Magnetoinductive waves in attenuating media

The capability of magnetic induction to transmit signals in attenuating environments has recently gained significant research interest. The wave aspect—magnetoinductive (MI) waves—has been proposed for numerous applications in RF-challenging environments, such as underground/underwater wireless networks, body area networks, and in-vivo medical diagnosis and treatment applications, to name but a few, where conventional electromagnetic waves have a number of limitations, most notably losses. To date, the effects of eddy currents inside the dissipative medium have not been characterised analytically. Here we propose a comprehensive circuit model of coupled resonators in a homogeneous dissipative medium, that takes into account all the electromagnetic effects of eddy currents, and, thereby, derive a general dispersion equation for the MI waves. We also report laboratory experiments to confirm our findings. Our work will serve as a fundamental model for design and analysis of every system employing MI waves or more generally, magnetically-coupled circuits in attenuating media.

Notation: Recall that the subscripts 1 and 2 denote coil 1 and 2, the superscript denote the layers in Fig. S1b, r, l, w and N are the mean radius, thickness, width and number of turns of the coil respectively, as shown in Fig. S1a, d and ∆d are the vertical distance and horizontal displacement between two coils, h is the separation from the surface of a coil to the surface of its respective insulating box, c = d − (h 1 + h 2 ) − (t 1 + t 2 )/2 is the thickness of conductive layer between two coils, J i is the Bessel function of the first kind, of the i-th order, t is the time variable.
Here, the superposition principle is applied to reduce the complexity. The field solutions for a coil are formulated in the absence of the other. Similar to most eddy current problems, cylindrical coordinates are employed in this case to facilitate the boundary conditions. All the media are assumed to be linear, isotropic and homogeneous. In this case, we derive the differential equation of the magnetic vector potential A (n) as 1 : where [X] T denotes the transpose matrix of a matrix [X], ρ, ϕ, and z are the radial, azimuthal, and axial coordinates, ρ, ϕ, and z are the corresponding unit vectors in cylindrical coordinates, µ 0 is the permeability of free space, µ (n) , (n) , and σ (n) are the permeability, permittivity, and conductivity of the n-th region respectively, and t is the time variable. Basically, the induced eddy current density must be in the same direction as the magnetic vector potential. With the assumption of an axially symmetric conductor geometry, the induced eddy currents have only one spatial component along the azimuthal angle ϕ and so is the magnetic vector potential, i.e., A (n) (0, A (n) ϕ , 0) and ∂A (n) /∂ϕ = 0. Thus, the differential equation for the magnetic vector potential of coil 1 due to an applied voltage V 0 reduces to: 1ϕ is the vector potential of coil 1 in the region n, which for symmetry reasons only depends on the cylindrical coordinates ρ and z. To simplify the mathematical complexity, we define the magnetic vector potential in time-domain as the convolution of the Magnetic Vector Potential Impulse Response (MVPIR) function with the transient current in the respective coils 2 .
are the MVPIRs, Ψ 1 (ρ, z, t) is the MVPIR generated by a unit impulse current in coil 1 in free space, E 1 (ρ, z, t) is the MVPIR generated by eddy currents in the medium, and δ(x) is the Dirac delta function of variable x. The magnetic vector potential produced by a Dirac-coil (an infinitely thin coil with infinite conductivity) of radius r, is calculated in terms of an infinite integral of Bessel functions 3 : By substituting i(t) for δ(t) and taking the integral sum of all the Dirac-delta contributions over the volume of the coil 1 and then by multiplying the term by its appropriate turn density, the time-dependent unit impulse response of coil 1 can be determined as: Taking the Fourier Transform in t and Hankel Transform in ρ, the unit impulse response of coils can be rewritten in frequency domain and α-space: where ω and α are the transformed variables of t and ρ, respectively. Next, we derive the unit impulse responses induced by eddy currents in n-th region, E (n) 1 (r, z, t), from the field solutions: Applying the Fourier Transform and Hankel Transform, The solutions of equation (S8) can be expressed in the general form: is the separation constant in the n-th zone, and 1,2 (α, ω) are unknown constants. In this particular configuration, cf. Fig. S1b, we have: where µ m , m , and σ m are the medium's relative permeability, relative permittivity and conductivity, respectively. Finally, the solutions for MVPIR for coil 1 problem in each region are summarized below: (S11) We then determine the unknown coefficients in (S11) by applying the boundary conditions (S12) and (S13) from the continuity conditions for the electric and magnetic field components at the interfaces shown in (S14) where z n→n+1 is the location of the interface between consecutive zones n and n + 1. Once all the coefficients are known, every other physically observable electromagnetic quantity can be calculated directly from the magnetic vector potential.
Here, we define the complex self inductance term in the same manner as the self inductance.
Whereas the self inductance represents the interaction between the source magnetic field produced by a coil with itself to produce an EMF, the complex self inductance can be referred as the interaction of the secondary magnetic field generated by the eddy currents with the coil. On the other hand, since the magnetic vector potential in the region of the second coil is an aggregate of the primary field generated by the first coil and the secondary field generated by the eddy currents, cf. (S11), the complex mutual inductance is a summation of the original mutual inductance in free-space and the coupling between the second coil and the eddy currents generated by the first coil. In other words, it is the mutual inductance between two coils through the medium.
Consequently, we calculate the complex self inductance of coil 1 by integrating the eddycurrent-based MVPIRs arising from currents in coil 1 in the region containing the coil itself over its volume.
Likewise, we obtain the complex mutual inductance by integrating the eddy-current-based MVPIR arising from currents in a nearby coil in the region containing the secondary coil over its volume.
Applying integral transformations in 3, 4 for the mutual inductance of two coils with a horizontal offset, the complex mutual inductance can be calculated by: By proceeding analogously, the solutions for MVPIR for coil 2 problem in each region are obtained: (S17) where and X (n) 2 (α, ω) and Y (n) 2 (α, ω) can be found by solving the boundary conditions (S12) -(S14).
Finally, we can obtain the complex self inductance of coil 2 and the complex mutual inductance between coil 2 and coil 1 by using the following equations: 1B. Simulation Results The two coils are identical split ring resonators with the dimension: mean radius r 0 = 11 mm, length l = 5 mm, width w = 1 mm, and gap width g = 1 mm. Their complex self inductance and mutual inductance can be recast as: The medium now has relative permittivity of r = 78, relative permeability of µ r = 1, and variable conductivity.
where Z,R 0 , L 0 and C 0 are the self impedance, resistance, inductance, capacitance of the coils in free-space, respectively, and ∆L = ∆L − ∆L is the complex self inductance. In principle, the resonant frequency occurs when the inductive reactance cancels out the capacitive reactance.
Unlike the normal lumped elements which are idealised to be frequency-independent for a certain range of frequency, the complex Kirchhoff coefficients are instead frequency-dependent. Therefore, the resonant frequency of a coil immersed inside a conductive medium is the root of the following equation representing the total reactance equalling to zero: Hence, its Q-factor inside the medium,Q, is then obtained by: 2B. Frequency Splitting The resonance splitting occurs when the two magnetically-coupled circuits are placed in the strongly-coupled regime, i.e., the strength of the mutual coupling between the two resonators is larger than losses 7 . The current responses in the two coils then exhibit two peaks above and below the resonance frequency in the non-coupled case, namely the odd and even mode. To isolate the losses added by the dissipative medium, since the Q-factor of the resonators in free space is usually high, the undamped resonant frequencies for two coupled coils can be sought by ignoring the internal resistance of the coil (Z = jωL − 1/(jωC)). The natural response of the system can be written as: whereω 0 = 1/ (L 0 + ∆L ) C 0 is the angular frequency of the coils inside the medium. Equation (S27) can be rearranged as: Therefore,ω 0 2 /ω 2 is the eigenvalue of the matrix on the left hand side of (S28). After simple mathematical operations, the splitting frequency can be determined as: In free space, (S30) reduces to: Recall that Z 0 = R 0 + jX 0 , (S31) can be rewritten as: As discussed above, the splitting resonance are not very separate. The additional reflected loss from the receiving coil (ωM 0 ) 2 R 0 /(R 2 0 + X 2 0 ) is then approximately constant near the resonances.
The well-known symmetry of the current response in coupled magnetic circuits arises as a result.
where the term is usually referred as the loss from the feedback effect. Here, the numerator of the feedback loss is expressed: With the same argument for splitting resonances, the total energy loss due to the presence of the complex self inductance and the Ohmic loss,R 0 = R 0 + ω∆L might still be constant about the resonance. Nevertheless, from (S34), when the working frequency rises through the self resonancẽ ω 0 , the reactance of the coils,X 0 , crosses the zero line, changing from capacitive (negative value) to inductive (positive value). The progression of frequency indeed mitigates the feedback loss from the secondary coil. Therefore, the current in the transmitter exhibits an antisymmetric response with the lower resonance experiencing much higher damping than the upper resonance. Likewise, the current response in the receiver can be explained accordingly. In conjunction with the notch, the immersion depth modifies both complex coefficients.
However, when the coils are submerged with a depth of i v ≥ 60 mm, it plays a less important role as the mean absolute percentage errors in both the modulus and phase are less than 3% compared to the reference values for both complex Kirchhoff terms. The immersion level therefore should be larger than 60 mm for this particular configuration to corroborate the accuracy of the analytical model. As a result, the U-shaped supports used to carry the rods interconnecting the dielectric insulator have a height of h c = 100 mm to satisfy the immersion level conditions.  Resonator Characterization In this part, we report the resonator characterization and the effects of the insulating boxes. When insulating the split rings by placing them inside non-magnetic cavities, the cavity material having a dielectric constant different from that of vacuum may introduce a small capacitance to the circuits. Predictably, it leads to a small decrease in the self resonance of the rings. Fig. S8 shows the transmission-type experimental apparatus in order to measure the Q-factor and resonant frequency of each coil with and without dielectric cavities. This method is widely employed because of its better accuracy compared to other methods, given that the two measuring prboes are in loosly coupled regime 8,9 . Here, we placed a pair of weakly-coupling probes in close proximity to the coils under test, which were supported by a Balsa-wood fixture, and recorded the transmission coefficient S 21 .
(a) (b) Figure S8: The measurement set-up to measure the quality factor of the fabricated split ring resonators (a) alone and (b) in combination with their corresponding insulating cavities.
To increase the accuracy, the complex transmission coeffcient S 21 is fit to a Lorentzian curve using a nonlinear least-squares fit 9 : where the fitting parameters are as follows: f 0 is the resonant frequency, ∆f Lorent is the bandwidth, |S max | is the maximum magnitude, A 1 is the constant background, A 2 is the slope on the background, and A 3 is the skew. The Q-factor is then determined using the fit parameters f 0 and ∆f Lorent : A large number of coils are manufactured and tested. In this document, we only detailed the resonant frequency and Q-factor of the coils that were selected to build MI waveguides, as shown in Fig. S9. Figure S9: Resonant frequency, f 0 , and Q-factors, Q, of split ring resonators with and without their corresponding insulators calculated from the Lorentzian fit.
Experimental Apparatus Fig. S10 shows photographs of the actual experiment apparatus in the laboratory.