A Compact Thévenin Model for a Rectenna and Its Application to an RF Harvester with MPPT

This paper proposes a compact Thévenin model for a rectenna. This model is then applied to design a high-efficiency radio frequency harvester with a maximum power point tracker (MPPT). The rectenna under study consists of an L-matching network and a half-wave rectifier. The derived model is simpler and more compact than those suggested so far in the literature and includes explicit expressions of the Thévenin voltage (Voc) and resistance and of the power efficiency related with the parameters of the rectenna. The rectenna was implemented and characterized from −30 to −10 dBm at 808 MHz. Experimental results agree with the proposed model, showing a linear current–voltage relationship as well as a maximum efficiency at Voc/2, in particular 60% at −10 dBm, which is a remarkable value. An MPPT was also used at the rectenna output in order to automatically work at the maximum efficiency point, with an overall efficiency near 50% at −10 dBm. Further tests were performed using a nearby transmitting antenna for powering a sensor node with a power consumption of 4.2 µW.


Introduction
Radio frequency (RF) energy harvesting has been extensively proposed to power tiny devices such as RFID tags, autonomous sensors, or Internet of Things (IoT) nodes. RF energy can be harvested either from dedicated sources, such as in the case of RFID devices [1][2][3][4], or from the RF energy already present in the ambient environment and coming from unintentional sources such as TV, FM radio, cellular, or WiFi emitters [2,[5][6][7][8][9][10]. Figure 1 shows the block diagram of an RF harvester powering a sensor node. The rectenna (rectifying antenna) transforms the RF signal to a DC voltage and the maximum power point tracker (MPPT) provides the optimum load to the rectenna to transfer the maximum power to the sensor node.

Introduction
Radio frequency (RF) energy harvesting has been extensively proposed to power tiny devices such as RFID tags, autonomous sensors, or Internet of Things (IoT) nodes. RF energy can be harvested either from dedicated sources, such as in the case of RFID devices [1][2][3][4], or from the RF energy already present in the ambient environment and coming from unintentional sources such as TV, FM radio, cellular, or WiFi emitters [2,[5][6][7][8][9][10]. Figure 1 shows the block diagram of an RF harvester powering a sensor node. The rectenna (rectifying antenna) transforms the RF signal to a DC voltage and the maximum power point tracker (MPPT) provides the optimum load to the rectenna to transfer the maximum power to the sensor node. Rectenna MPPT Sensor node RF harvester Figure 1. Block diagram of a radio frequency (RF) harvester powering a sensor node. Figure 1. Block diagram of a radio frequency (RF) harvester powering a sensor node. The rectenna is composed of an antenna, an impedance matching network, and a rectifier. As the available power at the antenna decreases so does the generated voltage. Whenever this voltage is not high enough to properly bias the diodes of the rectifier, power efficiency severely decreases. Several techniques have been proposed to increase the efficiency at low power levels. One of them consists of using an L-matching network for boosting the voltage at the rectifier input [1,3,5,[10][11][12][13][14][15][16][17][18][19][20][21]. As for the MPPT, several works propose its use with rectennas using either commercial chips [6,7] or ad hoc designs [22][23][24][25].
With the aim of gaining more insight into the performance of the rectennas, different analytical models are proposed. However, the derived expressions, which in some cases seek to model the rectenna output as an equivalent Thévenin circuit, are rather complex and may require additional simulations or extensive calculations, which hide the influence of the different parameters of the rectenna on its performance [12,18,[26][27][28]. At the other extreme, the Thévenin parameters are sometimes inferred by experimental characterization [25,[29][30][31][32]. However, in these cases no relationship with the rectenna parameters is established.
Taking into account the previous limitations, this paper proposes a compact Thévenin model for the rectenna with the benefit of achieving manageable expressions of the Thévenin parameters as a function of the parameters of the rectenna so as to gain insight into its operation. In particular, the rectenna under study consists of an L-matching network and of a half-wave rectifier. The proposed model is then experimentally verified and the rectenna further tested in a high-efficency RF harvester with MPPT.
The paper, which continues and expands the work presented in [32], is organized as follows. Section 2 presents the rectenna and the derived Thévenin equivalent. Section 3 describes the MPPT and the sensor node. Section 4 presents the materials and methods and Section 5 provides the experimental results and discussions. Finally, Section 6 concludes the work. Complementarily, two appendices are included. Appendix A presents an analytical development useful for the derivation of the Thévenin equivalent and Appendix B shows simulations of the rectenna with and without the matching network. 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 m and an inductor L m ), a half-wave rectifier, and an output filtering capacitor (C o ). The antenna is modelled by a sinusoidal voltage source v a of amplitude V ap and frequency f o with a series radiation resistance R a . On the other hand, v in , Z in, and P in are, respectively, the sinusoidal voltage, impedance, and power at the input of the rectifier, i d is the diode current, and V o , I o , and P o are, respectively, the DC voltage, current, and power at the rectenna output. An equivalent resistance R o is defined as V o /I o . The rectenna is composed of an antenna, an impedance matching network, and a rectifier. As the available power at the antenna decreases so does the generated voltage. Whenever this voltage is not high enough to properly bias the diodes of the rectifier, power efficiency severely decreases. Several techniques have been proposed to increase the efficiency at low power levels. One of them consists of using an L-matching network for boosting the voltage at the rectifier input [1,3,5,[10][11][12][13][14][15][16][17][18][19][20][21]. As for the MPPT, several works propose its use with rectennas using either commercial chips [6,7] or ad hoc designs [22][23][24][25].

Rectenna and Its Thévenin Model
With the aim of gaining more insight into the performance of the rectennas, different analytical models are proposed. However, the derived expressions, which in some cases seek to model the rectenna output as an equivalent Thévenin circuit, are rather complex and may require additional simulations or extensive calculations, which hide the influence of the different parameters of the rectenna on its performance [12,18,[26][27][28]. At the other extreme, the Thévenin parameters are sometimes inferred by experimental characterization [25,[29][30][31][32]. However, in these cases no relationship with the rectenna parameters is established.
Taking into account the previous limitations, this paper proposes a compact Thévenin model for the rectenna with the benefit of achieving manageable expressions of the Thévenin parameters as a function of the parameters of the rectenna so as to gain insight into its operation. In particular, the rectenna under study consists of an L-matching network and of a half-wave rectifier. The proposed model is then experimentally verified and the rectenna further tested in a high-efficency RF harvester with MPPT.
The paper, which continues and expands the work presented in [32], is organized as follows. Section 2 presents the rectenna and the derived Thévenin equivalent. Section 3 describes the MPPT and the sensor node. Section 4 presents the materials and methods and Section 5 provides the experimental results and discussions. Finally, Section 6 concludes the work. Complementarily, two appendices are included. Appendix A presents an analytical development useful for the derivation of the Thévenin equivalent and Appendix B shows simulations of the rectenna with and without the matching network. Figure 2 shows the schematic circuit of the rectenna under study [33], which includes a highpass L-matching network (composed of a capacitor Cm and an inductor Lm), a half-wave rectifier, and an output filtering capacitor (Co). The antenna is modelled by a sinusoidal voltage source va of amplitude Vap and frequency fo with a series radiation resistance Ra. On the other hand, vin, Zin, and Pin are, respectively, the sinusoidal voltage, impedance, and power at the input of the rectifier, id is the diode current, and Vo, Io, and Po are, respectively, the DC voltage, current, and power at the rectenna output. An equivalent resistance Ro is defined as Vo/Io. The amplitude Vap is given by [12] as follows: The amplitude V ap is given by [12] as follows: where P av is the available power at the antenna. The matching network, at matching conditions, that is, Z m = R a (where Z m is defined in Figure 2), boosts the voltage at the input of the rectifier by a voltage gain, G t , given by [33] as follows:

Rectenna and Its Thévenin Model
where V inp is the voltage amplitude of v in and Q is the circuit quality factor given by: where ω o = 2πf o . On the other hand, the value of L m must comply: where C 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 Thévenin 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 p -C p ) of the coil, diode, and layout of the circuit. These parasitic elements are derived in Appendix A, where R p models the losses of the coil and diode and C p includes the parasitic capacitance of the diode, coil, and layout. where Pav is the available power at the antenna. The matching network, at matching conditions, that is, Zm = Ra (where Zm is defined in Figure 2), boosts the voltage at the input of the rectifier by a voltage gain, Gt, given by [33] as follows: where Vinp is the voltage amplitude of vin and Q is the circuit quality factor given by: where ωo = 2πfo. On the other hand, the value of Lm must comply: where Cp 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 Thévenin model is provided here. First, the left-hand equivalent circuit of Figure 3 accounts for the antenna, the matching network, and the parasitic elements (Rp-Cp) of the coil, diode, and layout of the circuit. These parasitic elements are derived in Appendix A, where Rp models the losses of the coil and diode and Cp includes the parasitic capacitance of the diode, coil, and layout. Analyzing the left-hand circuit of Figure 3 at fo, we can achieve the Thévenin equivalent represented by the right-hand circuit of Figure 3, where: Next, the Thévenin 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 Req and Ceq).
The diode is forward biased when vin, assumed sinusoidal, surpasses Vo. As a result, id is pulsed and is composed of the fundamental frequency (fo) as well as its harmonics and a DC component (Io). Impedance Zs (defined in the circuit) is zero at DC (due to the coil Lm) and is equal to Req at fo since Lm and Ceq form a parallel resonant circuit presenting an infinite impedance. On the other hand, at the harmonics of fo we have Zs<<Req (due to Ceq) whenever Q is high enough. Therefore, only the current at fo (iin) originates a voltage drop and vin will be sinusoidal, as assumed before. Thus, apart from boosting the voltage, the matching network ideally acts as an input band-pass filter that prevents any of the DC current and harmonics to flow through the antenna resistance and dissipate power. This Analyzing the left-hand circuit of Figure 3 at f o , we can achieve the Thévenin equivalent represented by the right-hand circuit of Figure 3, where: Next, the Thévenin 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 eq and C eq ). The diode is forward biased when v in , assumed sinusoidal, surpasses V o . As a result, i d is pulsed and is composed of the fundamental frequency (f o ) as well as its harmonics and a DC component (I o ). Impedance Z s (defined in the circuit) is zero at DC (due to the coil L m ) and is equal to R eq at f o since L m and C eq form a parallel resonant circuit presenting an infinite impedance. On the other hand, at the harmonics of f o we have Z s << R eq (due to C eq ) whenever Q is high enough. Therefore, only the current at f o (i in ) originates a voltage drop and v in will be sinusoidal, as assumed before. Thus, apart from boosting the voltage, the matching network ideally acts as an input band-pass filter that prevents any of the DC current and harmonics to flow through the antenna resistance and dissipate power. This leads to an ideal rectenna efficiency of 100%, assuming no losses in the circuit components and in the diode [34]. Contrariwise, when no matching network is present, maximum rectenna efficiency decreases to 46%, due to the additional losses at R a originated by the current harmonics generated by the diode pulsed current, as demonstrated in [35]. Appendix B confirms these results via simulations. Finally, the value of C o has to be much higher than the diode junction capacitance (C j ), as explained in Appendix A, to keep V o nearly constant, that is, with a low voltage ripple (∆V o ). This second condition leads to: leads to an ideal rectenna efficiency of 100%, assuming no losses in the circuit components and in the diode [34]. Contrariwise, when no matching network is present, maximum rectenna efficiency decreases to 46%, due to the additional losses at Ra originated by the current harmonics generated by the diode pulsed current, as demonstrated in [35]. Appendix B confirms these results via simulations. Finally, the value of Co has to be much higher than the diode junction capacitance (Cj), as explained in Appendix A, to keep Vo nearly constant, that is, with a low voltage ripple (ΔVo). This second condition leads to: assuming a fixed forward voltage drop Vγ at the diode and: where Veqp and Iinp are the amplitudes of veq and iin, respectively. Substituting (8) into (7) provides: On the other hand, for the right-hand circuit we have: Then, by equating powers, we obtain: and Pd is the average power dissipated across the diode. Thus, replacing (12) into (11) and using (7), we arrive at the following: Finally, using (13) in (9) and equating (9) and (10), we obtain the parameters of the Thévenin model: The left-hand circuit of Figure 4 leads to the equivalent Thévenin circuit of the rectenna, represented by the right-hand circuit of Figure 4, by linking their output voltage-current relationship. For the left-hand circuit, we have: assuming a fixed forward voltage drop V γ at the diode and: where V eqp and I inp are the amplitudes of v eq and i in , respectively. Substituting (8) into (7) provides: On the other hand, for the right-hand circuit we have: Then, by equating powers, we obtain: where and P d is the average power dissipated across the diode. Thus, replacing (12) into (11) and using (7), we arrive at the following: Finally, using (13) in (9) and equating (9) and (10), we obtain the parameters of the Thévenin model: where V eqp can be derived from v eq in (5), using V ap instead of v a , resulting in: Then, using (15) and R eq of (5) in (14), we have: Therefore, from (16), with an increasing P av and thus V ap , V oc increases whereas R T holds constant. Next, from (10), we can express I o as: and the output power P o over a load resistor R o can be simply calculated as: being the power efficiency of the rectenna as: Applying the maximum power transfer theorem, maximum power is extracted from the rectenna for V o = 0.5V oc , which is known as the maximum power point (MPP) voltage (V MPP ). From (19), the resulting efficiency is as: Thus, using (16) in (20), we arrive at: As can be seen from (21), η rect,max increases with increasing P av . Obviously, with no losses (R p = ∞ and V γ = 0) η rect,max = 1 is obtained. On the other hand, the dependence of η rect,max on G t is rather more complex. In [33], an optimum value of G t was derived arising from the trade-off between the losses introduced by the coil and that due to the voltage drop of the diode. This optimum gain leads, from (16), to a particular value of R T .

MPPT and Sensor Node
In general, a sensor node directly connected to the output of the rectenna will not provide an equivalent resistance R o = R T , at which the rectenna output operates at the MPP. Thus, an impedance matching stage (in addition to the matching network of the rectenna) is needed between the rectenna output and the sensor node, which can be implemented by a DC/DC converter. An MPPT, which consists of a DC/DC converter plus a tracking algorithm, can be used for automatically searching and settling that optimum value of R o , which also corresponds to V o = V MPP . Thus, the overall power efficiency of the RF harvester will be given by the following: where η MPPT is the efficiency of the MPPT and η rect = η rect,max since the MPPT biases the rectenna at the MPP. In this work, the fractional open circuit voltage (FOCV) MPPT technique is used, since it leads to simple and power efficient implementations. In this technique, the open circuit voltage (V oc ) of the energy transducer (a rectenna here) is first measured and a fraction k of V oc is used to operate at V MPP and thus achieve η rect,max . Taking into account the analysis in Section 2, a proper choice here is k = 0.5 (V o = V MPP = 0.5V oc ). Figure 5 presents the block diagram for the implementation of the FOCV MPPT technique, where C L , C REF , and C load are capacitors, R oc1 and R oc2 are resistors, S 1 and S 2 are switches, V load is the output voltage used to power the sensor node, and P load is the power transferred to the sensor node. The operation is the following. First, S 1 closes and S 2 opens (sampling period). For high values of R oc1 and R oc2 , the output of the rectenna can be considered as open and thus V o = V oc . The voltage divider formed by R oc1 and R oc2 fixes V MPP = kV oc , being k = 0.5 here (i.e., R oc1 = R oc2 ). The input capacitor (C L ) momentarily stores the incoming harvested energy. Secondly, S 1 opens and S 2 closes (regulation period). Thus, V MPP holds constant thanks to C REF , and the DC/DC converter settles V o around V MPP and transfers the harvested energy by the rectenna to the sensor node. (Vo = VMPP = 0.5Voc). Figure 5 presents the block diagram for the implementation of the FOCV MPPT technique, where CL, CREF, and Cload are capacitors, Roc1 and Roc2 are resistors, S1 and S2 are switches, Vload is the output voltage used to power the sensor node, and Pload is the power transferred to the sensor node. The operation is the following. First, S1 closes and S2 opens (sampling period). For high values of Roc1 and Roc2, the output of the rectenna can be considered as open and thus Vo = Voc. The voltage divider formed by Roc1 and Roc2 fixes VMPP = kVoc, being k = 0.5 here (i.e., ROC1 = ROC2). The input capacitor (CL) momentarily stores the incoming harvested energy. Secondly, S1 opens and S2 closes (regulation period). Thus, VMPP holds constant thanks to CREF, and the DC/DC converter settles Vo around VMPP and transfers the harvested energy by the rectenna to the sensor node. In order to periodically update Voc (i.e., a change in Pav changes Voc), the described sequence is periodically repeated, with the sampling period being much shorter than the regulation period. In this way, Vo will settle most of time at VMPP. To increase the efficiency at light loads, the DC/DC converter uses special control techniques such as pulse frequency modulation (PFM) or burst-mode [36].
Taking into account (22), Pload can be related with Pav as follows: The value of Pload and thus of Pav must be enough, in average, to power the sensor node, which usually includes a rechargeable storage unit. This unit accounts for the variability of Pav, gathering or providing energy whenever Pav is higher or lower than required. Storage units can be supercapacitors, batteries, or a combination of both [37]. On the other hand, the required value of Pload and thus of Pav can be reduced by operating the sensor node in sleep mode most of the time and minimizing its active time.

Materials and Methods
The rectenna shown in Figure 2 was implemented on a printed circuit board with Rogers substrate and with the following components: Cm = 0.5 pF (AVX, Fountain Inn, SC, USA), Lm = 27 nH (0603CS model, Coilcraft, Cary, IL, USA), Co = 1 nF, and a Schottky HSMS-2850 diode (Avago Technologies, San Jose, CA, USA) [33]. The selected value of Co comfortably accomplished Error! Reference source not found., in order to theoretically have a small ripple (below 1 mV) with the In order to periodically update V oc (i.e., a change in P av changes V oc ), the described sequence is periodically repeated, with the sampling period being much shorter than the regulation period. In this way, V o will settle most of time at V MPP . To increase the efficiency at light loads, the DC/DC converter uses special control techniques such as pulse frequency modulation (PFM) or burst-mode [36].
Taking into account (22), P load can be related with P av as follows: The value of P load and thus of P av must be enough, in average, to power the sensor node, which usually includes a rechargeable storage unit. This unit accounts for the variability of P av , gathering or providing energy whenever P av is higher or lower than required. Storage units can be supercapacitors, batteries, or a combination of both [37]. On the other hand, the required value of P load and thus of P av can be reduced by operating the sensor node in sleep mode most of the time and minimizing its active time.

Materials and Methods
The rectenna shown in Figure 2 was implemented on a printed circuit board with Rogers substrate and with the following components: C m = 0.5 pF (AVX, Fountain Inn, SC, USA), L m = 27 nH (0603CS model, Coilcraft, Cary, IL, USA), C o = 1 nF, and a Schottky HSMS-2850 diode (Avago Technologies, San Jose, CA, USA) [33]. The selected value of C o comfortably accomplished, in order to theoretically have a small ripple (below 1 mV) with the values of I o shown later in Section 5, as well as the condition stated in Appendix A (C o >> C j ). The circuit of Figure 2 was used for the rectenna characterization, where an RF generator (Agilent E4433B, Santa Clara, CA, USA) was connected at the input instead of the antenna and a Source Measure Unit (SMU, Agilent B2901A, Santa Clara, CA, USA) configured as a voltage sink (quadrant IV) at the output. The generator was set at a tuned optimal frequency of 808 MHz and at different values of P av (−30 dBm, −20 dBm, and −10 dBm). For each value of P av , the SMU was set at different values of V o while measuring P o . Then, η rect was obtained as P o /P av .
As for the FOCV MPPT, a BQ25504 chip (Texas Instruments, Dallas, TX, USA) was used, and in particular an evaluation board provided by the manufacturer. The chip contains a boost converter with PFM control and the board includes, in reference to Figure 5, C L = 4.8 µF (combination of two ceramic capacitors of 4.7 µF and 100 nF placed in parallel), C REF = 10 nF, and C load = 104.8 µF (combination of three ceramic capacitors of 100 µF, 4.7 µF, and 100 nF placed in parallel). The default values of R oc1 and R oc2 were modified to 10 MΩ in order to fix k = 0.5 (the default value is set to 0.78). The sampling and regulation periods are prefixed by the chip to 256 ms and 16 s, respectively. Then, the efficiency of the whole RF harvester (rectenna plus MPPT) was characterized by using the RF generator at the input of the rectenna and the SMU set at 3 V at the output of the MPPT (V load ). The RF generator was set at different values of P av , from −20 dBm to −5 dBm in steps of 1 dBm, and for each value the SMU measured the output power P load . Then, from (23), η T was estimated.
For demonstration purposes, the RF harvester including the MPPT was also employed to power a sensor node intended to upgrade a mechanical gas meter to a smart device [38]. For these tests, the node was programmed to stay in a standby mode, consuming 1.4 µA. The input power (P av ) was set to keep the voltage supply of the sensor node (V load ) at 3 V, thus P load = 4.2 µW. As for the RF harvester input, two configurations were used: (1) an RF generator and (2) a receiving monopole antenna. In the second case, another identical monopole antenna was connected to a nearby RF generator, jointly acting as a wireless energy transmitter. The antennas showed an insertion loss higher than 10 dB at 808 MHz. Figure 6 shows pictures of both setups. For demonstration purposes, the RF harvester including the MPPT was also employed to power a sensor node intended to upgrade a mechanical gas meter to a smart device [38]. For these tests, the node was programmed to stay in a standby mode, consuming 1.4 µA. The input power (Pav) was set to keep the voltage supply of the sensor node (Vload) at 3 V, thus Pload = 4.2 µW. As for the RF harvester input, two configurations were used: (1) an RF generator and (2) a receiving monopole antenna. In the second case, another identical monopole antenna was connected to a nearby RF generator, jointly acting as a wireless energy transmitter. The antennas showed an insertion loss higher than 10 dB at 808 MHz. Figure 6 shows pictures of both setups.

Experimental Results and Discussion
As for the proposed rectenna, Figure 7 shows the measured values (in dots) of Io (blue circles) and ηrect (red squares) as a function of Vo at different values of Pav. A least-squares fitting of (17) to the experimental data of Io was performed (blue continuous line) to obtain the Thévenin parameters (Voc and RT) at each power level, which are shown in Table 1. Calculated values of Vap, from (1), and of Veqp, from (15), are also included in Table 1. This fitting differs from that performed in [32], where the efficiency data (ηrect) were used instead, which leads to slight differences in the Thévenin parameters.

Experimental Results and Discussion
As for the proposed rectenna, Figure 7 shows the measured values (in dots) of I o (blue circles) and η rect (red squares) as a function of V o at different values of P av . A least-squares fitting of (17) to the experimental data of I o was performed (blue continuous line) to obtain the Thévenin parameters (V oc and R T ) at each power level, which are shown in Table 1. Calculated values of V ap , from (1), and of V eqp , from (15), are also included in Table 1. This fitting differs from that performed in [32], where the efficiency data (η rect ) were used instead, which leads to slight differences in the Thévenin parameters. The new fitting procedure was considered more convenient as both V oc and R T can be more easily inferred from the fitting curve. As can be seen, the fitting curves match well the experimental data, and more at the highest power of −10 dBm, which confirms that the rectenna can be well approximated by a Thévenin equivalent circuit. Then, V oc and R T were used to obtain η rect using (19), and the resulting curves are also represented in Figure 7 (red continuous line). The match with the experimental data is good, and again better at −10 dBm. obtained, which is within the range of values found in Table 1. The inferred values of RT moderately change with Pav due to the relative low value of Q, which limits the accuracy of the rectenna model proposed in Section 2. However, a higher value of Q, which could be obtained using a lower value of Cm and appropriately readjusting Lm, does not lead to the optimum gain Gt [33], thus decreasing the power efficiency. On the other hand, Voc in Table 1 increases with increasing Pav and thus Vap, which agrees with (16). The values of Voc can be estimated in advance, when necessary, from (16) by calculating Veqp from (15), shown in Table 1, and inferring a value of Vγ from the manufacturer data or from simulations.  From the measured data of ηrect (red squares in Figure 7), Table 2 shows the achieved ηrect,max and its corresponding voltage (VMPP,exp), as well as the experimental open circuit voltage (Voc,exp) of the rectenna. In Figure 7, ηrect,max, VMPP,exp, and Voc,exp are also marked for Pav = −20 dBm. As can be seen, ηrect,max increases with increasing Pav, ranging from 13.6% at −30 dBm to 60.3% at −10 dBm, which agrees with (21). The values of ηrect,max can be estimated in advance, when necessary, from (21) and inferring a value of Vγ from the manufacturer data or from simulations. One particular case is the upper limit, which would be achieved for Pav→∞ (or Vγ→0), in our case 74%. The resulting efficiencies (ηrect,max) are among the highest published in the literature for similar designs [33]. On the other hand, Voc from Table 1 nearly matches Voc,exp. Finally, VMPP,exp equates or nearly matches 0.5 Voc,exp, the regulated voltage at the input of the MPPT. Thus, the proposed and implemented MPPT will be able to extract the maximum power (or nearly) from the rectenna.   With C m = 0.5 pF, R a = 50 Ω, and f o = 808 MHz, Q = 7.88 results from (3), and G t = 3.97 from (2). Then, from (16) and assuming the value of R p = 9.21 kΩ derived in Appendix A, R T = 4.7 kΩ is obtained, which is within the range of values found in Table 1. The inferred values of R T moderately change with P av due to the relative low value of Q, which limits the accuracy of the rectenna model proposed in Section 2. However, a higher value of Q, which could be obtained using a lower value of C m and appropriately readjusting L m , does not lead to the optimum gain G t [33], thus decreasing the power efficiency. On the other hand, V oc in Table 1 increases with increasing P av and thus V ap , which agrees with (16). The values of V oc can be estimated in advance, when necessary, from (16) by calculating V eqp from (15), shown in Table 1, and inferring a value of V γ from the manufacturer data or from simulations.
From the measured data of η rect (red squares in Figure 7), Table 2 shows the achieved η rect,max and its corresponding voltage (V MPP,exp ), as well as the experimental open circuit voltage (V oc,exp ) of the rectenna. In Figure 7, η rect,max , V MPP,exp , and V oc,exp are also marked for P av = −20 dBm. As can be seen, η rect,max increases with increasing P av , ranging from 13.6% at −30 dBm to 60.3% at −10 dBm, which agrees with (21). The values of η rect,max can be estimated in advance, when necessary, from (21) and inferring a value of V γ from the manufacturer data or from simulations. One particular case is the upper limit, which would be achieved for P av →∞ (or V γ →0), in our case 74%. The resulting efficiencies (η rect,max ) are among the highest published in the literature for similar designs [33]. On the other hand, V oc from Table 1 nearly matches V oc,exp . Finally, V MPP,exp equates or nearly matches 0.5 V oc,exp , the regulated voltage at the input of the MPPT. Thus, the proposed and implemented MPPT will be able to extract the maximum power (or nearly) from the rectenna. As for the whole RF harvester (rectenna plus the MPPT), Figure 8 shows the experimental values of η T versus P av . At −20 dBm, η rect,max = 39.3% (Table 2) but η T = 6.5%, resulting, from (22), in η MPPT = 16.5%. This low value of η MPPT is due to both a low input voltage value (140 mV = 0.5 V oc,exp ) and a low value of P o (3.9 µW = η rect,max P av ). Contrariwise, at −10 dBm, η rect,max = 60.3% and η T = 48.6%, resulting in η MPPT = 80.6%, which agrees with the data from the BQ25504 chip's datasheet. At higher values of P av (−5 dBm), η T reached a value of 55.6%. Compared to [6], where a similar chip for the MPPT was used, η T is quite higher. 6%. Compared to [6], where a similar chip for the MPPT was used, ηT is quite higher.  When powering the sensor node, the required value of Pav was −17.6 dBm. This value fits well with (23), considering the corresponding efficiency in Figure 8 (≈24%). This performance was also tested with the antennas at a distance of 0.5 and 1 m. The power output of the remote RF generator was tuned at appropriate values so as to operate the node, resulting in 8.0 and 13.2 dBm, respectively. These values accounted for the respective link budgets.

Conclusions
This work proposed a compact Thévenin model for a rectenna and its application for designing a high-efficiency RF harvester. The rectenna under study consists of an L-matching network and a When powering the sensor node, the required value of P av was −17.6 dBm. This value fits well with (23), considering the corresponding efficiency in Figure 8 (≈24%). This performance was also tested with the antennas at a distance of 0.5 and 1 m. The power output of the remote RF generator was tuned at appropriate values so as to operate the node, resulting in 8.0 and 13.2 dBm, respectively. These values accounted for the respective link budgets.
The use of a matching network leads to sinusoidal voltage and current waveforms at f o , as mentioned in Section 2. Here, the equivalent circuit model at f o is derived from the manufacturer models of the diode (https://docs.broadcom.com/docs/AV02-1377EN) and inductor (https://www. coilcraft.com/pdfs/spice_0603cs.pdf) used for the implemented rectenna and reported in Section 4. This model will be used in the left-hand circuit of Figure 3.
First, the left-hand circuit in Figure A1 shows the equivalent linear circuit model of the diode, connected between node A and the V o node at Figure 2, together with the output capacitor C o , where R s is the parasitic series resistance, C j is the parasitic junction capacitance, and R j is the junction resistance. R j depends inversely on the bias current and makes only sense for modelling small current variations around a bias current. In our case, the diode current is pulsed and thus the inclusion of R j is not appropriate. Instead, a constant voltage drop will be assumed in Section 2 for the analysis of the circuit of Figure 4.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Parallel Circuit Model of the Inductor and Diode
The use of a matching network leads to sinusoidal voltage and current waveforms at fo, as mentioned in Section 2. Here, the equivalent circuit model at fo is derived from the manufacturer models of the diode (https://docs.broadcom.com/docs/AV02-1377EN) and inductor (https://www.coilcraft.com/pdfs/spice_0603cs.pdf) used for the implemented rectenna and reported in Section 4. This model will be used in the left-hand circuit of Figure 3.
First, the left-hand circuit in Figure A1 shows the equivalent linear circuit model of the diode, connected between node A and the Vo node at Figure 2, together with the output capacitor Co, where Rs is the parasitic series resistance, Cj is the parasitic junction capacitance, and Rj is the junction resistance. Rj depends inversely on the bias current and makes only sense for modelling small current variations around a bias current. In our case, the diode current is pulsed and thus the inclusion of Rj is not appropriate. Instead, a constant voltage drop will be assumed in Section 2 for the analysis of the circuit of Figure 4.

Rs Cj
Cjp Rsp Figure A1. Equivalent linear circuit model of the HSMS-2850 diode together with (left) the output capacitor Co and (right) its parallel equivalent circuit.
Using the series-to-parallel equivalent circuit transformation for the left-hand circuit of Figure A1 without Rj and considering Co >> Cj, the right-hand circuit results, where: Using the series-to-parallel equivalent circuit transformation for the left-hand circuit of Figure A1 without R j and considering C o >> C j , the right-hand circuit results, where: From the diode datasheet, R s = 25 Ω and C jo = 0.18 pF (C jo is C j at zero bias and will be the assumed value for C j hereafter). With f o = 808 MHz, R sp = 47.9 kΩ results from (A1) and C jp ≈ C j = 180 fF results from (A2). As for the inductor, connected between node A and ground at Figure 2, the left-hand circuit of Figure A2 shows the manufacturer model, where R v = k f o . Using the series-to-parallel circuit transformation, the circuit in the middle is obtained, where: From the coil datasheet, R 1 = 17 Ω, R 2 = 30 mΩ, C 1 = 49 fF, L m = 27 nH, and k = 5.75 × 10 −5 .
At 808 MHz, R v = 1.63 Ω and from (A3) to (A6), L mp ≈ L m = 27 nH, R vp = 11.5 kΩ, R 1p = 950 kΩ, and C 1p ≈ C 1 = 49 fF. Then, neglecting R 2 , since it is very small, the right-hand circuit of Figure A2 is obtained, where R Lp = R vp R 1p = 11.4 kΩ. Joining the right-hand circuits in Figures A1 and A2, and considering the parasitic capacitance from node A to ground arising from the layout (Clay), the circuit in Figure A3 is obtained, where: Joining the right-hand circuits in Figures A1 and A2, and considering the parasitic capacitance from node A to ground arising from the layout (C lay ), the circuit in Figure A3 is obtained, where: Joining the right-hand circuits in Figures A1 and A2, and considering the parasitic capacitance from node A to ground arising from the layout (Clay), the circuit in Figure A3  This equivalent circuit is used in the left-hand circuit of Figure 3.

Appendix B. Simulations of the Rectenna Efficiency with and without an L-Matching Network
In order to highlight the benefits of using a matching network, we report simulation results of the rectenna efficiency with and without an L-matching network. For the simulations, the ADS software (Version 2017, Keysight, Santa Rosa, CA, USA) was used. The simulated circuit with the matching network was that of Figure 2 with a resistor Ro connected at its output, in parallel with Co. Ideal components were used for the matching network with the values reported in Section 4. The diode was modelled without parasitic elements (null series resistance and junction capacitance) but with a saturation current of 3 µA (that corresponding to the HSMS-2850 diode used for the implemented rectenna). For the RF source, a frequency of 808 MHz (that used for the experiments) was used with Pav ranging from −10 to 50 dBm in steps of 20 dB. For the circuit without the matching This equivalent circuit is used in the left-hand circuit of Figure 3.

Appendix B. Simulations of the Rectenna Efficiency with and without an L-Matching Network
In order to highlight the benefits of using a matching network, we report simulation results of the rectenna efficiency with and without an L-matching network. For the simulations, the ADS software (Version 2017, Keysight, Santa Rosa, CA, USA) was used. The simulated circuit with the matching network was that of Figure 2 with a resistor R o connected at its output, in parallel with C o . Ideal components were used for the matching network with the values reported in Section 4. The diode was modelled without parasitic elements (null series resistance and junction capacitance) but with a saturation current of 3 µA (that corresponding to the HSMS-2850 diode used for the implemented rectenna). For the RF source, a frequency of 808 MHz (that used for the experiments) was used with P av ranging from −10 to 50 dBm in steps of 20 dB. For the circuit without the matching network, the right terminal of R a was directly connected to the diode anode (node A). For both circuits, a harmonic balance analysis was performed with P av and R o as sweeping parameters. Figure A4 shows the simulation results of the rectenna efficiency (η rect = P o /P av ) plotted against R o for different values of P av . The left graph shows the results for the circuit with the L-matching network. As can be seen, η rect steeply increases for increasing values of P av , achieving around 99% at P av = 50 dBm. At −10 dBm, efficiency is higher than the corresponding results of Figure 7 because ideal components were used for the simulation. On the other hand, the right graph shows the results for the circuit without the L-matching network. As can be seen, η rect steeply increases for increasing values of P av but now reaches a maximum value around 46%, as predicted theoretically in [35], due to the additional losses at R a originated by the current harmonics generated by the diode pulsed current. At −10 dBm, efficiency is lower than 4%. Therefore, the use of the matching network allows a notable increase in the rectenna efficiency, because it provides voltage gain and prevents any of the DC current and harmonics to flow through the antenna resistance. network, the right terminal of Ra was directly connected to the diode anode (node A). For both circuits, a harmonic balance analysis was performed with Pav and Ro as sweeping parameters. Figure A4 shows the simulation results of the rectenna efficiency (ηrect = Po/Pav) plotted against Ro for different values of Pav. The left graph shows the results for the circuit with the L-matching network. As can be seen, ηrect steeply increases for increasing values of Pav, achieving around 99% at Pav = 50 dBm. At −10 dBm, efficiency is higher than the corresponding results of Figure 7 because ideal components were used for the simulation. On the other hand, the right graph shows the results for the circuit without the L-matching network. As can be seen, ηrect steeply increases for increasing values of Pav but now reaches a maximum value around 46%, as predicted theoretically in [35], due to the additional losses at Ra originated by the current harmonics generated by the diode pulsed current. At −10 dBm, efficiency is lower than 4%. Therefore, the use of the matching network allows a notable increase in the rectenna efficiency, because it provides voltage gain and prevents any of the DC current and harmonics to flow through the antenna resistance.