## 2 Theoretical Frame and Materials
### Materials  
Even though a preinstallation length compensation is feasible (installing a fixed reactive series component), employing automatic adjustable tuning would more so be convenient in confining the transmission around an optimal point, in order to account for the inevitable occurrence of small variations in the loop (for example, the addition of one more PV module to a string, which would increase the loop physical length). For this purpose, a simple tuning circuit was designed which was able to show either capacitive or inductive impedance by means of a control voltage. The basic variable component used was a varicap diode whose capacity could be adjusted depending on the inverse voltage applied. The circuit is shown in Figure 4a, and it is composed of a toroidal transformer with the primary (L2) connected to the cable loop and the secondary (L1) connected to the variable capacitor (C) (voltage controlled). The actual implementation circuit for the voltage-controlled capacitor is shown in Figure 4b. Figure 3: The shift of the loop resonance frequency with the addition of a reactance in series: the modulus (**a**) and phase (**b**) of the scattering parameter-1 measured with a VNA connected to the raw cable loop and with the addition in series of a coil and three increasing values of capacitors. The network analysis of the circuit in Figure 4a leads to an impedance seen from the primary port:  
\[Z=jwL_{2}(1-w2\ L_{1}C\ (1-K2))\div(1-w2L_{1}C) \tag{11}\]  
where the mutual inductance coefficient in the transformer is expressed by:  
\[M=K\sqrt{L_{1}L_{2}} \tag{12}\]  
and \(K\) is the coupling coefficient, showing a parallel resonance at  
\[w_{1}=1\div\sqrt{L_{1}C} \tag{13}\]  
and a series resonance at  
\[w_{2}=w_{1}\div\sqrt{1-K2} \tag{14}\]  
This impedance has two inductive regions, (\(w<w_{1}\)) and (\(w>w_{2}\)), and one capacitive region: (\(w_{1}<w<w_{2}\)). In the centre of the capacitive region with a good coupling in the transformer (\(1-K2<<1\)), we have (\(1>>w2\ L_{1}C\ (1-K2\)) and (\(1<<w2L_{1}C\)), so  
\[Z\approx\ 1\div\left(jwC\frac{L_{1}}{L_{2}}\right) \tag{15}\]  
and our circuit works as a capacitor multiplier by the factor \(\frac{L_{1}}{L_{2}}\), which can be very useful for loops requiring high-capacitive compensation with values higher than the maximum achieved by the varicap diode, for example, for loops with high inductance in series (the inductive coupling of the transceivers). For loops with low series inductance (the capacitive coupling of transceivers), the usual length of a typical PV string and carrier frequencies over 5 MHz, our circuit is better used  
Figure 4: Schematic of tuning circuit in (**a**) its generic representation and (**b**) actual implementation, and (**c**) schematic of voltage doubler detection circuit. in the region around the higher frequency of resonance \(w_{2}\), where the impedance is close to zero, and for frequencies lower than \(w_{2}\) the impedance is capacitive, and for frequencies higher than \(w_{2}\) the impedance is inductive. Around this region and \((1\ll w2L_{1}C)\), the impedance can be expressed as:  
\[Z=jwL_{2}(1-K2)-jL_{2}\div(wL_{1}C) \tag{16}\]  
equivalent to a series LC circuit with  
\[L_{equ}=L_{2}(1-K2) \tag{17}\]  
and  
\[C_{equ}=\frac{L_{1}}{L_{2}}C \tag{18}\]  
resonant at \(w_{2}\). Since the capacitor \(C\) can be dynamically adjusted, connecting this circuit in series with the cable loop, we can add series inductance, capacitance or none of them depending on what is required by the loop, and the dynamic adjusting range can be selected with the value of \(L_{2}\) since it is a common factor in the expression of Z around \(w_{2}\). The possibility of the capacitance control varying a biasing inverse voltage on the varicap diode allows us to implement a microcontroller-based tuning system that will also require some kind of detection of the resonance condition in the loop.