## Trace Scaling

Imagine a microstrip trace (left side of Figure 1). Now scale up that trace, producing a new trace exactly *k* times larger in width, thickness, and height. It may surprise you to discover that this process of cross-sectional scaling does not change the per-unit-length values of trace capacitance, *C*, or trace inductance, *L*. The formulas for those quantities involve only ratios among width, thickness, and height. The absolute values don't matter. A six-mile-wide trace sitting atop a proportionately huge slab of fire-retardent board material (FR-4) enjoys the same exact capacitance per inch as its smaller cousin.

I mention this rule in case you encounter a PCB (printed-circuit-board) trace with dimensions smaller than the minimum values that your signal-integrity simulation tool allows. That scenario can easily happen in multichip-module work involving aluminum traces on silicon substrates. If that situation happens, use the rule of cross-sectional scaling to fake out your simulation tool using physical dimensions large enough to work with the tool but electrically identical to your original circuit.

The rule of cross-sectional scaling preserves the per-unit-length values of capacitance and inductance, but what about other parameters necessary for lossy-line calculations? You can handle them, too. The *G* parameter, conductance, describes the dielectric properties of the material surrounding your conductors. Specify the same dielectric material in your expanded circuit as the original, and you get the identical dc leakage and dielectric-loss performance. That part is easy. The *R* parameter represents the trace's resistance per unit length, and you must adjust it. Figure 1 illustrates the correct procedure. The trace on the left comprises a conductor with a resistivity of ρ ohm-meters. The trace on the right scales that resistivity by a factor of *k²*.

You can check the resistance-scaling procedure by thinking about the formula for the resistance of a long, thin copper bar. That formula equals the resistivity of copper, times the length of the bar, divided by the cross-sectional area of the bar. If you change the bar height and width by *k*, its cross-sectional area changes by *k²*, decreasing the resistance by *k²*. If you then multiply the resistivity of copper by *k²*, the total resistance reverts to its original value.

The resistivity-scaling procedure also properly handles skin-effect resistance. To understand this concept, you must remember that skin depth equals (2ρ/(2π*f*μ))^{½}, where ρ is the resistivity of the conductor, *f* is the frequency of operation, and μ is the magnetic permeability of the dielectric material. You must also know that what matters in a lossy-line problem is not the absolute value of skin depth but its value as a proportion of trace thickness. The skin-depth formula shows that scaling ρ by *k²* scales the skin depth by *k*, which renders the ratio of skin depth to trace thickness (also scaled by *k*) the same as in the original trace. That means that the onset frequency for the skin effect will be the same for your new, scaled, trace as in the original.

Please note that the scaling coefficient for resistivity as described here applies when you scale only the cross-sectional dimensions of the trace, but leave the length unchanged. If you are making a 3-D scaled model, with scaled lengths, then the correct scale factor for resistivity is *k*, not *k²*. The comments in this article apply only when you want to scale the cross-sectional dimensions of a trace for the purpose of getting around the minimum feature-size requirements of your signal-integrity simulation tool.

I do not know whether your simulator will allow you to make the required changes in resistivity. If it does, you can build a new layer stack with the same actual per-unit-length values of resistance, inductance, conductance, and capacitance as your original configuration, only at a physical scale acceptable to your simulator. Your simulator won't know the difference.