## 2 Theoretical Frame and Materials
### Theoretical Frame
Our starting point is the schematic in Figure 1.  
Figure 1: Schematic of the communications loop (signal path) excited by a current generator (I), with a perimeter d.  
The cable ring of perimeter d represents a simplified sketch of the signal path, including the PV cable line and bypass capacitors (with enough capacitance for presenting a very low impedance at our working frequency), where the I/O impedance of the transceivers located along the ring is neglected for a first approximation. It is excited by a current generator (which in practice could be the secondary binding of a transformer). A general expression of the current wave along this loop is:  
\[I(x)=I\,\left(A\ e^{-jkx}+B\ e^{jkx}\right) \tag{1}\]  
Representing the sum of two waves moving in opposite directions along the ring, where I is the current amplitude with its time dependency:  
\[I=I_{0}\ e^{-jwt} \tag{2}\]  
Here, x is a coordinate indicating the distance along the loop measured clockwise from the generator position and k is the wavevector depending on the frequency (f) and phase velocity (0.6 c) and A and B are constants (which could be complex) to be determined. The boundary conditions imposed are related to the continuity of the current at the generator thus:  
\[I(0)=I\ (A+B)=I(d)=I\,\left(A\ e^{-jkd}+B\ e^{jkd}\right)=I \tag{3}\]  
leading to  
\[A+B=1\ ;\ A\ e^{-jkd}+B\ e^{jkd}=1 \tag{4}\]  
In general, the solutions of this system of equations lead to (complex) values of A and B different from zero and thus to a stationary wave pattern for the current distribution. These solutions are not desirable to implement a communications system because stationary waves give rise to maximum and minimum current amplitude points along the loop, which means different reception levels since a transceiver could be located at any point in the ring. For the case where  
\[\text{kd}=\pi(1+2n) \tag{5}\]  
where n is an integer value, this system of equations is incompatible and there is no solution, representing a destructive interference between clockwise and anticlockwise waves. The optimal solutions for our purpose are those where A = 0 or B = 0, which leads to a travelling wave solution of the form:  
\[I(x)=I\ e^{-jkx}\ \text{or}\ I(x)=I\ e^{jkx} \tag{6}\]  
not presenting maximum or minimum current amplitude points. To obtain these kinds of solutions, it is mandatory that:  
\[kd=2\ \pi\ n \tag{7}\]  
that is to say, the perimeter of the loop is an integer multiple of the wavelength. However, even when the above condition is satisfied, there are stationary wave solutions (A and B \(\
eq\) zero) satisfying only the extra condition A + B = 1. In order to determine the actual spatial current distribution over a typical PV cable loop, we built a setup comprising 20 m of cable from our real test PV plant (Figure 6, show later) and measured and assessed the string as one single conductor line travelling in a straight fashion from the positive lead of one signal generator to the end of the PV array arrangement where it then turned around, coming back as a loop at a distance of one metre apart from the initiated point at the positive lead, finishing in the negative lead of the signal generator. The topology of the circuit is thus a loop-like one, but the shape is elongated from a circular one. The generator was adjusted for a sinusoidal signal of 9.142 MHz (determined as the signal corresponding to a wavelength equal to the loop length, fulfilling the above condition) and 10 volts of amplitude. The validation of the system in short wire installations is interesting, since for longer installation lengths resonance can be achieved at the same frequency by means of higher-order modes, where more than one wavelength is present in the line. Essentially, if it works for our 20 m loop, it will work also in loops which are 40 m, 60 m, etc., and the fine adjusting for intermediate lengths can be easily achieved by the tuning circuit designed (see Section 2.2).