A comprehensive 2-D inductance modeling approach for VLSI interconnects: Frequency-dependent extraction and compact circuit model synthesis

Although three-dimensional (3-D) partial inductance modeling costs have decreased with stable, sparse approximations of the inductance matrix [2], [20] and its inverse [3], [9], 3-D models are still intractable when applied to full chip timing or crosstalk analysis. The 3-D partial inductance matrix (or its inverse) is too large to be extracted or simulated when power-grid cross-sections are made wide to capture proximity effect and wires are discretized finely to capture skin effect. Fortunately, 3-D inductance models are unnecessary in VLSI interconnect analysis. Because return currents follow interconnect wires, long interconnect wires can be accurately modeled as two-dimensional (2-D) transmission lines and frequency-dependent loop impedances extracted using 2-D methods [29]. Furthermore, this frequency dependence can be approximated with compact circuit models for both uncoupled and coupled lines. Three-dimensional inductance models are only necessary to handle worst case effects such as simultaneous switching in the end regions. This paper begins by explaining and defending the 2-D modeling approach used in [8] and [28]. It then extends the extraction algorithm in [29] to efficiently include distant return paths. Finally, a novel synthesis technique is described that approximates the frequency-dependent series impedance of VLSI interconnects with compact circuit models suitable for timing and noise analysis.