Equation 1.
Abstract
Equivalent-circuit models are obtained from direct interpretations of electromagnetic field data. Improved intuitive understanding is achieved with direct visualization of wave propagation through 3D structures. Applications to MCMs and power-distribution networks.
Introduction
Modelling the inductance of a 3D structure is a tough challenge -- developing an intuitive understand is sometimes worse. Both are critical to achieving design success. Given today's increasing clock speeds, system complexities, and reduced time to market, accurate package models can not wait until prototype measurements can be made. If you wait, you are too late. And lab measurements may not lend correct insight into the actual details of the complex 3D electrodynamics deep within the structure; whereas the real-time FD-TD graphical images of pulses propagating through the 3D structures add invaluable insight and true understanding to the actual operation of the structure. The performance and reliability of today's high-performance systems are determined by the electrical properties of high-density impedance-controlled connectors, MCM packages, and the entire power-distribution network. Critical system-design decisions are made early on (long before any prototype is built) based on the assumed electrical operation of these key components -- their electrical models (and your understanding of the components actual operation) must be correct to achieve first-pass design success. Accurate equivalent-circuit models of complex 3D structures are readily constructed from simple first-principles interpretations of FD-TD field data. This note describes a professionally developed tool (dubbed "LC") that integrates a user-friendly graphical-user-interface with a FD-TD field-solver engine to easily build accurate equivalent-circuit models for, and visualize wave propagation through, arbitrarily-shaped 3D structures.
The tool is named LC because it delivers the equivalent-circuit parameters of inductance and capacitance for of 3D components. LC is a revolutionary tool for technology pioneers, it teaches the user about their technology enabling early and accurate characterization, understanding and application of key system technologies.
Tool Description and Methods
A user-friendly graphical user interface is required to enable the non-FD-TD expert to properly harness the vast amount of data generated by during an FD-TD simulation. The following summarizes the features in LC:
Model Generation
Seamless Simulation Control
Advanced Visual Post Processing
FD-TD as the computational engine
The FD-TD method provides full-wave numerical solutions to Maxwell's time-dependent curl equations. Time domain is the world of the packaging engineer. Modeling and visualizations in the time domain inherently enhances the intuitive understanding of the complex electrodynamics and the relationships between the temporal and spacial domains. Reflections are completely modeled and easily understood. Time domain field data permits the direct calculation of equivalent-circuit-model parameters using simple first-principle interpretations of the Electric and Magnetic field data. Capacitance from the Electric field, and inductance from the Magnetic field as show below. All field and parametric representations within LC can be plotted in the frequency domain using LC's built-in DFT capabilities.
3D Inductance Calculation Method
Direct calculation of the equivalent-circuit inductance (self or mutual) of any 3D path is obtained with a simple application of first-principles of electromagnetics as follows:
Inductance is the quantity that describes the relationship between magnetic flux and current. Self inductance is simply the ratio of magnetic flux generated, divided by the amount of current that created that flux. In other words:
Equation 1.
This is analogous to resistance, which is the quantity describing the ratio of induced voltage drop divided by the current that created that drop: R = V/I.
For a given electromagnetic pathway (such as Trace 1 shown in Figure 1) the actual value of inductance is determined by the amount of flux that builds up around the trace. The total amount of flux generated is entirely determined by geometries and spacings of the trace carrying the original signal current (I) and its proximity to the conductor that carries the resulting return-path current (i). Larger separations between the signal and return paths permit more flux generation per unit of current with results in larger inductance. (Since H-fields are conservative note that all of the flux lines are trapped between the two conductors, which complete the circuit, carrying the signal and return-path currents.)
With this method, the challenge of modeling 3D inductance simply reduces to the user properly defining the closed surface over which the flux integration is performed. Call this closed surface the flux net (S). In Figure 1, the incremental inductance for the segment of trace 1 from A to B is calculated using the flux net defined by the plane ABCD. The upper and lower edges of this flux net are defined by the edges of the conductors that carry the signal and return-path currents.
Figure 1. Trace over a solid ground plane
Mutual inductance between trace 1 and 2 is modeled by calculating the total amount of flux caught in ABCD when the signal current (I2) is sent down trace 2. The above illustrative example was simple and 2D in nature. Consider Figure 2 to see how this simple
Figure 2. Trace over a slotted ground plane
method is used to quantify the dramatic increase in effective trace inductance when a slot is cut in the ground plane. Now the return path current is diverted around the slot following the path C C' D' D. This new return path converts the simple planer flux net in Figure 1 to the hockey net shown in Figure 2. An exam ple simulation of the above structures uncovers a tripling of effective trace inductance due to the slot in the ground plane.
TABLE 1. LC Model of Slotted Ground Plane
Signal trace over: L C
Solid Ground Plane 0.87 nH 1.17 pF Slotted Ground Plane 3.59 nH 0.81 pF
Gauss's Law guarantees that the total flux caught in the flux net is independent of the shape of the net, permitting the user to define the geometrically simplest flux net between the signal and return-path conductors. Rectilinear flux nets are easily and naturally defined in the FD-TD grid
The intuitive understanding of this fairly simple 3D structure is enhanced by examining the time-domain snapshot of the ground-plane surface currents shown in Figure 3. Observe the return-path current diverting around the slot in the ground plane.
Figure 3. Surface currents in the slotted ground plane
Visualizations of the full-wave solutions to Maxwell's Equations enable the user to understand where the real return-path(S) currents will flow in complex 3D strucures. Understanding and modeling return-path current flow is paramount to acurate 3D inductance modeling. LC enables the packaging engineer to harness the power of Maxwell's Equations.
Figure 4. Magnetic flux calculation of a 1 amp pulse moving across
a solid and slotted ground plane
Figure 4. Electric charge calculation of a 1 amp pulse moving across
a solid and slotted ground plane
Applications
Design and Modeling of High-Speed Digital Interconnects
As system clock rates relentlessly march toward 1 GHz it has become clear that overall system performance is set by the propagation-delay and signal integrity of the chip-to-chip transmission paths. All components in this transmission path that may be carrying signals with 150 ps edge rates must be impedance controlled. Especially the high-density module-to-module edge connectors that now exceed 600 impedance-controlled signals per linear inch and the MCM package interfaces that feature over 1700 connections per square inch to the PCB. To insure the performance and reliability design goals are met, the Ground Rules must establish a reasonable electrical noise budget. There are two chief components to the noise budget; signal integrity and power distribution. Signal integrity issues revolve around the detailed electical characteristics of complex 3D high-speed digital transmission structures.
Modeling and Design of Low-Voltage Power Distribution Systems
Delivering reasonable clean low-voltage power throughout the system is the second critical aspect to the noise budget. The entire system from within the ICs to the DC-DC converters must be modeled and managed to achieve a successful design. A prior modelling is critical in this application. The power distribution system, whose design must be frozen early in the system definition, must be known correct long before the hardware is built. The large dI/dt current spikes caused by the CMOS simultaneously switching output requires low-inductance power-supply bussing with the correct balance of staged capacitance. Clean power is becoming even more critical as supply levels and logic swings continue to decrease thereby, tightening the noise budget even further.
Summary
The advantage of having a design tool such as LC available is that the developer will have a step function improvement in reliability. It allows early and accurate evaluation of key technologies which reduces the number of prototypes and therefore allows a shorter time to market. Ideally the product will be correct by design the first time. Overall the designer needs to see and understand to succeed. LC is an innovative and intuitive engineering tool that can educate and empower even the EM novice.
We currently compile and support LC on Cray Research supercomputers and the following workstations: SGI, HP, SUN, IBM.
Acknowledgments
Thank you Sir James Clerk Maxwell for making my job easier.
Cray Research retains intellectual property rights to the software and methods described herein.
Copyright © Cray Inc.