I recently asked a group of 90 digital-board designers in Boston how many power-supply voltages were in their latest designs. Three, four, and five-voltage systems were quite common. The most was eight. The worst of the lot chopped the power plane into a profusion of irregular zones, just like a map of the Balkans. Computing the dc-voltage drop across such a power-distribution network can be challenging.
Given a solid, infinite copper plane, the dc resistance in ohms between vias A and B is approximately:
RDC = (ρ/πt)ln(s/r)
- ρ is the dc resistivity of the planes (6.58×10-7 Ω-in. for pure electrodeposited copper),
- t is the thickness of the planes (0.00137 in. for 1-oz copper),
- s is the separation between vias A and B, and
- r is the radius of the drilled holes.
If the planes are finite, however, or irregularly shaped, the resistance changes. The change becomes especially pronounced when either via A or via B lies near the boundary of the region.
Fortunately, it's easy to measure the dc resistance between any two points within a region of arbitrary shape. Start with a blank pc board having solid copper on both sides. Cut it to the dimensions of your final product. Solder wires onto the board at the positions where you anticipate having vias.
For best accuracy, connect into the planes with wires having the same diameter as the diameter of your planned vias. Don't worry too much about the wire radius, though, because the radius of the connection point enters the resistance formula only inside a logarithm; it has little influence on the final result unless the vias are particularly close together. You can check the magnitude of the via-size effect by running the above formula first with one via size, then another.
Connect a power supply, and pump some current through your connections. A sensitive dc-voltage meter (something that measures millivolts) can then display voltage drops at arbitrary positions across the plane. The current source and separate voltage-measuring device together constitute a very accurate four-terminal resistance-measuring setup.
I just measured the resistance of an 11-in. by 17-in. sheet of 1-oz copper. I connected my source wire (+) very near one end of the sheet and the load wire (–) at the other. I used a small bench power supply to pump 4A through the plane. The voltage drop from end to end in this configuration was 8 mV, indicating a resistance of 2 mΩ. This compares with the 1.137 mΩ that the formula predicts. The difference between the reality and the formula is due to current crowding at the boundaries of the finite plane.
The voltages that this experiment measures scale exactly in proportion to the applied load current and inversely with the thickness of the plane. If you scale the length, breadth, and hole radius in your experiment all by the same factor and hold the thickness constant, the dc resistance does not change. These rules of scaling can help you make useful scale models of unusually large (or unusually small) power-supply regions.
If you don't have a band saw or jigsaw to cut out a pc-board pattern, you can score lines in the copper plating to form a pattern using an industrial window-cleaning razor or a box-cutter knife. Make sure you score a line wide enough to totally isolate one region from another. You can also easily measure finished boards.