Lumped-circuit model extraction for vias in multilayer substrates

Via interconnects in multilayer substrates, such as chip scale packaging, ball grid arrays, multichip modules, and printed circuit boards (PCB) can critically impact system performance. Lumped-circuit models for vias are usually established from their geometries to better understand the physics. This paper presents a procedure to extract these element values from a partial element equivalent circuit type method, denoted by CEMPIE. With a known physics-based circuit prototype, this approach calculates the element values from an extensive circuit net extracted by the CEMPIE method. Via inductances in a PCB power bus, including mutual inductances if multiple vias are present, are extracted in a systematic manner using this approach. A closed-form expression for via self inductance is further derived as a function of power plane dimensions, via diameter, power/ground layer separation, and via location. The expression can be used in practical designs for evaluating via inductance without the necessity of full-wave modeling, and, predicting power-bus impedance as well as effective frequency range of decoupling capacitors.