Verifying Edge Singularity

Hi Ppl,

Edge Singularity - is not always known phenomena and definitely
has large impact on understanding of resistors (at least at RFIC)

I spoke with Jim Rautio about that ,as an a support subject ,and we thougt
it's better to move this subject to this forum.

Here are the things we already written :

"

In order to understand loss mechanism in passive devices and to know how better simulate design

we encountered with the effect of edge singularity.



After I read some papers on the subject(one of them is by James C. Rautio – An investigation of microstrip Conductor Loss - Dec 2000)

there is still one not clear issue.



It’s well know fact that the charge has non uniform distribution along width of conductor ,with rectangle cross section,

(Maxwell solved this problem years ago) with no dependence on frequency.



We also know the fact that current in microstrip is I(x) = sigma(x) * Vphase



sigma(x) = charge distribution along width of conductor

Vphase = Phase velocity.



So why only at R ~ w * L this effect comes to be dominant (as it written at the paper of James C. Rautio mentioned above)

Why at DC it said to be that current is uniform across the section .Vphase supposed to be constant until very high frequencies,

where it rise again to velocity of light. And if it risen, for most dielectrics ,there is no big difference.

"

Jim's answer :

"

To see that there is no edge singularity at low frequency, imagine a planar resistor. The current is equal across the entire width of the resistor. There is no edge singularity. Of course, R >> wL in this case.



There is another way you can easily and directly verify that there is no edge singularity in this case. Take a few one quarter watt composition resistors (the kind with wires on each end). Place all the resistors side by side and connect them in parallel. Put a DC voltage across the resistors and measure the current in each resistor. The current in each resistor is the same. This is true no matter how you position the resistors.



Now, put a low frequency voltage source across the resistors. Measure the current in each resistor. The current in each resistor is still the same.



Of course, the resistors are not perfect. They also have some series inductance. So, increase the frequency of your voltage source until the inductance becomes important. Now, wL > R, and the current in all the resistors will not be equal. The current in resistors toward the center will be lower. The current in the resistors at the edge will be higher. This is because of mutual inductance between the resistors.



The same thing happens in microstrip, depending on which is most important, R per unit length, or L per unit length. At low frequency R is most important. At high frequency, L is most important.

The problem with your reasoning at low frequency is the sigma(x) is valid only for TEM transmission line. When there is resistance, there is voltage along the length of the line, by Ohm’s law. If there is voltage along the length of the line, then there is E field along the length of the line. Now it is no longer TEM.

"

Added after 21 minutes:

Jim the first question now comes to my head is about sigma(x)
You said that since this is not TEM mode there is problem with above quantity ,
but when I looking on charge distribution of microstrip(not TEM) line in Sonnet for all the frequincies it always the same distribution along the width so what wrong
with my conlclusion that the charge distribution is constant (again as Maxwell
solved) until ,as I understand, the freqeuncy rate will be higher that
relaxation time of electrons.

Re: Edge Singularity

Hi GoaGosha -- Sorry for long time to reply, I have been traveling all week (gave my Life of Maxwell presentation many times, regularly setting all time attendence records at MTT Chapter meetings).

As for your question, I am not familar with "electron relaxation times". If you want to get into electron specific phenomena (not needed for this problem), you should consider quantum electrodynamics (QED), a very difficult area. In Maxwell's equations, electric current is considered as a continuous mathematical field (i.e., a vector defined over a space). Of course, in reality, we know it to be the flow of billions of tiny electrons, but that makes no difference for this problem.

When we talk about a "charge distribution" in regards to a TEM line, we are talking about the quasi-static model. In other words if certain requirements are met, you can use the electrostatic charge distribution to obtain the Zo of a transmission line. It turns out that the electrostatic charge distribution equals the RF current distribution if these requirements are met.

The requirement is that the transmission line must be TEM. This means at least two things: 1) The dielectric must be exactly the same everywhere, and 2) the metal must be everywhere a perfect conductor.

Note that these two conditions are not sufficient. For example, rectangular waveguide meets these conditions, but it is not TEM. However, any transmission line that does not meet these two conditions is also not TEM.

Lossless stripline meets these two conditions and is TEM. Lossy microstrip fails both conditions. However, we can still use a static analysis as long as the conditions are almost true.

For typical microstrip situations, the fact that there are two dielectrics (air and substrate) only adds some dispersion. The dispersion becomes important only at high frequency. Thus the quasi-static result for Zo is not very good for high frequency.

For typical microstrip situations, R per unit length is much less than ωL per unit length. Thus, we can still use the quasi-static solution. However at low frequency, ωL goes to zero and R per unit length becomes important. Here too, the quasi static result fails once more.

For real world (lossy) stripline, the quasi-static result fails at high frequency, when skin effect loss starts increasing. It also fails at low frequency when R becomes more important than ωL.

Thus, in real world situations (lossy stripline, or microstrip), the RF current distribution is no longer independent of frequency and is not equal to the quasistatic charge distribution. In fact, it is the variation with frequency that is very important to include if you want a high accuracy solution. This variation with frequency that has been the topic of a lot of research over the last three decades or so.

If you take the Sonnet (or any EM analysis) low enough in frequency, you will see a large change in current distribution as the edge singularity goes away. Variation at high frequency can be very small, but it is still important because that is what causes dispersion.