Mathematical and Physical Principles of PI Technology

[Aziz]

Hi all,

I for one would focus to the X/R response relation in the decay time.
Setting the integration windows so that R reponse is greater than the X response.

Good job!
Aziz

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I used Corbyn's equation 6 and asked AI to calculate a Tau for both a clad US quarter and a pre-1965 silver US quarter. Here is what I got:

Modern Clad Quarter
T = (7.16 * (0.01213)^2) / 13.17
T = (7.16 * 0.00014713) / 13.17
T = 0.0010535 / 13.17
T = 0.0000800 seconds
--> T_clad ~ 80 microseconds

Pre-1965 Silver Quarter
T = (7.16 * (0.01213)^2) / 0.70
T = 0.0010535 / 0.70
T = 0.001505 seconds
--> T_silver ~ 1505 microseconds (1.5 ms)

Modern Clad Quarter: Utilizes a pure copper core sandwiched between resistive cupronickel skins (75% Cu, 25% Ni). The bulk resistivity of this composite core-and-cladding structure scales the effective material constant up to roughly k = 13.17 relative to gold.
Pre-1965 Silver Quarter: Composed of a highly conductive 90% silver alloy, yielding an incredibly low resistivity scaling factor of k = 0.70 relative to gold.

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Corbyn's cylinder model has a hole in the center, so it's not quite there, but it does give a good approximation, not too bad? What is needed is a flat volumetric disc model.


Smythe Flat-Disc Equation:
Tau = (mu_0 * sigma * a * t) / pi^2 [seconds]

Where:
* mu_0 = 4 * pi * 10^-7 H/m (Permeability of free space)
* sigma = 3.0 * 10^7 S/m (Composite clad bulk conductivity)
* a = 0.01213 m (Quarter outer radius)
* t = 0.00175 m (Quarter thickness)
* pi^2 = 9.86960

Step-by-Step Calculation:
Numerator = (4 * pi * 10^-7) * (3.0 * 10^7) * (0.01213) * (0.00175)

Note that 10^-7 and 10^7 cancel out perfectly:
Numerator = 4 * pi * 3.0 * 0.01213 * 0.00175
Numerator = 12 * pi * 0.0000212275
Numerator = 37.69911 * 0.0000212275
Numerator = 0.00080026 seconds

Denominator = pi^2 = 9.86960

Tau = 0.00080026 / 9.86960
Tau = 0.00008108 seconds
--> Tau_clad ~ 81.1 microseconds




The same result. That is a surprise. I smell fish.

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Smythe Flat-Disc Equation (Pre-1965 Silver Quarter):
Tau = (mu_0 * sigma * a * t) / pi^2 [seconds]

Where:
* mu_0 = 4 * pi * 10^-7 H/m (Permeability of free space)
* sigma = 5.1 * 10^7 S/m (90% Silver alloy bulk conductivity)
* a = 0.01213 m (Quarter outer radius)
* t = 0.00175 m (Quarter thickness)
* pi^2 = 9.86960

Step-by-Step Calculation:
Numerator = (4 * pi * 10^-7) * (5.1 * 10^7) * (0.01213) * (0.00175)

Note that 10^-7 and 10^7 cancel out perfectly:
Numerator = 4 * pi * 5.1 * 0.01213 * 0.00175
Numerator = 20.4 * pi * 0.0000212275
Numerator = 64.08849 * 0.0000212275
Numerator = 0.00136044 seconds

Denominator = pi^2 = 9.86960

Tau = 0.00136044 / 9.86960
Tau = 0.00013784 seconds
--> Tau_silver ~ 137.8 microseconds

This is vastly different from Corbyn's equation 6.

So I throw out Corbyn's tube cylinder model and use Smythe's formula, I guess. Corbyn uses Wheeler's solenoid approximations for a hollow pipe shell. This geometry maximizes the external flux linkage loop area. Smythe solves for a solid sheet where internal concentric current loops shrink toward the center, resulting in a much lower net self-inductance.

Corbyn's Equation 6 works beautifully for tracking the volumetric scaling of chunky, three-dimensional targets (like cans, boxes, or thick cylinders). However, for flat coins, Smythe's flat-disc derivative is required to accurately map real-world decay values without relying on massive empirical correction factors.

I have no idea what figure is actually accepted for a pre-1965 US quarter. Any comments?

Around 150 microseconds: https://www.geotech1.com/forums/foru...ion#post445303 , so the calculated 138 microseconds seems to be good app.

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In an actual experiment, to achieve the maximum possible coupling, the flat coin must be placed perpendicular to the direction of the magnetic flux lines (which means the flat face of the coin is parallel to the flat plane of the monocoil).

[Aziz]

Hi all,

don't blame me please.

Bessel functions may model such diffusion processes (eddy current diffusion).

https://en.wikipedia.org/wiki/Bessel_function

Look at the PDF. Fck me!
This is what Google AI says.

[Aziz]

Ok,

I have remembered the Bessel function.
I did remember Bessel only.

This is really mind blowing math. I have given up in the following thread to follow this.

https://www.geotech1.com/forums/forum/projects/project-challenge/425748-frequency-dependent-target-response

This was beyond my scope.

[dbanner]

Bessel functions emerge naturally when you try to solve Maxwell's equations in systems with cylindrical or circular symmetry. That is very interesting. Considering he(Bessel) was studying planetary motion, not eddy current decay in coins.


He was the first astronomer who determined reliable values for the distance from the Sun to another star by the method of parallax. Certain important mathematical functions were first studied systematically by Bessel and were named Bessel functions in his honor. Nice!



The Mathematical Link:
Bessel's Differential Equation

[Aziz]

Here is an another AI paper. This time from Claude Code Opus-5 (I have spent my last free prompt).
Kelvin functions appear here.

[Tinkerer]

Thank you Aziz for the Eddy current paper. I uploaded it to Claude and asked for a clarification of how this would apply to a square current wave PI. Here is the answer: Square wave eddy currents.docx

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VISUAL TIMELINE: EDDY CURRENT DECAY IN A SOLID CLAD US QUARTER


Orientation Reference:
The coin lies perfectly flat on the testing plane (parallel to the loop).
The primary magnetic field vectors pass straight vertically through the face,
hitting the coin's boundary profile at a clean 90-degree perpendicular angle.


FRAME 1: THE TRANSMITTER DRIVE PHASE (0 us to 3200 us)

The monocoil passes a steady 1A ramp current. A high-density vertical
flux matrix passes through the target area. A counter-acting "reverse"
eddy current loop forms strictly along the outer edge to oppose the flux.



Dense Magnetic Flux Lines

Strong Reverse Eddy Current circulating purely on the outer perimeter skin layers.


FRAME 2: THE INSTANT CUT-OFF SHOCK (3200 us to 3200.01 us)

The current source snaps shut with a crisp 10ns fall time. The primary field
collapses instantly. To conserve the crashing flux layout, a massive, peak-density
"forward" eddy current sheet materializes instantly along the boundary edge.

Peak Inductive Slew Rate:
Massive, saturated current rings locked onto the outer cladding.

FRAME 3: EARLY DECAY WINDOW (1 us to 10 us After Turn-Off)

The coil clears its own RLC parasitic footprint. The high structural resistance
of the outer cupronickel cladding skins (high nickel content) rapidly dissipates
the surface current energy, forcing the current loops to diffuse inward.


Surface skin currents rapidly burn out and diffuse inward (Resistive Skins Fading) to settle in the deeper core.


The coil clears its own RLC parasitic footprint. The high structural resistance
of the outer cupronickel cladding skins (high nickel content) rapidly dissipates
the surface current energy, forcing the current loops to diffuse inward.

Surface skin currents rapidly burn out and diffuse inward to settle in the deeper core.



FRAME 4: LATE DECAY WINDOW / STABLE MATRIX (10 us to 80+ us)

The resistive outer cladding is completely quiet. The remaining eddy currents
now circulate exclusively inside the high-conductivity solid copper core layer.
Because pure copper offers a very low-resistance path, these internal rings
glide slowly, decaying at the clear, stable 81.1 us Smythe time constant.
Outer cupronickel layers dead.
Core currents glide slowly, casting a readable secondary field.

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My take-away is that the initial non-linearity is due to eddy diffusion from the surface toward the core. this initial non-linearity is a direct consequence of the transient skin effect.
The current sheet must now diffuse radially from the outer perimeter inward toward the dead center along the wide diameter plane of the coin. This should hold true for all coins. Might the ~(1)tau diffusion time be consistent for all coins?

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AI conjured up this app for me. Seems quite good. It should give an idea of what results to expect when constructing DIY singular round (circular) coils (Multi-loop Mono-coils).

It agrees with the actual measured parameters contained in Tables C-1 and C-2 in "PI-Math" document.

And for magnet(enamel)copper wire, it is quite close to Qiaozhi's coil calculator (Maxwell/Wheeler) which I still use.

The results for Teflon wire looks reasonable as to what should be expected (lower CL(pF)), only way to find out would be to make a few coils and run tests.

https://drive.google.com/file/d/1DdhBR_2Ps7cfE3O2nXntY5BpzNHC1aHM/view?usp=sharing

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Playing around with AI to write python script is sure fun, I'll get over it soon though.

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Transmit side

Here I can generate the waveforms for any coil metric. I use "PI -Math" Fig.C-1 and Fig. C-2 as templates.