**1. Introduction**

The current decade has witnessed a rapid increase in the number of smart wireless devices, their influence over social and economic development has also been growing. Wireless devices have become increasingly compact, it has become much easier to integrate them into various environments, which in turn promotes development of the Internet

**Citation:** Eidaks, J.; Kusnins, R.; Babajans, R.; Cirjulina, D.; Semenjako, J.; Litvinenko, A. Fast and Accurate Approach to RF-DC Conversion Efficiency Estimation for Multi-Tone Signals. *Sensors* **2022**, *22*, 787. https://doi.org/10.3390/s22030787

Academic Editor: Zihuai Lin

Received: 6 December 2021 Accepted: 17 January 2022 Published: 20 January 2022

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of Things (IoT) and the underlying wireless sensor networks (WSNs). Smart cities [1], agriculture [2], and medicine [3] are just some areas where WSNs are employed to control smart environments via the IoT. The increasing use of WSNs has caused exponential growth in the number of autonomous individual sensor nodes (SN), which in turn poses poweringrelated challenges for the sensor networks. Battery power is the most common source for powering autonomous devices. Along with the increase of the number of autonomous devices used in the network, more time and attention are required to monitor the power level of every single device; the batteries should also be changed when necessary. However, devices situated in confined areas cannot be easily maintained, which may compromise the integrity of the WSN. Radio frequency (RF) wireless power transfer (WPT) offers a solution for preserving the integrity of the WSN during operation, providing control over the amount of energy each SN receives to perform its duties. The key benefits of using WPT for powering autonomous devices consist of a reduced need for batteries, which in its turn mitigates inconveniences related to powering of these devices, and the opportunity to maintain closer control over device energy levels. The use of RF allows transferring power to secluded SNs from a sufficient distance, it also allows for ambient energy harvesting.

The rectenna (receiving antenna paired with an RF–DC converter) with high power conversion efficiency (PCE) is the most important element of an efficient RF WPT. High PCE increases the amount of useful energy the autonomous device receives, which is particularly relevant in case of relatively long distances between the power transmitter and receiver, which cause reduction of the amount of received RF power. Over the years, many studies proposed various rectennas for WPT. Table 1 lists the key properties of the proposed rectennas ordered by frequencies and input powers. The table also includes the results of this study for comparison. The results will be further elaborated upon in this manuscript.


**Table 1.** Comparison of the experimentally studied rectennas.

1All instances of "single-tone" refer to an unmodulated carrier. 2 All instances of "multi-tone" refer to a sum of several subcarriers. 3 Relative permittivity εr = 3.4, the dielectric loss tangent tanδ = 0.0015. 4 Relative permittivity εr = 2.55, the dielectric loss tangent tanδ = 0.0018.

As seen from Table 1, different rectenna configurations have been proposed and studied. Rectennas that show PCE above 70% [5–9,11–13] use high RF input power (>15 dBm), which greatly limits the range of effective distances between the power transmitter and

the secluded SN if this received input power is to be achieved. Increase of the distance calls for increase of transmission power to maintain the required input power and PCE, which potentially exceeds power restrictions for the given frequencies. Studies [5,8] reached efficiencies over 90%. This can only be achieved with the receiver optimized for such high input powers (the RF–DC conversion is done using GaAs diodes), which is not optimal for practical applications in powering secluded SN using the given frequencies. The use of input power in the range around 0 dBm implies application of both SN and low-power technologies, such as RFID and E-ink [20,21]. This range of input RF power was less frequently addressed in literature than high and low (< −15 dBm) power ranges. Comparing rectennas in terms of frequencies, Table 1 demonstrates that rectennas were mainly developed for 2.45 GHz ISM frequency band. The use of high frequency also limits the effective distance between the transmitter and the SN. Sub-GHz ranges, such as 433 MHz (ISM) and 860 MHz (GSM-850), allow transferring of power to greater distances. Regarding the waveform of the power-carrying signal, rectennas listed in Table 1 mainly use a single-tone signal (an unmodulated carrier). However, studies, such as [11,15,19,22] and [23–25], reported an increase in PCE when multi-tone (formed by a sum of several subcarriers) power-carrying signals are used. The topology of the RF–DC circuit is another crucial parameter of rectenna design. The most common RF–DC topologies are presented in Figure 1: one-diode-based (half-wave rectifier), two-diode-based (voltage doubler), and four-diode-based (diode bridge rectifier) topologies. These topologies with slight variations were used in the studies listed in Table 1. Analyzing information in Table 1, it may be concluded that rectenna based on a voltage doubler RF–DC converter working at a sub-GHz frequency and multi-tone power-carrying signals proved to be the most well-balanced solution in terms of cost and efficiency for RF WPT applications targeted at powering SN and low power electronics.

**Figure 1.** One-diode-based rectifier (**a**), two-diode-based rectifier (**b**), diode bridge rectifier (**c**).

The considered studies mainly focused on enhancing performance of rectennas with experimental validation of results, aiming at development of reliable theoretical models for the WPT and RF–DC converters. Numerous theoretical models exist in the field of AC–DC [26] and DC–DC [27,28] converters, several modeling approaches have also been proposed over the years for RF–DC circuits. Development of an accurate computer model and its use in simulations is a feasible alternative to experimental studies of RF–DC power converters. In contrast to experiments, simulation is a more convenient and cost-effective solution, as it does not require fabrication of prototypes, especially when circuit design optimization is needed.

Despite recent advances in the field, the analysis of non-linear circuits not amenable to linearization is usually very time-consuming. This issue becomes even more pronounced when complex input waveforms are employed. Although transient analysis (TA) is a robust circuit analysis method [29], it is not suitable for analyzing RF–DC converters because long simulation times are required due to the presence of transients [30]. Furthermore, in case of narrow-band signals with periodic envelopes, the time step must be much smaller compared to the period of the carrier wave that leads to a very large number of iterations. Though some attempts have been made to speed-up the TA [31], the aforementioned restriction on the time step size considerably limits the performance of the method, as will be shown in this paper (see Section 2.5). Another widely used non-linear circuit analysis

method is the Volterra series method [32]. However, this method is mainly applied to weakly non-linear circuits, since for circuits with highly pronounced non-linearity the convergence is very slow. The harmonic balance (HB) method was initially proposed in [33] to solve problems in mechanical engineering, it has subsequently been adapted to treat non-linear circuits under sinusoidal excitation [34]. The issue of transients does not pose problems within HB, as this method allows computing the steady state response directly, involving the solving of a system of non-linear equations [35]. The system of equations can be reduced by partitioning the original circuit into linear and non-linear parts [36]. The resulting non-linear equations can be solved by means of Newton's method (NM) [37], or iteration relaxation method (IRM) [38,39], among others. The evaluation of the Jacobian matrix can be significantly accelerated using FFT algorithms [40] and the continuation method was developed to ensure convergence at high input powers [41]. The HB has also been extended to handle multi-tone input signals [42,43]. However, in such cases the Jacobian matrix is significantly larger, resulting in the high computational burden. This issue can be mitigated by exploiting useful properties of multidimensional FFT algorithms [44]. Over the last several decades, the method has found use in a number of applications, including the analysis of the behavior of both autonomous and nonautonomous oscillators [45–47]. Additionally, in an effort to reduce the simulation time, several extensions and modifications of the HB, as well as its multidimensional extensions, have been proposed, such as the hierarchical harmonic balance method [48], several parallel versions of the HB [49,50], the multi-level frequency decomposition-based HB [51], and the HB using the graph sparsification [52].

Although the methods mentioned above are accurate, they are highly computationally intensive. As a result, a number of approximate closed-form expression-based models have been proposed to analyze rectennas sharing a common load [53], single diode rectifiers [54,55], and Class-F rectifiers converters [56]. In [8], PCE up to 90% has been achieved for the input power range of 30–35 dBm at 2.4 GHz, using the SPICE model with the parameters obtained from experimental data by means of curve fitting. Similar results were obtained in [57] for a single shunt diode rectifier using an analytical model that also considers the effect of the transmission line. In [58], an approximate model was used to find PCE for multi-tone excitation with equally spaced frequencies. Unfortunately, the analytical models give only approximate results that may not be sufficient for the precise evaluation and circuit optimization, like in the case of [59], where the nonlinearity of the diodes and the possible influence of the PCB are not taken into account, resulting in a highly idealized theoretical model.

The method proposed in this paper allows for more computationally efficient treatment of RF–DC converters in the case of input signals with evenly spaced subcarriers. The method has been successfully validated experimentally, as it will be shown in Section 3. In contrast to the multidimensional HB method (MHB) that treats each subcarrier frequency as a fundamental frequency, the proposed approach requires only two fundamental frequencies. Thus, fewer harmonics are needed to approximate the voltages and currents, thereby significantly reducing CPU time.

The aforementioned studies of the rectennas and RF–DC converters focused largely on experimental research and design-specific modeling, paying limited attention to development of reliable and computationally effective models considering the influence of the PCB material for estimating the PCE, whose grea<sup>t</sup> importance has been comprehensively demonstrated in [18].

In the current paper, a novel theoretical approach to evaluating the PCE of a rectenna is introduced. The proposed approach offers the following advantages:


The validity and accuracy of the proposed approach were verified by measuring the PCE of a prototype RF–DC converter. A voltage doubler circuit with a sub-GHz carrier frequency was selected as a test case and a comprehensive analysis of the effect of multitone power-carrying signals with different peak-to-average power ratio (PAPR) levels on its PCE was conducted. To the best of the authors' knowledge, no exhaustive study of such combination of the circuit and signals has been reported in the literature thus far.

The paper is structured as follows: Section 2 describes the novel theoretical approach to PCE estimation and presents a comparative analysis of its performance against conventional methods with a voltage doubler circuit employed as a test object. Discussion and comparison of the results obtained by means of the proposed theoretical estimation approach and its experimental verification are presented in Section 3. Section 4 presents conclusions of the research.

#### **2. Development of a Realistic Model of RF–DC Conversion**

This section describes a computationally efficient theoretical approach (model) developed to estimate the PCE for RF–DC converter circuits. For the sake of completeness, the general case of the circuit containing an arbitrary number of diodes is considered. A voltage doubler-based RF–DC converter circuit illustrated in Figure 2 used to validate the approach (see Section 3) can be viewed as a special case. The approach is adapted to power-carrying signals with periodic envelopes. The spectra of such signals comprise harmonics whose frequencies can be expressed as linear combinations of two fundamental frequencies only. This property allows for the employment of a two-dimensional FFT algorithm, which accelerates computation. Performing PCE estimation in shorter times is particularly important, since converter optimization involving PCE calculation for various circuit configurations is tremendously time-consuming, especially for a large number of carriers.

**Figure 2.** Voltage doubler circuit topology.

Unfortunately, due to the sufficiently high complexity of circuit PCB layout and high operating frequency, some of the existing and extensively used non-linear circuit methods fall short of expectations. For instance, despite numerous advantages, the TA is not suitable for the analysis of converters driven by a multi-carrier signal for several reasons. First, quite a large ratio of the period of the envelope to that of the carrier wave (in the present study, it is in the order of 1000) leads to a large number of iterations needed to calculate at least one period of the output voltage. Second, the presence of a filtering capacitor causes transients; therefore, many periods have to be computed until the steady state is reached. Third, for the equivalent circuit of the PCB to be valid over a frequency range encompassing at least 7–10 harmonics of the carrier wave, it must possess quite a complicated topology that is difficult to handle [60]. Therefore, TAs have been abandoned in favor of their frequency (or time-frequency) domain counterparts, such as the HB.

The HB relies upon Fourier series representation of circuit voltages (currents) and leverages some useful properties of well-established FFT algorithms leading to reduced consumption of computational resources. However, the method is not well-suited for multi-tone excitation. To tackle this issue, the authors propose to employ a two-frequency harmonic balance (2F-HB) method described in this paper. In contrast to its conventional counterpart, the 2F-HB exploits the fact that the spectra of the circuit currents and voltages consist of a number of sub-bands centered at integer multiples of the carrier frequency. Furthermore, each sub-band contains harmonics that are equally spaced. This property of the spectrum enables one to leverage the power of 2D versions of FFT [61] to achieve a substantial reduction in CPU time.

Experimental studies and simulations using RF–DC converter circuit models that do not consider the effect of the PCB show large discrepancies between the experimental and theoretical results [62]. Discrepancies are generally caused by the fact that the contribution of the PCB is either completely neglected, or its effect is only partially accounted for via some approximations. The proposed approach, in contrast, considers the contribution of the PCB through the calculation of the Y parameters obtained by means of full-wave numerical analysis. More precisely, the PCB is treated as a multi-port network formed from the original circuit by disconnecting discrete circuit components, as illustrated in Figure 3. The main advantage of this approach is that the accuracy of the PCE estimation depends solely on the accuracy of the 3D model. It should be noted that the approach is by no means perfect—3D models are typically idealized, neglecting some imperfections of real-world circuits. Nevertheless, it provides more accurate PCE estimation for PCBs having a complex layout, such as the one studied herein. Regarding the nonlinearity of the circuit, the proposed approach utilizes the standard SPICE diode model [63], as it describes the behavior of Schottky diodes with reasonable accuracy. Furthermore, the model closely approximates the diode breakdown behavior, which is particularly important, since the diodes under study possess quite low breakdown voltages (in the order of 2–4 V).

**Figure 3.** An equivalent circuit of PCB with equivalent two-port networks of linear devices connected to its linear ports.

It is noteworthy that the approach can be integrated into existing non-linear circuit simulators employing the HB or its extended multi-tone version, namely, the MHB. In the case of simulators using the MHB, only the subroutines responsible for the evaluation of the Jacobian entries have to be replaced or modified. Specifically, the approach proposed in this work requires the use of 2D-FFT and its inverse algorithms to perform time-tofrequency and reverse transformations of the non-linear element voltages (diode voltages). Regarding solvers capable of handling multi-tone signals driven non-linear circuits, only minor modifications in the existing codes are required. In fact, the proposed method can be

viewed as a two-dimensional MHB where one of the fundamental frequencies is that of the carrier wave, whereas the other is the subcarrier separation frequency.

#### *2.1. Two-Frequency Harmonic Balance Method*

As mentioned previously, the conventional HB is not a good candidate for handling multi-tone excitation, since FFT algorithms require uniform spectra, thus, a large number of harmonics should be considered. More specifically, all harmonics up to a specific order must be used for the approximation. In contrast, in the case of multi-tone excitation, the spectrum is not uniform—it consists of a number of sub-bands formed from the nonlinear conversion products. Therefore, it would not be wise to consider the harmonics between the subbands with negligibly small amplitudes. On the other hand, neglecting these harmonics prohibits the use of FFT, thereby reducing the computational efficiency. To overcome this issue, a multidimensional extension of the harmonic balance method (MHB) has been proposed [42–44]. The method approximates voltages (currents) with the truncated multidimensional Fourier series [64], enabling the use of multidimensional FFT algorithms (NFFT) to speed up calculations. While the MHB can be used to analyze multi-tone signalpowered RF–DC converters, the CPU time grows rapidly with the number of subcarriers. To mitigate this problem, the 2F-HB was developed and validated on a voltage doubler circuit.

The 2F-HB handles multi-tone signals in a more time-efficient way, since it requires fewer voltage (current) phasors than the HB and MHB and thus outperforms them. The proposed method relies upon the approximation of the voltage across each circuit element by a truncated two-dimensional extension of the Fourier series of the form:

$$v\_m(t) \Rightarrow v\_m(t\_1, t\_2) = \sum\_{n\_1 = -N\_1/2}^{N\_1/2} \sum\_{n\_2 = -N\_2/2}^{N\_2/2} \hat{V}\_{n\_1, n\_2}^{(m)} \mathbf{e}^{j n\_1 \omega\_1 t\_1} \mathbf{e}^{j n\_2 \omega\_2 t\_2},\tag{1}$$

where *V*(*m*) *n*1,*n*2 are the phasors of voltage *vm*(*t*), *ω*1 denotes the carrier frequency (CF) and *ω*2— the subcarrier separation frequency (CSF), *N*1 and *N*2 determine the numbers of harmonics of *ω*1 and *ω*2, respectively, used to approximate the voltage.

The circuit currents are approximated in the same way. The main benefit of using Equation (1) is that it yields a compact equation system, owing to derivative-free relations between the linear element voltage and current phasors. The introduction of time variables *t*1 and *t*2 associated with *ω*1 and *ω*2, respectively, allows evaluating the Jacobian matrix, which will be discussed further, in a considerably more time-efficient manner via the use of 2D-FFT [65].

The voltage (current) phasors can be found by solving a system of circuit equations derived by applying nodal analysis to the equivalent circuit (EC) obtained by reducing the linear sub-circuit to a mesh network. The equation for the *n*-th node of the EC is:

$$i\_{\sum n} = i\_n(\upsilon\_n) + \mathbf{Y}\_n \upsilon\_n + \sum\_{m=1, m \neq n}^{M} \mathbf{Y}\_{nm} \upsilon\_m + i\_{\text{eq},n} = \mathbf{0},\tag{2}$$

where *Yn* denotes an operator transforming the phasors according to the self-admittance of the *n*-th node, *Ynm* is the mutual admittance operator for the *n*-th and *m*-th nodes, *in*(*vn*) is the current through the *n*-th diode, *<sup>i</sup>*eq,*n* is an equivalent current representing the effect of independent sources contained in the circuit, *<sup>i</sup>*∑,*<sup>n</sup>* is the total current at the *n*-th node, and *M* is the number of circuit nodes.

NM is used to solve the system of equations obtained by collecting Equation (2) for all nodes, iteratively constructing and solving the following systems of linear equations:

$$\mathbf{J}^{(l)}\Delta\mathbf{v}^{(l)} = -\mathbf{\hat{r}}^{(l)},\tag{3}$$

where ˆ **J** (*l*) is the Jacobian matrix, **r**ˆ (*l*) is the residual vector calculated at the *l*-th iteration of the NM, and Δ**v**(*l*) is the phasor correction vector.

The Jacobian matrix entries are 2D-FFT transformed partial derivatives of each *<sup>i</sup>*∑,*<sup>n</sup>* with respect to the real and imaginary parts of each *V*(*m*)(*l*) *n*1,*n*2 . The residual vector contains 2D-FFT transformed *i* (*l*) ∑,*n*. It is worth noting that the Jacobian matrix can be transformed column-wise using 2D-FFT algorithms. Alternatively, the 2D-FFT algorithm needs to be run only once to evaluate the first column of the Jacobian matrix, while the other columns can be obtained by applying cyclic shifts to the entries of the first column. Equation system (3) can be solved by using either a plain linear equation solver [66] or various iterative methods, e.g., Krylov subspace methods [67].
