Requirement-Driven Magnetic Beamforming for MIMO Wireless Power Transfer Optimization

Motivation

  1. For the general MIMO setup, MultiSpot doesn’t consider peak current/voltage constraints. Yang et al. studied the magnetic beamforming problem in an MIMO MRC-WPT system to find all the boundary points of the multi-user power region with the peak current and voltage constraints for each TX. They simplified the problem by ignoring the mutual inductance among RXs.
  2. These researches lack the ability to control the power distribution among receivers, and could not be applied to the scenario when different RXs have different requirements of power which can be denoted as weight factors.

Contribution

  1. We formulate the requirement-driven magnetic beamforming design in the MIMO MRC-WPT system as a weighted sum-power maximization problem. We consider the peak current/voltage constraints, and provides a high efficient, provable optimal solution.

  2. We discuss the WSPMax problem with limited power budget where the peak current/voltage constraints are ignorable. We introduce a close-form theoretical bound of the WSPMax problem, and demonstrate that the power transfer efficiency maximization problem can be solved through TX-only solution without any feedback from the RXs, despite the fact that the beamforming is impacted by the interactions between TXs and RXs.

Methods

Proof of $ \phi_s = - \lambda_{Y,s} $

$ \phi_s = \frac{\vec{x_s^{\ast}} X_2 \vec{x_s}}{\vec{x_s^{\ast}} X_1 \vec{x_s}} $, $ \vec{Y} = - X_1^{-1} X_2 $, $ \lambda_s \vec{x_s} = \vec{Y} \vec{x_s} $.

$$ \begin{aligned} \phi_s &= \frac{\vec{x_s^{\ast}} X_2 \vec{x_s}}{\vec{x_ s^{\ast}} X_1 \vec{x_s}} \\ &= \frac{\lambda_s \vec{x_s^{\ast}} X_2 \vec{x_s}}{\vec{x_s^{\ast}} X_1 \lambda_s \vec{x_s}} \\ &= \frac{\lambda_s \vec{x_s^{\ast}} X_2 \vec{x_s}}{\vec{x_s^{\ast}} X_1 \vec{Y} \vec{x_s}} \\ &= \frac{\lambda_s \vec{x_s^{\ast}} X_2 \vec{x_s}}{\vec{x_s^{\ast}} X_1 (- X_1^{-1} X_2) \vec{x_s}} \\ &= -\lambda_s \end{aligned} $$

Proof of $ \lambda_{\ddot{Y}, s} = - \frac{\lambda_{\ddot{Z}, s}}{\lambda_{\ddot{Z}, s} + 1}$

$ \lambda_{\ddot{Y}, s} \vec{x_s} = \vec{\ddot{Y}} \vec{x_s} $, $ \vec{\ddot{Y}} = - (R_T + H^* R_R H)^{−1} (H^* R_R H) $, $ \lambda_{\ddot{Z}, s} \vec{x_s} = \vec{\ddot{Z}} \vec{x_s} $.

$$ \begin{aligned} \lambda_{\ddot{Y}, s} \vec{x_s} = \vec{\ddot{Y}} \vec{x_s} \\ &\Rightarrow \lambda_{\ddot{Y}, s} \vec{x_s} = - (R_T + H^* R_ RH)^{−1} (H^* R_R H) \vec{x_s} \\ &\Rightarrow \lambda_{\ddot{Y}, s} (R_T + H^* R_R H) \vec{x_s} = - (H^* R_R H) \vec{x_s} \\ &\Rightarrow \lambda_{\ddot{Y}, s} R_T^{-1} (R_T + H^* R_R H) \vec{x_s} = - R_T^{-1} (H^* R_R H) \vec{x_s} \\ &\Rightarrow \lambda_{\ddot{Y}, s} (I + \ddot{Z}) \vec{x_s} = - \ddot{Z} \vec{x_s} \\ &\Rightarrow (\lambda_{\ddot{Y}, s} + 1) \ddot{Z} \vec{x_s} = - \lambda_{\ddot{Y}, s} \vec{x_s} \\ &\Rightarrow \ddot{Z} \vec{x_s} = - \frac{\lambda_{\ddot{Y}, s}}{(\lambda_{\ddot{Y}, s} + 1)} \vec{x_s} \\ \end{aligned} $$

Hence, $ \lambda_{\ddot{Y}, s} = - \frac{\lambda_{\ddot{Z}, s}}{\lambda_{\ddot{Z}, s} + 1}$.

Reflections

  1. Introduction When discussing studies involving multiple TXs and/or multiple RXs, the authors could explain the rationale behind using MIMO, such as the efficiency of beamforming, rather than simply listing the three types. Providing this context enhances the reader’s understanding of the advantages and practical implications of MIMO systems.

  2. Methods In solving the WSPMax problem with peak current/voltage constraints, the authors assumed that a controller exists for communication with all TXs/RXs. This provides a foundation for designing the communication system in the furture work.

  3. Experiments The merit of this study lies in the fact that the authors not only evaluated the performance of their algorithm but also verified its convergence properties.

    However, for the performance evaluation, the authors set $ w_q = 1 $ for each $ RX_q $ in a context unrelated to specific requirements. This approach might have been adopted due to the challenges associated with assigning different weights to different $ RX $ instances.