*Article* **Fast and Accurate Approach to RF-DC Conversion Efficiency Estimation for Multi-Tone Signals**

**Janis Eidaks, Romans Kusnins, Ruslans Babajans \*, Darja Cirjulina, Janis Semenjako and Anna Litvinenko**

> Institute of Radioelectronics, Riga Technical University, Kalku St. 1, LV-1050 Riga, Latvia; janis.eidaks@rtu.lv (J.E.); romans.kusnins@rtu.lv (R.K.); darja.cirjulina@rtu.lv (D.C.); janis.semenako@rtu.lv (J.S.); anna.litvinenko@rtu.lv (A.L.)

**\*** Correspondence: ruslans.babajans@rtu.lv

**Abstract:** The paper presents a computationally efficient and accurate numerical approach to evaluating RF–DC power conversion efficiency (PCE) for energy harvesting circuits in the case of multi-tone power-carrying signal with periodic envelopes. This type of signal has recently received considerable attention in the literature. It has been shown that their use may result in a higher PCE than the conventional sine wave signal for low to medium input power levels. This reason motivated the authors to develop a fast and accurate two-frequency harmonic balance method (2F-HB), as fast PCE calculation might appreciably expedite the converter circuit optimization process. In order to demonstrate the computational efficiency of the 2F-HB, a comparative study is performed. The results of this study show that the 2F-HB significantly outperforms such extensively used methods as the transient analysis (TA), the harmonic balance method (HB), and the multidimensional harmonic balance method (MHB). The method also outperforms the commercially available non-linear circuit simulator Keysight ADS employing both HB and MHB. Furthermore, the proposed method can be readily integrated into commonly used commercially available non-linear circuit simulation software, including the Keysight ADS, Ansys HFSS, just to name a few—minor modifications are required. In addition, to increase the correctness and reliability of the proposed method, the influence of PCB is considered by calculating Y parameters of its 3D model. The widely employed voltage doubler-based RF–DC converter for energy harvesting and wireless power transfer (WPT) in sub-GHz diapason is chosen to validate the proposed method experimentally. This RF–DC converter is chosen for its simplicity and capability to provide sufficiently high PCE. The measurements of the PCE for a voltage doubler prototype employing different multi-tone waveform signals were performed in laboratory conditions. Various combinations of the matching circuit element values were considered to find the optimal one in both—theoretical model and experimental prototype. The measured PCE is in very good agreemen<sup>t</sup> with the PCE calculated numerically, which attests to the validity of the proposed approach. The proposed PCE estimation method is not limited to one selected RF–DC conversion circuit and can also be applied to other circuits and frequency bands. The comparison of the PCE obtained by means of the proposed approach and the measured one shows very good agreemen<sup>t</sup> between them. The PCE estimation error reaches as low as 0.37%, and the maximal estimation error is 32.65%.

**Keywords:** wireless power transfer; energy harvesting; power conversion efficiency; single diode rectifier; voltage doubler; harmonic balance method; autonomous sensor node; wireless sensor network; multi-tone signal; full-wave simulations of PCB
