I. INTRODUCTION
The exploitation of evanescent waves in the near field for power transfer purpose has gained a lot of interest recently [^{1}]. In order to prevent radiation, the drivers are highQ electrically small antennas (ESA). Most of their energy remains stored in the surrounding near field and power transfer occurs primarily via induction. Nonetheless, such inductive power transmission is efficient only for distances smaller than the diameter of the antennas being also quite sensible to misalignment [^{2}]. Electromagnetic MTM slabs presenting effective negative permeability can be used as nearfield lenses to improve the overall efficiency of such inductive power transmission systems by enhancing the inductive coupling between the antennas. It has been called MTMenhanced coupling [^{3}][^{5}].
Following the work previously presented in [^{6}], this paper describes a transmissionline based approach of the interaction between a MNG MTM and a magnetic link. By introducing the concept of virtual magnetic transmission line (VMGTL) to describe the magnetic circuit and by calculating an equivalent virtual impedance to represent the MTM slab, the gain mechanism of those lenses can be regarded as a nonFoster impedance matching. Foster's reactance theorem states that any lossless passive oneport network has a reactance (or susceptance) that increases with frequency [^{7}]. According to [^{8}], nonFoster circuits can provide negative resistance, negative inductance and negative capacitance (meaning that their slope varies decreases with frequency). As it will be further shown, a passive MNG MTM behaves as a bandlimited negative impedance described by a negative resistor in series with a negative inductor. The negative inductance implies on reduction of the amount of net energy stored in the magnetic circuit while the negative resistance implies on potential gain across the MTM.
This work is organized as follows: In Section II, a generalized theory of transmission lines is presented in order to include a magnetic coupled system. In Section III, equivalent virtual impedance for the proposed MTM slab is obtained. In Section IV and V, the analytic calculations and numerical results are presented and compared to demonstrate that a magnetic link can be described as VMGTL and that the gain associated with the MTM is due to a better impedance matching between the drivers. The results are supported by experimental evidence.
II. Generalized theory of transmission lines
A. Physical and virtual transmission lines
Most of the classical literature on TL is dedicated to electric transmission lines (ELTL), this is to say, the usage of two or multiple electric conductors (twowire TL, coaxial cable, stripline and so on) to guide electromagnetic (EM) waves. Only recently, a broader concept of TLs has been proposed. As it has been demonstrated in [^{9}], timevarying magnetic fluxes can guide EM fields like electric currents in ELTLs. In [^{10}], it is proposed a generalization for TL theory that includes the fluxguided modes.
Fluxes and charges have the same dimension. It means that a timevarying flux will act as a sort of “virtual current” driving the fields:
where ψ_{m} and ψ_{e} are, respectively, the magnetic and the electric fluxes, B and D the magnetic and electric flux densities, J_{m,d} and J_{e,d} the magnetic and electric displacement current densities, and I_{m,d} and I_{e,d} the total magnetic and the electric displacement currents.
Based on that fact, the phenomena of induction can be effectively described by TL equations by taking the displacement currents, the fluxes and the reluctance of the medium as virtual currents, virtual charges and virtual impedances, respectively. The TLs employing displacement currents to guide the EM fields are defined as virtual TLs while those employing conduction currents are defined as Physical TLs.
B. Propagation equations for VMGTL
For all purpose, a VMGTL consists on a magnetic circuit between two insulated drivers with no physical charge flowing between them and interacting only by means of inductive coupling. The transmitter is the magnetomotive force (MMF) source that generates the timevarying magnetic flux (or the displacement current) of the magnetic link while the receiver is the transducer that converts the magnetic flux to physical charges at the level of the load (see Fig. 1). In this work, the load is assumed to be a 50 Ohms resistance.
The total MMF of the virtual line is a real magnetic potential V_{m} because fluxes going in opposite directions possess inverted magnetic potentials. Consequently, there must be a uniform magnetic field stored between them. By integration, the total V_{m} of the VMGTL can be obtained:
Since the magnitude of the magnetic flux and the magnetic reluctance perceived by it are functions of the angular frequency ω, V_{m} is frequencydependent. As it is shown in Fig. 2, the vector product of the magnetic field stored between the incoming and ongoing fluxes and the electric circulation generated by the time derivative of the flux is the very powerflow mechanism of the link [^{11}], which implies it is frequencydependent as well:
The magnetic potential V_{m} and the magnetic virtual current I_{m,d} of the VMGTL follow propagation equations analogous to the ones of conventional TLs:
where Z_{m}, Y_{m},
Assuming the simplifying hypothesis of a magnetic flux confined in an invariant transversal section, the real and the imaginary parts of the magnetic reluctance
where
From Eq. (5) and (6), it can be seen that the term
The characteristic electrical impedance Z_{0} of the VMGTL, its complex propagation parameter γ, its load reflection coefficient Γ and its input impedance Z_{in} can be determined by:
The magnetic potential V_{m} and the virtual current I_{m,d} are associated with the electric potential V_{e} and the electric current I_{e} in the terminals through the following relationship:
Note that the magnetic and electric potentials and currents are dimensionally inverted. Since VMGTLs support transverse electromagnetic (TEM) modes as shown in [^{9}], the propagation is a onedimensional Helmholtz problem [^{13}]:
C. Propagation equations for virtual magnetic transmission lines in free space
If the flux is not channeled from the source to the load by a ferromagnetic material, the flux leakage of the VMGTL must be taken into consideration. Starting from the result presented in the previous section, the flux leakage is assumed to affect mainly the complex propagation constant γ. In order to compensate that, a numerically determined factor δ is introduced to adjust the phase variation and the attenuation of both the magnetic current and the magnetic potential of the VMGTL:
where κ_{m} is the magnetic coupling coefficient:
Hence, the overall gain introduced by the MTM can be defined directly from the power flow by calculating the difference between the transmission coefficient S_{21} of circuit with and without the MTM slab:
III. EQUIVALENT VIRTUAL IMPEDANCE FOR A MUNEGATIVE METAMATERIAL
In this section, the analytical representation of the virtual impedance of MNG MTM is compared with the one obtained through numerical simulation using the Sparameters to estimate the equivalent μ_{r} of the structure [^{14}]. The numerical results were obtained on Keysight ADS 2016 using the Method of Moments (MoM). The proposed MTM to synthesize a MNG slab is the same model exploited in [^{15}]. Its main characteristics are presented in Table I. Using the Lorentz model for EM materials, the relative permeability μ_{r} of the MTM can be estimated [^{16}]:
where ξ is the damping ratio of the system and F is the coupling coefficient between adjacent cells of the lattice. Note that μ_{r,MTM} will tend to 1 out of the resonance.
Main parameters  Value 

L_{cell}  240 nH 
f _{0}  32.4 MHz 

100 pF 
Q_{cell}  245 
Number of cells  49 
Periodicity  2.3 cm 
In Fig. 3, it can be seen that μ_{r} becomes negative close to the resonance frequency in both the analytical model (f_{0} = 32.4 MHz) and the numerical simulation (f_{0} = 32.4 MHz). Although the overall behavior of the model and simulation are the same, the magnitudes, the Qfactor and the resonance frequency differs, which is attended due to the model's simplifications that underestimate the slab losses.
Using the effective value of μ_{r}, the equivalent inductance of the MTM can be determined:
Then, the MTM equivalent virtual impedance is given by:
Since the slope of Z_{MTM} does not increase monotonically with the frequency, it can be said to possess a bandlimited nonFoster behavior. As it is shown in Fig. 4, bellow the resonance the MTM is totally invisible to the virtual line (Z_{MTM} = 0). Around the resonance, it is perceived by the magnetic link as a negative resistance or an electric potential gain (Re{Z_{MTM}} < 0) and a negative reactance or a capacitance (Im{Z_{MTM}} < 0). Finally, above the resonance it is seen as a purely positive reactance or an inductance (Z_{MTM} ∈ Im^{+}).
When the MTM slab is seen as a negative resistance, it tends to enhance (to restore) the magnetic current of the VMGTL. As a negative reactance, it acts similarly to a negative inductance converting the stored magnetic energy into the complementary electric one, which reduces the total inductance of the circuit. Finally, as a pure positive reactance, it acts as conventional inductance and stores the surrounding magnetic field. From this comes the conclusion that the observed gain at the resonance is actually taken its surplus of energy from the attenuation of the higher frequencies modes that are being “trapped” by the MTM.
In the presence of the MTM slab, Eq. (6) can be rewritten as:
which is equal to a characteristic impedance Z_{0} and complex propagation parameter γ given by:
So, both Z_{0} and γ will not be altered below the resonance, it will tend to diminish around the resonance and it will tend to increase above the resonance.
IV. APPLICATION
In order to validate the proposed model, the obtained results using the VMGTL formalism are compared with both numerical and experimental ones. The experimental data were obtained using a Vector Network Analyzer. The transmission and reflection coefficients were simulated on Keysight ADS 2016 using the MoM. The transversal magnetic field components were simulated on Keysight EMPro 2015 using the FiniteDifference TimeDomain (FDTD) method. The analytical results were a direct application of the virtual line model presented in section II.
The proposed freespace magnetic link consists of two loop antennas of radius r = 5 cm made of copper wire with diameter p = 1 mm assisted by the MTM described in section III. The loops operate far from selfresonance. It is assumed that the internal resistance of the source and the stray capacitance and the radiation resistance of the antennas are negligible. The secondary driver is connected to a load R_{L} = 50 Ohms. The drivers are separated by a distance D = 15 cm. The amplitude of the voltage source is
The selfinductance and the selfcapacitance of the loops can be estimated by [^{17}], [^{18}]:
The distributed inductance and capacitance of the line can be approximated by:
For loops magnetically coupled through the air:
Once the air has no intrinsic material loss, the reluctance of the VMGTL is purely real:
The coupling coefficient between the antennas can be estimated by [^{19}]:
V. RESULTS AND DISCUSSION
As it is shown in Fig. 5 and Fig. 6, the magnetic potential difference between the fluxes inside the link gives rise to a transversal component of the magnetic field H_{⊥}. When the MTM is activated line in Fig. 6, the distribution of H_{⊥} and consequently the distribution of V_{m} becomes much more uniform along the line.
As shown in Fig. 7 and Fig. 8, the homogenization of the potential distribution inside the magnetic link is equivalent to an improved impedance matching in the TL theory point of view.
Another important aspect of a better distribution of V_{m} concerns the accuracy of the model. As it can be seen in Fig. 9 and Fig. 10, the model can describe the overall behavior of the magnetic link with and without the MTM. Nonetheless, it is much more accurate when the MTM is inside the magnetic link. The enhancement of the accuracy of the model is a consequence of the magnetic flux focusing induced by the MTM. It makes more realistic the hypothesis of magnetic flux confined to a path with constant transversal section.
Concerning the gain introduced by the MTM, the analytical prediction is quite optimistic in comparison with numerical simulation (see Table II). It is a consequence of the simplifications adopted by the model. However, even if the magnitude of the gain is overestimated by the model, the overall behavior of the system predicted by it shows a good agreement with the experimental one (see Fig. 11).
At last, it must be stressed out that the relative position of the MTM to the transmitting driver has not modified the transmission gain. This result reinforces that the interaction between the magnetic link and the MTM slab has a TLlike behavior.
VI. CONCLUSION
In this work, it has been proposed the usage of a generalized TL theory to include the phenomenon of induction following the previous efforts presented by [^{9}]. The concept of virtual currents and charges were introduced to well distinguish between fluxguided EM modes from chargeguided EM modes. It had been demonstrated that freespace magnetic coupling can also be described as virtual TL and that a MTM can be interpreted as a virtual impedance added to this circuit.
It has been shown that the amplification effect is not contradictory with the law of conservation of energy since it results from the exchange of energy between the reactive modes in the near field of the loops. The MTM attenuates the higherfrequency modes and transfers the surplus power to the resonant one. It means that the MTM acts as a bandlimited gain medium around its resonance.
Since the transmission improvement is a consequence of the reduction of the virtualline reactive power, any element interacting with the magnetic link causing the very same effect in terms of the impedance seen by the VMGTL (i.e., a resonant coil) would produce a similar transmission gain as the one presented by this particular MTM. This fact seems to confirm the results presented in [^{20}].
Finally, the analytical model was verified through numerical and experimental results of a given magnetic link. The model confirms that the MTM gain comes from the improvement of the impedance matching between the transmitter and the receiver due to the nonFoster behavior of its equivalent virtual impedance close to its operating frequency. Although the model successfully predicts the behavior of the system and the underline mechanism of enhancement of power transfer due to the MTM slab, it is not accurate concerning the magnitude of these results. This lack of accuracy is mainly due to the nondeterministic nature of the magnetic potential of nonconfined VMGTL. In the future, the introduction of a probabilistic description of the magnetic link would solve the accuracy related problems.