There are two major steps to perform when constructing a hybrid model for a given MF communication system: 1) determine adequate sizes for all full-wave sections, capable of capturing sufficient detail of the near-field interactions within the system, and 2) determine the TL equations that best represent the uniform MF propagation occurring in the coal mine communication network. In other words, “what is the correct ratio of full-wave sections to TL equation sections?” This is important since the smaller and less frequent the full-wave sections are, the smaller the resulting computational cost will be.

Starting with the determination of the full-wave sections, consider two examples of particular interest for analyzing and optimizing the survivability of a communication network. First, consider a TL excited by a transmitting antenna. The length of the full-wave section in the vicinity of the transmitter must be adequately long in order to capture the correct antenna-to-line coupling. A hybrid model with this type of full-wave section could be used to determine how best to excite an MF communication network in an underground mine. That is, because the antenna coupling section is modeled using full-wave techniques, it would be straightforward to move the antenna, change its orientation, or change its position and then use the TL equations within the hybrid approach to calculate the impressed currents on the remainder of the communication network. For the second important example, consider a broken TL due to a collapsed tunnel. Again, it is extremely important to use a full-wave region sufficiently large enough to accurately model the power losses associated with the broken TL.

Based on the experience gained from this work, there is no generalizable procedure that can answer how much of a given MF communication network should be modeled using full-wave sections and how much can be modeled using TL equations. In fact, this highlights the motivation behind the hybrid approach. That is to say, if the designer could trivially model and predict how various antennas would couple to a given communication network and how that network would respond to extreme environmental changes induced by a mine collapse, then analytical expressions, such as TL equations, would suffice alone. Unfortunately, this modeling task is not so straightforward. Hence, choosing the appropriate full-wave sections must be done on a case-by-case basis.

To show how this was typically done in this work, an example was constructed, which is outlined in Figure 2a, showing a twin-lead TL having a separation distance of 2 m and a wire radius of 1 mm (Z₀ = 911Ω). The TL, which was terminated at both ends with loads Z, was positioned near two small loop antennas (20-cm diameter) separated by a distance L that were displaced from the plane of the TL by a distance Δx. The objective in this example was to determine what length of the TL near the two antennas must be modeled using MoM in order to accurately calculate the power transferred from T_x, operating at 500 kHz, to R_x. First, for a given L and Δx, the power transfer was calculated using strictly moment method (i.e., no TL equations) to establish a baseline. Then, a hybrid model was created starting with full-wave sections 2 m long, symmetrically positioned about the antennas’ respective positions. Next, the power transfer for shorted, matched, and open terminations was calculated. If the power transfers were within 10% accuracy of their full-wave-only counterparts, then the goal was met. If the power transfer error for any termination type was greater than 10%, the full-wave section of the hybrid model was increased by 2 m, and the power transfers were calculated again. This process was subsequently continued until the accuracy goal was met and was performed for three different L values (0.7, 1, and 2 km), each with 11 different Δx values (linearly spaced from 0 to 50 m). The result from this experiment is shown in Figure 2b, where it is clear that as the antenna was positioned further from the TL, the length of the full-wave section must be increased for a fixed accuracy criterion. In all cases investigated, the required full-wave section length was subwavelength (λ₀ ≈ 600 m). It is also apparent that this relationship was independent of L due to the identical TL characteristics in each case.

Note that although this example suggested rather short full-wave sections were required to model accurate antenna coupling, all of the examples explored in the remainder of this paper made use of longer (~ λ/4 to λ/2) full-wave sections. The main reason for this was so that when presenting hybrid model calculations, the variations in line current magnitude and phase could be easily identified and compared with the full-wave equivalent models.

In order to apply TL equations to the hybrid models, the propagation constant and characteristic impedance associated with the system must be calculated beforehand. With these data, TL equations may be formulated to calculate the equivalent loading and excitation on the full-wave sections associated with the sources and loads (e.g., transmitters and receivers), respectively. The TL propagation constant and characteristic impedance can be calculated using either numerical or analytical techniques. Typically, closed-form solutions are preferred due to the ease with which calculations can be made; however, they are not always available for the diverse range of geometries and materials commonly encountered in coal mines. Many times, only approximate solutions or more complex modal equations can be formulated. In this respect, it can be most straightforward to use available commercial codes to model prescribed geometries and calculate the associated propagation characteristics. For example, when considering low-loss TL, it can be shown that the propagation phase and attenuation constants are approximated by [36]

Here, L is the length of the TL, m is a nonnegative integer, and A=Zsc/Zoc, where Z_sc and Z_oc are the input impedances for shorted and open terminations, respectively. Finally, the characteristic impedance can be calculated by [47]

This way, the parameters required for the TL equations of the hybrid approach can be solved by calculating only the input impedances corresponding to open and shorted terminations. This technique, when applied to modeling performed with commercial codes, provides an efficient alternative when straightforward analytic methods are not available. Once propagation characteristics have been calculated, TL equations can be implemented in the hybrid approach and subsequently applied to optimization procedures. The remainder of this paper will concentrate on examples were the hybrid approach was applied.

Ultimately, it is desired to model and optimize full-scale mines with the hybrid approach. Before this is done, it is instructive to demonstrate the efficiency and accuracy of the hybrid approach for a less complex geometry. The initial example considered demonstrates the accurate modeling of a TL T-intersection in free space, where Figure 3a details the geometry that was used. A 1-W ideal power source, operating at 500 kHz (free-space wavelength, λ₀ ≈ 600 m) was used to feed the network. The twin-lead TLs were assigned a separation distance of 2 m and a wire radius of 1 mm. Each branch of the T-intersection was chosen to be 500 m long and was terminated with a load Z, which, for TL analysis, was chosen to be open, shorted, and matched. As shown in Figure 3a, three different hybrid models, i.e., Hybrids 1, 2, and 3, were considered and corresponded to utilizing ideal TL sections that were 100, 300, and 400 m long, respectively. The three hybrid models and the original full-wave model were compared in order to demonstrate the accuracy and increasingly reduced computational demands of the hybrid approach when larger and larger ideal TL sections were utilized, which effectively reduced the size of matrices used in MoM calculations. Figure 3b compares the calculated current profiles of the full-wave and hybrid models along the X- and Y-axis-aligned branches. The current calculations, which were normalized to the source current, indicate that the hybrid approach agreed well with its full-wave counterpart. Note that where current values have been omitted indicates the location of the analytic ideal TL sections. Finally, using an 8-core Intel E5620, hybrid models 1, 2, and 3 reduced simulation times by approximately 4, 20, and 80 times, respectively, compared with the equivalent full-wave model.

In the previous example, the computational time required by the hybrid model depended only on the size of full-wave sections used for the source and load regions. Subsequently, the hybrid approach allows networks to increase in size without discernible change in computational times, whereas an equivalent full-wave model using MoM would require more computational time and resources as the physical size of the model increases. This not only makes the hybrid approach attractive for large problems but also facilitates modeling of full-scale mines, whereas using MoM would not be feasible, even with notable computer resources.

In the next example, this fact was exploited and tested. Specifically, a TL 27 λ₀ long at 500 kHz (i.e., 16.2 km) was placed between two parallel plates of lossy earth in order to test the hybrid model subject to environmental loading, which is an important consideration when modeling MF communication networks in coal mines. The TL possessed the same separation and wire radius as in the previous example. A 10-cm radius loop antenna with a wire radius of 1 mm was used for both the transmitting (T_x) and receiving (R_x) antennas. T_x was fed by a 1-W power source operating at 500 kHz. This geometry is illustrated in Figure 4a. For this example, the environmentally-loaded TL equations were constructed using moment method models in conjunction with the techniques discussed in [36]. Currents along the full-wave section surrounding R_x were calculated for shorted and open terminations and are compared for the hybrid and full-wave models in Figure 4b. This example clearly demonstrates the accuracy of the hybrid approach, even over very large distances of propagation. Because of this large distance, computational times for the full-wave and hybrid models were 5540 and 6.9 s, respectively, on an 8-core Intel E5630. This equated to a factor of 800 reduction in computation time.

In order to account for the various possible MF propagation modes in coal mines, it is also important to consider those that use the surrounding earth material as a current path. This may occur when either a multiple wire system is experiencing CM propagation or a single wire conductor is placed within the coal mine tunnel. In the following sections, the modeling of a single wire conductor placed close (i.e., much less than one wavelength) to a lossy earth material is considered. This geometry, as shown in Figure 5, can equivalently represent multiple wire conductors facilitating in-phase propagation (CM propagation) and subsequently using the earth as the current return path. As predicted in the literature [46], [48, 49], at MF, a quasi-TEM mode is readily established between a single conductor and earth material return path. Because of this characteristic, the modeling of a conductor positioned near a single earth layer should be a good approximation to an identical conductor placed nearest to one (out of four) of the boundaries within a mine tunnel (more on this and its importance shortly). This is a reasonable application since, in practice, conductors would usually be supported by ceilings or floors. As will be shown, with this approximation, the TL analysis is significantly simplified. However, it is worth noting that this approximation may not hold as well when the conductor is placed in a corner of the tunnel cross section, effectively placing it close to two intersecting earth surfaces. Developing accurate and efficient models for this scenario is a topic of future consideration.

Propagation with earth return has been a topic of interest for almost a century. First reported in 1926, Carson [49] derived approximate TL solutions under the assumption that the wires are thin, placed in close proximity to the ground, and that the ground return dissipation factor is characteristic of good conductors (i.e., σ/εω ≥ 1). In effect, Carson’s approximation provides straightforward calculations but requires low frequencies of operation and/or the return path to have high conductivity, which may not be the case for all mine materials of interest. Significant improvement was made in 1972, when the analysis of J. R. Wait [46] provided the exact modal equation for all ranges of both electrical and geometric parameters. Wait’s model showed that the propagation constant, i.e., jγ, could be written in terms of the standard TL modal equation given by

(4)jγ=ZY, where Z and Y are the equivalent series impedance and shunt admittance, respectively, and are functions of γ.

The difficulty with Wait’s approach is that γ is a function of itself and therefore requires advanced numerical methods to compute solutions in all but the simplest cases. To avoid the challenges associated with solving the exact modal equation and obtain intuition for the system, attempts to improve on Carson’s original approximation have been reported, which introduce corrective terms that account for displacement currents associated with lossy dielectrics. The approximation used here was reported in 1996 by D’Amore and Sarto [48], which is formulated in the Appendix. When compared with Wait’s exact modal (4), this approximation is remarkably simple to use, so long as the appropriate limiting conditions are met, providing an accurate means to perform rapid computations of the propagation constant and characteristic impedance.

Next, a hybrid model was constructed using the D’Amore approximation outlined in the Appendix. The example considered was a wire placed 20 cm above an earth/air interface. The lossy earth return was assigned material parameters of σ = 0.01 S/m and ε = 10. The 5-km-long copper wire had a 1-mm radius and was terminated into the ground, normal to the air/earth surface boundary. The grounding rod was 1.5 m long and had a radius of 1 mm. At one end of the TL, there was an ideal voltage source operating at 500 kHz. At the opposite end, there was a load Z. The corresponding geometry is illustrated in Figure 6a. In this hybrid model, a 4.4-km-long section between the source and the load was replaced by TL equations constructed using the D’Amore approximation. Effectively, these TL equations were used to properly load the source section and excite the load section. Line currents were calculated and are compared in Figure 6b for the load Z designated to be both open and short. It is extremely important to note that there was an additional resistance associated with the grounding rod geometry (i.e., length and diameter) beneath the earth’s surface, which can be seen from the reduced SWR of the so-called shorted currents. To account for this in the hybrid model, the resistance was approximated using the low-frequency power system equations for grounding rod resistance discussed in [50], which is given by

(5)Rterm=12πσllog(2la) where σ is the ground conductivity, l is the length of the grounding rod, and a is the radius of the grounding rod. It is important to note that this equation assumes that the earth material surrounding the grounding rod is characteristic of a good conductor. This is often the case for MF propagation in coal mines; however, an alternative approach would be required for low-conductivity earth materials.

The current calculations in Figure 6b indicate only a small current error between the hybrid and full-wave models—approximately an error of 2 dB and 15 degrees in phase. Again, note that the omission of currents in the hybrid model indicates where TL equations have been applied. The good agreement of current calculations for both termination types further indicates the accuracy of the D’Amore approximation for these geometries when excited with MF. Furthermore, using an 8-core Intel E5620, the hybrid models were simulated in 7.4 s. The full-wave counterpart was completed in 440 s, which corresponded to a factor of 60 reduction in computation time.

As was previously suggested, the D’Amore approximation can be effectively utilized due to the TEM nature of MF waves in coal mines, where most of the EM energy is confined between the wire conductor(s) and the nearest tunnel boundaries. The next example considered employing the hybrid technique to support this argument. In this example, we made use of the fact that if a wire conductor is positioned closer to one tunnel boundary than the others, such as is shown in Figure 7b, then the EM environment should be approximately equivalent to that shown in Figure 7a, which is the geometry that was just previously considered for a hybrid model.

In the next example, a 1-km-long wire conductor was positioned 50 cm above a mine tunnel floor. The tunnel cross section was assigned typical coal mine dimensions with a height of 2 m and a width of 6 m; however, due to the approximation used here, these dimensions were not expected to greatly affect the propagation characteristics. That is, only the mine floor material parameters and distance the wire was positioned from that surface would dictate the EM properties of the TL. An ideal voltage source operating at 500 kHz was positioned opposite to either a short or open termination. This geometry is illustrated in Figure 7c. Currents for the hybrid and full-wave models were calculated for both termination types and two different mine floor conductivities (ε_g = 10): σ_g = 1e−2 S/m and 1e−3 S/m. The corresponding currents that were calculated are compared in Figure 8. There was some disagreement in the calculated currents, likely due to additional loading from the other tunnel boundaries; however, overall, the results demonstrate the accuracy of the hybrid approach for mine tunnels where a single conductor is placed close to one of the boundaries. Again, it is important to note that this assumption will need further consideration and revision in geometries where the wire conductor is placed equally close to two intersecting surfaces. Furthermore, to add complexity to the system, the coal seam (i.e., tunnel walls) and conductive rock layers (i.e., ceiling and floor) generally have conductivities that differ by one to several orders of magnitude [27]. This is a topic to be considered in future work, but remember it is always possible to construct accurate TL models by applying the techniques formulated in [36] to representative full-wave models. To conclude this example with calculation time comparisons, the full-wave and hybrid models were simulated in 3800 and 720 s, respectively, on a 12-core Intel E5645. This equated to a factor of 5.2 reduction in the computation time. It is very important to note that for this example, the memory requirements were 37.7 and 4.6 GB for the full-wave and hybrid models, respectively. Hence, for typical computers, the coal mine tunnel length is very limited using full-wave techniques alone; only the hybrid approach can allow for substantial line lengths required to model large-scale coal mine communication networks. Furthermore, optimizing such networks without the hybrid method would be infeasible all together.