp('This is the Fitness of the "bestfit" function for iteration');
disp(i);
disp(FitnessG);
meck(i+1)=FitnessG;
for n=1:Element
FinalDs(i+1,n)=OAr(n);
end
disp('Values for E−patch for iteration');disp(i);
disp(OAr);
!rm out.txt
i=i+1;

206

Store_db1(i+1)=(db1)

%Write and update a output file
fid=fopen(['Rectangular_patch_values_1.2_freq_LD_1_5.txt'],'w+');
fprintf(fid,'Max1
Min1
Max2
Min2
Max3
Min3
Max4
Min4
\n')
fprintf(fid,' %.2f
%.2f
%.2f
%.2f
%.2f
%
.3f
%.3f
%.3f
', [ Max1; Min1; Max2; Min2; Max3;
Min3; Max4; Min4;])
fprintf(fid,'\n Iteration
patch Width
patch Length
Probe X
dielectric height
1.2')
for j=1:i

for n=1:Element
if n==1
fprintf(fid,'\n
%.3f
%.3f
', [j; FinalDs(j,n)]);
else
fprintf(fid,'
%.3f
',FinalDs(j,n));
end
if n==Element
fprintf(fid,' %.3f ' , [Store_db1(j+1)]);
else
end
end
end
fclose(fid);
end

meck(i+1)=FitnessG;
plotness=1:length(meck);
figure,plot(plotness,meck);
h = legend('Final Fitness');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 9 expriements')
ylabel('Fitness')
title('Fitness vs Runs')

figure,plot(plotness,Store_db1);
h = legend('db1');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 9 expriements')
ylabel('db1 values')

207

title('db1 values')

208

Figure F.1: |S11 | as a function of Taguchi iteration number for rectangular patches
optimizing at 1.5 and 1.2 GHz.

209

Appendix G: Matlab code for
Rosenbrock Taguchi optimization
close all
clear all
[filename pathname] = uigetfile('*.txt','Choose the OA file from
online or generated code MUST BE ACII Formated!');
OA=dlmread([pathname filename]);
SizeofOA=size(OA);
Element = 2;% input number of Elements to be tested
if SizeofOA(2) == Element
else
newColummnsize=SizeofOA(2)−Element;
for n=1:newColummnsize % reduces the columm size for expriment
OA(:,SizeofOA(2))=[];
SizeofOA=size(OA);
end
end

% Level Design User specifies level
Level=0:2; %Input by user assuming 3 levels
lengthflevel=length(Level);
Max1=10; % Input by User
Min1=0; % Input by User
Max2=10; % Input by User
Min2=0; % Input by User
rr=.99; % Reduction rate used by User
i=0;
diffrentlevels=0:1; % Because not all levels will have the same
range. Specified here is how many are different
A=2; %changes the level times

210

Ld1= (Max1 − Min1)/(lengthflevel+1); % first specifed level differnce
first interation
Ldi1=Ld1*rr^i;
Ld2= (Max2 − Min2)/(lengthflevel+1); % first specifed level differnce
first interation
Ldi2=Ld2*rr^i;
while (Ldi1/Ld1)>.01;
clear LevelOA2 LevelOA3 LevelOA1
Level=0:2;
clear n b g c

Ld1= (Max1 − Min1)/(lengthflevel+1); % first specifed level differnce
first interation
Ldi1=(Ld1*A)*rr^i;
Ld2= (Max2 − Min2)/(lengthflevel+1); % first specifed level differnce
first interation
Ldi2=(Ld2*A)*rr^i;

for n=1:SizeofOA(2)

if i==0
switch n
case 1
Level(2)= (Max1−Min1)/2 + Min1;% Middle of range
Level(1)=Level(2)−Ldi1;
Level(3)=Level(2)+Ldi1;
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 2
Level(2)=(Max2−Min2)/2 + Min2; %Middle of range
Level(1)=Level(2)−Ldi2;
Level(3)=Level(2)+Ldi2;
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
end
else
switch n
case 1

211

Level(2)= OAr(n);
Level(1)=Level(2)−Ldi1;
Level(3)=Level(2)+Ldi1;
if Level(1)≤Min1
Level(1)=Min1;
else
end
if Level(3)≥Max1
Level(3)=Max1;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
case 2
Level(2)= OAr(n);
Level(1)=Level(2)−Ldi2;
Level(3)=Level(2)+Ldi2;
if Level(1)≤Min2
Level(1)=Min2;
else
end
if Level(3)≥Max2
Level(3)=Max2;
else
end
LevelOA1(n)=Level(1);
LevelOA2(n)=Level(2);
LevelOA3(n)=Level(3);
end

end

end

countoflevel=0;

clear LevelOA

for n=1:SizeofOA(1) % Filling up a Level based OA
for b=1:SizeofOA(2)

212

if OA(n,b)==0
LevelOA(n,b)=LevelOA1(b);
elseif OA(n,b)==1
LevelOA(n,b)=LevelOA2(b);
else
LevelOA(n,b)=LevelOA3(b);
end
end

end
clear FitnessF
for n=1:SizeofOA(1)

%Fitness=Fitness+(LevelOA(n,g))^2−10*cos(2*pi*(LevelOA(n,g)))+10;
%Fitness=1−abs(sin(pi*(LevelOA(n,1)−3))/(pi*(LevelOA(n,1)−3)))*abs
(sin(pi*(LevelOA(n,2)−3))/(pi*(LevelOA(n,2)−3)));
%Fitness=exp((LevelOA(n,1)*sin(4*LevelOA(n,1))+1.1*LevelOA(n,2)*
sin(2*LevelOA(n,2))));
%
Fitness=20+((LevelOA(n,1)*sin(4*LevelOA(n,1))+1.1*LevelOA(n,2)*
sin(2*LevelOA(n,2))));
%Fitness=exp((1−LevelOA(n,1))^2+100*(LevelOA(n,2)−(LevelOA(n,1))^2)^2
_;

FitnessG=(1−LevelOA(n,1))^2+100*(LevelOA(n,2)−(LevelOA(n,1))^2)^2;
FitnessF(n,1)=FitnessG;
end
clear StoN
StoN=−20*log10((FitnessF)); % Signal to Noise Ratio
clear FinalResponse
%Build Response table

for n=1:SizeofOA(2)
response=0;
response2=0;
response3=0;
countoflevel=0;
for h=1:SizeofOA(1)
if OA(h,n)==0
response=response+StoN(h);
countoflevel=countoflevel+1;

213

elseif OA(h,n)==1
response2=response2+StoN(h);
elseif OA(h,n)==2
response3=response3+StoN(h);
end
end
for c=1:lengthflevel
if c==1
FinalResponse(c,n)=response/countoflevel;
elseif c==2
FinalResponse(c,n)=response2/countoflevel;
elseif c==3
FinalResponse(c,n)=response3/countoflevel;
end
end
end
clear Optparameters

clear OptMat
Optparameters=max(FinalResponse);
for n=1:SizeofOA(2) % Matrix use to single out the good s/n's
for c=1:lengthflevel
if Optparameters(n)==FinalResponse(c,n)
OptMat(c,n)=FinalResponse(c,n);
else
OptMat(c,n)=pi;
end
end
end
clear OAr
for n=1:SizeofOA(2) % matrix used to specify final levels that are
optimal
for c=1:lengthflevel
if OptMat(c,n)==pi;
else
switch c
case 1

214

OAr(n)=LevelOA1(n);
case 2
OAr(n)=LevelOA2(n);
case 3
OAr(n)=LevelOA3(n);
end
end
end
end
% The design for this Fitness Function Raguchi for use only for
Taguchi

%FitnessG=1−abs(sin(pi*(OAr(1)−3))/(pi*(OAr(1)−3)))*abs(sin(pi*(OAr
(2)−3))/(pi*(OAr(2)−3)));
FitnessG=(1−OAr(1))^2+100*(OAr(2)−(OAr(1))^2)^2;
meck(i+1)=(FitnessG);
if FitnessG == 0
Store(i+1)=OAr(1);
Store2(i+1)=OAr(2);
break
else
i=i+1;
Store(i+1)=OAr(1);
Store2(i+1)=OAr(2);
end
end
x=[0:.01:10];
y=[0:.01:10];
ta=length(x);

%
%
% for bah=1:ta
%
%
for cah=1:ta
%
meckfitness(cah,bah)=x(bah)*sin(4*x(bah))+1.1*y(cah)*sin(2*
y(cah));
%
%
end
%
% end
fid=fopen(['Taguchi_Opt_varibles_LD_2_Rosenbrock.txt'],'w+');

215

for j=1:i

fprintf(fid,' %.9f
%.9f
', [meck(j); Store(j); Store2(j)]);

end
fclose(fid);
%figure;
%
% mesh(x,y,meckfitness)
% xlabel('x values')
% ylabel('y values')
% title('global minimun')

meck(i+1)=FitnessG;
plotness=1:length(meck);
figure,plot(plotness,meck);
h = legend('Final Fitness');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 9 expriements')
ylabel('Fitness')
title('Fitness vs Runs')

figure,plot(plotness,Store);
h = legend('x values');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 9 expriements')
ylabel('x values')
title('x values')
figure,plot(plotness,Store2);
h = legend('y vaules');
set(h,'Interpreter','none')
axis auto
xlabel('Number of Runs of 9 expriements')
ylabel('y values')
title('y values')

216

%.9f

\n

Appendix H: Matlab code for gain
measurements at MSU engineering
building
clear all
close all
%converter for radtion plots
R=1.90500; %seperation distance in meters from antennas
c=3e8; %constant speed of light
f=1300E6; %frequency of operation
lambda=c/f; %wavelenght
%Pr= ; %recieved power
%Pt= ; %transmitted power
S21_antenna= −22.367187500 ; %magnatidue of s21 in dB of antenna
S21_cable= .0340366; %magnatidude of s21 in dB of cable
Gt_1300=1/2*(S21_antenna−S21_cable)+ 10*log10((4*pi*R)/lambda);
Gt_1300_lol=1/2*(S21_antenna−S21_cable+ 20*log10((4*pi*R)/lambda));
R=1.90500; %seperation distance in meters from antennas
c=3e8; %constant speed of light
f=1227.5E6; %frequency of operation
lambda=c/f; %wavelenght
%Pr= ; %recieved power
%Pt= ; %transmitted power
S21_antenna= −22.298828125; %magnatidue of s21 in dB of antenna
S21_cable= .029809952; %magnatidude of s21 in dB of cable
Gt_1227=1/2*(S21_antenna−S21_cable)+ 10*log10((4*pi*R)/lambda);
R=1.90500; %seperation distance in meters from antennas
c=3e8; %constant speed of light
f=1575E6; %frequency of operation
lambda=c/f; %wavelenght
%Pr= ; %recieved power
%Pt= ; %transmitted power
S21_antenna= −25.266601562; %magnatidue of s21 in dB of antenna
S21_cable= .040971756; %magnatidude of s21 in dB of cable
Gt_1575=1/2*(S21_antenna−S21_cable)+ 10*log10((4*pi*R)/lambda);

217

R=1.90500; %seperation distance in meters from antennas
c=3e8; %constant speed of light
f=1640E6; %frequency of operation
lambda=c/f; %wavelenght
%Pr= ; %recieved power
%Pt= ; %transmitted power
S21_antenna= −25.250976562; %magnatidue of s21 in dB of antenna
S21_cable= .046300888; %magnatidude of s21 in dB of cable
Gt_1640=1/2*(S21_antenna−S21_cable)+ 10*log10((4*pi*R)/lambda);
Horn_gain_upstairs=[.11 .01 .022 .023; .037 .01 .009 .007; .04 .004
.009 .005; .015 .008 .0032 .002];
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%

Horn_gain_upstairs_1300=
Horn_gain_upstairs_1575=
Horn_gain_upstairs_1640=
Z_X_cross_plane_1227_cylinder.txt
Z_Y_cross_plane_1227_cylinder.txt
Z_X_cross_plane_1575_cylinder.txt
Z_Y_cross_plane_1575_cylinder.txt
Z_X_plane_1227_cylinder.txt
Z_Y_co_plane_1227_cylinder.txt
Z_X_plane_1575_cylinder.txt
Z_Y_co_plane_1575_cylinder.txt

%converter for radtion plots
[degree,gain]= textread(['Z_X_co_polar_1300.txt'], '%f%f');
l=length(degree);
j=1;
n=0;
fid=fopen(['Z_X_co_ploar_1300_cylinder_FINAL.txt'],'w+');
for n=1:100
y=101−n;
fprintf(fid,' \n
))]);

%f

%f

', [−degree(y); (10*log10(gain(n+90)

j=j+1;
end
for n=1:100

218

fprintf(fid,' \n
;

%f

%f

', [degree(n); (10*log10(gain(n)))])

j=j+1;
end

fclose(fid);

[degree,gain]= textread(['Z_X_co_ploar_1300_cylinder_FINAL.txt'], '%f
%f');

l=length(degree);
j=1;
n=0;
fid=fopen(['Z_X_co_ploar_1300_cylinder_FINAL_102610.txt'],'w+');
for n=1:200
y=92−n;
fprintf(fid,' \n %f
%f ', [degree(n); 10*log10((10^(Gt_1300
/10))*(10^(gain(n)/10))/(Horn_gain_upstairs(2,1)))]);
j=j+1;
end
fclose(fid);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[degree,gain]= textread(['Z_Y_co_polar_1300.txt'], '%f%f');
l=length(degree);
j=1;
n=0;
fid=fopen(['Z_Y_co_ploar_1300_cylinder_FINAL.txt'],'w+');
for n=1:100
y=101−n;
fprintf(fid,' \n
))]);

%f

%f

', [−degree(y); (10*log10(gain(n+90)

j=j+1;
end
for n=1:100

219

fprintf(fid,' \n
;

%f

%f

', [degree(n); (10*log10(gain(n)))])

j=j+1;
end

fclose(fid);

[degree,gain]= textread(['Z_Y_co_ploar_1300_cylinder_FINAL.txt'], '%f
%f');

l=length(degree);
j=1;
n=0;
fid=fopen(['Z_Y_co_ploar_1300_cylinder_FINAL_102610.txt'],'w+');
for n=1:200
y=92−n;
fprintf(fid,' \n %f
%f ', [degree(n); 10*log10((10^(Gt_1300
/10))*(10^(gain(n)/10))/(Horn_gain_upstairs(2,1)))]);
j=j+1;
end
fclose(fid);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[degree,gain]= textread(['Z_Y_co_polar_1640.txt'], '%f%f');
l=length(degree);
j=1;
n=0;
fid=fopen(['Z_Y_co_ploar_1640_cylinder_FINAL.txt'],'w+');
for n=1:100
y=101−n;
fprintf(fid,' \n
))]);

%f

%f

', [−degree(y); (10*log10(gain(n+90)

j=j+1;
end
for n=1:100

220

fprintf(fid,' \n
;

%f

%f

', [degree(n); (10*log10(gain(n)))])

j=j+1;
end

fclose(fid);

[degree,gain]= textread(['Z_Y_co_ploar_1640_cylinder_FINAL.txt'], '%f
%f');

l=length(degree);
j=1;
n=0;
fid=fopen(['Z_Y_co_ploar_1640_cylinder_FINAL_102610.txt'],'w+');
for n=1:200
y=92−n;
fprintf(fid,' \n %f
%f ', [degree(n); 10*log10((10^(Gt_1640
/10))*(10^(gain(n)/10))/(Horn_gain_upstairs(4,1)))]);
j=j+1;
end
fclose(fid);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[degree,gain]= textread(['Z_X_co_polar_1640.txt'], '%f%f');
l=length(degree);
j=1;
n=0;
fid=fopen(['Z_X_co_ploar_1640_cylinder_FINAL.txt'],'w+');
for n=1:100
y=101−n;
fprintf(fid,' \n
))]);

%f

%f

', [−degree(y); (10*log10(gain(n+90)

j=j+1;

221

end
for n=1:100

fprintf(fid,' \n
;

%f

%f

', [degree(n); (10*log10(gain(n)))])

j=j+1;
end

fclose(fid);

[degree,gain]= textread(['Z_X_co_ploar_1640_cylinder_FINAL.txt'], '%f
%f');

l=length(degree);
j=1;
n=0;
fid=fopen(['Z_X_co_ploar_1640_cylinder_FINAL_102610.txt'],'w+');
for n=1:200
y=92−n;
fprintf(fid,' \n %f
%f ', [degree(n); 10*log10((10^(Gt_1640
/10))*(10^(gain(n)/10))/(Horn_gain_upstairs(4,1)))]);
j=j+1;
end
fclose(fid);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[degree,gain]= textread(['Z_X_cross_polar_1300.txt'], '%f%f');
l=length(degree);
j=1;
n=0;
fid=fopen(['Z_X_cross_ploar_1300_cylinder_FINAL.txt'],'w+');
for n=1:100
y=101−n;

222

fprintf(fid,' \n
))]);

%f

%f

', [−degree(y); (10*log10(gain(n+90)

%f

%f

', [degree(n); (10*log10(gain(n)))])

j=j+1;
end
for n=1:100

fprintf(fid,' \n
;

j=j+1;
end

fclose(fid);

[degree,gain]= textread(['Z_X_cross_ploar_1300_cylinder_FINAL.txt'],
'%f%f');

l=length(degree);
j=1;
n=0;
fid=fopen(['Z_X_cross_ploar_1300_cylinder_FINAL_102610.txt'],'w+');
for n=1:200
y=92−n;
fprintf(fid,' \n %f
%f ', [degree(n); 10*log10((10^(Gt_1300
/10))*(10^(gain(n)/10))/(Horn_gain_upstairs(2,1)))]);
j=j+1;
end
fclose(fid);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[degree,gain]= textread(['Z_Y_cross_polar_1300.txt'], '%f%f');

223

l=length(degree);
j=1;
n=0;
fid=fopen(['Z_Y_cross_ploar_1300_cylinder_FINAL.txt'],'w+');
for n=1:100
y=101−n;
fprintf(fid,' \n
))]);

%f

%f

', [−degree(y); (10*log10(gain(n+90)

%f

%f

', [degree(n); (10*log10(gain(n)))])

j=j+1;
end
for n=1:100

fprintf(fid,' \n
;

j=j+1;
end

fclose(fid);

[degree,gain]= textread(['Z_Y_cross_ploar_1300_cylinder_FINAL.txt'],
'%f%f');

l=length(degree);
j=1;
n=0;
fid=fopen(['Z_Y_cross_ploar_1300_cylinder_FINAL_102610.txt'],'w+');
for n=1:200
y=92−n;
fprintf(fid,' \n %f
%f ', [degree(n); 10*log10((10^(Gt_1300
/10))*(10^(gain(n)/10))/(Horn_gain_upstairs(2,1)))]);
j=j+1;
end
fclose(fid);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

224

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[degree,gain]= textread(['Z_X_cross_polar_1640.txt'], '%f%f');
l=length(degree);
j=1;
n=0;
fid=fopen(['Z_X_cross_ploar_1640_cylinder_FINAL.txt'],'w+');
for n=1:100
y=101−n;
fprintf(fid,' \n
))]);

%f

%f

', [−degree(y); (10*log10(gain(n+90)

%f

%f

', [degree(n); (10*log10(gain(n)))])

j=j+1;
end
for n=1:100

fprintf(fid,' \n
;

j=j+1;
end

fclose(fid);

[degree,gain]= textread(['Z_X_cross_ploar_1640_cylinder_FINAL.txt'],
'%f%f');

l=length(degree);
j=1;
n=0;
fid=fopen(['Z_X_cross_ploar_1640_cylinder_FINAL_102610.txt'],'w+');
for n=1:200
y=92−n;
fprintf(fid,' \n %f
%f ', [degree(n); 10*log10((10^(Gt_1640
/10))*(10^(gain(n)/10))/(Horn_gain_upstairs(4,1)))]);

225

j=j+1;
end
fclose(fid);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

[degree,gain]= textread(['Z_Y_cross_polar_1640.txt'], '%f%f');
l=length(degree);
j=1;
n=0;
fid=fopen(['Z_Y_cross_ploar_1640_cylinder_FINAL.txt'],'w+');
for n=1:100
y=101−n;
fprintf(fid,' \n
))]);

%f

%f

', [−degree(y); (10*log10(gain(n+90)

%f

%f

', [degree(n); (10*log10(gain(n)))])

j=j+1;
end
for n=1:100

fprintf(fid,' \n
;

j=j+1;
end

fclose(fid);

[degree,gain]= textread(['Z_Y_cross_ploar_1640_cylinder_FINAL.txt'],
'%f%f');

l=length(degree);
j=1;
n=0;
fid=fopen(['Z_Y_cross_ploar_1640_cylinder_FINAL_102610.txt'],'w+');

226

for n=1:200
y=92−n;
fprintf(fid,' \n %f
%f ', [degree(n); 10*log10((10^(Gt_1640
/10))*(10^(gain(n)/10))/(Horn_gain_upstairs(4,1)))]);
j=j+1;
end
fclose(fid);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

227

BIBLIOGRAPHY

228

Bibliography
[1] Xiaoning Ye Yahya Rahmat-Samii Fan Yang, Xue-Xia Zhang. Wide-band eshaped patch antennas for wireless communications. IEEE Transactions on Antennas and Propagation, 49(7), July 2001.
[2] http://www.emtalk.com/mpacalc.php.
[3] Fan Yang Demir Veysel Elsherbeni Atef Wei-Chung Weng. Optimization using
taguchi method for electromagnetic applications. pages 1 –6, Nov. 2006.
[4] Chi Ho Chan, Sai Ho Yeung, Wing Shing Chan, and Kim Fung Man. Uwb sickleshape patch dipolar antenna with stable radiation pattern. pages 1993 –1996,
june 2007.
[5] A.A. Alshehri and A.R. Sebak. A novel uwb planar patch antenna for wireless
communications. pages 1 –4, july 2008.
[6] Nasimuddin, Zhi Ning Chen, T.S.P. See, and Xianming Qing. Multi-dielectric
layer multi-patches microstrip antenna for uwb applications. pages 1019 –1021,
oct. 2007.
[7] E. Chang, SA Long, and WF Richards. An experimental investigation of electrically thick rectangular microstrip antennas. IEEE transactions on antennas
and propagation, 34(6):767–772, 1986.
[8] Mook-Seng Leong Ban-Leong Ooi, Shen Qin. Novel design of broad-band stacked
patch antenna. Antennas and Propagation, IEEE Transactions on, 50(10):1391
– 1395, Oct 2002.
[9] Bulla G. Seraﬁn P.-Fernandez C.R. Monser-G. de Salles A.A.A. Pedra, A.C.O.
Bandwidth and size optimisation of a wide-band e-shaped patch antenna. pages
422 –426, 29 2007-Nov. 1 2007.
[10] Khalil K. Excell-P.S. See C.H. Abd-Alhameed, R.A. Simulation and measurement of broadband microstrip patch antenna for 3g wireless communications.
volume 2, pages 477 – 480 vol.2, March-3 April 2003.
[11] Chahine S.A. Kabalan K.Y.-El-Hajj A.-Chehab A. Bzeih, A. Empirical formulation and design of a broadband enhanced e-patch antenna. pages 1 –9, March
2007.
229

[12] Misran N. Take T.C.-Moniruzzaman-M. Islam, M.T. Optimization of microstrip
patch antenna using particle swarm optimization with curve ﬁtting. volume 02,
pages 711 –714, Aug. 2009.
[13] Cwik T. Rahmat-Samii-Y.-Manteghi M. Villegas, F.J. Parallel genetic-algorithm
optimization of a dual-band patch antenna for wireless communications. volume 1, pages 334 – 337 vol.1, 2002.
[14] A. Hedayat, N.J.A. Sloane, and J. Stufken. Orthogonal arrays: theory and applications. Springer Verlag, 1999.
[15] Ranjit K. Roy. Design of Experiments Using the Taguchi Approach. John Wiley
and Sons inc., 2001.
[16] CR Rao. Hypercubes of strength d leading to confounded designs in factorial
experiments. Bull. Calcutta Math. Soc, 38:67–78, 1946.
[17] C.R. Rao. Factorial experiments derivable from combinatorial arrangements of
arrays. Supplement to the Journal of the Royal Statistical Society, 9(1):128–139,
1947.
[18] C.R. Rao. On a class of arrangements. Proceedings of the Edinburgh Mathematical
Society, 8(03):119–125, 1949.
[19] Elsherbeni-A.Z. Wei-Chung Weng, Fan Yang. Linear antenna array synthesis
using taguchi’s method: A novel optimization technique in electromagnetics.
Antennas and Propagation, IEEE Transactions on, 55(3):723 –730, March 2007.
[20] Wei-Chung Weng, Fan Yang, and Atef Elsherbeni. Electromagnetics and antenna
optimization using taguchi’s method. Synthesis Lectures on Computational Electromagnetics, 2(1):1–94, 2007.
[21] Wei-Chung Weng, Fan Yang, V. Demir, and A. Elsherbeni. Electromagnetic
optimization using taguchi method: a case study of linear antenna array design.
pages 2063 –2066, july 2006.
[22] C.A. Balanis. Antenna theory. Wiley New York, 1997.
[23] L.C. Kempel, J.L. Volakis, and R.J. Sliva. Radiation by cavity-backed antennas
on a circular cylinder. Microwaves, Antennas and Propagation, IEE Proceedings
-, 142(3):233 –239, jun 1995.
[24] J.S. Dahele, R.J. Mitchell, K.M. Luk, and K.F. Lee. Eﬀect of curvature on
characteristics of rectangular patch antenna. Electronics Letters, 23(14):748 –
749, jul. 1987.
[25] N. J. A. Sloane. http://www2.research.att.com/ njas/oadir/.
[26] R.L. Haupt and SE Haupt. Practical genetic algorithms. Wiley-Interscience,
2004.
230