Series Resistance
6.2 Sheet Resistance of a Copper Layer
6.2 Sheet Resistance of a Copper Layer 149
sq
Len Len Len
R R n
A t w t w
= = = =
Where
= the bulk resistivity of copper = 1.68 uohm-cm Len = the length of the trace
t = the thickness of the copper trace, for 1 oz copper, 34 u w = the trace width
Rsq = the sheet resistance with units resistance per square n = number of squares down the trace
The last version of this relationship is incredibly powerful. It will dramatically simplify how we calculate the resistance of any trace etched from a copper layer.
The trace thickness will be exactly the same for every trace etched from the same copper layer. This means the ratio, /t, will be the same for every trace etched from the same layer. This figure of merit is given a special name, sheet resistance, Rsq.
The resistance of a trace is the (sheet resistance) x Len/w. If the trace is in the shape of a square, so that Len = w, the resistance from one edge to the other is literally just the sheet resistance. The trace can be 0.1 inches x 0.1 inches, or 10 inches x 10 inches. The ratio of Len/w is always the same.
This means that the resistance of any square-shaped trace etched from the same copper layer will have exactly the same resistance, equal to the sheet resistance. This is counterintuitive. How can a square 0.1 inches on a side have the same edge-to-edge resistance as a square 10 inches on a side?
If we double the length of a trace, the path length doubles and the resistance will double. But then if we also double the width to keep the square shape, the resistance will be cut in half since the cross- sectional area has doubled. These two-dimensional changes cancel out. Every square-shaped trace in the same sheet will have the same edge-to-edge resistance.
This property of every square cut from the same sheet having the same edge-to-edge resistance is a powerful principle that we will leverage to quickly estimate the resistance of traces.
The resistance of a square-shaped trace is a unique resistance which is a figure of merit for each layer of copper. We call the resistance of one square cut out of the same sheet the resistance per square, or the sheet resistance, Rsq.
This is the resistance from edge-to-edge of any square-shaped trace.
This is why the units of sheet resistance are ohms/square.
For 1 oz copper, the sheet resistance is
8 sq
1.7 10 m
R 0.5 m
t 34 um
− −
= = =
For ½ oz copper, the thickness is half that of 1 oz copper and the sheet resistance is twice this value, 1 mohm/sq. This is an easy number to remember.
In a 1 oz copper layer, each square is 0.5 mOhms. To find the total resistance of any uniform trace, we just count how many squares are down its length. This is n = Len/w. Each square’s resistance is in series so the total resistance of the trace is 0.5 mohm/sq x n.
A trace 10 mils wide and 1 inch long has 1 in/0.01 in = 100 squares down its length. Each square in a 1 oz copper sheet is 0.5
mohms/square, so the total resistance of the trace is 0.5 mohms/sq x 100 sq = 50 mOhms.
It really is as simple as that. Counting the squares in a trace is illustrated in Figure 6.3.
6.2 Sheet Resistance of a Copper Layer 151
Figure 6.3 The number of squares on a trace is how many times the width fits down the length.
Just at a glance, it’s possible to estimate the number of squares down a trace.
For a minimum width trace, 6 mil wide, the resistance for a 1-inch length is 0.5 mohm/sq x 1000 mils/6 mils = 0.5 x 170 = 85 mohm.
Knowing the resistance per length of a specific trace width is a handy figure of merit to remember. The resistance per length is just
sq Len
R R R = Len= w
For the case of a 6 mil wide trace length, the resistance per inch is
sq Len
R 0.5m / sq
R 83m / in
w 0.006in
= = =
Note that the units of squares in the sheet resistance is dimensionless and just disappears.
For a 20 mil wide trace, the resistance per length is just 0.5 mohm/sq /0.02 inches = 25 mohm/in.
We can use this estimate of 83 mohm/inch as a good measure of the resistance per inch of any signal trace. A trace 3 inches long would have a series resistance of 3 in x 83 mohm/in = 250 mohm of series resistance.
When calculating the number of squares, the most important
consideration is to use a consistent set of units for the length and the width. The number of squares is dimensionless.
Get in the habit of looking at the traces on your circuit board and quickly estimating the resistance of each trace. This will give you a feel for one of their electrical properties and what to expect for their performance.
To quickly estimate the resistance of a trace on your board, use the rules of thumb: for 1 oz copper, the series resistance of a 6 mil wide trace is 83 mohms/inch and, for a 20 mil
wide trace, it is 25 mohms/inch.
Watch this video and I will walk you through the concept of sheet resistance and why it is such a useful term to
characterize copper sheets and to estimate trace resistance.