Power Distribution Networks with On-Chip Decoupling Capacitors

Observing that the effective distance between the power and ground lines in a specific power delivery network is n + 1 times greater than the separation d between the lines making up the pair (see Fig. 15.9), from (15.3), the mutual inductance between the power and ground paths of the different voltage domains for a pseudo-paired power distribution grid with DSDG is
Substituting (D.1) (D.4) into (15.8), the mutual inductive coupling M prdII loop between the two current loops in a pseudo-paired power distribution grid with DSDG is
Simplifying (D.5) and considering that ln ? and ln k are approximately the same for different distances between the lines, M prdII loop is