MI: Channel Modeling for MetamaterialEnhanced
Magnetic Induction Communications
Abstract
Magnetic Induction (MI) communication technique has shown great potentials in complex and RFchallenging environments, such as underground and underwater, due to its advantage over EM wavebased techniques in penetrating lossy medium. However, the transmission distance of MI techniques is limited since magnetic field attenuates very fast in the near field. To this end, this paper proposes Metamaterialenhanced Magnetic Induction (MI) communication mechanism, where a MI coil antenna is enclosed by a metamaterial shell that can enhance the magnetic fields around the MI transceivers. As a result, the MI communication system can achieve tens of meters communication range by using pocketsized antennas. In this paper, an analytical channel model is developed to explore the fundamentals of the MI mechanism, in the aspects of communication range and channel capacity, and the susceptibility to various hostile and complex environments. The theoretical model is validated through the finite element simulation software, Comsol Multiphysics. Proofofconcept experiments are also conducted to validate the feasibility of MI.
I Introduction
Despite the presence of wireless connectivity in most terrestrial scenarios, there are still many hostile and complex environments that cannot be covered by existing wireless communication techniques, including underground, underwater, oil reservoirs, groundwater aquifers, nuclear plants, pipelines, tunnels, and concrete buildings. Wireless networks in such environments can enable important applications in environmental, industrial, homeland security, and military fields, such as monitoring and maintenance of groundwater and/or oil reservoirs [1], or damage assessment and mitigation in nuclear plants [2], among others. However, the harsh wireless channels prevent the direct usage of conventional electromagnetic (EM) wavebased techniques due to the high material absorption when penetrating lossy media.
Among potential solutions, the Magnetic Induction (MI) technique has shown great potentials in underground [3] and underwater [4] environments. In a MI communication system, the HF band magnetic field generated by a MI transmitter coil is utilized as the signal carrier [5]. Since most natural media have the same magnetic permeability as air, MI keeps the same performance in most materials. Even in lossy media like groundwater, the MI path loss caused by skin depth can be minimized since MI communication is realized within one wavelength from the transmitter [6]. In addition, MI does not suffer from the multipath fading problem in EM wavebased solutions [4]. However, MI systems depend on the magnetic field generated by the transceivers in the near field, which attenuates very fast. Consequently, the range of MI communication is very limited.
To this end, we introduce metamaterials to MI communications, which can manipulate and enhance the magnetic fields transmitted and received by MI transceivers. Metamaterials are artificial structures made of carefully designed building blocks, which can generate unique physical phenomenon such as backward waves and negative refraction index [7, 8]. The novel properties of metamaterials have been utilized in subwavelength imaging [9], wireless power transfer [10], and antenna miniaturization [11]. Since the key problem of the MI communication technique is the fast falloff rate in near field, we see great potentials in using metamaterials to enhance MIbased communications and finally achieve both extended medium penetration performance and practical communication ranges. To date, no efforts have explored the design of the metamaterial enhancement of MI communications in complex environments.
In this paper, we propose the Metamaterialenhanced Magnetic Induction (MI) communication mechanism for the aforementioned wireless applications in various environments that are structurally complex and challenging for RF wireless signals. By introducing the MI antennas (the MI coil antenna enclosed by a metamaterialenabled resonant sphere, as shown in Fig. 1), we show that the efficient wireless communication can be realized in lossy environments with good range. The whole communication process (starting from the transmitter, via the lossy transmission medium, and ending at the receiver) is investigated as an integrated system. We develop an analytical channel model that quantitatively captures the unique interactions among MI transceivers, the metamaterialenabled resonant structure, and complex environments, which are not observed in existing metamaterial applications. The proposed MI mechanism and the channel model are validated by both the Finite Element Method (FEM) software, i.e., Comsol Multiphysics [12], and proofofconcept experiments. Based on the derived channel model, we confirm the feasibility of achieving tens of meters communication range in MI systems by using pocketsized antennas. The developed channel model bridges the communication system optimization with the metamaterial device design.
The reminder of this paper is organized as follows. Related works are presented in Section II. Then, in Section III, an analytical channel model for MI communication is developed to characterize how the metamaterial sphere works in MI systems in lossy media. Next, the channel characteristics of MI communication including the pointtopoint and MI waveguide communication are discussed in Section IV. Finally, this paper is concluded in Section V.
Ii Related Work
MI techniques have been utilized in many complex environments. In [13, 14, 15], voice and low data rate communications have been established by MI in underground mines. In [4, 16, 17], MI communication is realized in lossy underwater environment, where very large coil antennas are utilized. In [3, 5, 18], MI is introduced to wireless underground sensor networks, where wirelessly networked sensor devices are buried in soil medium. In [19], MI is utilized to transmit both data and power into human body for medical applications. Besides theoretical research, many commercial MI systems have also been developed for mining safety and undersea surveillance [20, 21, 22], among others. Despite of the advantages, the existing MI communication systems have very limited ranges due to the fast falloff in near field unless very large coil antennas are used. To extend the very limited range, waveguide structures [23] can be utilized. In [5], we show underground MI communication range can be significantly extended by passive relay coils, i.e., the MI waveguides. However, existing MI waveguides require very high density of relay coils, which prevents practical implementation.
Metamaterials have been utilized in a wide range of applications, such as the metamaterial cloak, metamaterial enhanced MRI [9], and metamaterial antenna [11]. Among the various research thrusts in metamaterials, two areas are most relevant to our work, including wireless energy transfer and RF antenna miniaturization. (i) In [10], a metamaterial slab is introduced to increase the wireless energy transfer efficiency. In [24, 25], the metamaterial enhancement in energy transfer is validated in both theoretical analysis and experiments. Different from the single frequency wireless energy transfer, the MI communication system proposed in this paper requires the wireless signals occupy a significant bandwidth to carry information with high date rate. Moreover, existing metamaterialenhanced wireless energy transfer systems need a large metamaterial slab (much larger than the coil itself). The charging range is too short for communication systems. Therefore, a technological breakthrough is required to realize the MI communication. (ii) In [26], metamaterials are introduced to the field of RF antenna miniaturization. In [27, 28], an electrical dipole antenna enclosed in a metamaterial shell is investigated in deep subwavelength range. The far field propagating wave and the radiated power from the electrical dipole can be dramatically amplified in lossless air medium. In contrast, the MI communication discussed in this paper focuses on the near field EM components, especially the magnetic field around magnetic dipole (i.e., coil), which needs a major reexamination on the metamaterial resonant structure. More importantly, MI communication is designed to operate in complex media, which dramatically change the condition and the properties of metamaterial resonance. In addition, when comparing performances with conventional antennas, existing works, such as [27], use the same dipole moments. However, since MI in lossy medium can have very large frequencydependent resistance, the metamaterial antenna used in MI can have dramatically different dipole moments from conventional antennas when the input power are the same.
Iii Modeling and Analysis of MI Communications
In this section, an analytical channel model of the proposed MI communication technique is developed for complex and RFchallenging environments. Specifically, metamaterials are introduced to enhance both the wireless communications using pointtopoint MI and MI waveguide. The MI waveguide is actually a sequence of pointtopoint pairs. Hence it shares the same theoretical foundation as pointtopoint MI. Therefore, we first developed the path loss model for pointtopoint MI, which can be easily extended to MI waveguide. The discussion on the optimal metamaterial shell configuration is also universal for both settings.
In the following analysis, we use boldface lowercase letters for vectors and boldface capital letters for matrices. For a vector , we use to denote its magnitude and a unit vector to denote its direction. For a matrix , denotes its transpose and denotes its determinant. For a complex number, we use and to denote the real and the imaginary parts, respectively. If there is no special notation, the considered lossy medium in this paper is underground soil medium.
Iiia Enhancing MI Communication using Metamaterials
The original MI communication is accomplished by using a transmitter MI coil and a receiver MI coil [5]. Instead of using the widely used propagating EM waves in the far field, the MI communication utilizes the magnetic field generated from the transmitter MI coil in the near field. Modulated by digital signals, such magnetic field can induce the current that carries signals at the receiver MI coil, which complete the wireless communication.
In the MI system, the MI communication is enhanced by enclosing the original MI coils with metamaterial spherical shells. The near field EM components can be manipulated and possibly enhanced by letting waves travel through the metamaterial layer. In this subsection, we initiate the analysis by discussing how the metamaterial sphere influences the original MI communication mechanism.
According to our previous work in [5], the channel of the original MI communications (as well as the MI waveguide) can be modeled by the equivalent circuits shown in Fig. 2, where is the coil’s resistance; is the coil’s real selfinductance; is the compensation capacitor used to tune the circuit to be resonant; is the receiver load, is the source’s voltage, is the mutual inductance between two adjacent coils. The ideal power source has no impedance, which is consistent with the following Comsol FEM simulations where the source is also ideal. In order to compensate the real inductance and achieve circuit resonance, the compensation capacitor is utilized, where is the resonant frequency of the coil. It should be noted that for original MI, the imaginary part of the selfinductance 0 since there is no strong metamaterial amplification.
In MI, the equivalent circuit models are still valid but need a major modification. In particular, the metamaterial sphere can dramatically change the properties of all the selfinductances and mutual inductances in Fig. 2. On the one hand, the mutual inductance is expected to be dramatically increased in MI. Since the design objective of the metamaterial sphere is to amplify the near field components of EM waves, more magnetic flux goes through the transmitter and receiver coils so that the mutual inductance is increased. On the other hand, while the selfinductance is real in the original MI, it becomes a frequencydependent complex number in MI, i.e., where and are real positive numbers. The imaginary part of (i.e., ) forms the frequencydependent resistance in MI antenna, which comes from two unique sources in MI: the metamaterials and the complex environments. Firstly, on the resonant metamaterial sphere that is very close to the MI coil itself, significant eddy currents are induced on the metallic components. The eddy current generates a secondary magnetic field that opposes the primary field, which reduces the current in the MI coil and equivalently increases the impedance. Secondly, the lossy medium also contribute to the imaginary part of the selfinductance due to the induced eddy current. Since the impedance of an inductor is , where is the angular frequency, the updated impedance with imaginary inductance is . Accordingly, the compensation capacitor becomes to achieve the magnetic resonance.
Once the reactance is canceled at both transmitter and receiver, the receiver load is matched with the coil resistance and the additional loss , i.e., . Different from the EM wavebased wireless systems, the transmitter and receiver in MI and MI are closely coupled to each other so that the impudence matching are done in an integrated transmitterreceiver system. Or in another word, receiver is part of the loads in transmitter while the transmitter is the source in the receiver.
Since the range of a wireless communication system is mainly determined by the channel path loss, we pay special attention to investigate the path loss in MI. It should be noted that the formulation of other important parameters in MI, including the channel bandwidth and channel capacity, can also be derived based on the path loss analysis, as shown in Section IIIC and Section IV. Based on the equivalent circuit model in Fig. 2 and above discussions, the general path loss formula in MI channel between two transceivers (pointtopoint communication) can be expressed as
(1a)  
(1b) 
where the load resistance is designed to maximize (1a), which equals . To elucidate the physics better, (1a) can be approximated by (1b) at the resonant frequency. The precondition is that is much smaller than . This approximation is practical since is very large due to the resonance of the metamaterial shell. Moreover, since we consider loose coupling for long distance communication, the mutual inductance is very weak. As a result, is much smaller than , so that the precondition holds.
Similarly, the MI waveguide is formed by adding the metamaterial sphere on each MI coil (including the transmitter, receiver, and relays) in an MI waveguide. Consider that there are relaying coils in the waveguide. The first coil is the transmitter and the last one is the receiver. All the relays are well placed along a line with equal interval . Based on the equivalent circuit for MI waveguide shown in Fig. 3, the general path loss formula can be approximately expressed as
(2) 
According to (1b) and (2), the path loss in both MI pointtopoint communication and MI waveguide are strong functions of the mutual inductance , coil resistance , and the frequencydependent resistance brought by the metamaterials and the propagation medium. Since is a constant value, only and can be manipulated by designing the metamaterial sphere. To reduce the path loss and enlarge the communication range in MI, a straightforward strategy is to increase the mutual inductance and decrease the frequencydependent resistance , according to (1b) and (2). However, such strategy cannot be easily applied since the metamaterial shell can amplify both and simultaneously. If we reduce , we might lost the gain of . To investigate this tradeoff, the fraction can be used to define a new metric for MI communications, i.e., the Inductance Gain (IG), to characterize the benefit from metamaterial shell. We denote the inductance gain as , which is:
(3) 
where and are the inductances by using metamaterial shell, is the mutual inductance without using metamaterial shell. Note that without metamaterial, is relatively small and can be neglected here (unless in high conductive medium such as seawater and under ocean environments, which is out of the scope of this paper).
It’s worth mentioning that when comparing with the coil without metamaterial shell, we set the coil’s radius as (the outer radius of the metamaterial shell) instead of (coil radius inside the shell) for fairness. Then, we denote the resistance of the smaller coil inside the metamaterial shell as and the resistance of the larger original MI coil as . In the following sections, the optimization objective of the metamaterial sphere design is to maximize the inductance gain in (3).
IiiB Modeling the Metamaterialmanipulated EM Field in MI
According to the general framework of MI and the metamaterial enhancement strategy discussed in the previous subsection, the mutual inductance and the complex selfinductance play important roles in MI communications. To quantitatively characterize the influence of the metamaterial sphere on the mutual and selfinductance, in this subsection, we investigate and model the metamaterialmanipulated electromagnetic field around both the MI transmitter and receiver. The field model is then validated by the FEM simulations. Finally, a proofofconcept experiment is discussed to confirm the feasibility of the MI in real implementations.
It should be noted that we focus on the MI pointtopoint communication in this subsection. The performance of the MI waveguide can be easily derived based on the analysis of pointtopoint communication. Moreover, the orientation of the coil inside a metamaterial shell can affect the system performance especially when two coils are perpendicular to each other. This problem can be solved by the tridirectional coil antenna [1] that mounts three concentric and orthogonal signal coils together in both the transmitter and receiver. As each of the three concentric coils covers one direction in the Cartesian coordinate, the entire 3D space is covered. Then, no matter how the transmitter or receiver moves and rotates during deployment or operation, reliable wireless channel can be maintained. The tridirectional coil structure can be easily inserted into the metamaterial sphere in MI discussed in this paper. The performance is also easy to model by simply adding the fields from the three coils. However, such analysis would add unnecessary complexity and defocus the key point in this paper. Hence, we only consider coaxiallyplaced coils in the following analysis. Detailed discussion on the MI tridirectional coil antennas can be found in [1, 29, 30].
IiiB1 EM Field around MI Antenna
Consider the MI pointtopoint communication between a MI transmitter and a MI receiver, as shown in Fig. 1. The coil is located at the center of the metamaterial shell. We define the space inside and outside the shell as the first and third layer, and the shell itself is the second layer. In the first and second layers, there are standing waves while the third layer has traveling wave. Hence, spherical Bessel function of the first kind and spherical Neumann function are used to construct the solution in the first two layers. Due to the singularity of spherical Newmann function, only spherical Bessel function of the first kind is used in the first layer. Spherical Hankel function is utilized in the third layer. For the layer, the wavenumber , where , is the vacuum permittivity, is the relative permittivity, , is the vacuum permeability, is the relative permeability. Also, is the antenna radius, is the antenna current, is the distance from the origin, is the wave impedance, and and stand for magnetic field and electric field, respectively. The timedependance is assumed. In lossy medium, the conductivity has significant impact on the MI communication.
Following classical electromagnetic theory, we first construct the general solution to wave equations in spherical coordination. Then based on boundary conditions, we can obtain the complete solution. Notice that, since the magnetic dipole’s radius and its wire radius are much smaller than the wavelength, the antenna only radiates TE. Moreover, the metamaterial shell is much smaller than the wavelength, we only need to consider the first order mode [31]. Thus, the unknown magnetic fields in each layer can be expressed as
(4a)  
(4b)  
(4c) 
where is the unknown coefficient; is spherical Bessel function of the first kind and order 1, and is spherical Neumann function of order 1, is spherical Hankel function of the second kind and order 1, and the prime symbol denotes derivative.
According to Maxwell equations, the normal component of the magnetic flux () and the tangential component of the magnetic field () should be continuous at the boundary. Then by adding the excitation source and rearranging (4), we can obtain the unknown coefficients by
(5) 
where ; is a coefficient matrix and is the excitation vector. The detailed expressions for and are provided in Appendix A. After solving (5), by substituting the unknown coefficients , , , and into (4), the intensity of magnetic field around the MI transmitter (outside the metamaterial sphere), i.e., and in (4c), can be derived. The magnetic field intensity around the MI coil (inside the metamaterial sphere), i.e., and in (4a), can also be derived.
Similar as the transmitter, the magnetic field in each layer of the receiver can be expressed by (4). The difference is that the magnetic field in the third layer is the scattered field from the receiving shell. In order to distinguish the transmitter and receiver, the unknown coefficients are for each layer at receiver side. By substituting with in (4), we can obtain the magnetic field intensity inside the receiver’s shell. Based on boundary conditions, can be determined by
(6) 
where ; and similarly detailed expressions for and are provided in Appendix A.
Once the solutions of the coefficients , , , and are derived, the magnetic field intensity distribution around the receiver (inside the metamaterial sphere) can be expressed in the same format given in (4) (replacing with ).
IiiB2 Validation using FEM Simulations
In this subsection, we verify the developed field model via FEM simulation in Comsol Multiphysics [12]. The system configuration is set as follows. The overall size of the metamaterial sphere is set to be 10 cm in diameter, i.e., m. Such antenna size can be fit in many wireless devices. Similar to most MI communication systems, the operating frequency is set at 10 MHz. A single negative metamaterial layer is used. Without loss of generality, we set the relative permeability of the metamaterial to be , i.e., . The transmission medium outside the sphere is considered to be soil with the relative permeability , permittivity , and conductivity mS/m.
The maximum inductance gain (IG) can be obtained by finding the optimal thickness of the metamaterial layer, i.e., , and the permeability of the infilling inside the sphere, i.e., (See details in Section IIIC). Since we only need to validate the field model derived in this subsection, we directly use the optimal values: m and . In addition, the size of the MI coil is supposed to be as large as possible [5]. Theoretically we can set . However, as approaches , it will cause strong effect on the boundary. As suggested in [32], . Thus, we set m. For fair comparison, the radius of the coil without metamaterial shell is set to be cm (the same as ). It should be noted that the above parameters of the metamaterial sphere are practical. As demonstrated in [33, 34], it is possible to fabricate low loss metamaterial with unit size less than at around 10 MHz. The metamaterial sphere thickness used in this paper (=0.025 m) is , which is well above the threshold.
It should be noted that the intrinsic loss effect in metamaterial is also considered. As reported in [10], the measured loss is 0.05. In this paper, we consider has three levels of losses, i.e., high loss, low loss, and no loss. The corresponding parameters are: high loss , low loss and no loss . Comsol simulation model is shown in Fig. 4. It’s an axissymmetric model where the coordinate is cylindrical. AC/DC module is utilized here and the distance between two coils is 5 m. All the parameters in the simulations are summarized in Table I. Different from [27], when comparing the performances, we consider both the MI antenna and the original MI antenna have the same transmission power instead of the same antenna current. As discussed in Section IIIA, the MI coil has additional frequencydependent resistance from the imaginary selfinductance, which can consume significant power. For the considered four scenarios, the input impedance highly depends on the additional resistance (). Therefore, to make the comparison fair, we set the transmission power in (1b) as 1 W for all of the four scenarios.
5  10 MHz  

0.025 m  0.05 m  a  0.015 m  
0.047  2 mS/m  1 W  
1  10.005j  10.05j 

is the relative permeability in the second layer.
As shown in Fig. 5 and Fig. 6, the magnetic field intensity derived by the theoretical field model has good match with the FEM simulation results at both transmitter and receiver side. On the one hand, we observe that by using the metamaterial shell with optimal parameters, the magnetic field intensity can be increased by more than 1 order of magnitude, compared with the original MI system. On the other hand, the metamaterial loss can dramatically reduce the gain brought by the metamaterial sphere. Also, notice that the receiver side has larger gain than transmitter side. The reason is the magnetic field is amplified again by the receiver’s metamaterial shell. In Fig. 8, the enhancement on the magnetic field by no loss MI is visually shown by the Comsol simulation. The two configurations have the same input power. The coil with metamaterial shell can generate much stronger magnetic field. Also, the amplification at the receiver side is obvious. In contrast, the original MI receiving coil has very weak field which almost cannot be seen from the figure.
In addition, the effect of a ferrite core at the receiver is discussed since this strategy is widely used to improve MI’s performance. As shown in Fig. 7, we consider that there is a spherical core inside the receiving MI coil. The radius of the core is the same as the outer radius of metamaterial shell, i.e., 0.05 m. The relative permeability of the core is 200. Configurations of MI without ferrite core and MI are the same as previous discussions. Note that, due to the high permeability, the magnetic field (A/m) inside the core is very small. However, since the ferrite core has large permeability, we obtain a large magnetic flux intensity (). As shown in Fig. 7, the enhancement from the ferrite core in the original MI is much smaller than that from MI, which proofs the more significant enhancement of MI.
IiiB3 Experimental Validation
To validate the predicted MI enhancement, a proofofconcept prototype of metamaterial sphere is designed and implemented. As shown in Fig. 9, the ideal spherical shell is approximated by a 36face polyhedron. The diameter of the polyhedron is approximately 10 cm. Each face of the polyhedron forms a metamaterial unit, which is a 6turn coil with 1.5 cm radius and loaded with a variable capacitor. An 8turn MI coil antenna with 2.5 cm radius is fixed in the center of the polyhedron. By tuning the variable capacitor, the fabricated metamaterial shell can achieve the resonance at 15.5 MHz (i.e., achieves the negative magnetic permeability at 15.5 MHz). The equivalent circuit for transmitter and receiver is shown in Fig. 11. Thanks to the lossless environment, can be neglected. As a result, MI and MI have the same equivalent circuit. The inductance is still compensated by a capacitor to make the circuit resonant. The difference from the theoretical model is that the source is not ideal (it has resistance ) and also the receiver load is fixed.
Based on the equivalent circuit and loose coupling assumption, (1a) for both MI and MI can be updated according to the experimental configuration:
(7) 
In (7), while is the same in both MI and MI, the only difference is . Although and are fixed so that we cannot change to match with (as in the theoretical model), the performance difference between MI and MI keeps the same. Matching the resistance can only increase , which is the same for MI and MI. The differences on power ratios between MI and MI are still the same, i.e., the enhancement of MI keeps the same regardless the resistance is matched or not.
The experimental setup is shown in Fig. 9. The MI transmitter is enclosed by the metamaterial shell while the MI receiver is an original MI coil antenna (8turn, 2.5 cm radius). The Agilent 8753E RF network analyzer is used to measure the parameter of the prototype. The transmitter coil (inside the metamaterial shell) and the original MI receiver coil (the wire loop) are connected to the two ports of the network analyzer, respectively. On the one port, the network analyzer feeds the transmitter coil with 14 to 17 MHz signals. On the other port, the signal received by the MI receiver coil is input to the network analyzer to display the measurements. For comparison, we also conduct the same experiment for a transceiver pair without the metamaterial shell.
Fig. 11 gives the received signal strength of the MI communication with and without the metamaterial shell, i.e., MI and MI, respectively. The receiver is placed 10 cm away from the MI transmitter. Around the resonance frequency, i.e., 15.5 MHz, more than 20 dB enhancement is observed when the metamaterial shell is used, which is consistent with the theoretical and the simulation prediction. Therefore, the concept of metamaterial enhanced MI communication can be proved by this initial prototype and experiment. It should be noted that, the experiment results are not directly compared with the numerical results in this paper. The reason is that currently it is still an open issue to model a fabricated metamaterial. Hence, we cannot exactly determine the values of metamaterial thickness, effective radius, and effective permeability, which prevent a directly and appletoapple comparison between the experiments and the numerical results.
IiiC Analytical MI Channel Model
Based on the field model derived in the last subsection, the selfinductance and the mutual inductance as well as other channel parameters in the MI communication can be calculated. However, the field model requires complicated calculations, such as the inverse of the matrix of Bessel functions. Therefore, the model is limited to numerical results and can not provide analytical insights on the metamaterial enhancement mechanism, not to say the optimization of the system. To this end, we develop the analytical MI channel model with an explicit and tractable expressions of selfinductance, mutual inductance, and path loss, as well as bandwidth and capacity. Then based on the analytical model, the resonance condition as well as the optimal configuration of MI communications are investigated.
IiiC1 Deriving Explicit Expressions for Analytical Channel Model
We start the investigation by calculating the MI selfinductance and mutual inductance based on the developed field model. Due to the influence of the metamaterial shell, the selfinductance consists of two parts: one is the original inductance generated by the coil and the other one is the inductance contributed by the metamaterial enhancement:
(8) 
where is the magnetic flux through the transmit coil; is the area of the coil; is the orientation of the coil; and is the coil’s selfinductance without the shell, which can be approximated by , where is the wire radius [35]. Similarly, the mutual inductance is the magnetic flux through the receive coil over the current in the transmit coil, which can be expressed as
(9) 
Note that, the reradiated field from the receiver coil is considered in this mutual inductance since it is bidirectional.
According to (8) and (9), the key coefficients that determines the selfinductance and mutual inductance are and . To derive , is also needed to be calculated. Those coefficients need to be derived through (5) and (6), which require the calculation of the inverse of a matrix consisting of different types of Bessel functions. To derive tractable channel model, such functions need to be simplified. Fortunately, since the shell and the antenna are electrically small, i.e. , and can be simplified as
(10) 
(11) 
where , , , , all are real positive numbers, and is a function of distance which is determined by the antenna pattern. Here is the approximation of , which is
(12)  
where is the asymptotic approximation of all the wavenumbers and is the asymptotic approximation of all the radii. The detailed deductions for this approximation is provided in Appendix B.
In Fig. 14, the accuracy of the approximation is numerically evaluated. The system configuration and parameters are the same as Section IIIB, where the no loss case of is used. We increase continuously. As shown in the figure, the approximation has good match with the exact numerical results. Hence, the self and mutual inductance in MI communications can be accurately and explicitly expressed by (10), (11), and (12). Then the channel path loss of the MI pointtopoint and waveguide communication can be derived by substituting and into (1b) and (2). Moreover, the bandwidth and the channel capacity can also be calculated based on the path loss formula, which are discussed with the MI channel analysis in Section IV.
It should be noted that the determinant changes its sign at the resonant point (=0.025 m). Such change causes the negative selfinductance in MI, which has not been observed in existing works. We will discuss this unique property in next subsection.
IiiC2 Optimal Configuration of MI Communications
The transmitter and receiver are connected by mutual inductance . According to (11), can be maximized if is zero (i.e., is very small). There exists an optimal metamaterial sphere thickness that can greatly reduce the value of . As a result, the mutual inductance can be significantly increased. Then the magnetic field intensity around both MI transmitter and receiver can be dramatically enhanced. The condition to achieve such enhanced peak is to find that makes . The solution to can be developed as
(13) 
If the metamaterial sphere satisfies (13), it achieves the metamaterial resonance. Such resonance cannot be achieved if is positive since , which necessitates the usage of the metamaterials as the second layer of the sphere. Also, if we use a ferrite core for coil antenna to improve the performance, can be increased to more than 100. By adjusting we can still find the resonance configuration.
This resonance condition is also observed in other metamaterial antenna designs [27] and [36], where it appears that the resonance is the optimal operating mode since it amplifies the radiated power in the far field to the maximum extent. Any deviation from the resonance can significantly deteriorate the antenna’s performance. In contrast, the MI communications depend on the near field where the radiated power is not as important as in the far field communications. Moreover, in lossy media, the resonance not only maximizes the mutual inductance but also maximizes the frequencydependent resistance . Therefore, the role of metamaterial resonance in MI communications needs a major reexamination.
The frequencydependent resistance comes from the imaginary part of the selfinductance . Hence, we investigate the effect of metamaterial resonance on the selfinductance given in (10). Under the resonant condition in (13), can be updated as
(14) 
In (14), the absolute value of the second term is maximized since is the minimum value of . If the transmission medium is lossless, such as the air medium in most existing works, the second term in (14) is real since the wavenumber of the medium is real (i.e., ). Therefore, the selfinductance is real, which can be compensated by the capacitor. Hence, even the selfinductance is maximized, no additional loss is introduced to the MI system. However, in the lossy medium considered in this paper, the wavenumber becomes complex. Consequently, the imaginary part of (i.e., the frequencydependent resistance) in (14) is also maximized when resonant, which causes significant loss in MI.
Fig. 14 shows the total resistance () of a MI coil in the soil medium as a function of the sphere thickness , based on both the developed model and FEM simulations. As predicted, the coil resistance is extremely large when the sphere is resonant ( m). Hence, in MI, the resonance condition amplify both the mutual inductance and . As the sphere thickness moves away from the resonant condition, approximates to 0. As a result, the frequencydependent resistance disappears and only the coil wire resistance is left, i.e., . Fig. 14 shows the calculated and simulated inductance (i.e., the real part of ). Similarly to the imaginary part in Fig. 14, the real part of is dramatically amplified at the resonance point.
According to (3), the inductance gain between the MI transceivers is in fact determined by the ratio . The effect of resonance on is not clear yet since both and are maximized at the resonance point. However, according to (8) and (9), is inversely propositional to while is inversely propositional to . Considering that the resonance condition is in fact , is more significantly amplified than when resonant. Hence, we can conclude that the metamaterial resonance is still the optimal operation status in MI. However, due to the same resonance effect of the frequencydependent resistance (which incurs loss), the system performance does not deteriorate as fast as existing metamaterial antennas when deviates away from the optimal value. Hence, the MI system is not very sensitive to the size deviations, which is favorable in practical device fabrication.
Before numerically investigating the effects of resonance on the inductance gain , we first investigate an interesting observation in Fig. 14, where the inductance becomes negative when is a little smaller than 0.025 m (the resonance condition). To find out the reason of the negative selfinductance, we analyze the under the nonresonant condition. When the resonant condition (13) is not satisfied, the first term in (12) becomes dominant. Then can be expressed as
(15) 
where and . From (15), we observe that the imaginary part of disappears if the metamaterial sphere is not resonant, which is consistent with the results shown in Fig. 14 and Fig. 14. An interesting observation is that the real can be negative.
Fig. 15 shows the value of the numerator and denominator of in (15) as a function of the sphere thickness . When the metamaterial sphere is not resonant, the denominator can be either positive or negative: if m, while if m, . Since does not change its sign, have different signs in the two regions. As a result, the magnetic field generated by the coil should change its direction both inside and outside the shell. In Fig. 16 and Fig. 16, we simulate the direction of magnetic field in Comsol. We observe that when m and m, the magnetic field have different directions, which validates that existence of negative selfinductance in MI.
As shown in Fig. 14, the real and negative selfinductance appears in a region on side of the resonance point (when m), where the second term (negative) in (15) has a larger absolute value than (positive). When the sphere thickness becomes even smaller, the negative inductance is compensated by so that the total selfinductance becomes positive again. On the other side of the resonance point (when m), the selfinductance is always positive.
Although the negative real selfinductance does not influence the metamaterial enhancement, it may incur significant loss in the MI coil circuit if not well designed. In MI transceiver, there are two types of resonance: the resonance in the metamaterial sphere and the resonance in the MI coil. The metamaterial sphere resonance is achieved by selecting optimal sphere thickness while the MI coil resonance is achieved by using compensation capacitor to cancel the impedance caused by the selfinductance. The negative real selfinductance cannot be compensated by capacitors, which incurs significant loss in the MI coil circuit. Fig. 17 shows the inductance gain as a function of the sphere thickness if the negative selfinductance is not compensated. We observe a significant performance deterioration on the one side of the resonance point (when m).
Since metamaterial is an effective medium, it’s challenging to guarantee that the thickness exactly equals to the optimal value. If the fabricated is slightly smaller than the resonance point, significant performance drop can be incurred by the negative selfinductance. Two strategies can be adopted to address this problem. First, the negative selfinductance can be canceled if we match it with a positive inductor. Fig. 17 shows the inductance gain as a function of the sphere thickness in the ideal case: if the selfinductance has a positive value, a capacitor is added to the coil circuit to compensate it; while if the inductance is negative, the capacitor is replaced with a positive inductor. We observe that the big drop in the inductance gain disappear. This solution requires the precise knowledge of the fabricated metamaterial sphere to determine whether to use compensation capacitor or compensation inductor. Second, a much simpler way to address the negative selfinductance problem is to fabricate the metamaterial sphere a little bit thicker than the optimal resonance point. As shown in Fig. 17, no drop of gain appears in the region that m. Moreover, as discussed previously, the metamaterial enhancement in MI is not sensitive to the size deviation. Hence, a reliable MI system with good inductive gain can be derived if we design the sphere thickness slightly larger than the resonance value.
Iv Channel Characteristics of MI Communications
Based on the analytical model derived in Section III, we investigate the channel characteristics of the MI communication through both numerical analysis and FEM simulation in this section. The path loss, communication range, bandwidth, and channel capacity of both the MI pointtopoint and the MI waveguide communications are quantitatively analyzed in various environments. If not specially specified, the default system and environment parameters used in this section are the same as Section IIIB.3.
Iva Pointtopoint MI Communication
Fig. 18 shows the pointtopoint MI path loss as a function of the communication distance through both theoretical calculation and FEM simulation. Similar to Section IIIB.3, three levels of metamaterial loss are compared, including the no loss (), the low loss (), and the high loss (). To test the system robustness to the practical fabrication, three metamaterial sphere thicknesses are compared, including the resonance size ( m) as well as two larger sizes ( m and m). The sphere that is thinner than the resonance size is not considered due to the negative selfinductance problem discussed in Section IIIC.2. The FEM simulation only shows the path loss within the distance of 5 m due to the high computation complexity of the high resolution simulation. Consistent with the analysis in Fig. 17, resonant metamaterial sphere achieves the lowest path loss. As the thickness deviates from resonant radius, the gain introduced by metamaterial sphere gradually decreases. However, even is increased by 5 mm, the path loss of MI is still 30 dB lower than without the shell at 10 m distance when there is no loss in metamaterial which is shown in Fig. 18(c). Moreover, Fig. 18 shows that higher metamaterial loss can dramatically increases the MI path loss, which is consistent with the field analysis in Fig. 5 and Fig. 6. However, the influence of the metamaterial loss become less significant when the sphere thickness deviates from the resonant size, since both the thickness deviation and the metamaterial loss prevent the resonance in metamaterial sphere. It should be noted that there are many ways to reduce the metamaterial loss, such as geometric tailoring [37] and high inductancetocapacitance ratio [38]. In addition, by using the active metamaterials, such loss becomes even controllable [39].
Fig. 19 shows the theoretical and simulated frequency response of the same MI pointtopoint system, where the 3 dB bandwidth can be read from the curves. Since metamaterials are dispersive, we can only realize at a narrow band. In order to conduct a more practical analysis, here we consider the Drude model [36] to model such dispersion, where the permeability is a function of frequency:
(16) 
where and are the plasma and damping frequency, respectively. In this paper, is set as rad/s, and is set as 0, rad/s and rad/s for no loss, low loss, and high loss at 10 MHz, respectively. The derived from (16) is used in both the theoretical model and Comsol Multiphysics and the results are shown in Fig. 19. We observe that the bandwidth in MI communication is much narrow than the original MI system, which is due to the strong resonance introduced by metamaterials, especially in the no loss case. As the metamaterial loss increases or the sphere thickness deviates from the resonance size, the system bandwidth increases. Hence, there exists a tradeoff between the low path loss and high bandwidth in MI.
Since the objective of the MI communication system is to achieve a high data rate within a long transmission distance, the Shannon Capacity [40, 41] is used as the metric to evaluate the overall performance of the MI system:
(17) 
where is the resonant frequency; is the 3 dB bandwidth; is signal to noise ratio, where is the transmission power density, is the antilogarithm of (1b), and is the noise power density. Since the bandwidth in MI is very small (in the order of KHz), the noise power density can be considered as a constant. Similarly, the density of is also a constant within the bandwidth. Fig. 20 shows the channel capacity of the pointtopoint MI system with different metamaterial loss and sphere thickness. We set the transmission power as 10 dBm. It has been reported in [42] that the power of underwater magnetic noise at MHz band is around 140 dBm. However, considering the underwater environment has relatively low background noise level, we set the noise power as 100 dBm in this paper to guarantee the performance in much worse scenarios. We observe that the MI system can reach the communication range of almost 30 m with kbps level data rate, which doubles the range of the original MI system. Even with metamaterial loss, the range can still exceed 20 m. In the near region, the bandwidth imposes a strong constraint on the capacity since the path loss is low enough. As the distance becomes larger, path loss plays a more important role and the advantages of MI become obvious.
IvB MI in Other Complex Environments
Even most of the natural materials have the same permeabilities, their permittivities and conductivities can be dramatically different. Hence, we evaluate the performance of MI in other complex environments in the envisioned applications, including concrete and water. Different from soil, concrete has lower conductivity, while water has much larger permittivity and conductivity. In the numerical results, we consider concrete’s relative permeability, relative permittivity ,and conductivity as 1, 4.5, and 0.1 mS/m, respectively. Water’s relative permeability, relative permittivity, and conductivity are 1, 80.1, and 10 mS/m, respectively. In addition, the metamaterial has low loss and the shell has resonant inner radius ( m). The path loss, bandwidth, and channel capacity of MI in concrete and water are shown in Fig. 21. We observe that MI performs much better than conventional MI in both concrete and water, in aspects of communication range and channel capacity. In particular, with 0.1 mS/m conductivity in concrete, MI can achieve 100 kbps data rate at 40 m, while the original MI can only transmit in the same data rate within 10 m. If the conductivity in the medium is even lower, the communication range and data rate of the MI system can be further increased.
IvC MI Waveguide
The MI waveguide can be formed when multiple MI devices are placed along a line and the interdistance between adjacent devices is small enough. For example, in the application of wireless sensor networks, many MI sensor nodes can be densely deployed. Between the MI transmitter and receiver, multiple MI nodes exist and form a MI waveguide along the transmission path. Based on the MI channel model derived in Section III, we evaluate the performance of MI waveguide in this subsection.
Fig. 22(a) shows the path loss, bandwidth, and channel capacity of the MI waveguide and the original MI waveguide. The thickness of the metamaterial sphere is fixed at the resonance size m and the no loss case is considered. Other configurations are the same as the pointtopoint MI. The interval between adjacent MI device is 1 m. According to Fig. 22(a), the MI waveguide can further increase the communication range compared with the pointtopoint MI. Compared with the original MI waveguide, MI waveguide has much lower path loss but also much narrower bandwidth due to the joint resonant effects of multiple MI devices. According to the channel capacity given in Fig. 22(c), without metamaterial, the larger coil formed waveguide cannot reach a communication range larger than 15 m. In contrast, the MI waveguide achieves the communication range of more than 40 m with the data rate at kbps level.
Fig. 23 shows the Comsol simulations of the magnetic fields of the pointtopoint MI communication and the MI waveguide. It’s clear that with the help of the three passive relays, the magnetic field at receiver of the MI waveguide is much larger than the pointtopoint case. As a result, the signal power at the receiver in MI waveguide can be increased.
V Conclusion
In this paper, the metamaterialenhanced magnetic induction (MI) communication mechanism is proposed for wireless applications in complex environments. An analytical channel model is developed to lay the foundation of MI communications and networking under the impacts from lossy transmission medium. The channel model reveals unique properties of MI communications, including the negative selfinductance and frequencydependent resistance, which provides principle and guidelines in the joint design of communication systems and metamaterial antennas. The proposed MI mechanism and the channel model are validated and evaluated by using both the FEM simulations and proofofconcept experiments. The results of this paper confirm the feasibility of achieving tens of meters communication range in MI systems by using pocketsized antennas.
Appendix
Va Magnetic Field around Receiver
The excitation source is the coil. Without metamaterial shell, the radiated fields can be expressed as [43],
(18) 
The magnetic field inside and scattered by the shell can be expressed by (4). Also, the radiated magnetic field can be found in (18). By enforcing the boundary conditions and rearranging the items we can find
(19) 
and (shown on the top of next page).
(20) 
The difference between the transmit coil and receive coil is the excitation source. As shown in Fig. 1, the magnetic field generated by the transmit coil is scattered on the second sphere. According to Mie theory, multiple mode decomposition is required to find the exact solution. Since the size of the metamaterial shell is much smaller than the signal wavelength in the envisioned applications (MHz band signal with pocketsized device), the Rayleigh approximation can be applied [44]. When a spherical scatter is much smaller than the wavelength, the first order of the Mie solution can be a good approximation to calculated the magnetic field.
Therefore, the format of the EM field intensity inside the receiver is the same as that around the transmitter. However, the coefficients in the formulas are different and need to be determined by the new boundary conditions. Since the shell is much smaller than wavelength, all the incoming magnetic fields on the shell can be assumed to have the same magnitude . can be obtained from field in the third layer in (4c), i.e., and . As shown in Fig. 1, we build a new spherical coordination whose origin is the center of the receiver and the magnetic field is along axis. Then, the magnetic field is decomposed along and direction, so that and , where is the angle between the incoming magnetic field and .
VB Subwavelength Approximation
For those special functions, if , , , , , , and . In the above approximations, we only keep the dominant real part and dominant imaginary part in the functions.
In addition, we consider there is no loss in the first layer and the wavenumber is real. According to the effective parameter analysis of the metamaterials in [45], the wavenumber in the second layer () is pure imaginary since the metamaterial adopted in this paper only has negative permeability. Moreover, since the environment is lossy (complex permittivity), the wavenumber of the propagation medium () is a complex number. Thus, , , , all are real positive numbers,
By using the above approximations, (20) can be simplified as
(22) 
where
(23a)  
(23b) 
Moreover, can be simplified as
(24) 
With the simplified , the target coefficients and can be explicitly formulated by using (22) and (24). Then by substituting the solutions of and into (8) and (9), we derive the explicit expressions of the selfinductance (10) and mutual inductance (11) in MI communications.
Based on (22),