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Effective Range and Optimal Frequency of Through-the-Earth Magnetic Induction Communication
Honglei Ma, Erwu Liu, Wei Ni, Jun Zhu, Zhijun Fang, Rui Wang, Xinyu Qu
TL;DR
Previous TTE MIC range studies often overlooked eddy losses and underground permittivity, although effective range is vital for deep underground communication. This paper derives closed-form expressions for effective range and optimal carrier frequency while accounting for underground conductivity and permittivity. The analysis and simulations identify carrier-frequency and antenna-radius optimization as effective ways to extend MIC range.
Problem
Previous MIC range studies often overlooked eddy losses and underground permittivity, complicating effective-range analysis for TTE communication.
Method
The paper derives closed-form expressions for effective MIC range and the carrier frequency that maximizes it, accounting for underground conductivity and permittivity.
Results
Optimizing carrier frequency and antenna radius proved effective for extending MIC range, with the optimal frequency potentially increasing range multifold.
Takeaways & Limitations
Carrier-frequency optimization can enhance MIC range without additional software or hardware costs, while antenna-radius optimization involves larger devices and higher deployment costs.
Abstract
from arXiv · showhide
Magnetic induction communication (MIC) is a promising technology for through-the-earth (TTE) communication. Previous studies on the MIC range have often overlooked the impact of eddy losses caused by underground materials. For TTE MIC, significant eddy losses complicate the analysis of the effective MIC range, which is vital for optimizing performance but has never been addressed in the literature. Accounting for the conductivity and permittivity of the underground medium, this paper derives the effective MIC range in TTE MIC, along with a closed-from expression that predicts the optimal carrier frequency to maximize this range. Finite element simulations validate the analysis, demonstrating that the optimal carrier frequency can significantly enhance the MIC range. It is also revealed that optimizing the antenna radius is effective in extending the MIC range for TTE and vehicle MIC applications.
I. INTRODUCTION
TTE communication supports deep underground links but faces severe propagation loss, making effective MIC range a central performance concern. This paper accounts for underground-material effects and derives closed-form range and frequency results for improving TTE MIC.
- TTE communication provides underground links exceeding tens of meters for tunnels, mining, robotics, subway, and resource exploration.
- Previous MIC studies often overlooked eddy currents from underground materials, which complicate channel-power-gain and range evaluation in TTE scenarios.Underground permittivity further complicates the analysis, while 1–30 MHz operation offers a shorter near-field range.
- Earlier TTE studies used numerical or closed-form approaches but overlooked coil-resonance circuit loss and underground-medium eddy loss.
- Most TTE studies adopted 10 kHz empirically, motivating analysis of a carrier frequency that maximizes MIC range across underground environments.
- The paper derives a closed-form effective MIC range and a closed-form prediction of the optimal carrier frequency while accounting for eddy loss and medium permittivity.
- The MIC link model represents transmit and receive coils through their inductances, radii, turns, matching capacities, impedances, mutual inductance, and underground-medium propagation.
PN CSDJ2
The effective MIC range is obtained by solving the mixed-field range equation. The resulting expression uses the Lambert-W inverse and is constrained by a near-field upper bound.
- The effective MIC range is obtained by applying the Lambert-W inverse function to the mixed-field range equation and solving for distance.
- The resulting range solution is subject to a near-field upper bound.
III. OPTIMAL EFFECTIVE MIC RANGE
The paper analyzes how carrier frequency and underground-material conductivity affect effective MIC range. It identifies frequency optimization as a feasible strategy for extending range in TTE and IoV settings.
- The analysis derives the optimal carrier frequency for enhancing MIC range and examines the impact of underground-material conductivity.
- Optimizing carrier frequency is presented as the most feasible range-extension strategy because waveguides and active relays are challenging in limited, dynamic TTE and IoV environments.The paper also states that higher frequency facilitates increased data rate, while coil-size adjustment creates deployment challenges.
A. Optimal Frequency
The paper derives a closed-form approach for predicting the optimal carrier frequency, addressing the non-convex frequency dependence of effective MIC range. The analysis identifies conditions for one or two locally optimal frequencies and shows that underground material properties affect the optimum.
- The effective MIC range is non-convex in carrier frequency because of the Lambert-W function and underground permittivity.
- A proposition based on Lambert-W properties derives the optimal carrier frequency for maximizing MIC distance.
- There can be one or two locally optimal frequencies: a first stationary point and the infinite-frequency endpoint, which lacks practical physical meaning.
- When the first optimum does not exist, effective MIC range increases monotonically with frequency; otherwise, substantial frequency deviations can reduce range by severalfold.
- The first optimal frequency lies within the Fresnel region under stated transmit-power conditions.
- Underground conductivity and permittivity significantly influence the optimal frequency, making the empirical 10 kHz choice suitable only for limited scenarios.
B. Effect of Underground Materials Conductivity
The analysis shows that underground conductivity reduces the effective TTE MIC range. This establishes conductivity as a direct material constraint on communication distance.
- The derivative of effective MIC range with respect to underground conductivity is negative for all conductivity values.
- The effective range of TTE MIC decreases as the conductivity of underground media increases.
IV. SIMULATION AND NUMERICAL RESULTS
The simulation section verifies the derived range expression and examines how frequency, transmit power, coil size, and medium conductivity affect MIC performance. It uses homogeneous underground-media assumptions and vehicle-scale antenna parameters.
- The study verifies the effective MIC range expression and evaluates frequency, transmit power, coil size, and medium conductivity through simulation.
- The underground environment is modeled as homogeneous, with permeability and permittivity set to specified values from prior work.
- The simulation uses vehicle-scale coils with 15 transmit turns, 0.6 m transmit radius, 30 receive turns, and 0.4 m receive radius.
- The assumed transmitting power spectral density is 5 W / 450 Hz, ambient noise power spectral density is −103 dBm / 2000 Hz, and the SNR threshold is 4.7748.
A. Effective MIC range
Finite-element and analytical results validate the effective MIC range formulation, especially in the near-field regime. Frequency sweeps show an optimal range near resonance and substantial sensitivity to frequency selection outside the optimum.
- FEM and theoretical effective-range results agree closely at shorter distances, while their discrepancy grows as radiation-field effects become important.
- The closed-form range expression is presented for single- and multi-layered conductive media, with FEM curves from COMSOL compared against analytical calculations.
- Analytical and numerical solutions validate the predicted optimal-frequency conditions, with peaks slightly right-shifted from the corresponding predicted frequencies.
- All observed peaks lie within the Fresnel region, where radiation effects can extend MIC range by up to about 33% for configuration L6.
- For typical configurations, the second optimal frequency approaches infinity, while an extreme configuration can instead produce a range that increases monotonically with frequency.
- At the designed 10 kHz resonance frequency, the system achieves maximum MIC range; across the 3-dB bandwidth, range decreases by about 7%, from 109.5 m to 102 m at band edges.
C. Influence of Underground Medium
The paper evaluates how underground conductivity and optimization choices affect effective MIC range. Simulations show that conductivity reduces range, while frequency and coil-radius optimization are more effective than transmit-power increases, subject to size and cost trade-offs.
- Simulation setup: FEM simulations evaluate conductivity effects and deviations from the closed-form range solution in multilayered underground media.The modeled soil spans 0–220 m with layered conductivities whose series equivalent is approximately 0.01 S/m.
- Conductivity effects: Higher underground conductivity remarkably decreases the effective TTE MIC range, with medium effects intensifying at higher carrier frequencies.This FEM result validates the analysis of conductivity effects.
- Optimization comparisons: Increasing transmit power from 0.001 W to 1,000 W increases MIC range by only several times, indicating energetic inefficiency.By contrast, frequency adjustment is described as energy-efficient and without significant software or hardware costs.
- Optimization comparisons: Increasing coil radius initially expands MIC range significantly, but gains diminish with further enlargement, especially at higher frequencies.An optimal coil size therefore exists, while excessive radius increases device size, deployment burden, and cost.
- Conclusions: Optimizing antenna radius and carrier frequency is identified as more effective for enhancing MIC range than optimizing transmit power.Carrier-frequency optimization avoids additional software or hardware costs, whereas larger antennas can substantially increase device size and cost.
APPENDIX A PROOF OF PROPOSITION 1
The appendix proves properties of the effective-range function by analyzing intersections between two Lambert-W-related functions. These properties bound stationary points and characterize when optimal frequencies are unique, multiple, or absent.
- Stationary-point formulation: The optimal frequency f* is obtained by solving d′(f)=0, where d′(f) is the derivative of effective MIC range with respect to frequency.The proof analyzes stationary points through this derivative condition.
- Intersection analysis: Because the relevant function intersections number at most two, d(f) has at most two stationary points.The stationary points are analyzed through intersections between W(A(f)) and R(f).
- Function properties: A(f) is monotonically increasing and concave downward, while R(f) is monotonically increasing and convex.These properties support the subsequent intersection-count argument.
- Optimal-frequency cases: The stationary points correspond to maximum and minimum values, indicating at most two locally optimal frequencies.The proof also identifies a unique stationary point under a sufficient condition.
- Boundary behavior: When ǫf increases beyond a threshold, the number of intersections and stationary points becomes zero.In the limiting case where d(f) increases monotonically, the right endpoint ∞ becomes the unique locally optimal frequency.