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38 The PCB Design Magazine • October 2017 say down to one percent, we need a ratio of ten to twenty. If we take the tighter accuracy esti- mate of twenty, the signal passing through this example network should have a rise time of 20 * 150 ps = 3 ns, or longer. This limits the band- width to approximately 100 MHz. If we need wider bandwidth, we can proportionally scale the L and C values. For instance, using L = 0.75 nH and C = 0.3 pF will produce a 50-ohm interconnect model with a bandwidth of 1 GHz, but if we need the same 450 ps end-to-end delay, we will need to use ten times more, namely thirty cascaded seg- ments. This model is so simple that any circuit simulator can take it, but to achieve high ac- curacy and high bandwidth at the same time, we may end up with a lot of circuit nodes and considerable run time. Of course, for the ideal lossless case we have in Figure 1, we could just use a single SPICE Tline element with the re- quired characteristic impedance and total delay, but that way we would lose the horizontal con- nectivity [1] . Package or connector pins, traces and cables are one-dimensional transmission lines in the sense that most of the signal will only propa- gate along the conductor. Power planes are two-dimensional transmission lines: any signal (mostly the power noise) can and will propagate in any direction on the plane. To model this two-dimensional nature, we can take the ladder equivalent circuit from Figure 1 and turn it into a two-dimensional grid circuit. Figure 2 shows the conceptual circuit diagram. Each subcircuit in the grid can be one LC segment from Figure 1 with appropriate L and C values. The calculation of the subcircuit element values is explained in [2] . If we have plane out- lines or slots/cutouts in the planes that cannot be easily approximated with rectangular shapes, the same grid concept can be extended to utilize variable-size grid cells [3] . The simulation mod- els based on grid equivalent circuits will provide results at each of the circuit nodes; in between circuit nodes we need to rely on interpolation. Alternately, for simple plane shapes, analytical formulas can also be used to simulate the self and transfer impedance of planes at any arbi- trary point [4] . Of course, today we have a number of com- mercial 2.5D tools available that can simulate two-dimensional structures [5] , [6] , [7] . We could also use 3D tools, though simulating electrically very large structures that way is not really feasible. Turning our attention back to causality: the model in Figure 1 is fully causal, but it is not only band limited, it represents only lossless in- terconnects. This means no series resistance rep- resenting the conductor losses, no parallel con- ductance to capture the dielectric losses, and as a result, the capacitance and inductance values are frequency independent constants. If we CAUSAL POWER PLANE MODELS Figure 2: A two-dimensional SPICE grid network to simulate a pair of rectangular planes.

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