Power Distribution Networks with On-Chip Decoupling Capacitors

Assuming the separation between the pairs is n times larger than the distance between the power and ground lines inside each pair d (see Fig. 15.8), from (15.3), the mutual inductance between the power and ground paths of the different voltage domains for a fully paired power distribution grid with DSDG is
Substituting (C.1) (C4) into (15.8), the mutual inductive coupling M prdI loop between the two current loops in a fully paired power distribution grid with DSDG is
Simplifying (C.5) and considering that ln ? and ln k are approximately the same for different distances between the lines, M prdI loop is