Analytical Formulas for Wave Propagation at Low Frequencies in Half-Spaces Due to a Horizontal Magnetic Dipole

The Sommerfeld Integrals (SIs) are re-visited in the presence of half-spaces under the assumption of $\gamma _{0}\approx {\mathrm i}\lambda $ being valid for low-frequency propagation in order to investigate the near-field due to a Horizontal Magnetic Dipole (HMD). Analytical expressions are derived for the excited waves over a planar homogeneous conducting ground or sea surface, and a numerical Finite Element Method (FEM) approach is also exploited to validate its correctness and effectiveness. The proposed method is additionally verified by late released experimental data, and it is shown that the results for various simplified expressions qualitatively corroborate the conclusions found in published papers. The proposed formulas can be used as a valuable tool for the study of the properties of near-field propagations from a loop antenna.


I. INTRODUCTION
Although the National Bureau of Standards (NBS) formula recommended in 1967 [1] has been widely used for near-field measurement, which covers various frequency bands of the overall range from 30 Hz to 1000 MHz, and its recommended practice was given in 2012 [2], it seems that an effective general solution describing the near-field of a calibration antenna device is still needed to derive for providing with higher accuracies, driven by the widely applications, i.e. at some Radio Frequency Identification (RFID) frequency bands [3]- [6].
Applications of some lower RFID bands are usually found for instrument calibrations, such as the Low Frequency (LF, 120 − 150 kHz) band in [7]- [9] and High Frequency (HF, at center frequency of 13.56 MHz) band in [10]- [12], respectively. From a direct inter-comparison of experimental measurement with analytical results by Chinese researchers in [13], the authors have claimed the importance to obtain more accurate formulas, rather than segmented expressions as The associate editor coordinating the review of this manuscript and approving it for publication was Xiaokang Yin . defined for various ''quasi-static'' distance ranges, to express the near-field strength decaying in some calibration systems.
It is desirable of two independent approaches at least of measuring any physical quantity, i.e. the Standard-Antenna Method (SAM) and the Standard-Field Method (SFM), consisting of generating and computing the voltage and desired component of a standard field, respectively [1]. The former consists of measuring radiation in near zone by the component of the field being evaluated, and computing the value of the field in terms of this voltage. The second method consists of generating and computing the desired component of a standard field and form of a transmitting antenna, its current distribution, the distance from the transmitting antenna to the point at which the field component is being evaluated, and the effect, if any, of the ground.
Both methods require rigorous expressions to simulate a radiating device of the magnetic field for calculating the amplitude of the magnetic field induced in a standard receiving antenna. Applications are found, for example, in studies on variation of field strength in underground mine area on the size of Very Low Frequency (VLF: 3 ∼ 30 kHz) loop antennas laid on the Earth surface or mountain [14]- [16], covering the dimensions and form of the standard antenna, and its orientation with respect to the field vector. This old subject had been developed theoretically by investigators over the years, including the case of excitation at some height above the earth due to the launch activity of a space borne loop antenna [17], [18].
In real-life scenarios, it is of great importance to idealize the radiation source as modelled by a Hertzian HMD, especially in the estimation electromagnetic radiation by some the seismic activity, and the propagation characteristics of the Super Low Frequency (SLF: 30 ∼ 300 Hz) / Extremely Low Frequency (ELF: 3 ∼ 30 Hz) electromagnetic waves on the ground which would be released by lightening discharges. A recent study by Wang et al. strengthened the value of this research, which applied a speeding numerical convergence algorithm for precisely evaluation [19].
Among the techniques by which the desired component of field strength can be evaluated, two methods are mostly applied: 1) full wave simulation, being effective to solve the problem exploited in [20], [21] in calculation of the VLF / SLF electric wave propagating among the Earth-ionosphere cavity; and 2) deriving of the SIs [22], as an efficient method to evaluate wave propagation in half-spaces, which had been first released in 1909, and then, developed by Bańos and Wesley [23], Bańos [24], King et al. [25], Margetis [26], Margetis and Wu [27] and Collin [28] over the recent decades, respectively.
In the present study, the authors will re-visit the SIs by a good approximation with γ 0 ≈ iλ under low-frequency condition, and derive the analytical solution for near-field propagation of a HMD source along the boundary of planar air-todielectric interface over a lossy medium. The study has been innovated by our latest publication [29], where a simple but efficient solution has been obtained under this assumption, in order to investigate the characteristics of near-field propagation due to a Vertical Magnetic Dipole (VMD) source.
Even though studies on this problem has been frequently addressed in the last decades analytically [30]- [32], excellent performance providing with more accuracies as exhibited in [29] would ensure the advantage of the present approach. Aside from the mathematical derivations, the present case of a HMD is different from a VMD excitation in a twofold fashion: one is about the diverse propagation behaviours for field types, and another is because of the different propagation directions. This makes the focus here is on deriving for the field component in the air-space, while the previous study is for wave penetrating in dielectric medium.
Based on above analysis, computation and discussion are carried out by equipment of the derived formulas for the nearfield propagation from a HMD antenna over the planar sea surface and the time dependence e iωt is suppressed throughout the analysis. The results are compared with the evaluations by the ''quasi-static'' formula, which had been exploited in the recent publications [35], [36], the NBS formula, with experimental data, as well as the numerical solution to prove its correctness and effectiveness. This general approach would open a new way to obtain the simple formulas for evaluating SIs for the study of the characteristics of near-field which propagates along a planar lossy ground or sea surface, and also provide with some help and assistance to the researcher or engineer who has interest in relevant study.

A. SOMMERFELD INTEGRALS IN THE PRESENCE OF HALF-SPACES FOR FIELD COMPONENTS INDUCED BY A HMD SOURCE
The geometry and notations are shown in Fig. 1, in which the upper half-space (z ≥ 0) is Region 0 occupied by air, and the lower half-space (z < 0) is Region 1 being characterized as a lossy ground or sea. For each region, the wave number is defined by k 0 = ω/c and k 1 = √ iωµ 0 σ , respectively, in which µ 0 (the value of µ 0 is 4π × 10 −7 N/A 2 ) and ω are permeability constant and angular frequency in free space and σ is the conductivity of ground or seawater medium, respectively.
Approaching to the boundary of two regions, a couple of loop antennas are mounted at the height distances of h and z above the ground, standing for transmitting and receiving devices, respectively. In Region 0, the SIs expressing for the electromagnetic field generated by a HMD source can be used for evaluating low-frequency near-field propagation of the instrument in a cylindrical coordinate (ρ, ϕ, z), which can be described as follows: in which It is noted that the square root in (7) is taken in the first quadrant, and S is the area of the transmitting loop with square meters.

B. ANALYTICAL FORMULAS EXPRESSING FIELD COMPONENTS NEAR THE BOUNDARY OF AIR-TO-DIELECTRIC INTERFACE
In the applied scenarios, the radial distance ρ between the transmitter and receiver being subjected k 0 ρ 1 satisfies the near-field condition at the operated low frequencies for various ''quasi-static'' distances, and the ratio of the wave numbers ensures k 0 /|k 1 | 1 for lossy medium. Usually, both the heights (h and z) of the transmitting and receiving antennas are much less than the horizontal distance ρ, i.e. h ρ and z ρ. The condition of k 0 ρ 1 ensures that the main contribution of the Sommerfeld's integrals from (1) to (6) is from the path of integration where λ k 0 . Therefore, the following approximations are exploited in derivations.

1) AT THE ''NEAR-ZONE'' RANGE
It is seen the antenna of the instrument will receive the horizontal component of the magnetic field generated by a standard small loop in order to calibrate the instrument. In this case, two approximations can be inferred from γ 0 ≈ iλ, summarized as follows: i) It leads to ii) By neglecting terms of higher order k 2 0 γ 1 /k 2 1 γ 0 , the factor Q and k 0 − k 1 /(MN ) may be reduced to Thus, the SIs defined by integrals from (1) to (6) can be simplified by using the conditions from (11) to (14), rewritten as follows: where The formulas for evaluating above expressions are developed by solving the separated integrals accordingly. In monograph by King et al. [25], explicit formulas are given for evaluating SIs from F 1 to F 6 , which are listed in Sec. appendix. The rest integrals are to be considered, focusing on finding analytical solutions of the following integrals from F 7 to F 11 .
They are 3) SIMPLIFICATION OF (γ 0 − γ 1 ) AT LOW FREQUENCIES In accord to [29], an efficient simplification is found to be valid for radio wave propagation at low frequencies.
Use is made by follow, employed as Eq. (18) in [29], then 1 M has 1 M = inferred from (9) with consideration of in which is applied.

4) ANALYTICAL FORMULAS FOR INTEGRALS FROM F 9 TO F 11
For the three magnetic components H 0z , H 0ρ and H 0ϕ , the integrals from F 9 to F 11 in equations from (18) to (20) are defined, accordingly Substituting (29) into (30), then, replacing the function (γ 1 − γ 0 ) in the integrands of equations from (31) to (33), it yields, 2ρ 3 e ik 0 r (34) VOLUME 8, 2020 by which the formulas are obtained where integrals are partitioned by several integrations addressed in [25]. Specifically, practice is made by: D4, D1, I D and I A for F 9 (d 1 ); D9, D13, I K and I Q for F 10 (d 1 ); D11, D14, I M and I R for F 11 (d 1 ), respectively. Based on the above analysis, the analytical formulas have been derived readily for near-field propagation of a HMD (representing for a loop antenna) in the presence of halfspaces.

5) THE APPLIED LOW FREQUENCY RANGES AND SIMPLIFIED EXPRESSIONS
It should be clarified that the applied operating low frequencies are subjected to below restriction [29]: to guarantee the effectiveness of (27). Generally speaking, the amplitude of a low frequency electromagnetic wave near the ground depends on the ground conductivity, the operating frequency, and the propagation distance.
When the antennas of the transmitter and the receiver are located on the ground, we have d m = 0 and r m = ρ. It is seen that the high-order terms of k 0 ρ and k 1 ρ can be neglected. Specifically, the six field components in Region 0 can be derived readily. The derived formulas are subjected to the following simplified form under the assumption, written as follows: in which the simplified integrals can be solved correspondingly, as follows: 2ρ 3 e ik 0 ρ , (50) with z ≈ h = 0 and d 0 = d 1 = 0.

C. SIMPLIFIED EXPRESSIONS OF NEAR-FIELD PROPAGATION OF A HMD ANTENNA
At short propagation distances, the derived formulas can be simplified defined by equations from (48) to (52) according to the following approximations: In this case, the propagation distance is assumed to be in the region of k 0 ρ 1, where the amplitude of the magnetic field probably be approximated by the static field with k 0 ρ = ρ c ω → 0. Thus, the integrals are solved accordingly, as follows: ρ 3 , respectively. The applied range had previously been defined by ''quasi-near range'' in [30]. In this case, the propagation distance is assumed to in the region of |k 1 ρ| 1 yielding e ik 1 ρ ≈ 0, for the reason that the parameter of m(k 1 ρ) will be great.
It is noted that the propagation distance with in Region 1 is subjected to the value of the skin depth, defined by follow: 3

) CURVE C
Considering that F 7 ∼ F 11 can be reduced to the final approximated formulas for the six components of the electromagnetic field, summarization is found in Tab. 1. Specifically, it can be deduced that the resulted formula C for each component would be in consistent with the quasi-static field results for A.

III. COMPUTATION
To validate the effectiveness of the derived formulas, the values of magnitudes for the magnetic field component H 0ϕ are  2-D numerical modeling of near-field propagation by FEM, which is generated by a loop antenna exposed in the air (the excitation is taken by Ida = 1) with (a) the realistic model, and (b) meshing grids by arbitrary triangular elements with PML boundary, respectively. plotted in Fig. 2 by various comparable results, in which the discrete measured data are taken from a recent near-field experiment released in [13] conducted by Chinese researcher and the rest curve lines are obtained by analytical formulas by using the NBS's formula in [1] with the radiating radius is taken as R 0 = d 2 + R 2 1 + R 2 2 , and the approximated solutions by A, B, C, respectively.

A. NEAR-FIELD CALIBRATIONS
This experiment data are taken from recorded in literature [13], in which the conductivity is measured as 0.12 S/m by GMS-07e. The height of the transmitting loop is 0.3 m (R 1 = 0.2 m; 16 turns) and the receiving rod is placed on the ground, i.e. z = 0. The sensor of the receiver (R 2 =  The values of E 0z , E 0ρ and E 0ϕ expressing for the electric components (in dB) in near-zone from a HMD source computed by various formulas (marked by A, B and C, respectively) under different near-field approximations of low-frequency propagation at the operating frequency f = 50 kHz with d = h = 0 + , and the relative permittivity and conductivity of lossy medium are taken as ε = 80, σ = 4 S/m, respectively, and ϕ = π/2, ϕ = π/2, ϕ = 0, respectively. 0.03775 m; 1.2 m length). The excitations are taken as chosen by I = 0.0828, 0.524, 0.262 amperes at various operating frequencies f = 3, 10, 30 kHz, respectively. It can be seen from Fig. 2 that the calculated results of derived formulas are basically consistent with the measured data and within a reasonable error range.

B. THE FULL-WAVE SIMULATION
Moreover, a full-wave simulation of the presented model in Fig. 3 (a) is given by a FEM (Finite Element Method). The meshing grids are chosen aribitrary by triangular elements as shown in Fig. 3 (b). The comparison of time consuming for the full-wave simulation and analytical solution are given in Tab.2. Calculation in Fig. 4 give the amplitudes of the field components taken by the numerical method. However, the accuracies are also subjected to different mesh sizes, as illustrated in Figs.4 (a) and 4 (b). The result in Fig. 4 reveals some discrepancies in magnitudes of field components where the error is subjected to different mesh sizes. It is noted that the calculated results with the numerical method are not stable, however, the analytical formulas proposed in this paper does not have such a problem.

C. THE ADVANTAGES
With same conditions, we compared the analytical formula, and the corresponding various simplified expressions found in published papers in Tab. 1 expressed as A, B, C, respectively. The values of magnitudes to the propagation distance ρ are computed and plotted in Fig. 5 and Fig. 6, respectively, at the operating frequency f = 50 kHz and the conductivity and the relative permittivity of seawater are chosen as σ = 4 S/m, and ε = 80, respectively. It is seen that the strength of electromagnetic components decays as the propagation distances increase in different rates. It should be pointed out that the quasi-static formulas are believed to be accurate for a long time even though the available expressions are not complete for all components. For example, the formula for the component E 0ρ is not available for evaluation [30]. In the FIGURE 6. The values of H 0z , H 0ρ and H 0ϕ expressing for the magnetic components (in dB) in near-zone from a HMD source computed by various formulas (marked by A, B and C, respectively) under different near-field approximations of low-frequency propagation at the operating frequency f = 50 kHz with d = h = 0 + , and the relative permittivity and conductivity of lossy medium are taken as ε = 80, σ = 4 S/m, respectively, and ϕ = 0, ϕ = 0, ϕ = π/2, respectively. comparisons from Fig. 5 and Fig.6 by the derived formulas with that by quasi-static approach addressed in 1964 by Durrani, the derived formulas are in accord with the available results by the quasi-static formulas to prove its effectiveness and correctness.
When the observation point is chosen arbitrarily on thex−ẑ plane, the distributions of the three components strength for ELF wave are depicted in Figs. 7 (a), (b) and (c), respectively. As an extension of the study, the Poynting trajectories are also plotted in Fig. 7 (d), at the operating frequency f = 50 kHz. In the computations, the conductivity and relative dielectric constant of the rock medium is chosen as approximately σ = 10 −4 S/m, and ε = 4 in the left column, respectively, and the conductivity and relative dielectric constant of seawater medium are taken as σ = 4 S/m, and ε = 80 in the right column, respectively.
The plotted distribution with Poynting trajectory in Fig.7 depicted the energy flow near the surface of the lossy ground, which shows that the ground condition has slight influence to the near-field propagation performance

IV. CONCLUSION
The study is to derive the SIs in half-spaces due to a HMD, and it gives some illustrative numerical evaluations by the obtained formulas, which could be seen as a valuable tool for the study of the properties of near-field propagation. The resulted magnetic field strength coincides with those obtained by using the NBS's formulas, FEM solution, and also with the measured experiment.
Providing with approximated results under different conditions, it may summarized in Tab. 1. We compared the properties of near-field propagation at low frequencies by various propagation distance conditions: 1) the ''quasi-static'' field region is defined by k 0 ρ 1, typically, when the horizontal distance ρ between the observer and the source dipole is less than 0.1 times of the skin depth δ of the lossy medium; 2) when it subjected to |k 1 ρ| 1, the lossy ground medium behaves as a perfect conductive reflector; and 3) for a more general case, we propose a approximation of |k 0 k 1 |/γ 2 0 1 is valid for near-field propagations at a wide frequency-band of low frequencies, which corresponds to conductivity of the ground medium, respectively. In future work, the proposed method may be utilized in more applications, such as the application environment of deep underground exploration, or submarine prospection and communication.

APPENDIX
In evaluation of the Fourier-Bessel representations of integrals from F 1 to F 6 , the following expressions of identities have been derived previously in the monograph [25] by King et al, written by:   with m = 0, 1 and n = 0, 1, respectively.