# Experiments in Optimization Applied to Antenna Impedance Matching

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1 Experiments in Optimization Applied to Antenna Impedance Matching William Brooks, WB6YVK

2 The Intersection of Two Evolving Disciplines Optimization Mathematics Learning from Nature. ham radio antennas Antenna Q Physics Antenna Q is part of a search for an efficient, but electrically small antenna. 2

3 Presentation Outline Introductions The basic L, T, and Pi circuits What governs antenna impedance bandwidth Brief overview of well-known impedance matching techniques Simple impedance matching circuit topologies Introduction to Nature-Inspired Optimization Algorithms Introduction of ZNET Optimization Algorithm Examples results using monopole, dipole, delta loop, and Yagi antennas; Resonant and non-resonant antennas, multi-band equalizers, circuit topology pruning algorithm. Lumped LC equalizer circuits, and distributed element (transmission line) equalizer circuits. Lessons Learned (so far), and future developments. 3

4 Scope and Objective This is a work in progress.. The scope of this presentation is intended for amateur radio enthusiasts. Rigorous antenna Q, Maxwell, Kuroda, Chu, Fano, Gewertz, Hurwitz, Brune, Youla, Gauss, Newton, Powell, Hessian, Jacobian, et al equations will be avoided.. But, the above are helpful to understand the problem. The objective of these experiments is; Learn something new about mathematical optimization techniques. 4

5 Preliminaries, Definitions, Single Match vs. Double Match and Clarifications All examples shown here are Single Match; the source is 50+j0 Ohms. Singly-Terminated Networks vs. Doubly Terminated Networks All examples shown here are Doubly-Terminated Networks Circuit diagrams direction; Many RF circuit design textbooks draw circuits from source left to load right, but Many Smith Chart tools (e.g. SimSmith) draw circuits from load left to source right. Both will be shown here. 5

6 An 80 meter ½ λ Inverted V Wire Antenna Computer Model of Antenna Physical Geometry 6

7 4NEC2 Computation Results using NEC 4.1 Engine 7

8 Unmatched ½ λ Inverted V Wire Dipole Frequency Sweep of R,X, Z, and Phase 8

9 ½ λ Inverted V Wire Dipole Frequency Sweep of Antenna VSWR and Reflection Coefficient Computed using NEC4.1 from Lawrence Livermore National Laboratory. 9

10 Equalizer Block Diagram 50 Ohms P input P Load P as Complex Antenna Equalizer Load P input refl P load refl Z in equalizer TPG = Transducer Power Gain TTT = P LLLL P aa Z in antenna An ideal, lossless equalizer P Load = P as Thomas R. Cuthbert, Jr., PhD, Broadband Direct-Coupled and Matching RF Networks, TRCPEP Publications, Greenwood, AK, (1999), pp.2-4. An excellent description and comparison of power gain, transducer gain, and available gain terms is available Power, Transducer, Available, and Insertion Gains Defined from Maury Microwave, 10

11 Enter the L Tuning Equalizer source antenna source antenna R source > R antenna low pass L R source < R antenna source antenna source antenna high pass L R source < R antenna R source > R antenna The L can also be two inductors, or two capacitors. Selection of L circuit topology governed by impedance values (location on Smith Chart), VA3IUL offers a graphic summation of L circuits and applicable Smith Chart zones at 11

12 L Equalizer Added to 80 meter ½ λ Inverted V Wire Dipole Antenna antenna alone antenna with L 12

13 Smith Chart of L Equalizer Added to 80 meter ½ λ Inverted V Wire Dipole Antenna (L equalizer at radio, then 100 feet LMR400 coax to antenna) 13

14 Smith Chart of L Equalizer Added to 80 meter ½ λ Inverted V Wire Dipole Antenna (100 feet LMR400 coax from radio, L equalizer at antenna) 14

15 Common Amateur Radio Antenna Equalizers 3 element, High-Pass T circuit source antenna VSWR < 2 achieved over a narrow span of the RF spectrum; adjustable tuning required. 15

16 A T is Two L s R s R s R L C1 C2 R I Q 1 (Virtual R) Q 2 R L R I Q 1 = R I R S 1 R s (1 + Q 12 ) 1 R I (1 + Q 22 ) R s Q 2 = R I R L 1 Q 0 = Q 1 + Q 2 2 > 1 2 R L R S 1 If R L > R S R L The impedance of the virtual R is greater than the impedance at either end, i.e. step up, then step down. B.K. Chung, Q-based design method for T network impedance matching, Microelectronics Journal 37 (2006)

17 Common Amateur Radio Antenna Equalizers 3 element, Low-Pass Pi Tuner circuit source antenna VSWR < 2 achieved over a narrow span of the RF spectrum; adjustable tuning required. 17

18 A Pi is Two L s R s R s R L Q 1 Q 2 R I (Virtual R) R L Q 1 = R S R I 1 R L Q 2 = R L R I 1 R s 1 R S (1 + Q 12 ) R I (1 + Q 22 ) R I The impedance of the virtual R is less than the impedance at either end, i.e. step down, then step up. 18

19 R, X, Z, and Phase of 80 meter ½ λ Inverted V Wire Dipole Antenna Plus Equalizer Networks Low pass L High pass T Low pass Pi 19

20 Initial Questions What governs the impedance match bandwidth of an antenna across a whole frequency band? How to design such an antenna impedance equalizer? 20

21 What Governs the Impedance Match? The Many Types of Q; Antenna Q, Q ant, Match Bandwidth, Q BW, L & C component Q u, Number of equalizer circuit elements. Randy Rhea created a CD Q from A to Z available from 21

22 Antenna Q and Impedance (active subjects in recent literature) 22

23 Some Representative Impedance Matching Design Approaches Smith Chart Graphical Approach Fano Classical method Simplified Real Frequency Technique (SRFT) H.J. Carlin, B.S. Yarman, W.K. Chen, P.L.D. Abrie; Curve fitting for Hilbert Transform, Gewertz polynomial manipulations, Brune circuit synthesis. Direct Search Minimax Optimization Algorithms Local vs. global search of solution space, Constrained vs. unconstrained variables, Deterministic vs. stochastic algorithms. 23

24 Some Representative Impedance Matching Design Approaches Smith Chart Graphical Approach Fano Classical method Simplified Real Frequency Technique (SRFT) H.J. Carlin, B.S. Yarman, W.K. Chen, P.L.D. Abrie; Curve fitting for Hilbert Transform, Gewertz polynomial manipulations, Brune circuit synthesis. Direct Search Minimax Optimization Algorithms Local vs. global search of solution space, Constrained vs. unconstrained variables, Deterministic vs. stochastic algorithms. 24

25 Smith Chart Graphical Approach Wilfred N. Caron, 1989 Philip H. Smith, 1995 Anthony A. R. Townsend, The Smith Chart and its Applications, is available online at no cost from 25

26 Simple L for Single Frequency Match Q = R aaaaaaa R SSSSSS 1 VSWR=2 circle Constant Q circle Q = Q= Initial Antenna impedance 26

27 Increasing the Number of Circuit Elements R.W.P. King, Transmission Line Theory, McGraw-Hill, NY (1955) provides details of impedance matching using the Smith Chart. 27

28 Multi-L Tuning Can get more of the antenna inside the VSWR=2 boundary circle. Q=0.400 VSWR=2 28

29 Frequency Sweep of Multi-L Tuning antenna alone Antenna + multiple L s 29

30 Some Representative Impedance Matching Design Approaches Smith Chart Graphical Approach Fano Classical method Simplified Real Frequency Technique (SRFT) H.J. Carlin, B.S. Yarman, W.K. Chen, P.L.D. Abrie; Curve fitting for Hilbert Transform, Gewertz polynomial manipulations, Brune circuit synthesis. Direct Search Minimax Optimization Algorithms Local vs. global search of solution space, Constrained vs. unconstrained variables, Deterministic vs. stochastic algorithms. 30

31 Famous Papers by R. Fano (1948) and H. Wheeler (1950) Wideband Impedance Matching,

32 Return Loss and Q factors with Number of Circuit Elements Number of equalizer circuit elements Return Loss VSWR=1.92 VSWR=1.22 VSWR=1.07 M. Gustafsson, Bandwidth, Q Factor, and Resonance Models of Antennas, Progress In Electromagnetics Research, PIER 62, 1 20,

33 33

34 Some Representative Impedance Matching Design Approaches Smith Chart Graphical Approach Fano Classical method Simplified Real Frequency Technique (SRFT) H.J. Carlin, B.S. Yarman, W.K. Chen, P.L.D. Abrie; Nonlinear least squares curve fitting for Hilbert Transform, Gewertz polynomial manipulations, Brune circuit synthesis; Direct Search Minimax Optimization Algorithms Local vs. global search of solution space, Constrained vs. unconstrained variables, Deterministic vs. stochastic algorithms. 34

35 SRFT Beginnings Herbert J. Carlin, Cornell University 35

36 36

37 SRFT First Steps 37

38 Curve Fitting the R values 38

39 SRFT Circuit Design Hilbert Transform of resistance curve fit yields reactance function. Then, some polynomial manipulation yields Impedance function. After some more polynomial manipulation, impedance function becomes; Z p = 0.144p p p p p p p p p p p After some long division and Brune circuit synthesis becomes; 50 Ohms 11.1 nh nh nh pf 746 pf 219 pf Short monopole 12.6:1 transformer 39

40 SRFT Techniques Textbooks B.S. Yarman, 2010 B.S. Yarman, 2008 (MATLAB) (MATLAB) Complete updated MATLAB source code for Real Frequency Technique can be downloaded from 40

41 SRFT Techniques Textbooks (FORTRAN) W.K. Chen University of Illinois at Chicago 41

42 SRFT Techniques and Other Textbooks A history of impedance matching techniques by Thomas R. Cuthbert is available online at no cost at; Impedance_Matching In addition, there are other analytical techniques, e.g. H-Infinity by W. UC San Diego. P.L.D. Abrie, 2009 (.exe) 42

43 Results Using SRFT Magnitude VSWR =2 Boundary Phase Angle 43

44 MATLAB RF Toolbox SRFT Optimization Results TPG Results Using MATLAB Toolbox Retrieved from Designing Broadband Matching Networks (Part 1: Antenna), 44

45 Yarman s SRFT Circuit Solution for the Short Monopole LPP6 circuit topology 12.66:1 transformer 45

46 Yarman s SRFT Short Monopole Solution VSWR 20 MHz to 100 MHz 46

47 47

48 SRFT Limitations (2012) Also see Accuracy and Stability of Numerical Algorithms by Nicholas J. Higham, Published by SIAM (Society for Industrial and Applied Mathematics), Philadelphia (1996). 48

49 49

50 Some Representative Impedance Matching Design Approaches Smith Chart Graphical Approach Fano Classical method Simplified Real Frequency Technique (SRFT) H.J. Carlin, B.S. Yarman, W.K. Chen, P.L.D. Abrie; Curve fitting for Hilbert Transform, Gewertz polynomial manipulations, Brune circuit synthesis. Direct Search Optimization Algorithms Local vs. global search of solution space, Constrained vs. unconstrained variables, Deterministic vs. stochastic algorithms. 50

51 Some Representative Optimization Algorithms Nelder-Mead Hooke-Jeeves Powell Fletcher Fletcher-Reeves Newton-Raphson Levenberg-Marquardt Multi-Directional Search Implicit Filtering * Quasi-Newton Gauss-Newton Dividing Rectangles (DIRECT) Generalized Reduced Gradient (GRG2) Classical Source code available from Ant Colony Artificial Bee Colony Bat Algorithm Cuckoo Search Differential Evolution Firefly Wolf Search Central Force Optimization Gravitational Search Particle Swarm Simulated Annealing Wind Driven Shuffled Complex Evolution Swarm-Firefly DEPSO Invasive Weed Nature-Inspired 51

52 Direct Search Optimization Algorithms 52

53 Finding the Global Minimum Easy in a Smooth Convex Landscape 53

54 Finding the Global Minimum Not So Easy in a Not So Smooth Landscape 54

55 VSWR Measurement Data from AIM 4170

56 Measurement Noise in Measured VSWR

57 Finding the Global Minimum Easy to Miss the Minimum 57

58 But..which algorithm? 58

59 80 meter wire ½ λ Inverted V Matching Li used a genetic algorithm, reported 6 hours of computation time to complete an optimization run for a 3 element Pi equalizer circuit. 59

60 A Minimum in a L Network Landscape Minimum Iyer also used a genetic algorithm, reported 4 hours of computation time was required to complete an optimization run for a L-T, 5 element equalizer circuit. 60

61 Introduce ZNET Design Tool Accepts industry standard Touchstone file format.s1p files that represent load (antenna) electrical impedance data; frequency, resistance, and reactance. File data from any of; EM or circuit simulation design tools (e.g. NEC, SPICE, FDTD, etc.), or SWR meter or VNA measurements (e.g. AIM 4170), or TDR measurements with RLGC parameter extraction software. User selects canonical circuit topology e.g. low-pass, band-pass, high-pass, PiLPiC etc., ZNET uses real-life components with User-supplied unloaded component Q u data. All components are fixed values, i.e. no variable components to tune. ZNET performs circuit optimization, modifies circuit topology if needed, i.e. includes a circuit pruning algorithm, provides simulated results with component values, performance metrics, and graphs. 61

62 ZNET Design Approach Modular Software Common front end; User Interface, Antenna data file input, Circuit characteristics inputs from User (e.g. circuit topology, constraints, component unloaded Q values, RF power, etc.), Solution report outputs to User (component values, circuit, graphs, comparative results metrics), Optimization back-end that plugs in ; Plug in any optimization algorithm. Ability to use existing optimization codes when available MATLAB, Python, C/C++, FORTRAN. 62

63 ZNET Software Overview Exploits computations performed in linear algebra vector-matrix form. Uses Nature-Inspired Direct Search; Nonlinear minimax optimization of multiple variables, derivative-free, Direct Search Global search by extending to unconstrained optimization. A.R. Conn, K. Scheinberg, L.N. Vicente, Introduction to Derivative-Free Optimization, Society for Industrial and Applied Mathematics [SIAM], (2009). 63

64 ZNET Testing and Verification or, How Will I Know This is a Good Solution? Also see X.-S. Yang, Test problems in optimization, in: Engineering Optimization: An Introduction with Metaheuristic Applications (Eds Xin-She Yang), John Wiley & Sons, (2010). 64

65 Verification of Optimization Algorithm Performance Using Known Benchmark Problems and Known Solutions Rosenbrock s Valley, a.k.a. banana function 65

66 Rosenbrock Optimization Minimum (1,1) 66

67 Verification of Optimization Algorithm Performance Using Known Benchmark Problems Rastrigin s function 67

68 Verification of Optimization Algorithm Performance Using Known Benchmark Problems Ackley s function 68

69 Why Grid or Random Initial Search is Problematic N. Andrei, An Unconstrained Optimization Test Functions Collection Advanced Modeling and Optimization, Volume 10, Number 1,

70 Example Equalizer Circuit Topologies and Nomenclature Low Pass Equalizer Circuit Topology. Low Pass Parallel to Load 8 elements. LPP8. radio antenna Band Pass Equalizer Circuit Topology. Band Pass Parallel to Load 8 resonators. BPP8. radio antenna 70

71 Not Grid or Random Initial Values 100 points Random Distribution Halton Sequence B.L. Robertson, Direct Search Methods for Nonsmooth Problems Using Global Optimization Techniques, Ph.D. dissertation, University of Canterbury, Christchurch, New Zealand,

72 Freq R X ZNET Accepts FRX & Touchstone.s1p Formats ****** ******* ******* Data represents antenna impedance as a function of frequency. Data can be acquired from VNA or SWR meter, or be calculated using antenna simulation software. Spreadsheet can be used to convert data files types, and to compute initial performance metrics. 72

73 An 80 meter ½ λ Inverted V Wire Antenna Computer Model of Antenna Physical Geometry 73

74 ZNET High-Pass T and 80 meter Inverted V Wire Dipole Input Screen 74

75 ZNET High-Pass T and 80 meter Inverted V Wire Dipole Output Screen 75

76 An 80 meter ½ λ Inverted V Wire Antenna with High Pass T Equalizer Circuit Topology 76

77 An 80 meter ½ λ Inverted V Wire Antenna with High Pass T Equalizer Circuit Topology 77

78 An 80 meter ½ λ Inverted V Wire Antenna with High Pass T Equalizer Circuit Topology and Path Note the path taken 78

79 An 80 meter ½ λ Inverted V Wire Antenna with Low Pass Pi Equalizer Circuit Topology 79

80 An 80 meter ½ λ Inverted V Wire Antenna with Low Pass Pi Smith Chart Note the path taken. 80

81 Using Standard Value Capacitors Stripline Capacitor 81

82 Excel Solver Selects Capacitor Combination Use Solver to select minimum number of standard value capacitors to achieve desired capacitance. Solver is bundled with Microsoft Excel. LibreOffice Calc has a similar Solver, also includes a solver DEPSO. A faster solver OpenSolver is available at no cost from 82

83 DEPSO Optimizer DEPSO is bundled with the Calc application in LibreOffice 83

84 Using Standard Capacitors 1481 pf 2384 pf = =

85 80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology ZNET Solution 85

86 80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology ZNET Solution P llll = db 86

87 80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology Smith Chart 87

88 80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology 88

89 80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology 1000 Watt Power Handling Note the RF voltages and currents in each component. 89

90 80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology 1000 Watt Power Handling Note the RF power dissipated by each component. 90

91 Optimum Corner Frequency for Passband ω 2 = 1+ ω 0 2Q bb 2Q bb 2 +1 T.R. Cuthbert, Broadband Direct-Coupled and Matching RF Networks, TRCPEP, (1999), pg19. 91

92 80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology Increased to Optimum Passband Example: 80 meter band; MHz ω 0 = ω 1 ω 2 = 3.5MMM 4.0MMM = MHz (geometric centric frequency) Q bb = = ω 2 ω 0 = 1+ ( ) = new passband ω 1 = 3.500/1.069 = MMM ω 2 = = MMM 92

93 80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology Increased to Optimum Flat Passband P llll = db Note: % bandwidth increased From 13.4% to 26.8%, i.e. (x2) 93

94 80 meter ½ λ Inverted V Wire on 80 meters using BPP4 Circuit Topology 94

95 80 meter ½ λ Inverted V Wire on 80 meters using BPP4 Circuit Topology VSWR P llll = db 95

96 80 meter ½ λ Inverted V Wire on 80 meters using BPP4 Circuit Topology Smith Chart 96

97 80 meter ½ λ Inverted V Wire on 80 meters using HPS4 Circuit Topology 97

98 80 meter ½ λ Inverted V Wire on 80 meters using HPS4 Circuit Topology Antenna VSWR Before and After Equalizer VSWR 98

99 80 meter ½ λ Inverted V Wire on 80 meters using HPS4 Circuit Topology Smith Chart 99

100 Li s Genetic Algorithm Solution (required 6 hours computer time) 100

101 Li s Genetic Algorithm Solution Smith Chart 101

102 ZNET Solution for Li s Dipole 102

103 ZNET Solution for Li s Dipole using LPS4 Circuit Topology 103

104 ZNET Solution for Li s Dipole Using LPS4 Circuit Topology Smith Chart 104

105 ZNET Solution to Li s Dipole using LPS6 Circuit Topology 105

106 ZNET Solution to Li s Dipole using LPS6 Circuit Topology 106

107 ZNET Solution to Li s Dipole using LPS6 Circuit Topology Smith Chart Required seconds computation time. 107

108 Cascade Serial Transmission Line (CASTL) Impedance Match See L.N. Dworsky, Modern Transmission Line Theory and Applications, Wiley, NY (1979). 108

109 80 meter ½ λ Inverted V Wire on 80 meters using CASTL Circuit Topology VSWR CASTL circuit 109

110 80 meter ½ λ Inverted V Wire on 80 meters using CASTL Circuit Topology Smith Chart Note the path. 110

111 80 meter ½ λ Inverted V Wire on 80 meters using CASTL Circuit Topology 1000 Watt Power Handling 111

112 Homebrew Parallel Wire Transmission Lines Parallel Wire Transmission Line ATLC2 computed these images. ATLC2 was developed by KQ6QV. (see URL below.) Current distribution for closely spaced two conductor transmission lines. E Field from two-wire transmission line cross-section. Spreadsheet computes spacing d for a required impedance for a given width wire diameter and spacing dielectric (e.g. air for lowest loss). See 112

113 Homebrew Parallel Plate Transmission Lines and Stubs Parallel Plate Transmission Lines and Stubs Spreadsheet computes spacing d for a required impedance for a given width w and spacing dielectric (e.g. air for lowest loss). w d Parallel Plate Transmission Line (TEM mode) Optimizer WB6YVK MKS Units inches center frequency = MHz 2.12E+07 Line Impedance Plate Width = 12.7 mm Zo = i Ohms Plate Conductivity = S/m aluminum Ro = Ohms Spacing = mm Xo = Ohms Dielectric constant of spacer = 2.3 PTFE magnetic constant (u) = E-06 H/m electric constant (e) = E-12 F/m velocity = E+08 m/sec vel factor = RLGC Results Shorted stub as an inductor R' = 2.42E-01 Ohms/m Q stub = L' = E-07 H/m G' = 1.00E-10 mhos/m C' = 4.04E-11 F/m Ro = Ohms Xo = Ohms Zo magnitude = Ohms alpha' = 9.69E

114 ATLC2 Example: 150 Ohm Parallel Plate Transmission Line 114

115 80 meter ½ λ Inverted V Wire on 80 meters shorter CASTL Circuit Topology #4 using non-standard transmission lines and stubs 115

116 80 meter ½ λ Inverted V Wire on 80 meters shorter CASTL Circuit Topology #4 using non-standard transmission lines and stubs 116

117 80 meter ½ λ Wire Folded Dipole 117

118 80 meter ½ λ Wire Folded Dipole using LPS6 Circuit Topology Note that no transformer is needed. 118

119 80 meter ½ λ Wire Folded Dipole using LPS6 Circuit Topology 119

120 40 meter ¼ λ Monopole 120

121 40 meter ¼ λ Monopole using ZNET Solution LPS4 Circuit Topology 121

122 40 meter ¼ λ Monopole using LPS4 Circuit Topology 122

123 40 meter ¼ λ Monopole using LPS6 Circuit Topology VSWR 123

124 40 meter ¼ λ Monopole using LPS6 Circuit Topology Smith Chart 124

125 6 meter ¼ λ Ground Plane (Arrow GP52) 125

126 6 meter ¼ λ Ground Plane (Arrow GP52) using LPS4 Circuit Topology VSWR 126

127 6 meter ¼ λ Ground Plane (Arrow GP52) using LPS4 Circuit Topology Smith Chart 127

128 Shortened ¼ λ 6 meter Ground Plane (Arrow GP52) using LPS4 Circuit Topology VSWR 128

129 10 foot Ground Plane on 6 meters using LPP4 Circuit Topology VSWR 129

130 432 MHz Yagi Antenna Antenna design was computer optimized for MHz by the commercial manufacturer. 130

131 432 MHz Yagi Antenna using LPS6 Circuit Topology Smith Chart SWR = 2 circle ZNET Matched antenna Frequency Sweep from 400 to 450 MHz. Optimized antenna as supplied by the manufacturer.. 131

132 Equal Response vs. 50 Ohms Achieving an equal response (flat across frequency band) and meeting 50 Ohms are two different goals. Often, a transformer would be useful to shift the equalized impedance to 50 Ohms..but often requires an inconvenient transformer ratio. Enter the Norton transform. 132

133 R L >R S Enter Impedance and Admittance Inverters J-Inverters L C -L -L -C -C R S >R L -L K-Inverters -L -C -C L C 133

134 Implementing Norton Transforms -L -L radio -C -C antenna The negative valued components are absorbed by the BP resonators. 134

135 Norton Transforms As Transformers radio antenna There is a limitation on the achievable transformation ratio, n 2, see Abrie s textbook for details. 135

136 80 meter ½ λ Inverted V Wire using Capacitor Pi and Inductor L 136

137 80 meter ½ λ Inverted V Wire using Capacitor Pi and Inductor L 137

138 Using L T & Low Pass Pi Transformers XFMRPI Circuit Topology 138

139 80 meter ½ λ Inverted V Wire using WMB3 Circuit Topology T Pi T 139

140 80 meter ½ λ Inverted V Wire using WMB3 Circuit Topology VSWR P llll = db 140

141 80 meter ½ λ Inverted V Wire using WMB3 Circuit Topology Smith Chart 141

142 Non-resonant Antennas Monopole antennas, 20 feet tall monopole, 10 feet tall monopole, Flat top dipole antennas, 100ft wide (short) 80 meter wire V dipole, 20 feet width dipole, 10 feet width dipole. Multi-band equalizers using single equalizer circuit on a nonresonant antenna. 142

143 20 Foot Tall Monopole HF Antenna Computer model image. Commercial HF antenna. 143

144 20 foot Monopole on 20 meters using LPP4 Circuit Topology 144

145 20 foot Monopole on 20 meters using LPP4 Circuit Topology VSWR P llll = db 145

146 20 foot Monopole on 20 meters using LPP4 Circuit Topology Smith Chart Note the path taken 146

147 20 foot Monopole on 20 meters using LPS6 Circuit Topology VSWR P llll = db 147

148 20 foot Monopole on 20 meters using LPS6 Circuit Topology Smith Chart 148

149 20 foot Monopole on 40 meters using LPS6 Circuit Topology VSWR P llll = db 149

150 20 foot Monopole on 40 meters using LPS6 Circuit Topology Smith Chart 150

151 20 foot Monopole on 20 meters using WMB2 Circuit Topology 151

152 20 foot Monopole on 20 meters using WMB2 Circuit Topology VSWR P llll = db 152

153 20 foot Monopole on 20 meters using WMB2 Circuit Topology Smith Chart 153

154 Example ZNET Solution 10 foot Monopole using 15m BPS2 Circuit 154

155 10 foot Monopole on 15 meters using BPS2 Circuit Topology VSWR 155

156 10 foot Monopole on 15 meters using BPS2 Circuit Topology Smith Chart 156

157 10 foot Monopole on 15 meters using LPS6 Circuit Topology VSWR P llll = db 157

158 80m Inverted V Wire Dipole 100 foot (short) using LPS8 Circuit Topology VSWR 158

159 80m Inverted V Wire Dipole 100 foot (short) using LPS8 Circuit Topology Smith Chart 159

160 80m Inverted V Wire Dipole 100 foot (short) using LPS8 Circuit Topology VSWR P llll = db 160

161 80m Inverted V Wire Dipole 100 foot (short) using LPS8 Circuit Topology (no transmission line from radio) 161

162 80m Inverted V Wire Dipole 100 foot (short) using LPS8 Circuit Topology (including 100 feet LMR400 from radio) 162

163 80m Inverted V Wire Dipole 100 foot (short) using WMB3 Circuit Topology Smith Chart 163

164 80m Inverted V Wire Dipole 100 foot (short) VSWR using WMB3 Circuit Topology 164

165 80m Inverted V Wire Dipole 100 foot (short) using WMB3 Circuit Topology 165

166 20 foot Flat Top Dipole 166

167 20 foot Flat Top Dipole No Matching 20m through 10m 167

168 20 foot Flat Top Dipole No Matching 20m through 10m 168

169 20 foot Flat Top Dipole on 15 meters CASTL Circuit using non-standard transmission lines 169

170 20 foot Flat Top Dipole on 15 meters CASTL Circuit using non-standard transmission lines VSWR 170

171 20 foot Flat Top Dipole on 15 meters CASTL Circuit using non-standard transmission lines Smith Chart Note the path. 171

172 20 foot Flat Top Dipole 1000 Watt Power Handling on 15 meters CASTL Circuit using non-standard transmission lines 172

173 20 foot Flat Top Dipole on 15 meters CASTL Circuit using non-standard transmission lines and stubs 173

174 20 foot Flat Top Dipole 1000 Watt Power Handling on 15 meters CASTL Circuit transmission lines and stubs 174

175 20 foot Flat Top Dipole on 15 meters CASTL Circuit transmission lines and stubs Smith Chart Note the path. 175

176 20 foot Flat Top Dipole on 15 meters CASTL Circuit using Commercially Available Transmission Lines 176

177 Multi-Band Equalizer Single Equalizer Circuit Fixed value elements All lumped L & C elements Single non-resonant antenna 177

178 20 foot Flat Top Dipole 17m through 15m Continuous Match using LPS8 Circuit Topology Smith Chart 178

179 20 foot Flat Top Dipole 17m and 15m Dual Band Match using Single LPS6 Circuit Topology Smith Chart 179

180 20 foot Flat Top Dipole 17m and 15m Dual Band Match using Single LPS6 Circuit Topology VSWR 180

181 20 foot Flat Top Dipole 17m and 15m Dual Band Match using Single LPS10 Circuit Topology VSWR 181

182 20 foot Flat Top Dipole 17m, 15m, & 20 MHz WWV Match using Single LPS8 Circuit Topology VSWR 20 MHz WWV 182

183 20 foot Flat Top Dipole 4 Band Match using Single LPS10 Circuit Topology 17m, 15m, 12m, & 10m Bands VSWR 183

184 20 foot Flat Top Dipole 4 Band Match using Single LPS10 Circuit Topology 17m, 15m, 12m, & 10m Bands VSWR 184

185 20ft Flat Top Dipole on 4 bands 17m, 15m, 12m, & 10m bands Smith Chart 185

186 ZNET Circuit Topology Modification Algorithm ZNET has a circuit topology modification algorithm that can automatically prune circuit elements to improve the impedance match. 186

187 Eliminating Components using Bounded Constraints High-Pass T circuit topology example. Non-resonant antenna example. 10 foot monopole across 15 meter band. Compute using bounded optimization constraints, Compute with extended optimization constraints, Add a circuit element pruning optimization algorithm. 187

188 C1 Upper Optimization Boundary de-normalized values C1 is a candidate for removal from equalizer circuit. normalized values 188

189 T Circuit Solution Including C1 C1 value at upper optimization boundary. 189

190 T Circuit Solution VSWR with C1 190

191 T Circuit Solution with C1 Removed 191

192 T Circuit Solution VSWR without C1 192

193 Extended C1 Upper Boundary Increased C1 upper boundary value x1000 so that remaining components (i.e. L2 & C3) can be optimized. 193

194 T Circuit Solution with Extended Boundary 194

195 T Circuit Solution VSWR with Extended Boundary 195

196 T Circuit Extended Solution Over Wider Frequency Span 196

197 10ft 15m Monopole with High Pass T with Circuit Element Pruning Automatic circuit element pruning algorithm will automatically identify circuit elements to remove, delete them, and re-compute the optimum values for surviving circuit elements. 197

198 Results of Circuit Element Pruning Algorithm 198

199 VSWR Results of Circuit Element Pruning Algorithm 199

200 40 meter ½ λ Inverted V Wire Dipole LPP4 no pruning 200

201 40 meter ½ λ Inverted V Wire Dipole with Circuit Element Pruning Results of Circuit Element Pruning Algorithm, L4 deleted, remaining element values re-optimized. 201

202 10 foot Monopole on 20m Example Note the high VSWR 202

203 10 foot Monopole 20 meters using BPP2 Circuit Topology 203

204 10 foot Monopole 20 meters using BPP2 Topology AFTER Pruning 204

205 10ft Monopole on 20 meters using BPP4 Circuit Topology and Pruning Algorithm ZNET algorithm with pruning returned to simple L circuit topology again. 205

206 Component Tolerances Study Vary L & C element values 206

207 Component Tolerances Study 40m ½ λ Inverted V Wire Dipole using LPS6 Circuit Topology Smith Chart Initial results 207

208 Component Tolerances Study 40m ½ λ Inverted V Wire Dipole using LPS6 Circuit Topology Smith Chart All 6 LC components simultaneously varied +/- 5%. 208

209 Component Tolerances Study 40m ½ λ Inverted V Wire Dipole using LPS6 Circuit Topology Smith Chart All 6 LC components simultaneously varied +/- 2%. 209

210 Component Tolerances Study 40m ½ λ Inverted V Wire Dipole using LPS4 Circuit Topology Smith Chart 210

211 Component Tolerances Study 40m ½ λ Inverted V Wire Dipole using LPS4 Circuit Topology Smith Chart All 4 LC components simultaneously varied +/- 5%. 211

212 Component Tolerances Study 40m ½ λ Inverted V Wire Dipole using LPS4 Circuit Topology Smith Chart All 4 LC components simultaneously varied +/- 2%. 212

213 Component Tolerances Study 20 foot Dipole on 15m using CASTL Circuit Topology Smith Chart Zo of both transmission lines simultaneously varied +/- 5%. 213

214 20 foot Flat Top Dipole on 15 meters CASTL Circuit transmission lines and stubs Smith Chart Zo of transmission lines & stubs simultaneously varied +/- 5%. 214

215 Revisit Yarman s Short Monopole Example ZNET solution using Yarman s LPS6 circuit topology. Changed circuit element values, Changed transformer from a 12.66:1 transformer to a standard 16:1 transformer (ala a transmission line transformer). 215

216 Yarman s Monopole Example ZNET Circuit 16:1 transformer Using same LPP6 circuit topology as Yarman, but changed transformer to standard 16:1, Recomputed circuit element values using ZNET. 216

217 Yarman s Monopole Example ZNET Solution VSWR 217

218 Yarman s Monopole Example SRFT and ZNET VSWR Comparison SRFT VSWR ZNET VSWR 218

219 Antenna Height Variations Vary the height above ground of 40 m ½ λ inverted V wire antenna using same LPS6 equalizer circuit; Apex 25 feet Apex 30 feet (design height) Apex 35 feet 219

220 40 meter ½ λ Inverted V Wire Dipole versus Apex Height Using Same Equalizer Circuit 25 feet 30 feet (design height) 35 feet 220

221 15 meter wire delta loop hardware test 221

222 15 meter wire delta antenna test VSWR before equalizer 222

223 15 meter wire delta antenna Smith Chart 223

224 QRP Test Equalizer at 15 meter Delta Loop Feedpoint 224

225 VSWR of QRP T Equalizer Using Low-Cost NTE Ceramic Capacitors 225

226 meter delta antenna VSWR measured at ZNET equalizer input at antenna feedpoint Using ATC Hi-Q capacitors, no coax pigtail, and no feedline choke. Measured with HP VNA, S11 data used to compute VSWR. VSWR Frequency (MHz) 226

227 Next Steps in ZNET Experiments Improve the accuracy of the L and C component loss models, add ability to design optimum inductor geometry, Improve the circuit topology modification capability, Merge transmission line models to enable mixed element design optimization, Add GUI interface, make user friendly, Test & Evaluate with benchmark problems and hardware prototypes, Extend optimization to antenna design (G/Q optimization objective). 227

228 Also see 228

229 229

230 230

231 Extending ZNET 231

232 Available Impedance Matching Software Applications OptiMatch Optenni AnTune Zmatch Wmatch (C++ source code) Ematch D.B. Miron, Small Antenna Design. Newnes, (2006). (MATLAB source code) M. Bakr, Nonlinear Optimization in Electrical Engineering with Applications in MATLAB, IET, London (2013). (MATLAB source code) 232

233 Interesting Antenna Design Optimization Software Tools 4NEC2 (simplex and genetic algorithm optimizers) Nikiml s Antenna pages (Differential Evolution) NEC Lab /documentos/index.htm The Applied Computational Electromagnetics Society [ACES] 233

234 Thank You Questions? 234

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