Notice of Retraction: A Closed-Form Solution for the Impedance Calculation of Grid Power Distribution Network

Abstract

In this paper, we propose a closed-form solution for the impedance calculation of grid power distribution network (PDN). Through the comparison of the equivalent circuit of the unit cell between the grid PDN and plane PDN, the complex three-dimensional (3-D) grid PDN is equivalent to a simple 2-D plane PDN with the same size. The impedance of this equivalent plane PDN can be calculated by the closed-form method. In order to further increase the accuracy of the proposed method, a modified modal function is proposed. And the original double series summation formula is reduced to a single series summation formula to accelerate the calculation. The proposed closed-form formula is scalable, and it can be used for grid PDNs with the sizes from micrometer to centimeter. To verify its accuracy, a micrometer grid PDN is calculated with the proposed closed-form formula up to 40 GHz, and a centimeter grid PDN is fabricated and measured for 10 MHz–1 GHz. The results obtained by the proposed method show good agreements with that obtained by 3-D full-wave simulation and measurement. Finally, the advantage of the proposed closed-form method for the analysis of the grid PDN modal voltage and current distributions and decoupling capacitor placement is demonstrated.