5.6 Logic and Voltage Drops
The signal levels that arrive at a logic gate can vary for a number of reasons. We have already considered the voltage drop in the driver logic. In Section 4.12, we discussed how skin effect increases the trace resistance. The voltage drop in traces can best be understood by first considering the resistance of a bar of metal.
The dc resistance of a bar of metal having a cross-section A and a thickness h is ρs/A, where s is the length of the bar and ρ is the resistivity. If the bar of metal is a square then the square has a resistance of
5.1 
When the thickness h of the bar is small compared to width and length, the conductor takes on the form of a sheet of metal. When the sheet of metal is used on a circuit board, it is referred to as a ground or power plane.
A sheet of metal can be characterized by the resistance of a square of that material. The term used is ohms-per-square or Ω2. The assumption that must be made is that the current flows uniformly across the square between opposite edges.
The term two-ounce copper means that two ounces of copper are deposited on 1 ft2 of material. Two-ounce copper has a resistance of 24.2 μΩ2 at dc. At 10 kHz, the resistance doubles to about 48.4 μΩ. Above 10 kHz, the resistance increases proportional to the square root of frequency. At 1 GHz, the resistance increases by a factor of 316 to about 0.015 Ω2. For a trace that is 5 mil wide and that is 1 in long, the resistance is computed by considering that there are 200 squares in series. The resistance is therefore 3.0 ohm. Because skin effect keeps the current near the surface, the resistance at 1 GHz for 2 oz or 1/2 oz copper is essentially the same.
The current on the ground plane spreads out depending on the height of the trace above the ground plane. For a 5-mil trace, 10 mil above the conducting surface, the width of the current path in the ground plane is approximately 15 mil. The resistance of this return path will be about 1.2 ohm/in. Thus, the total resistance in the signal path would be about 4.2 ohm/in. A trace that is 10 mil wide will reduce this resistance to about 2.5 ohm/in. Since skin effect is proportional to the square root of frequency, the resistance at 100 MHz would be reduced by a factor of about 3.
The resistivity of copper is 1.72 μΩ-cm. This is low compared to solder plating that has a value of 13.5 μΩ-cm. For plated traces, skin effect forces current to use the plating, which raises the resistance. To reduce this resistance, it is possible to plate outer layer traces with gold. Note that gold has a resistivity about equal to copper. Gold plating has been used in analog (microwave) circuits, but it is not generally used in digital designs. If the traces are embedded in a dielectric then plating is not needed (embedded microstrip or stripline). The areas that must still be plated are through-holes, pads, and test points.