#
Analytical Optimal Load Calculation of RF Energy Rectifiers Based on a Simplified Rectifying Model^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Analytical Rectification Model

#### 2.1. Half-Wave Rectification Model

#### 2.2. N-Stage Voltage-Multiplier Rectification Model

## 3. Calculation of Optimal Load Resistance

#### 3.1. Problem Formulation

#### 3.2. Closed-Form Approximations for Low Input Power

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

## 4. Validation and Discussions

#### 4.1. Simulation Setup and Results

#### 4.2. Measurement Setup

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Diode equivalent circuit [25].

**Figure 3.**Optimal load calculated by transient simulation (blue), numerical solution of the analytical model (red), and closed-form solutions (yellow and purple), when (

**a**) ${N}_{f}=1$, and (

**b**) ${N}_{f}=4$ with ${\Delta}_{f}=1$ MHz, with half-wave, 1-, and 2-stage voltage-multipliers using Skyworks SMS7630. The results under different frequency bands (

**a1**,

**b1**) 400 MHz, (

**a2**,

**b2**) 900 MHz, and (

**a3**,

**b3**) 2.4 GHz are shown, too.

**Figure 4.**Optimal load calculated by transient simulation (blue), numerical solution of the analytical model (red), and closed-form solutions (yellow and purple) when (

**a**) ${N}_{f}=1$, and (

**b**) ${N}_{f}=4$ with ${\Delta}_{f}=1$ MHz, simulated with half-wave, 1-, and 2-stages voltage-multipliers using Avago HSMS285x. The results under different frequency bands (

**a1**,

**b1**) 400 MHz, (

**a2**,

**b2**) 900 MHz, and (

**a3**,

**b3**) 2.4 GHz are shown, too.

**Figure 8.**Comparison between measurement, simulation, and analytical model of the one-stage multiplier. Output DC voltage with (

**a**) ${N}_{f}=1$ and (

**c**) ${N}_{f}=4$; output DC power with (

**b**) ${N}_{f}=1$ and (

**d**) ${N}_{f}=4$.

**Figure 9.**Comparison between measurement, simulation, and analytical model of the two-stage multiplier. Output DC voltage with (

**a**) ${N}_{f}=1$ and (

**c**) ${N}_{f}=4$; output DC power with (

**b**) ${N}_{f}=1$ and (

**d**) ${N}_{f}=4$.

**Figure 10.**Optimal loads calculated by simulation, numerical solution of the analytical model, and measurement with (

**a**) ${N}_{f}=1$ and (

**b**) ${N}_{f}=4$. Note that the measured results with the lowest input tone amplitude ${v}_{A}$ tend to be outliers since the rectifier’s output voltage is close to the noise floor of the oscilloscope.

${\mathit{i}}_{\mathit{s}}$ | ${\mathit{R}}_{\mathit{S}}$ | N | ${\mathit{C}}_{\mathit{J}0}$ | M | ${\mathit{V}}_{\mathit{J}}$ | ${\mathit{L}}_{\mathit{S}}$ | ${\mathit{C}}_{\mathit{P}}$ | |
---|---|---|---|---|---|---|---|---|

SMS7630 | 5 $\mathsf{\mu}$A | 20 $\Omega $ | $1.05$ | 0.14 pF | $0.4$ | 0.51 V | 0.05 nH | 0.005 pF |

HSMS285x | 3 $\mathsf{\mu}$A | 25 $\Omega $ | $1.06$ | 0.18 pF | $0.5$ | 0.35 V | 2 nH | 0.08 pF |

R1 | R2 | R3 | R4 | R5 | R6 | R7$/$R8 | R7 | |

Value [$\Omega $] | 16.2 | 100.3 | 328 | 558 | 822 | 1.2 k | 1.99 k | 3.29 k |

R8 | R9 | R11 $/$ R14 | R10 | R11 | R12 | R13 | R14 | |

Value [$\Omega $] | 5.1 k | 7.49 k | 9.63 k | 11.97 k | 14.96 k | 18.01 k | 22 k | 26 k |

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**MDPI and ACS Style**

Yao, L.; Dolmans, G.; Romme, J.
Analytical Optimal Load Calculation of RF Energy Rectifiers Based on a Simplified Rectifying Model. *Sensors* **2021**, *21*, 8038.
https://doi.org/10.3390/s21238038

**AMA Style**

Yao L, Dolmans G, Romme J.
Analytical Optimal Load Calculation of RF Energy Rectifiers Based on a Simplified Rectifying Model. *Sensors*. 2021; 21(23):8038.
https://doi.org/10.3390/s21238038

**Chicago/Turabian Style**

Yao, Lichen, Guido Dolmans, and Jac Romme.
2021. "Analytical Optimal Load Calculation of RF Energy Rectifiers Based on a Simplified Rectifying Model" *Sensors* 21, no. 23: 8038.
https://doi.org/10.3390/s21238038