Mathematical and Physical Principles of PI Technology

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I am starting this thread to serve as a dedicated space for deep-diving into the foundational physics and mathematical modeling behind Pulse Induction (PI) technology. While there are plenty of excellent threads on specific DIY builds and schematics, I wanted a place where we can collectively break down the core principles that govern PI performance from the ground up.
My goal is to pull together and synthesize the wealth of information scattered across the web. I intend to aggregate technical theory from existing forum threads, academic articles, and all relevant patents to build a truly comprehensive repository here.
Let's demystify the math behind the magic. Looking forward to your insights!

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Here is detailed article on 'Optimizing Target Responses' that models and analyzes how non-ideal RLC networks behave during pulse induction turn-on and turn-off events."

This was a series of posts written by Carl-NC consisting of 5 parts from this thread https://www.geotech1.com/forums/foru...rget-responses. I have compiled them into a single document.

Note on formatting: Please be aware that some of the mathematical characters and subscripts may not have rendered with perfect accuracy due to character encoding conversion from the original. Refer to the original thread.

[Carl-NC]

Do you have ITMD3?

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Hi Carl, no, I don't. I'll eventually have get a copy of course. I almost forgot about ITMD3, I've been taken up on other things of late.

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Ferric toes (Eric Foster) had reported some data using a Hocking Conductivity meter:https://www.geotech1.com/forums/foru...settling/page3

A US 1889 silver dollar reads 84. A 1964 half dollar 92. A silver quarter 90. A copper cent 52. A dime 52., and the humble nickel 5.5. By comparison, a Whites Electronics commemorative 1oz medallion 0.999 fine silver reads 101. The lower figures for the silver coins are due to small quantities of other metals alloyed to make the coins harder than the base metal, and wear better. Pure silver should read 105.

With the help of AI, these results were converted to eddy current decay time constants.

coin conductivity and time constant comparison.zip


Example Calculation for the US Nickel:

• Hocking Meter Reading = 5.5 % IACS
• Diameter = 21.21 mm
• Thickness = 1.95 mm

Formula:
Tau = 0.0911 * (Hocking Reading) * (Diameter in mm) * (Thickness in mm)
Calculation:
Tau = 0.0911 * 5.5 * 21.21 * 1.95 = 20.72 microseconds


Where does the 0.0911 number come from?
It is a "shortcut constant" created by combining all the unchanging physical math rules together. The result is Tau in microseconds.

The steps were:
Step 1: Convert the Hocking Meter Reading to Conductivity
Step 2: Establish the Coin's Dimensions
Estimate Tau Based on Field Orientation

Disclaimer: These Tau values are theoretical estimates for comparative use only. Actual decay times will vary based on coin wear, exact metal composition, and testing equipment setup.

[Tinkerer]

With a PI metal detector the target response amplitude varies very much, depending on the size of the target, the distance from the coil and the material of the target itself. Which is the best way to "normalize" the target amplitude, so that we can look only at the time constants of the targets and analyse the decay curve of the targets to

The clean way to do this is to work in the log domain, not to literally "normalize" the amplitude by dividing curves down to a common scale. Here's why that's the right move and what it buys you.
The physics. For a target dominated by simple eddy-current decay, the voltage you sample after TX blanking is well modeled as
V(t) ≈ A · e^(−t/τ)
A is a big, messy multiplicative factor — it bundles target size, distance/coupling (mutual inductance falls off steeply with range and orientation), and any gain in your front end. τ is what actually carries the material/geometry signature (for a simple conductor, τ ≈ μ₀σd²/k, with k a shape constant), and it's largely decoupled from A. Because A only scales the curve and never changes its shape, the natural way to strip it out is to take the log:
ln V(t) = ln A − t/τ
That's a straight line in t, with intercept ln A (all the "how big/how close/how conductive-in-an-absolute-sense" information) and slope −1/τ (the shape you actually want). So the "normalization" isn't a division step you do to the raw curve — it falls out for free once you log-transform and fit a line. Amplitude becomes the intercept, τ becomes the slope, and they're orthogonal by construction.
Why this beats a naive amplitude-normalize. If you instead divide every sample by, say, the first gate (V(t)/V(t₀)), you're using one noisy sample as your entire reference, and multiplicative noise on that one point corrupts every point downstream of it. Doing it in log space and fitting a slope by least squares over many gates averages the noise down instead of propagating it from a single point — with your 94-channel filter bank you have far more gates than the classic 2-point "early/late ratio" scheme most commercial PI detectors use (that ratio is literally the 2-point special case of this same idea: ln(V_early/V_late)/(t_late − t_early) = −1/τ). You have enough resolution to do a proper weighted regression instead, which is a real advantage.
The complication worth designing for: not everything decays as a single exponential. Ferrous targets and mineralized ground exhibit magnetic viscosity (Néel/after-effect) relaxation, which looks much more like a power law than an exponential — its signature is a straight line in ln V vs ln t (log-log), not ln V vs t (log-linear). So a robust front end should fit both models per target and use which one fits better (plus the residual curvature) as an extra discriminant axis — eddy-current-only vs. viscous/ferrous vs. mixed — completely independent of amplitude, since both fits are done in log space. I generated a quick synthetic-target figure to make this concrete:
Presented files
Presented files
Left panel is what you get straight off the ADC — six wildly different-looking curves, dominated by A. Middle panel is the same data after ln(V): the two copper targets (same τ = 60 µs, ~8× different amplitude from size/distance) collapse onto parallel lines — only the vertical offset (intercept, i.e. ln A) differs, the slope is identical. Right panel shows the same data in ln V vs ln t — the ferrous targets (nail, iron junk, magnetic-viscosity decay) straighten out there instead, revealing themselves as a different physical decay law rather than just "a weird τ."
A few implementation notes specific to your architecture:
Do the synchronous bipolar-pulse averaging (your PRF/mains cancellation) before the log transform, in the linear domain — log(mean(V)) ≠ mean(log(V)), and averaging after logging biases the estimate, especially at low SNR near the tail where it matters most. Guard the regression against samples that dip below ~3× noise floor (visible in the figure as where each curve just stops) — this is a real dynamic-range limit, not an artifact, and it means your usable decay window is itself target-dependent (short τ or weak/deep targets give you fewer usable gates, which argues for an adaptive-window, SNR-weighted least-squares fit rather than a fixed set of gates). And keep the intercept (ln A) around as a separate output rather than discarding it — it's your depth/size/signal-strength estimate, useful for gain ranging and depth indication even though you deliberately exclude it from the τ-based target ID axis.
One physical caveat worth flagging: this removes the amplitude confound (size/distance/gain), but size and conductivity are still coupled inside τ itself (τ ∝ σd²), so a small very-conductive target and a larger less-conductive one can land at a similar τ — that's an inherent physics limit of single-parameter time-constant discrimination, not something normalization can fix. With 94 gates you have the SNR headroom to go further than a single τ if you want — fitting for curvature (deviation from a single exponential, indicating a multi-mode/composite target) or a two-exponential model via Prony's method / matrix-pencil on the log-decay residual — which is a natural next step given the front-end bandwidth you've already built.

[Carl-NC]

Well done.

[Carl-NC]

Here is Appendix C (PI Math) from ITMD3.

PI-Math.pdf

I read your compilation document and found an error I made; Page 3 paragraph following Fig 1.6:

"During TX turn-off T20 clearly has a stronger initial eddy current than T20 (d) ..."

should say:

"During TX turn-off T20 clearly has a stronger initial eddy current than T100 (d) ..."

I fixed the error in the original post.

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Here is Appendix C (PI Math) from ITMD3.

[ATTACH]n452393[/ATTACH]

I read your compilation document and found an error I made; Page 3 paragraph following Fig 1.6:

"During TX turn-off T20 clearly has a stronger initial eddy current than T20 (d) ..."

should say:

"During TX turn-off T20 clearly has a stronger initial eddy current than T100 (d) ..."

I fixed the error in the original post.

Yes. Only now I see it.

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Classical Single-Period / Mono-Pulse Architecture

TX Turn-On Response Analysis

Reference: "PI Math" page 553

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The foundational laws, and Lumped element theory.

It is good to know some calculas also. I used this book.

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An AI generated app for generating all the results.

The app was generated using the Eq C-2 from "PI math"



Coil Inductance Calculation
The app implements the Kirchhoff-Stefan continuous multi-loop circular bundle formulation.


Link to the app:

https://drive.google.com/file/d/1wJtfBP653kQq_3C7UPsKs8FCZxB4W0_z/view?usp=sharing

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I've been doing a little research. Real-world buried metal targets are three-dimensional volumetric objects, not infinitely thin wire loops. A simple single-loop target represents a first-order target. It behaves like a single RL circuit with one distinct time constant, producing a pure exponential decay.

On the other hand, a solid cylinder, for example, is a higher-order target. Because of the way eddy currents diffuse through its thickness (skin effect), a solid cylinder actually possesses infinitely many overlapping time constants, decaying as a sum of multiple exponentials.

Geometry, volume, conductivity, and magnetic permeability. Every real-world metal object buried in the ground possesses all four of these physical properties. Together, these characteristics dictate exactly how strong the target’s response will be and how long its eddy currents will take to decay.

I suppose that when trying to model certain targets by running simulations, for better accuracy, you can model it as a ladder network of nested RL loops. You can string together several nested stages to create an inductive-resistive delay line.

It is interesting to note that the first stage is magnetically coupled directly to your search coil and each successive stage is coupled to the previous stage and not directly to the search coil. I wonder if this coupling method is correct?

I suppose for standard pulse induction (PI) coin detectors with sample delays greater than 10–15 μs, it makes no practical difference at all.

[boilcoil]

You are probably right.
I assume all this occurs under the influence of electrodynamic forces and skin effect. Generally speaking, electrodynamic forces tend to expand the current loop within its own plane while compressing it along its axis. This may explain why the eddy currents initially appear only along the coin's periphery (electrodynamics) and on its surface (skin effect). Most likely, at the very beginning, two current loops form on opposite sides of the coin's rim - driven by the rapid rate of change in magnetic flux and the dominance of the skin effect—before merging into a single loop (electrodynamics) and subsequently spreading throughout the coin's volume.
I think this is not a one-dimensional process, and simulating it would be extremely complex.

[Carl-NC]

Consider a step magnetic field applied to a metal cylinder. At t=0+, eddy currents are distributed only across the top surface of the cylinder. Immediately, the eddies begin to diffuse downward into the depth of the cylinder, but also radially diffuse outward toward the perimeter. After about 1 tau, the eddy diffusion is done and then it's all decay, and closely follows a single tau decay. Here is the step response of a silver US quarter:


The dashed line is the single-tau (146us) decay. The initial non-linearity is due to eddy diffusion and is done at about 146us, which happens to be the tau of the quarter.

[boilcoil]

Thank you Carl, for the clarifications.