experiments in antenna impedance matching · •many smith chart tools (e.g. simsmith ) draw...
TRANSCRIPT
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Experiments in Optimization Applied to
Antenna Impedance Matching
William Brooks, WB6YVK [email protected]
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The Intersection of Two Evolving Disciplines
Optimization Mathematics
Antenna “Q” Physics
Antenna Q is part of a search for an efficient, but electrically small antenna.
Learning from Nature.
ham radio
antennas
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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.
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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.
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Preliminaries, Definitions, and Clarifications
Single Match vs. Double Match • 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.
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An 80 meter ½ λ Inverted V Wire Antenna
Computer Model of Antenna Physical Geometry 6
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4NEC2 Computation Results using NEC 4.1 Engine
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Unmatched ½ λ Inverted “V” Wire Dipole Frequency Sweep of R,X, Z, and Phase
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½ λ Inverted “V” Wire Dipole Frequency Sweep of Antenna VSWR and Reflection Coefficient
Computed using NEC4.1 from Lawrence Livermore National Laboratory. 9
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Equalizer Block Diagram
𝑇𝑇𝑇 =𝑇𝐿𝐿𝐿𝐿𝑇𝐿𝑎
TPG = Transducer Power Gain
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, http://www.maurymw.com.
An ideal, “lossless” equalizer PLoad = PaS
Equalizer
50 Ohms
PaSPLoad
Complex Antenna Load
Pinput
Pinput refl Pload refl
Zin antennaZin equalizer
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Enter the “L” Tuning Equalizer source source
source source
antenna antenna
antenna 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 www.qsl.net/va3iul/.
“low pass L”
“high pass L”
Rsource > Rantenna
Rsource > Rantenna
Rsource< Rantenna
Rsource< Rantenna
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L Equalizer Added to 80 meter ½ λ Inverted V Wire Dipole Antenna
antenna alone
antenna with L
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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)
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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)
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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.
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A “T” is Two “L”’s
Q1 Q2
𝑄1 =𝑅𝐼𝑅𝑎− 1
𝑄2 =𝑅𝐼𝑅𝐿− 1
𝑅𝑠(1 + 𝑄12)
1𝑅𝐼(1 + 𝑄2
2)
C1 C2
𝑄0 =𝑄1 + 𝑄2
2 >12
𝑅𝐿𝑅𝑎− 1 If RL > RS
B.K. Chung, “Q-based design method for T network impedance matching”, Microelectronics Journal 37 (2006) 1007–1011.
Rs Rs
RL RL
(Virtual R)RI
Rs
RI
RL
The impedance of the “virtual R” is greater than the impedance at either end, i.e. “step up”, then “step down”.
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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.
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A “Pi” is Two “L”s
The impedance of the “virtual R” is less than the impedance at either end, i.e. “step down”, then “step up”.
Q1 Q2
𝑄1 =𝑅𝑎𝑅𝐼− 1
𝑄2 =𝑅𝐿𝑅𝐼− 1
1𝑅𝑎(1 + 𝑄1
2) 𝑅𝐼(1 + 𝑄2
2)
(Virtual R)
RsRs
RLRLRI
Rs
RI
RL
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R, X, Z, and Phase of 80 meter ½ λ Inverted V Wire Dipole Antenna Plus Equalizer Networks
Low pass L Low pass Pi High pass T
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Initial Questions
What governs the impedance match bandwidth of an antenna across a whole frequency band?
How to design such an antenna impedance equalizer?
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What Governs the Impedance Match?
• Antenna Q, “Qant”, • Match Bandwidth, “QBW”, • L & C component Qu, • Number of equalizer
circuit elements.
The Many Types of Q;
Randy Rhea created a CD “Q from A to Z” available from http://www.scitechpub.com.
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Antenna Q and Impedance (active subjects in recent literature)
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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.
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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.
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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 http://www.ie.itcr.ac.cr/acotoc/Ingenieria/TEM%20II/Material%20Vario/Smith_Chart_Book_Complete.pdf.
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Simple L for Single Frequency Match
𝑄 =𝑅𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑅𝑆𝑆𝑆𝑆𝑆𝑎
− 1
Q =80.688
50− 1
Q= 0.783
Constant “Q” circle
Initial Antenna impedance
VSWR=2 circle
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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
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Multi-L Tuning
Can get more of the antenna inside the VSWR=2 boundary circle.
VSWR=2
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Q=0.400
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Frequency Sweep of Multi-L Tuning
antenna alone
Antenna + multiple “L’s”
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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.
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Famous Papers by R. Fano (1948) and H. Wheeler (1950)
“Wideband Impedance Matching”, 1950
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Return Loss and Q factors with Number of Circuit Elements
M. Gustafsson, “Bandwidth, Q Factor, and Resonance Models of Antennas”, Progress In Electromagnetics Research, PIER 62, 1–20, 2006.
Number of equalizer circuit elements
Return Loss
VSWR=1.92
VSWR=1.22
VSWR=1.07
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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.
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SRFT Beginnings Herbert J. Carlin, Cornell University
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SRFT First Steps
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Curve Fitting the R values
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SRFT Circuit Design
𝑍 𝑝 =0.144𝑝5 + 0.35𝑝4 + 0.42𝑝3 + 0.404𝑝2 + 0.19𝑝1 + 0.517𝑝6 + 2.47𝑝5 + 3.98𝑝4 + 5.73𝑝3 + 4.06𝑝2 + 2.5𝑝 + 0.655
After some more polynomial manipulation, impedance function becomes;
12.6:1 transformer
50 Ohms
Shortmonopole
219 pF746 pF538.9 pF
11.1 nH 9.277 nH 2.453 nH
After some long division and Brune circuit synthesis becomes;
Hilbert Transform of resistance curve fit yields reactance function. Then, some polynomial manipulation yields Impedance function.
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SRFT Techniques Textbooks
B.S. Yarman, 2010 B.S. Yarman, 2008
http://www.siddikyarman.com/?author=1 Complete updated MATLAB source code for Real Frequency Technique can be downloaded from
(MATLAB) (MATLAB)
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W.K. Chen University of Illinois at Chicago
1988 2000 (FORTRAN)
1993
SRFT Techniques Textbooks
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SRFT Techniques and Other Textbooks
P.L.D. Abrie, 2009
(.exe)
http://ethw.org/History_of_Broadband_ Impedance_Matching
A history of impedance matching techniques by Thomas R. Cuthbert is available online at no cost at;
In addition, there are other analytical techniques, e.g. “H-Infinity” by W. Helton @ UC San Diego.
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Results Using SRFT
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VSWR =2 Boundary
Magnitude
Phase Angle
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MATLAB RF Toolbox SRFT Optimization Results
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TPG Results Using MATLAB Toolbox
Retrieved from “Designing Broadband Matching Networks (Part 1: Antenna)”, http://www.mathworks.com/help/rf/examples/designing-broadband-matching-networks-part-1-antenna.html
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Yarman’s SRFT Circuit Solution for the Short Monopole
12.66:1 transformer
“LPP6 circuit topology”
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Yarman’s SRFT Short Monopole Solution VSWR 20 MHz to 100 MHz
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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).
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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.
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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) 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
http://www4.ncsu.edu/~ctk/iffco.html
Nature-Inspired Classical
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Direct Search Optimization Algorithms
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Finding the Global Minimum Easy in a Smooth Convex Landscape
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Finding the Global Minimum Not So Easy in a Not So Smooth Landscape
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VSWR Measurement Data from AIM 4170
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Measurement “Noise” in Measured VSWR
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Finding the Global Minimum Easy to Miss the Minimum
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But……..which algorithm?
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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.
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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.
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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 “Qu” 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.
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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.
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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).
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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).
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Verification of Optimization Algorithm Performance Using Known “Benchmark” Problems
and Known Solutions
Rosenbrock’s Valley, a.k.a. “banana function” 65
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Rosenbrock Optimization Minimum
(1,1)
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Verification of Optimization Algorithm Performance Using Known “Benchmark” Problems
Rastrigin’s function 67
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Verification of Optimization Algorithm Performance Using Known “Benchmark” Problems
Ackley’s function 68
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Why Grid or Random Initial Search is Problematic
N. Andrei, “An Unconstrained Optimization Test Functions Collection” Advanced Modeling and Optimization, Volume 10, Number 1, 2008.
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Example Equalizer Circuit Topologies and Nomenclature
Low Pass Equalizer Circuit Topology. Low Pass Parallel to Load 8 elements….”LPP8”.
Band Pass Equalizer Circuit Topology. Band Pass Parallel to Load 8 resonators….”BPP8”.
radio antenna
radio antenna
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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, 2010.
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ZNET Accepts FRX & Touchstone .s1p Formats
Freq R X 3.5000 63.9374 -85.0664 3.5125 64.7067 -81.0424 3.5250 65.4829 -77.0252 3.5375 66.2661 -73.0144 3.5500 67.0562 -69.01 3.5625 67.8533 -65.0117 3.5750 68.6575 -61.0194 3.5875 69.4689 -57.0327 3.6000 70.2874 -53.0517 ****** ******* ******* 3.9250 94.2681 49.0936 3.9375 95.302 52.9923 3.9500 96.3448 56.8904 3.9625 97.3965 60.7882 3.9750 98.4573 64.6857 3.9875 99.5273 68.5832 4.0000 100.606 72.4809
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.
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An 80 meter ½ λ Inverted V Wire Antenna
Computer Model of Antenna Physical Geometry 73
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ZNET High-Pass “T” and 80 meter Inverted V Wire Dipole Input Screen
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ZNET High-Pass “T” and 80 meter Inverted V Wire Dipole Output Screen
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An 80 meter ½ λ Inverted V Wire Antenna with High Pass “T” Equalizer Circuit Topology
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An 80 meter ½ λ Inverted V Wire Antenna with High Pass “T” Equalizer Circuit Topology
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An 80 meter ½ λ Inverted V Wire Antenna with High Pass “T” Equalizer Circuit Topology
and Path
Note the path taken
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An 80 meter ½ λ Inverted V Wire Antenna with Low Pass “Pi” Equalizer Circuit Topology
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An 80 meter ½ λ Inverted V Wire Antenna with Low Pass “Pi” Smith Chart
Note the path taken.
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Using Standard Value Capacitors
Stripline Capacitor
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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 http://opensolver.org/
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DEPSO Optimizer
DEPSO is bundled with the “Calc” application in LibreOffice www.libreoffice.org 83
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Using Standard Capacitors 1481 pF 2384 pF
1000 + 390 + 91= 1481 2200 + 160 + 24 = 2384
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80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology ZNET Solution
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80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology ZNET Solution
∆𝑇𝑙𝐿𝐿𝐿 = 0.145 dB
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80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology Smith Chart
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80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology
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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.
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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.
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Optimum Corner Frequency for Passband
𝜔2𝜔0
= 1+ 2𝑄𝑏𝑏2+1
2𝑄𝑏𝑏
T.R. Cuthbert, Broadband Direct-Coupled and Matching RF Networks, TRCPEP, (1999), pg19. 91
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Example: 80 meter band; 3.5 – 4.0 MHz 𝜔0 = 𝜔1 × 𝜔2 = 3.5𝑀𝑀𝑀 × 4.0𝑀𝑀𝑀 = 3.741 MHz (geometric centric frequency)
𝑄𝑏𝑏 =3.741
4.000 − 3.500 = 7.482 𝜔2𝜔0
= 1+ (2×7.482)2+12×7.482
= 1.069 new passband 𝜔1 = 3.500/1.069 = 3.274 𝑀𝑀𝑀 𝜔2 = 4.00 × 1.069 = 4.276 𝑀𝑀𝑀
80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology
Increased to Optimum Passband
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80 meter ½ λ Inverted V Wire on 80 meters using LPS4 Circuit Topology
Increased to Optimum Flat Passband
Note: % bandwidth increased From 13.4% to 26.8%, i.e. (x2)
= 0.082 dB ∆𝑇𝑙𝐿𝐿𝐿
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80 meter ½ λ Inverted V Wire on 80 meters using BPP4 Circuit Topology
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80 meter ½ λ Inverted V Wire on 80 meters using BPP4 Circuit Topology VSWR
= 0.207 dB ∆𝑇𝑙𝐿𝐿𝐿
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80 meter ½ λ Inverted V Wire on 80 meters using BPP4 Circuit Topology Smith Chart
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80 meter ½ λ Inverted V Wire on 80 meters using HPS4 Circuit Topology
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80 meter ½ λ Inverted V Wire on 80 meters using HPS4 Circuit Topology
Antenna VSWR Before and After Equalizer VSWR
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80 meter ½ λ Inverted V Wire on 80 meters using HPS4 Circuit Topology Smith Chart
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Li’s Genetic Algorithm Solution (required 6 hours computer time)
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Li’s Genetic Algorithm Solution Smith Chart
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ZNET Solution for Li’s Dipole
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ZNET Solution for Li’s Dipole using LPS4 Circuit Topology
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ZNET Solution for Li’s Dipole Using LPS4 Circuit Topology Smith Chart
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ZNET Solution to Li’s Dipole using LPS6 Circuit Topology
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ZNET Solution to Li’s Dipole using LPS6 Circuit Topology
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ZNET Solution to Li’s Dipole using LPS6 Circuit Topology Smith Chart
Required 11.95 seconds computation time.
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Cascade Serial Transmission Line (CASTL) Impedance Match
See L.N. Dworsky, Modern Transmission Line Theory and Applications, Wiley, NY (1979). 108
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80 meter ½ λ Inverted V Wire on 80 meters using CASTL Circuit Topology VSWR
CASTL circuit
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80 meter ½ λ Inverted V Wire on 80 meters using CASTL Circuit Topology Smith Chart
Note the path.
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80 meter ½ λ Inverted V Wire on 80 meters using CASTL Circuit Topology 1000 Watt Power Handling
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Homebrew Parallel Wire Transmission Lines
Parallel Wire Transmission Line
See http://www.hdtvprimer.com/KQ6QV/HomePage.html
ATLC2 computed these images. ATLC2 was developed by KQ6QV. (see URL below.)
E Field from two-wire transmission line cross-section.
Current distribution for closely spaced two conductor transmission lines.
Spreadsheet computes spacing “d” for a required impedance for a given width “wire diameter” and spacing dielectric (e.g. air for lowest loss).
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Homebrew Parallel Plate Transmission Lines and Stubs
Parallel Plate Transmission Lines and Stubs
w
d
Spreadsheet computes spacing “d” for a required impedance for a given width “w” and spacing dielectric (e.g. air for lowest loss).
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Parallel Plate Transmission Line (TEM mode) Optimizer WB6YVKMKS Units inches
center frequency = 21.225 MHz 2.12E+07 Line ImpedancePlate Width = 12.7 mm 0.0127 0.500 Zo = 124.999929334706-0.179690404275512i Ohms
Plate Conductivity = 35400000 S/m aluminum Ro = 125.00 OhmsSpacing = 6.386248872 mm 0.006386249 0.251 Xo = -0.18 Ohms
Dielectric constant of spacer = 2.3 PTFEmagnetic constant (u) = 1.25664E-06 H/m
electric constant (e) = 8.84194E-12 F/m velocity = 1.9781E+08 m/secvel factor = 0.659
RLGC Results Shorted stub as an inductorR' = 2.42E-01 Ohms/m Q stub = 458.7L' = 6.31905E-07 H/mG' = 1.00E-10 mhos/mC' = 4.04E-11 F/m
Ro = 124.9999293 OhmsXo = -0.1796904 Ohms
Zo magnitude = 125.0000585 Ohmsalpha' = 9.69E-04
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ATLC2 Example: 150 Ohm Parallel Plate Transmission Line
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80 meter ½ λ Inverted V Wire on 80 meters shorter CASTL Circuit Topology #4
using non-standard transmission lines and stubs
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80 meter ½ λ Inverted V Wire on 80 meters shorter CASTL Circuit Topology #4
using non-standard transmission lines and stubs
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80 meter ½ λ Wire Folded Dipole
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80 meter ½ λ Wire Folded Dipole using LPS6 Circuit Topology
Note that no transformer is needed.
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80 meter ½ λ Wire Folded Dipole using LPS6 Circuit Topology
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40 meter ¼ λ Monopole
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40 meter ¼ λ Monopole using ZNET Solution LPS4 Circuit Topology
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40 meter ¼ λ Monopole using LPS4 Circuit Topology
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40 meter ¼ λ Monopole using LPS6 Circuit Topology VSWR
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40 meter ¼ λ Monopole using LPS6 Circuit Topology Smith Chart
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6 meter ¼ λ Ground Plane (Arrow GP52)
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6 meter ¼ λ Ground Plane (Arrow GP52) using LPS4 Circuit Topology VSWR
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6 meter ¼ λ Ground Plane (Arrow GP52) using LPS4 Circuit Topology Smith Chart
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Shortened ¼ λ 6 meter Ground Plane (Arrow GP52) using LPS4 Circuit Topology VSWR
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10 foot Ground Plane on 6 meters using LPP4 Circuit Topology VSWR
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432 MHz Yagi Antenna Antenna design was “computer optimized” for 420-450 MHz by the commercial manufacturer.
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432 MHz Yagi Antenna using LPS6 Circuit Topology Smith Chart
Frequency Sweep from 400 to 450 MHz.
ZNET Matched antenna
SWR = 2 circle
“Optimized” antenna as supplied by the manufacturer..
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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….
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Enter Impedance and Admittance Inverters
L
-L -L
C
-C -C
-C -C
C
-L-L
L
RL>RS
RS>RL
J-Inverters
K-Inverters
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Implementing Norton Transforms
The negative valued components are “absorbed” by the BP resonators.
-L -L
-C -C radio antenna
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Norton Transforms As Transformers
antennaradio
There is a limitation on the achievable transformation ratio, n2, see Abrie’s textbook for details.
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80 meter ½ λ Inverted V Wire using Capacitor “Pi” and Inductor “L”
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80 meter ½ λ Inverted V Wire using Capacitor “Pi” and Inductor “L”
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Using L “T” & Low Pass Pi Transformers “XFMRPI” Circuit Topology
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80 meter ½ λ Inverted V Wire using WMB3 Circuit Topology T T Pi
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80 meter ½ λ Inverted V Wire using WMB3 Circuit Topology
VSWR
= 0.154 dB ∆𝑇𝑙𝐿𝐿𝐿
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80 meter ½ λ Inverted V Wire using WMB3 Circuit Topology Smith Chart
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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 non-resonant antenna.
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20 Foot Tall Monopole HF Antenna
Computer model image.
Commercial HF antenna.
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20 foot Monopole on 20 meters using LPP4 Circuit Topology
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20 foot Monopole on 20 meters using LPP4 Circuit Topology VSWR
∆𝑇𝑙𝐿𝐿𝐿 = 0.012 dB
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20 foot Monopole on 20 meters using LPP4 Circuit Topology Smith Chart
Note the path taken
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20 foot Monopole on 20 meters using LPS6 Circuit Topology VSWR
= 0.020 dB ∆𝑇𝑙𝐿𝐿𝐿
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20 foot Monopole on 20 meters using LPS6 Circuit Topology Smith Chart
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20 foot Monopole on 40 meters using LPS6 Circuit Topology
VSWR
= 0.663 dB ∆𝑇𝑙𝐿𝐿𝐿
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20 foot Monopole on 40 meters using LPS6 Circuit Topology Smith Chart
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20 foot Monopole on 20 meters using WMB2 Circuit Topology
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20 foot Monopole on 20 meters using WMB2 Circuit Topology VSWR
= 0.044 dB ∆𝑇𝑙𝐿𝐿𝐿
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20 foot Monopole on 20 meters using WMB2 Circuit Topology Smith Chart
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Example ZNET Solution 10 foot Monopole using 15m BPS2 Circuit
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10 foot Monopole on 15 meters using BPS2 Circuit Topology VSWR
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10 foot Monopole on 15 meters using BPS2 Circuit Topology Smith Chart
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10 foot Monopole on 15 meters using LPS6 Circuit Topology VSWR
= 0.014 dB ∆𝑇𝑙𝐿𝐿𝐿
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80m Inverted V Wire Dipole 100 foot (short) using LPS8 Circuit Topology VSWR
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80m Inverted V Wire Dipole 100 foot (short) using LPS8 Circuit Topology Smith Chart
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80m Inverted V Wire Dipole 100 foot (short) using LPS8 Circuit Topology VSWR
= 0.782 dB ∆𝑇𝑙𝐿𝐿𝐿
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80m Inverted V Wire Dipole 100 foot (short) using LPS8 Circuit Topology
(no transmission line from radio)
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80m Inverted V Wire Dipole 100 foot (short) using LPS8 Circuit Topology
(including 100 feet LMR400 from radio)
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80m Inverted V Wire Dipole 100 foot (short) using WMB3 Circuit Topology Smith Chart
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80m Inverted V Wire Dipole 100 foot (short) VSWR using WMB3 Circuit Topology
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80m Inverted V Wire Dipole 100 foot (short) using WMB3 Circuit Topology
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20 foot Flat Top Dipole
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20 foot Flat Top Dipole No Matching 20m through 10m
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20 foot Flat Top Dipole No Matching 20m through 10m
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20 foot Flat Top Dipole on 15 meters CASTL Circuit
using non-standard transmission lines
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20 foot Flat Top Dipole on 15 meters CASTL Circuit
using non-standard transmission lines VSWR
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20 foot Flat Top Dipole on 15 meters CASTL Circuit
using non-standard transmission lines Smith Chart
Note the path.
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20 foot Flat Top Dipole 1000 Watt Power Handling on 15 meters CASTL Circuit
using non-standard transmission lines
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20 foot Flat Top Dipole on 15 meters CASTL Circuit
using non-standard transmission lines and stubs
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20 foot Flat Top Dipole 1000 Watt Power Handling on 15 meters CASTL Circuit
transmission lines and stubs
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20 foot Flat Top Dipole on 15 meters CASTL Circuit
transmission lines and stubs Smith Chart
Note the path.
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20 foot Flat Top Dipole on 15 meters CASTL Circuit using
Commercially Available Transmission Lines
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Multi-Band Equalizer
Single Equalizer Circuit •Fixed value elements •All lumped L & C elements
Single non-resonant antenna
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20 foot Flat Top Dipole 17m through 15m Continuous Match
using LPS8 Circuit Topology Smith Chart
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20 foot Flat Top Dipole 17m and 15m Dual Band Match using
Single LPS6 Circuit Topology Smith Chart
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20 foot Flat Top Dipole 17m and 15m Dual Band Match using
Single LPS6 Circuit Topology VSWR
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20 foot Flat Top Dipole 17m and 15m Dual Band Match using Single LPS10 Circuit Topology VSWR
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20 foot Flat Top Dipole 17m, 15m, & 20 MHz WWV Match using
Single LPS8 Circuit Topology VSWR
20 MHz WWV
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20 foot Flat Top Dipole 4 Band Match using Single LPS10 Circuit Topology
17m, 15m, 12m, & 10m Bands VSWR
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20 foot Flat Top Dipole 4 Band Match using Single LPS10 Circuit Topology
17m, 15m, 12m, & 10m Bands VSWR
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20ft Flat Top Dipole on 4 bands 17m, 15m, 12m, & 10m bands Smith Chart
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ZNET Circuit Topology Modification Algorithm
ZNET has a circuit topology modification algorithm that can automatically “prune” circuit elements to improve the impedance match.
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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.
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C1 Value @ Upper Optimization Boundary
C1 is a candidate for removal from equalizer circuit.
normalized values
de-normalized values
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“T” Circuit Solution Including C1
C1 value at upper optimization boundary.
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“T” Circuit Solution VSWR with C1
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“T” Circuit Solution with C1 Removed
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“T” Circuit Solution VSWR without C1
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Extended C1 Upper Boundary
Increased C1 upper boundary value x1000 so that remaining components (i.e.
L2 & C3) can be optimized.
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“T” Circuit Solution with Extended Boundary
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“T” Circuit Solution VSWR with Extended Boundary
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“T” Circuit Extended Solution Over Wider Frequency Span
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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.
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Results of Circuit Element “Pruning” Algorithm
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VSWR Results of Circuit Element “Pruning” Algorithm
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40 meter ½ λ Inverted V Wire Dipole LPP4 no pruning
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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.
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10 foot Monopole on 20m Example
Note the high VSWR
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10 foot Monopole 20 meters using BPP2 Circuit Topology
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10 foot Monopole 20 meters using BPP2 Topology AFTER Pruning
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10ft Monopole on 20 meters using BPP4 Circuit Topology and Pruning Algorithm
ZNET algorithm with pruning returned to simple L circuit
topology again.
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Component Tolerances Study
Vary L & C element values
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Component Tolerances Study 40m ½ λ Inverted V Wire Dipole
using LPS6 Circuit Topology Smith Chart
Initial results
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Component Tolerances Study 40m ½ λ Inverted V Wire Dipole
using LPS6 Circuit Topology Smith Chart
All 6 LC components simultaneously varied +/- 5%.
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Component Tolerances Study 40m ½ λ Inverted V Wire Dipole
using LPS6 Circuit Topology Smith Chart
All 6 LC components simultaneously varied +/- 2%.
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Component Tolerances Study 40m ½ λ Inverted V Wire Dipole
using LPS4 Circuit Topology Smith Chart
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Component Tolerances Study 40m ½ λ Inverted V Wire Dipole
using LPS4 Circuit Topology Smith Chart
All 4 LC components simultaneously varied +/- 5%.
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Component Tolerances Study 40m ½ λ Inverted V Wire Dipole
using LPS4 Circuit Topology Smith Chart
All 4 LC components simultaneously varied +/- 2%.
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Component Tolerances Study 20 foot Dipole on 15m
using CASTL Circuit Topology Smith Chart
Zo of both transmission lines simultaneously varied +/- 5%.
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20 foot Flat Top Dipole on 15 meters CASTL Circuit
transmission lines and stubs Smith Chart
Zo of transmission lines & stubs simultaneously varied +/- 5%.
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Revisit Yarman’s Short Monopole Example
• Changed circuit element values,
• Changed transformer from a 12.66:1 transformer to a standard 16:1 transformer (ala a transmission line transformer).
ZNET solution
using Yarman’s
LPS6 circuit
topology.
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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.
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Yarman’s Monopole Example ZNET Solution VSWR
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Yarman’s Monopole Example SRFT and ZNET VSWR Comparison
SRFT VSWR ZNET VSWR
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Antenna Height Variations
• Vary the height above ground of 40 m ½ λ inverted “V” wire antenna using same LPS6 equalizer circuit; – Apex height @ 25 feet – Apex height @ 30 feet (design height) – Apex height @ 35 feet
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40 meter ½ λ Inverted “V” Wire Dipole versus Apex Height Using Same Equalizer Circuit
30 feet (design height)
25 feet
35 feet
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15 meter wire delta loop hardware test
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15 meter wire delta antenna test VSWR before equalizer
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15 meter wire delta antenna Smith Chart
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QRP Test Equalizer at 15 meter Delta Loop Feedpoint
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VSWR of QRP “T” Equalizer Using Low-Cost NTE Ceramic Capacitors
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Frequency (MHz)
VSW
R15 meter delta antenna VSWR
measured at ZNET equalizer inputat antenna feedpoint
20 20.25 20.5 20.75 21 21.25 21.5 21.75 22 22.25 22.51
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
Using ATC Hi-Q capacitors, no coax pigtail, and no feedline choke. Measured with HP VNA, S11 data used to compute VSWR.
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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).
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Also see http://www.eit.lth.se/index.php?uhpuid=scd.mgu&hpuid=139&L=1 228
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Extending ZNET
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Available Impedance Matching Software Applications
• OptiMatch http://www.microwavesoftware.com/optiad.html
• Optenni http://www.optenni.com/
• AnTune http://www.antune.net/index.html
• Zmatch http://www.nuhertz.com/software/zmatch
• Wmatch http://www.mikehutt.com/imn.htm (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)
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Interesting Antenna Design Optimization Software Tools
• 4NEC2 (simplex and genetic algorithm optimizers) – http://www.qsl.net/4nec2/
• Nikiml’s Antenna pages (Differential Evolution) – http://clients.teksavvy.com/~nickm/scripts.html
• NEC Lab – http://www.ingenierias.ugto.mx/profesores/sledesma
/documentos/index.htm • The Applied Computational Electromagnetics
Society [ACES] – http://aces-society.org/software.php
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Thank You
• Questions ?
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