Wave Articulation Matrix

Sean Logan KG7ARW


Parameters for Finite Element Analysis
Cylinders as quarter-wave resonators (The yellow cylinder is not actually there. It is shown to demonstrate the relationship between the wavelengths.)


Welcome. Please find information regarding an experimental electromagnetic resonator. Using the Golden Ratio (1.61803...) as the base of logarithms gives rise to a Geometric Series of sidebands when the waves heterodyne. The following paper describes the technology in more detail.

Golden Ratio Heterodynes (5 May 2019)
Euclid's Extreme and Mean Ratio in RF Sidebands

Experimental Data

The initial set of measurements were made with the structure excited such that the Electric Field flowed in circles through the circumferences of the cylinders. That is, the Electric Field formed loops coaxial with the axis of symmetry of the structure. These measurements are documented here:
Measurements preformed in a shielded anechoic chamber at PSU. Frequency sweeps with VNA. Spectrum Analyzer measurements with wideband noise source. Discussion of experimental data. Calculated vs. measured frequencies of resonance.

Experimental Data, (18 Jan 2019)

We may also consider the mathematical Dual of this situation: Exciting the structure such that the Magnetic Field flows in circles through the circumference of the cylinders. That is, the Magnetic Field forms loops coaxial with the axis of symmetry of the structure. In othere words, look at the structure as a Magnetic Circuit, rather than an Electric Circuit. The reslts of such experiments are forthcoming.

Design Specifications

Dimensions, Decimal, GIF file
Dimensions, Algebraic, 4 element
Dimensions, Algebraic, 8 element
Dimensions, Decimal, .dxf CAD file
Dimensions, Decimal, text file
Plastic fixture for holding cylinders, PNG file
Plastic fixture for holding cylinders, DXF file
Cylinders, 4+1 element array
Cylinder Dimensions, Algebraic. 4+1 element array.
Cylinder Dimensions, Decimal. 4+1 element array.
Hyperbola Equation

Wiring Diagrams

The purpose of the circuitry is to create, and amplify, a signal made up of a Geometric Series of frequency components.
Block Diagram
Unbalanced Mixer Schematic
Class A Amplifier Schematic
GNU Radio file for creating a Geometric Series of Frequencies
Wiring Diagram 4
Wiring Diagram 7
Wiring Diagram 2
Time Varying Voltage (Electric Field) on Central Cylinder. The capacitance between the central cylinder and the next cylinder is like the coupling cap between the plate and the antenna.

3D Renderings

Perspective Views 1
Perspective Views 2
Perspective Views 3

Math and Theory

curl B = dE/dt + J
Golden Ratio base Logarithms
How the waves heterodyne (text explanation)
How the waves heterodyne (gif)
Heterodynes in the frequency domain
Physical geometry derived from wave equation
Physical geometry derived from wave equation (2)
Golden Rectangle construction

Rectangles rolled up to make the cylinders
The whole structure fits within a Hyperbolic Horn, which is the shape of a vortex in water
A Golden Spiral can be thought of as a wave with an exponential envelope.
If the period of the wave shortens by a factor each cycle, then the wave has a hyperbolic envelope.
This image shows the meaning of the logarithm taken in the previous image.
Here you can see the relationship between the period of a given cycle of the wave, and all subsequent periods.
Exponential vs. Hyperbolic growth. Hyperbolas reach an asymptote, whereas exponential curves always remain finite.
The geometry of the structure can be derived from a spiral wrapped around a hyperbolic horn.
Simpler form of the wave equation, with the asymptote at z=0. If this was a sound wave, it would sound like a chirp.

Fun Animations

CERN Open Hardware License v1.2

Copyright Sean Logan 2011-2019.

This documentation describes Open Hardware and is licensed under the CERN OHL v. 1.2.

You may redistribute and modify this documentation under the terms of the CERN OHL v.1.2. (http://ohwr.org/cernohl). This documentation is distributed WITHOUT ANY EXPRESS OR IMPLIED WARRANTY, INCLUDING OF MERCHANTABILITY, SATISFACTORY QUALITY AND FITNESS FOR A PARTICULAR PURPOSE. Please see the CERN OHL v.1.2 for applicable conditions.


Et Cetera

Circumference, wavelength and resonance calculations

Other Configurations



1. Lange, Christian. "L’Angolo aureo" June, 2002
2. Campbell, Rick. Designing and Building Transistor Linear Power Ampliers. QST. February, 2009. Part 2
3. Freeman, Jeffrey Phillips. "An In-depth Look at Duals and Their Circuits" 28 September 2020.