In this chapter, we focus on passive wideband modeling of RLCM circuits. We propose a new passive wideband reduction and realization framework for general passive high-order RLCM circuits. Our method starts with large RLCM circuits, which are extracted by existing geometry extraction tools like FastCap [83] and FastHenry [59] under some relaxation conditions of the full-wave Maxwell equations (like electro-quasi-static for FastCap or magneto-quasi-static for FastHenry) instead of measured or simulated data. It is our ultimate goal that we can obtain the compact models directly from complex interconnect geometry without measurement or full-wave simulations. The method presented in this chapter is called hierarchical model order reduction, HMOR, which is based on the general frequency-domain hierarchical model reduction algorithm [121, 122, 124] and an improved VPEC (vector potential equivalent circuit) [134] model for self and mutual inductance, which can be easily sparsified and is hierarchical-reduction friendly.

The HMOR method achieves passive wideband modeling of RLC circuits via multi-point expension and the convex programming based passivity enforcement method. In this section, we will show that the frequency-domain hierarchical reduction is equivalent to implicit moment-matching around s = 0, and that the existing hierarchical reduction method by one-point expansion [121, 124] is numerically stable for general tree-structured circuits.