## 2 Theoretical Frame and Materials
### Theoretical Frame
The current measurements were made in a low-invasive fashion at different points along the loop, sensing the AC magnetic field associated by means of a toroidal ferrite core surrounding the cable, with a 30-turn enamelled cable wiring. This way, an induced AC voltage proportional to the AC current in the line can be measured with an oscilloscope across the terminals of the wiring. The high input impedance of the oscilloscope guarantees that a very low current will flow through the wiring, so a negligible inductance was added to the loop during measurements. Figure 2 shows the results of these measurements taken every metre along the loop, and it shows a clear stationary wave pattern. These results agree with the literature about the current distribution on loop antennas, the theory of which is close to the one here considered. The difference between the maximum (100%) and minimum (21%) current amplitude levels from the measurements represents 13.55 dB, which can be managed easily by an automatic gain control (AGC) within receptors. There are two possibilities for fully exploiting the advantages of a resonant loop for communications avoiding significant differences in the signal levels along the loop: (I) Find a simple way to excite only the travelling wave mode over a one-wavelength ring (optimal). (II) Work at carrier frequencies low enough in a fashion that the loop length is a fraction of % of the wavelength or less. This is the slower option, but it still leads to carrier frequencies over 2 MHz (assuming a propagation speed of 0.6 c and a loop length of 20 m), significantly higher than the average previously reported. Regarding solution I, some authors have faced the theory of travelling wave solutions for loop antennas [18], compared with standing wave solutions [19], and even have determined the necessary conditions for exciting the travelling wave modes [11]. All this work, however, is focused on antenna design and consequently looks for the best radiation properties. For our application, radiation losses are not desired, since they mean power losses and could affect the electromagnetic compatibility of the system. Fortunately, the drastic reduction in the loop area due to the particular shape of our setup will prevent the system from great radiation losses, since radiation integrals obtain contributions of opposite current elements much closer to each other [20]. The travelling wave or quasitravelling  
Figure 2: Current amplitude distribution along the loop (normalised (%); measurement points and fitting to a stationary wave envelope function). wave mode of propagation for our system could be achieved by fulfilling some of the conditions exposed in [18; 21]. However, these conditions are not easily applicable in a practical environment, so the insertion in series of a nonreciprocal device is proposed. These kinds of devices will attenuate the waves travelling in one direction, leaving unaffected ones travelling in the opposite direction, leading to a travelling wave propagation mode. Some of the possible practical devices that could accomplish this task are magnetic-circulator-based isolators or unity gain amplifiers referenced to Earth in such a way that waves arriving towards the output will find very low impedances to Earth, being attenuated, but the ones arriving towards the input will find very high impedances and will progress to the output almost unaffected. In this work, we focus on resonance control, and isolator insertion is left for future research. Regarding solution II, if we work at wavelengths four times the length of the loop or bigger, the phase changes along the loop in \(\pi/4\) radians maximum. For a loop length of \(\gamma\) wavelength, the general solution for its current wave is:  
Figure 1:0  
where kx goes from 0 to \(\pi/2\) along the loop length, so the maximum change in the current wave amplitude is:  
Figure 1:1  
This value is small enough to avoid AGC systems in receptors and simplifies the electronics, allowing carrier frequencies over 2 MHz as explained before.