## 4 Circuit  
This section is focused on the discussion of the matching and the conditioning circuits. The matching circuit is implemented via an \(L\) matching network, while the conditioning circuit makes use of a half-wave Cockcroft-Walton (HWCW) multiplier. Unlike a simple rectifier circuit, the Cockcroft-Walton multiplier is also capable of elevating the output voltage while rectifying the input signal [22]. This is of great interest in energy harvesting applications, where the input voltages are very low. The HWCW is a multiplier circuit widely used in power electronics [23] in the generation of high DC voltages. Therefore, it has typically been implemented in particle accelerators [24], X-ray machines [25], and cathode ray tube televisions [26].  
Two circuits constitute a single stage of a HWCW multiplier [27]. The first circuit (marked in orange in Figure 12a) is a positive diode clamping configuration, also known as Villard circuit [23]. It is responsible of elevating the DC level of the AC input waveform. The second circuit (marked in blue in Figure 12a) is a half-wave rectifier with a smoothing capacitor. The diode \(D\) is in charge of rectifying the AC signal with the DC component to twice the input voltage peak, and the capacitor \(C\) considerably reduces the output voltage ripple \(\delta V\). The HWCW circuit and its operation principle are depicted in Figure 12. The greater the number of stages \(N\) placed in the circuit, the higher the output voltage is [23].  
Figure 11: Circuit model of the Archimedean spiral antenna.  
Figure 12: \(N\)-stage half-wave Cockcroft-Walton multiplier (**a**) and its working principle (**b**).  
Two circuit elements constitute a HWCW multiplier circuit. The diode is the most critical element, as it has to switch at frequencies of GHz and has to consume a very reduced amount of power. The rectifying efficiency of the multiplier depends mostly on the forward voltage drop \(V_{F}\) of the diode. Since there are two diodes per stage, the output voltage reduces by \(2 NV_{F}\) from the ideal value. Thus, it is preferable to use a single stage when the input power is low. A Schottky diode very suitable for this situation is the HSMS-2822 [28]. The HSMS-2822 package contains two series HSMS-2820 diodes, specifically designed for low power operation at frequencies below 4 GHz. According to the datasheet [28], the HSMS-2822 diodes can provide currents of 0.1 mA with a voltage drop of 0.22 V (@ 25 \({}^{\circ}\)C). The capacitor must be chosen in order to reduce the output ripple. The higher the capacitance value is, the lower the voltage ripple at the output. However, the parasitics of the capacitor play an important role. These are related with the concept of self-resonant frequency (SRF). The higher the SRF, the lower the effect of the parasitics in the circuit is. The parasitic of a capacitor is modeled as a series parasitic inductance, while the parasitic of an inductor is modeled as a shunt capacitor. Both are LC circuits, so the SRF is  
\[SRF\left[\mathrm{Hz}\right]=\frac{1}{2\pi\sqrt{LC}} \tag{2}\]  
Thus, the parasitic value can be calculated by measuring the SRF in the laboratory and using Equation (2). In [29], a 33 pF capacitor is used as a tradeoff value between the output ripple and the SRF.