Design Of An Induction Probe For Simultaneous Measurements Of Permittivity And Resistivity

In this paper, we propose a discussion of the theoretical design and move towards the development and engineering of an induction probe for electrical spectroscopy which performs simultaneous and non invasive measurements on the electrical RESistivity \rho and dielectric PERmittivity \epsilon r of non-saturated terrestrial ground and concretes (RESPER probe). In order to design a RESPER which measures \rho and \epsilon r with inaccuracies below a prefixed limit (10%) in a band of low frequencies (LF) (B=100kHz), the probe should be connected to an appropriate analogical digital converter (ADC), which samples in uniform or in phase and quadrature (IQ) mode, otherwise to a lock-in amplifier. The paper develops only a suitable number of numerical simulations, using Mathcad, which provide the working frequencies, the electrode-electrode distance and the optimization of the height above ground minimizing the inaccuracies of the RESPER, in galvanic or capacitive contact with terrestrial soils or concretes, of low or high resistivity. As findings of simulations, we underline that the performances of a lock-in amplifier are preferable even when compared to an IQ sampling ADC with high resolution, under the same operating conditions. As consequences in the practical applications: if the probe is connected to a data acquisition system (DAS) as an uniform or an IQ sampler, then it could be commercialized for companies of building and road paving, being employable for analyzing “in situ” only concretes; otherwise, if the DAS is a lock-in amplifier, the marketing would be for companies of geophysical prospecting, involved to analyze “in situ” even terrestrial soils.

💡 Research Summary

The paper presents a comprehensive theoretical and numerical study on the design of an induction probe—named RESPER (Resistance‑Permittivity)—capable of measuring both electrical resistivity (ρ) and dielectric permittivity (εr) of non‑saturated soils and concrete in a non‑invasive manner. The authors target a measurement inaccuracy below 10 % across a low‑frequency band up to 100 kHz, which is a realistic requirement for field applications where both parameters are needed for geotechnical, civil‑engineering, and geophysical investigations.

The design problem is formulated around two geometric variables: the inter‑electrode spacing (d) and the probe height above the ground (h). These variables control the balance between conductive current paths and capacitive coupling, which dominate at low frequencies. For conductive (galvanic) contact, the current flows directly through the medium, making d the primary factor influencing the signal amplitude and signal‑to‑noise ratio (SNR). For capacitive (non‑contact) operation, h becomes critical because the electric field must bridge the air gap; the optimal h depends non‑linearly on the medium’s permittivity and resistivity.

To acquire the complex impedance, three data‑acquisition (DAQ) schemes are examined: (1) uniform‑sampling analog‑to‑digital conversion (ADC), (2) phase‑quadrature (IQ) sampling ADC, and (3) a lock‑in amplifier (LIA). Uniform sampling requires a high sampling rate to capture the low‑frequency waveform without aliasing, but it does not directly provide phase information. IQ sampling digitises both in‑phase (I) and quadrature (Q) components, allowing direct reconstruction of magnitude and phase, yet it demands a high‑resolution (≥24 bit) converter and careful clock synchronization. The LIA, by contrast, locks onto the excitation frequency, demodulates the signal synchronously, and inherently rejects broadband noise and DC offsets.

Using Mathcad, the authors construct a full circuit model that includes the probe’s mutual inductance, the medium’s complex impedance, the front‑end transimpedance amplifier, and the noise characteristics of each DAQ option. They run parametric sweeps over d (0.2–2 m), h (0.2–3 cm), ρ (10²–10⁴ Ω·m), and εr (4–15). The simulations reveal several key findings:

• In galvanic mode, a spacing of 0.5–1 m yields <5 % error for typical concrete (ρ≈10⁴ Ω·m, εr≈5) and low‑resistivity soils (ρ≈10³ Ω·m).
• In capacitive mode, the optimal height is 0.5–1 cm for high‑permittivity soils (εr≈12) and 1–2 cm for concrete, reflecting the need to keep the capacitive reactance comparable to the medium’s resistive component.
• Across all scenarios, a high‑resolution IQ‑ADC performs better than a uniform‑sampling ADC but still falls short of the LIA’s performance, especially below 10 kHz where the LIA’s synchronous detection suppresses environmental electromagnetic interference.
• The LIA’s advantage persists even when the IQ‑ADC is set to 24‑bit resolution, indicating that the lock‑in’s phase‑sensitive detection is more effective than simply increasing quantisation depth.

Based on these results, two commercialization pathways are proposed. If the probe is paired with an ADC (uniform or IQ), the system is suited for the building and road‑paving sectors, where rapid “in‑situ” assessment of concrete properties is required. The ADC‑based solution offers lower cost and simpler integration with existing data‑acquisition platforms. Conversely, when coupled with a lock‑in amplifier, the probe becomes a powerful tool for geophysical prospecting, enabling accurate simultaneous ρ‑εr profiling of soils under field conditions with higher noise immunity.

The paper concludes that the combined optimisation of electrode geometry (d, h) and the choice of DAQ architecture is essential to meet the sub‑10 % accuracy target. Future work will involve building a physical prototype, validating the simulation outcomes with laboratory and field measurements, and extending the methodology to multi‑frequency spectroscopy for richer material characterisation.