# A Compact Thevenin Model for a Rectenna and Its Application to an RF Harvester with MPPT
## 2 Rectenna and Its Thevenin Model  
Figure 2 shows the schematic circuit of the rectenna under study [33], which includes a high-pass L-matching network (composed of a capacitor \(C_{\rm m}\) and an inductor \(L_{\rm m}\)), a half-wave rectifier, and an output filtering capacitor (\(C_{\rm o}\)). The antenna is modelled by a sinusoidal voltage source \(v_{\rm a}\) of amplitude \(V_{\rm ap}\) and frequency \(f_{\rm o}\) with a series radiation resistance \(R_{\rm a}\). On the other hand, \(v_{\rm in}\), \(Z_{\rm in}\), and \(P_{\rm in}\) are, respectively, the sinusoidal voltage, impedance, and power at the input of the rectifier, \(i_{\rm d}\) is the diode current, and \(V_{\rm o}\), \(I_{\rm o}\), and \(P_{\rm o}\) are, respectively, the DC voltage, current, and power at the rectenna output. An equivalent resistance \(R_{\rm o}\) is defined as \(V_{\rm o}/I_{\rm o}\). The amplitude \(V_{\rm ap}\) is given by [12] as follows:  
\[V_{\rm ap}=2\sqrt{2R_{\rm a}P_{\rm av}}, \tag{1}\]  
Figure 2: Schematic circuit of the rectenna under study. where \(P_{\rm av}\) is the available power at the antenna. The matching network, at matching conditions, that is, \(Z_{\rm m}=R_{\rm a}\) (where \(Z_{\rm m}\) is defined in Figure 2), boosts the voltage at the input of the rectifier by a voltage gain, \(G_{\rm t}\), given by [33] as follows:  
\[G_{\rm t}=\frac{V_{\rm imp}}{V_{\rm ap}}=\frac{1}{2}\sqrt{(1+Q^{2})}, \tag{2}\]  
where \(V_{\rm imp}\) is the voltage amplitude of \(v_{\rm in}\) and \(Q\) is the circuit quality factor given by:  
\[Q=\frac{1}{\omega_{\rm o}C_{\rm m}R_{\rm a}}, \tag{3}\]  
where \(\omega_{\rm o}=2\pi f_{\rm o}\). On the other hand, the value of \(L_{\rm m}\) must comply:  
\[L_{m}=\frac{1}{\omega_{\rm o}^{2}}\frac{1}{C_{\rm p}+C_{\rm m}Q^{2}/(1+Q^{2})}, \tag{4}\]  
where \(C_{\rm p}\) models the parasitic capacitance between node A and ground. To ease the analysis of the proposed rectenna and also gain more insight into its performance, a compact Thevenin model is provided here. First, the left-hand equivalent circuit of Figure 3 accounts for the antenna, the matching network, and the parasitic elements (\(R_{\rm p}\)-\(C_{\rm p}\)) of the coil, diode, and layout of the circuit. These parasitic elements are derived in Appendix A, where \(R_{\rm p}\) models the losses of the coil and diode and \(C_{\rm p}\) includes the parasitic capacitance of the diode, coil, and layout. Analyzing the left-hand circuit of Figure 3 at \(f_{\rm o}\), we can achieve the Thevenin equivalent represented by the right-hand circuit of Figure 3, where:  
\[ \begin{array}{l}v_{\rm eq}=2G_{\rm t}v_{\rm a}\frac{R_{\rm p}}{4G_{\rm t}^{ 2}R_{\rm a}+R_{\rm p}},\\ R_{\rm eq}=\left(4G_{\rm t}^{2}R_{\rm a}\right)\|R_{\rm p},\\ C_{\rm eq}=C_{\rm m}\frac{Q^{2}}{1+Q^{2}}+C_{\rm p}.\end{array}  \tag{5}\]  
Next, the Thevenin equivalent of Figure 3 is linked to the next stage of the rectenna, the rectifier, resulting in the left-hand circuit of Figure 4, where the diode does not include its parasitic elements since they have been already considered in the previous derivation (they are included in \(R_{\rm eq}\) and \(C_{\rm eq}\)). The diode is forward biased when \(v_{\rm in}\), assumed sinusoidal, surpasses \(V_{\rm o}\). As a result, \(i_{\rm d}\) is pulsed and is composed of the fundamental frequency (\(f_{\rm o}\)) as well as its harmonics and a DC component (\(I_{\rm o}\)). Impedance \(Z_{\rm s}\) (defined in the circuit) is zero at DC (due to the coil \(L_{\rm m}\)) and is equal to \(R_{\rm eq}\) at \(f_{\rm o}\) since \(L_{\rm m}\) and \(C_{\rm eq}\) form a parallel resonant circuit presenting an infinite impedance. On the other hand, at the harmonics of \(f_{\rm o}\) we have \(Z_{\rm s}<<R_{\rm eq}\) (due to \(C_{\rm eq}\)) whenever \(Q\) is high enough. Therefore, only the current at \(f_{\rm o}\) (\(i_{\rm in}\)) originates a voltage drop and \(v_{\rm in}\) will be sinusoidal, as assumed before.