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New for April 2007! Loss due to substrate conductivity in microstrip was never really a problem for engineers... until we started considering making stuff using silicon as a dielectric! This is an important topic for the MEMS industry.
Silicon is a semiconductor, as opposed to a semi-insulator that GaAs is, or an insulator like alumina. Silicon comes in many flavors, its bulk resistivity can be less than 1 ohm-cm, to as high as 10,000 ohm-cm in special cases.
We don't have a neat, closed form equation for this. So we used Sonnet to analyze the problem. Note that EDA software such as ADS has no way to account for substrate conductivity in their microstrip model. But Sonnet cranks through 100 GHz of analysis on this problem in ten seconds on a $400 Dell Dimension 4700, it's truly EM analysis on steroids.
This page is obviously under construction, but here's three plots for a 1mm line at various resistivities:
Here's 1000 ohm cm, or 0.1 Siemens/m. This is a damn good transmission line, almost lossless, but 1000 ohm-cm silicon is hard to come by!
Here's the result at 100 ohm cm. Now we have loss on the order of 0.5 dB/mm.
Here's the result at 10 ohm-cm (10 S/m). Now the loss is going to kill you!
Time for a Microwaves101 rule of thumb!
In order to use silicon as a substrate, you need resistivity at least 100 ohm-cm or the loss is going to eat your lunch.
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