## 2 Theoretical Limitations of the Rectifier's Efficiency  
The rectification process is a nonlinear process with some characteristics that limit the efficiency, especially when operating at high frequency. The input power wave signal of the RF rectifier can be described using linear combinations of the wave complex variable \(\overline{a}_{in}\) defined in Equation (4):  
\[\overline{a}_{in}=\frac{1}{2}(\frac{\overline{V}_{in}}{\sqrt{Z_{rec}}}+\sqrt{ Z_{rec}\,\bar{I}_{in}}) \tag{4}\]  
where \(\overline{V}_{in}\) and \(\overline{I}_{in}\) are the voltage and current complex variables at the rectifier input port, and \(Z_{rec}\) is the rectifier input impedance (in general a complex number). The rectifier input-power complex variable can be obtained from the \(\overline{a}_{in}\) variable following Equation (5).  
\[\overline{P}_{in}=\frac{1}{2}\overline{a}_{in}\overline{a}_{in}^{*} \tag{5}\]  
The input power (\(P_{in}\)) is obtained by averaging the corresponding temporal expression of \(\overline{P}_{in}\) over a period. It can be shown that only the DC, \(\omega_{o}\), and the harmonics contribute to the \(P_{in}\) averaged value. The power dispersion over the harmonic spectrum in the rectification process is described by the Fourier series of the rectifier wave. Equations (6) and (7) show the Fourier series of a half- (\(f_{hw}\)) and full-rectified (\(f_{frw}\)) waves of a sinusoidal signal with amplitude \(A\) and angular frequency \(\omega_{o}\).  
\[f_{frw}=\frac{A}{\pi}+\frac{A}{2}\sin(\omega_{o}t)-\frac{2A}{\pi}\sum_{n=1}^{ \infty}\frac{\cos(2n\omega_{o}t)}{4n^{2}-1} \tag{6}\]  
\[f_{frw}=\frac{2A}{\pi}-\frac{4A}{\pi}\sum_{n=1}^{\infty}\frac{\cos(2n\omega_{o }t)}{4n^{2}-1} \tag{7}\]  
The DC power component directly delivers a fraction of the input power to the load. The rest of the input power is spitted in the harmonic signal spectrum. Impedance matching networks are designed to guarantee that the input wave at the operating frequency is not reflected. However, RF systems are not matched for all frequencies, and harmonic components different from the matched wave will be mainly reflected when \(S_{11}\leq-3\) dB. The numerical evaluation of the DC and the fundamental frequency points out a 67% limit for the half-wave rectifier and a 64% for the full-wave rectifier in a matched system operating at the fundamental frequency (see Table 2).  
Therefore, in the case of a perfect impedance matching at the input signal frequency, a similar performance is expected for the half-wave and full-wave rectifier (67% and 64%, respectively). Depending on the return loses at the second harmonic, these efficiencies can be increased up to 82% in the case of the half-wave rectifier and up to 94% in the case of the full-wave rectifier.  
This maximum efficiency values are degraded by the loses in the lumped elements at low input power and by the diode breakdown voltage \(\Psi_{\text{RF}}\) at high input power [4], as observed in Figures 2-4 in the following section. In the case of high input power, the maximum DC power across the diode is limited by the reverse breakdown voltage in such a manner that \(\text{V}_{\text{DC diode max}}=\frac{\Psi_{\text{RF}}}{2}\). Consequently, the maximum DC power that a rectifier can deliver to the load is given by \(\text{P}_{\text{DCMAX}}=\frac{\Psi_{\text{RF}}^{2}}{4\Psi_{\text{RF}}}\). This limitation of the output power produces a quick efficiency drop, as \(P_{in}\) grows beyond some specific value that may vary depending on the rectifier's architecture [32].