Multiple and non-existent Barnes-Hut center-of-charge (CoC) solution in lossy layered substrates: Half space analytic and numerical study

For rapid extraction of capacitance in lossless substrates Barnes-Hut algorithm is an effective technique for evaluating N charges in O(N log N) operations. When substrates are lossy the hierarchical application of the CoC approximation, apart from resulting in unique CoC, can result in cases like there does not exist CoC or there exist multiple CoC solutions. In this paper we analytically and numerically study CoC approximation in lossy half space media.