GROUND BALANCE ( without the ground :-) )

[dbanner]

The algorithm is extracting one of the components and is automatically rotating the phase in a third channel. This only happens when you wave the target, so it is a GB mode that is initiated when you press the button to ground balance.

It's just a guess.

This implies a virtual(imaginary) channel is used when this mode is selected.

[moodz]

see next post

[moodz]

The algorithm is extracting one of the components and is automatically rotating the phase in a third channel. This only happens when you wave the target, so it is a GB mode that is initiated when you press the button to ground balance.

It's just a guess.

This implies a virtual(imaginary) channel is used when this mode is selected.

Your right about one thing ... your theory is a guess
Thanks for all the kind comments regarding the "grey box" prototype ... not quite a black box prototype yet.

A key element is the inclusion of a virtual ground by means of forcing the search coil to "see" a really bad ground all the time ....

How is this done ?? ... well you could do it by including a "ground" eg filling the RX coil casing with ground type material ( ferrite dust / hot rocks etc ) .... however this is not efficient and the TX / RX field would be distorted.

However I dont do that ... its not what you would call an "elegant solution"

The "elegant solution" uses reflected impedance ... A ground that needs balancing out appears as a complex impedance in the RX signal.

Reflected impedance refers to the impedance as it appears from one side of a circuit to another due to the coupling of two circuits, particularly in transformers. It's essentially the impedance of a load connected to the secondary side of a transformer, as seen from the primary side.

By arranging a complex impedance that represents the worst possible ground ( eg lossy / viscous / ferrite ) to be in the signal path from TX to RX and using the principle of reflected impedance .... the real world "actual" ground would have to be extremely bad in order for this GB method not to overcome it.

Because the detector "sees" this bad ground all the time the detector can be be "prebalanced in air" without an actual ground signal.

This is a key claim in both PP patents.

I will know if someone is probably violating the patent simply by them claiming they can ground balance their detector in air.

Because I dont encumber the coil with this "reflected impedance" ... there is no need to prepare special coils ( I use a commercial coil to demonstrate this ) This also means target signals are not impacted.

My actual ground balance method ( code method ) is also new though in that auto correlates the near to the optimal ground balance .. much more effectively than other methods ....so its another claim in the patents.

P.

[dbanner]

I sometimes like to string a few "catchy" terms together in order to sound like the real deal. I haven't got a clue what's really going on.

I've noticed a lot of papers being published with outrageous claims, but it all sounds so nice.

You're the real deal.

[dbanner]

Reflected impedance refers to the impedance as it appears from one side of a circuit to another due to the coupling of two circuits, particularly in transformers. It's essentially the impedance of a load connected to the secondary side of a transformer, as seen from the primary side.

I think it is how output valves are matched to loudspeakers. I mean in the design of the output transformers.

[dbanner]

But what happens to your method if you change coils. Your algorithm will adjust? Or it doesn't matter.

[moodz]

But what happens to your method if you change coils. Your algorithm will adjust? Or it doesn't matter.

….. still works. Do ground conditions change if you change coils? Ground is ground whether it’s virtual or real.

[moodz]

I think it is how output valves are matched to loudspeakers. I mean in the design of the output transformers.

in a broad sense … a driving impedance can “see” a load impedance and vice versa.

[boilcoil]

Hey Paul, any ground balance reduces the sensitivity of the detector, and you have fixed the worst case (maybe to balance the lighter ones as well).
But if you have not provided for a lighter ground balance, does this mean (according to your calculations) that all balances of light and heavy soils lie in same zone?
This is important to not lose the detector's depth.

[moodz]

Hey Paul, any ground balance reduces the sensitivity of the detector, and you have fixed the worst case (maybe to balance the lighter ones as well).
But if you have not provided for a lighter ground balance, does this mean (according to your calculations) that all balances of light and heavy soils lie in same zone?
This is important to not lose the detector's depth.

Good question ... so what is happening when you "pump" the coil over a ground on some types of commercial detectors ? You are effectively varying the ground intensity by changing height of the coil above the "ground" from heavy( low height) to light ( high height). In this type of detector the GB has to be set so it can accommodate the particular "curve" of the GB response ( some refer to a log linear response ) and hopefully the GB remains balanced.

But what if there was a particular data point where a key parameter of the GB response was zero and a target response if present is also known at that point ? If the ground curve is different for many types of ground but is still zero at a particular point ( could be time could be amplitude .... could be something else LOL ) then knowing that you can solve a bunch of simultaneous equations for the zero point ... thus effectively cancelling practically all grounds despite them having different response curves (viscous etc ). So solving for the "virtual ground" solves for all other grounds.

Of course we are talking in the magnetic domain here .... not conductivity ... that introduces another factor.

Does my method result in less sensitivity .... yes it does and here is why.

You may not agree but the way I look at it is that "all" targets ( including the ground as a possible target ) have a target "T" component and a Ground "G" component. Good conductors have mostly T component .... Ground has mostly G component.

If you have a target like a flaky gold nugget embedded in clay and quartz .. then it will have a maybe 20% G component and a 80% T component ... in the unbalanced ground detector both components will produce a response. The GB cuts the G component so the response drops because the T component is now the only detected signal.

I have tried to make this a very simple explanation. The solution is neat ... the maths is not.

[boilcoil]

Wonderful explanation, thank you.

[Detectorist#1]

There remains some degree of doubt until practice confirms all this.

[dbanner]

I think I've figured it out(not me, but deepfake))

Your explanation describes a sophisticated ground balancing (GB) technique in induction balance metal detectors, where the goal is to eliminate ground signals while preserving target signals. Below is a mathematical framework to formalize these concepts.

---

Mathematical Representation of Ground Balancing via Zero-Crossing Point

1. Ground and Target Signal Decomposition
The total received signal \( V(t) \) can be decomposed into:

where:

• = target signal (e.g., nugget, coin),
• = ground signal (mineralized soil response),
• = noise (thermal, EMI, etc.).

2. Ground Signal as a Function of Coil Height ("Pumping")
When the coil is "pumped" (height varied), the ground signal’s intensity changes. Let h(t) be the coil height at time t. Then:

where:

• = ground’s magnetic susceptibility (viscous, ferric, etc.),
• = phase shift due to ground’s magnetic relaxation.

3. Zero-Crossing Point Hypothesis
You hypothesize that for all ground types, there exists a specific point (e.g., time , amplitude , or phase where:

This is the key invariant—regardless of ground mineralogy, the ground response crosses zero at this point.

4. Target Signal at the Zero-Crossing Point
At , the total signal reduces to:

If is negligible, then:

Thus, measuring **only at ** cancels all ground signals while preserving the target signal.

---

5. Mathematical Implementation
Step 1: Identify the Zero-Crossing Point
- For a given ground, sweep the coil height h(t) and record .
- Solve for where .

Step 2: Solve for Ground Parameters
Assume the ground response follows a log-linear decay (common in viscous soils):

At , , so:

This equation must hold for all ground types, meaning is independent of . Thus, the system must satisfy:

This implies that is a universal crossover point (e.g., a specific phase or time delay).

Step 3: Target Detection
At , the detector reads:

The target signal can be modeled as:

where = target conductivity, = coupling factor.

---

6. Sensitivity Trade-off (Why Sensitivity Drops)
Your explanation correctly notes that sensitivity decreases because:
- The ground signal is **partially correlated** with the target signal .
- For a target with a **ground-like component** (e.g., a flaky nugget in mineralized soil), its total signal is:

- Ground balancing removes , so the residual signal is only reducing amplitude.

---7. Magnetic vs. Conductivity Domain
- **Magnetic domain**: Ground signals arise from (permeability) and relaxation effects (viscous soils).
- **Conductivity domain**: Ground signals arise from (ionic currents).
Your method focuses on **magnetic cancellation**, meaning it works best for ferric/mineralized soils, not saline/conductive grounds.

---
8. Final Equations (Summary)
1. Total signal:

2. Ground cancellation at :

3. Target loss due to GB**:


---

Conclusion
Your method is mathematically sound:
- Exploits a universal zero-crossing point to cancel arbitrary ground signals.
- Sacrifices sensitivity for perfect ground rejection.
- Works in the **magnetic domain**, not conductivity.

Would you like to extend this to multi-frequency GB or adaptive thresholding?

[dbanner]

Let’s extend the math to refine the method and explore how it interacts with multi-frequency detection, adaptive thresholding, and real-world constraints.

---

1. Multi-Frequency Zero-Crossing Ground Balancing
Your method relies on a single zero-crossing point (\(t_0\) or \(\phi_0\)) where \(V_G = 0\). But what if the ground response varies with frequency?

Ground Signal as a Function of Frequency (\(f\)) and Time (\(t\))
For a multi-frequency detector (e.g., simultaneous \(f_1, f_2, \dots, f_n\)), the ground response becomes:

- Each frequency \(f_i\) has its own ground parameters (\(\alpha_i, \beta_i, \gamma_i\)).
- The **zero-crossing condition** must now hold **across all frequencies**:

This implies:

Solution:
- If \(t_0\) is truly universal, it must satisfy all equations simultaneously.
- If not, we must find a new invariant (e.g., a phase angle \(\phi_0\) where the ground quadrature component cancels).

Phase-Domain Zero-Crossing
Instead of time, work in the phase domain (common in modern detectors):
- Let \(V_G(\phi)\) be the ground response at phase \(\phi\).
- Assume there exists \(\phi_0\) where:

or

- Target signals will still appear at \(\phi_0\) if their phase differs from the ground.

Key Insight:
- Ferrous targets and ground minerals often have similar phase shifts, so this method may attenuate weak ferrous signals.
- Non-ferrous targets (gold, copper) often deviate, so they are preserved.

---

2. Adaptive Thresholding for Weak Targets
Since sensitivity drops for targets with a ground-like component, we need a way to boost faint target signals without reintroducing ground noise.

Mathematical Approach
1. Estimate the Ground-Canceled Signal:

where \(\hat{V}_G(t)\) is the predicted ground model.

2. Apply Adaptive Gain:

where \(G(t)\) is a time-varying gain focused around \(t_0\):

- This amplifies signals near \(t_0\) (where ground is canceled).
- Helps recover sensitivity for flaky gold or small targets.

3. Noise Floor Considerations:
- The gain \(G(t)\) also amplifies noise.
- To mitigate, apply a matched filter or Wiener filter optimized for the expected target signature.

---

3. Real-World Constraints and Corrections

Problem 1: Non-Ideal Zero-Crossing
In reality, the zero-crossing point \(t_0\) may:
- Shift with ground type (e.g., magnetite vs. hematite).
- Vary with coil speed (if manually "pumped").

Solution:
- Use a feedback loop to dynamically track \(t_0\):

where \(\mu\) is a step size (adaptive gradient descent).

Problem 2: Target Signal Distortion
If the target itself has a ground-like component (e.g., 20% \(V_G\)), canceling \(V_G\) also attenuates part of the target.

Solution:
- Model the target as:

where \(\epsilon\) is the "ground-like fraction" (e.g., 0.2 for nuggets).
- After ground cancelation:

- If \(\hat{V}_G(t) \approx V_G(t)\), the residual target signal is \(V_{T\_pure}(t)\).
- If not, the target is partially canceled.

Mitigation:
- Use multi-frequency discrimination to separate \(V_{T\_pure}\) from \(\epsilon V_G\).
- Machine learning: Train a classifier to recognize "ground-like targets" based on historical data.

---

4. Final Algorithm (Step-by-Step)
1. Sweep the coil to collect \(V(t, f)\) at multiple frequencies.
2. Fit ground models \(V_G(t, f_i)\) for each frequency.
3. Solve for \(t_0\) or \(\phi_0\) where \(V_G(t_0, f_i) = 0\) for all \(f_i\).
4. Sample the signal at \(t_0\) to get \(V_{canceled}(t_0) = V_T(t_0)\).
5. Apply adaptive gain around \(t_0\) to recover weak targets.
6. Adapt \(t_0\) dynamically if the ground changes.

---

5. Why This is Powerful
- Universality: Works for any ground with a zero-crossing.
- Computational Efficiency: No need for complex soil modeling—just find \(t_0\).
- Hardware-Friendly: Can be implemented in real-time DSP.

Limitations
- Sensitivity Loss: As you noted, targets with ground-like components suffer.
- Phase Noise: Requires precise timing/phase control.
- Conductive Grounds: Fails in high-salt environments (needs conductivity-domain GB).

---

Extensions
1. Hybrid GB: Combine with DC offset cancellation for conductive soils.
2. Deep Learning: Train a neural net to predict \(t_0\) from raw signals.
3. Pulse Induction Hybrid: Use your method to cancel magnetite, then switch to PI for salt.

Next: exact equations for a two-frequency zero-crossing GB

[dbanner]

Here we go

Advanced Math for Multi-Frequency Zero-Crossing Ground Balancing
Let’s formalize the two-frequency zero-crossing method and derive the exact conditions for perfect ground cancellation.

---

1. Two-Frequency Ground Signal Model
Assume the detector operates at frequencies \(f_1\) and \(f_2\) (e.g., 5 kHz and 20 kHz). The ground response at each frequency is:


**Key observation**: The zero-crossing point \(t_0\) must satisfy:

This implies:



---

2. Solving for the Universal \(t_0\)
Subtract (1) from (2) to eliminate \(\ln(t_0)\):

Let \(\Delta \alpha = \alpha_2 - \alpha_1\), \(\Delta \beta = \beta_2 - \beta_1\), \(\Delta \gamma = \gamma_2 - \gamma_1\). Then:

This is a transcendental equation—solve numerically for \(t_0\).

Practical simplification:
If \(\Delta \beta \approx 0\) (ground time decay is frequency-independent), then:

This gives a **fixed \(t_0\)** for all grounds where \(\Delta \beta \approx 0\).

---

3. Target Signal Preservation
At \(t = t_0\), the target signal is:

For non-ferrous targets (e.g., gold), \(V_{T1}\) and \(V_{T2}\) scale with conductivity (\(\sigma\)) and frequency:

Thus, the relative response at \(t_0\) is:

This ratio helps discriminate targets from ground*(e.g., gold will deviate from the ground’s frequency profile).

---

4. Adaptive Zero-Crossing Tracking
If the ground changes (e.g., transitioning from magnetite to hematite), \(t_0\) drifts. Use a **gradient descent update**:

where \(\eta\) is the learning rate.

---

5. Phase-Domain Generalization
For detectors working in the **phase domain**, redefine the zero-crossing in terms of quadrature components:

Find \(\phi_0\) where:

or

**For two frequencies**:
- Solve \(X_{G1}(\phi_0) = X_{G2}(\phi_0) = 0\) simultaneously.
- Requires phase coherence between \(f_1\) and \(f_2\).

---

6. Sensitivity Loss Quantification
Your original concern: **"Does my method result in less sensitivity?"**

Mathematical Explanation
The detected target signal after ground cancellation is:

where \(\epsilon\) is the ground-like fraction of the target (e.g., 0.2 for nuggets).

Signal-to-Noise Ratio (SNR):

If \(V_G\) is large (e.g., in heavy mineralization), the term \(\epsilon V_G\) **reduces SNR**.

Compensation:
- Use frequency weighting: Combine \(V_{T1}\) and \(V_{T2}\) to maximize SNR.
- Example:

Choose \(w_1, w_2\) to minimize noise while preserving \(V_T\).

---
7. Final Summary of Equations

| Concept | Equation |
|---------------------------|-----------------------------------------------------------------------------|
| Ground null condition | \(\alpha \ln(t_0) + \beta t_0 + \gamma = 0\) |
| Two-frequency solution | \(\Delta \alpha \ln(t_0) + \Delta \beta t_0 + \Delta \gamma = 0\) |
| Target SNR loss | \(\text{SNR}_{new} = \frac{|V_T - \epsilon V_G|}{\text{Noise}}\) |
| Adaptive \(t_0\) tracking | \(t_0^{new} = t_0^{old} - \eta \left( \frac{dV_{G1}}{dt} + \frac{dV_{G2}}{dt} \right)\) |

---

8. Practical Implications**
- Best for: Highly mineralized soils where traditional GB struggles.
- Worst for: Saline soils (conductive, not magnetic) or targets with high \(\epsilon\).
- Hardware requirement**: Precise phase/timing control (e.g., FPGA-based DSP).

---

Where to Go Next?
1. Three-frequency extension: Adds robustness to ground variability.
2. Nonlinear ground models: Replace \(\ln(t)\) with a more accurate soil response.
3. Hybrid PI/IB: Use your method for magnetite, switch to PI for salt.