We have a steel pipe which is charged negatively. The Coulomb Potential in the space around the pipe is everywhere negative. We are going to vary the magnitude of the Coulomb Potential, without changing its sign. The sign will always remain negative.
We can think of the magnitude of the Coulomb Potential as a function of time, g(t). We could vary the Potential as a simple sine wave:
$$ g(t) = -1 - \sin{( \omega t)} $$
Or we might want a signal that has a particular spectral content. A signal composed of certain frequency components. A special spectrum of energy.
I want a signal made up of a Geometric Sequence of frequencies. The ratio between successive terms in the Sequence we shall call Alpha.
$$ \alpha = \frac{1 + \sqrt{5}}{2} $$ $$ \alpha = 1.61803... $$
In 300 BC, Euclid called this the "Extreme and Mean Ratio". Around the time of the Renaissance, people called it the Divine Proportion, or the Golden Ratio. It has an interesting property when it comes to the non-linear mixing of waves.
$$ g(t) = -Bias - ... - \sin{( \alpha^{-3} \omega t )} - \sin{( \alpha^{-2} \omega t )} - \sin{( \alpha^{-1} \omega t )} - \sin{( \omega t )} - \sin{( \alpha \omega t )} - \sin{( \alpha^2 \omega t )} - \sin{( \alpha^3 \omega t )} - ... $$
$$ g(t) = - Bias - \sum_{n=-\infty}^{\infty}{ e^{j \alpha^{n} \omega t} } $$
Let's look at some waveforms.
One term in the Geometric Series:
$$ f = 1 kHz $$
Two terms in the Geometric Series:
$$ f + \frac{f}{a} $$
Four terms in the Geometric Series:
$$ f + \frac{f}{a} + \frac{f}{a^2} + \frac{f}{a^3} $$
On a Spectrum Analyzer, our Geometric Series of frequencies will look something like this. We may need a parallel LC circuit to rebuild the waves after the diode heterodynes them.
The terms in the Geometric Sequence get closer together as we approach zero. There are actually an infinite number of terms between one and zero. This is similar to the transformation 1/z in the Complex Plane.
We're going to use 741 op-amps and a diode to create a Geometric Sequence of frequencies, beginning with just two waves.
The three keys to the circuit are:
If you overlay the Complex Plane with a colored checkerboard, then look at the transformation w = 1/z you will see something like this:
It looks like four Smith Charts.
I wrote some Python code which allows you to adjust R, L and C, and see a plot of frequency versus impedance magnitude. This is very similar to a Bode Plot, except that we are plotting on a linear scale, not a logarithmic scale.
This is for a parallel LC circuit, in which the resistance is in series with the inductor. That is how inductors are in real life: they have some amount of resistance.
