Metal Roughness Is Weird
In 2008, I started working with Dr. Allen F. Horn, III, Associate Research Fellow of Rogers Corporation, on metal surface roughness. He was obtaining incredibly weird measurements. He was measuring substrate dielectric constants by means of various microstrip and stripline transmission lines using various substrate thicknesses and various metal foil roughnesses. The substrate material in all cases was LCP (liquid crystal polymer). Since it is homogeneous, it should yield exactly the same dielectric constant for all cases. He saw major dependency. The measured dielectric constant (inferred from measurement of S21 phase of a through line) appeared to vary by up to 15% by just changing the substrate thickness or the foil roughness. The dielectric constant should not be a function of either the substrate thickness or the metal foil laminated onto it. Weird.
When Al showed his results to me, the first thing we did was to go over his test methodology and try to see where it went wrong. You know the old saying, “In God we trust; all others must have proof.” Since we are both members of the “all others” group, we went over the methodology very carefully. It all checked out. Try as we might, we simply could not find a problem.
Then we looked at existing models of roughness. In the late 1940s, Samuel P. Morgan of Bell Labs reported analyses of loss for surfaces that have a square wave profile and a triangle wave profile. He found that roughness never increases loss by more than a factor of two over smooth metal. This limit is not seen in measurements. In fact, analysis results using the Morgan model seem only modestly related to measurement results.
Next, I recalled that smooth surface skin effect, which increases the equivalent surface resistance of metal with frequency, also generates an exactly equal amount of inductive surface reactance. We often forget this, but check back in your junior-year college electromagnetics textbook. Skin effect generates both a surface resistance and a surface reactance that are exactly equal. It looks like most microwave analysis tools have ignored the skin effect reactance. Sonnet, of course, includes both the resistance and reactance. So we compared a few Sonnet analyses to a few of Al’s measurements. Magically, things changed, and in the right direction.
For example, when we keep the substrate dielectric constant in the EM analysis and add skin effect loss, the extra skin effect inductance increases the S21 phase of the measured line. If we did not know about surface inductance, we would mistakenly think the increased S21 phase was due to an increased substrate dielectric constant. This looked hopeful. Unfortunately, the change in S21 phase was nowhere near enough. There had to be something else going on.
The skin effect inductance we calculated was for smooth metal. Actual metal is rough. So, perhaps we have extra inductance due to roughness. The solution now seems simple. Just increase the resistivity that we specify in the EM analysis. This increases the loss, increases the inductance, and increases the S21 phase. Good idea, until we tried it. In order to get the through line S21 magnitude to match at high frequency, we had to increase the metal resistance in the EM analysis by a factor of eight. This now means that the DC resistance of the line is eight times too big. Not good.
Then we looked at the calculated S21 phase. Higher metal resistance means increased skin effect resistance and inductance. The increased inductance did indeed increase S21 phase, but by only about ¼ of what we needed to match measurements. Another dead end.
After numerous discussions with Al, we came up with the idea that metal roughness must generate an excess inductance, above and beyond the skin effect inductance. Working back and forth, I developed a model that fit Al’s measurements perfectly. We can hit both the measured loss and the S21 phase using the same dielectric constant for all substrate thicknesses and the same loss model for all roughnesses. Nice!
So where does all that excess inductance (it is considerable) come from? Once you see it, it is obvious. The current flows into the metal and down and around a “valley” in the surface profile. The electric field cuts straight across the metal peaks surrounding the valley. This is a one turn inductor. There are billions and billions of these tiny inductors saturating the surface of all rough metals. This extra inductance decreases the traveling wave velocity, increasing S21 phase. If we did not know about this extra inductance, we would think that the dielectric constant had changed. In reality, the dielectric constant is indeed constant. The increased S21 phase is due to the excess surface roughness inductance.
Al and I have published several papers on this topic, including one that won a best paper award at DesignCon 2010, “Effect of conductor profile on the insertion loss, phase constant, and dispersion in thin high frequency transmission lines,” and another paper at IMS2010, “Conductor Profile Effects on the Propagation Constant of Microstrip Transmission Lines.” Figure 7 in this second paper is key, illustrating the mechanism of the excess surface roughness inductance for, we think, the first time ever. Go to our website, www.sonnetsoftware.com/resources/technical-references-sonnet.html, if you would like to download a copy.
Sometimes life is rough, we just have to deal with it. The same is true for metals.