units of R
ac
are often reported in ohms per square. Note that the "per square" terminology
refers simply to a square of any size (i.e., length = width), since the resistance of one
geometric square of board trace at a given thickness is independent of the size of the square.
Thus, the surface resistance can be estimated simply by counting how many squares can be
fit geometrically into the area under examination and multiplying that number by R
s
times the
square root of frequency. The concept of dc resistance per square is illustrated in
Figure 4.8
.
Remember, in interconnect simulation it is very important to account for both the conductor
and the ground return path resistance.
Table 4.1
shows the skin effect properties of typical
materials.
Table 4.1: Frequency-Dependent Properties of Typical Metals
Metal
Resistivity
(
· m)
Skin Depth
(m)
Surface Resistivity, R
s
(
·
/square)
Copper (300
K)
1.72 × 10
-8
0.066F
-1/2
2.61 × 10
-7
Copper (77
K)
5.55 × 10
-9
0.037F
-1/2
1.5 × 10
-7
Brass
6.36 × 10
-8
0.127F
-1/2
5.01 × 10
-7
Silver
1.62 × 10
-8
0.0642F
-1/2
2.52 × 10
-7
Aluminum
2.68 × 10
-8
0.0826F
-1/2
3.26 × 10
-7
Source: Data from
Ramo et al. [1994]
.
Figure 4.8: Concept of ohms/square.
Effect of AC Losses on Signals.
There are two classic implementations of ac losses: those for digital designers and those for
microwave designers. The microwave designer is usually interested only in the ac resistance
in a frequency-domain simulation. This is easy to implement because most general
simulators, such as HSPICE, have a frequency-dependent resistor, which can be made to
vary with the root of the frequency [see
equation (4.11)
] and used in an equivalent circuit
constructed of cascaded LRC segments as in
Figure 2.4
.
The digital engineer, however, has a more difficult problem. Digital signals approximate
square waves and are subsequently wide band, which means that they contain many
frequency components. This is an important concept to understand. We demonstrate this
with the equation [
Selby, 1973
]
Summary :
Table 4.1: Frequency-Dependent Properties of Typical Metals Metal Resistivity ( · m) Skin Depth (m) Surface Resistivity, R s ( · /square) Copper (300 K) 1.72 × 10 -8 0.066F -1/2 2.61 × 10 -7 Copper (77 K) 5.55 × 10 -9 0.037F -1/2 1.5 × 10 -7 Brass 6.36 × 10 -8 0.127F -1/2 5.01 × 10 -7 Silver 1.62 × 10 -8 0.0642F -1/2 2.52 × 10 -7 Aluminum 2.68 × 10 -8 0.0826F -1/2 3.26 × 10 -7 Source: Data from Ramo et al.
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