# RE: [SI-LIST] : Trace Impedance Selection

From: John Spohnheimer ([email protected])
Date: Mon May 08 2000 - 16:04:23 PDT

While reading the various posts to the original question and the direction of some of the discussion, I can't help but think we may be confusing ourselves...

When I was first introduced to Network Analysis many years ago, one of the first points made was that Maxwell's equations (plus the Lorentz force equation) sum up all you need to know about E&M. Network Analysis is just a VERY useful approximation to the nasty differential equations that result when attempting to analyze real circuits. Since it is an approximation, you have to be aware of its assumptions, and when those assumptions might not apply. In particular, Network Analysis assumes that the size of the circuit is such that all pertinent phenomena can be represented by lumped elements, i.e. all circuit geometry sizes are much, much less than any wavelength of interest.

Well, that limitation was fine for our first course, but how do you handle real-life situations with time delays and distributed networks. That's where the T-line comes in. Not only does the T-line have some very nice electrical characteristics, but it also has some very convenient mathematical characteristics that extend the domain of network analysis. The T-line is a network analysis approximation that embodies certain characteristics that previously were only solvable with the full Maxwell's equations - hence a VERY useful approximation.

So far, so good. Where we seem to be running into difficulty is when we mix modes of our discussion. Specifically:

I claim that radiation from circuits is due to fundamental wave phenomenon - the full Maxwell's equations are required (at this point) to get believable results. Unfortunately, all the network components that we work with (including T-lines), have NO wave phenomenon associated with them. Attempting to mix both network analysis models with free-space wave propagation characteristics is, in my opinion, doomed to failure. Mixing models conceptually will almost assuredly lead to misunderstanding and weird, non-physical results.

A good example of this is the classic energy conservation problem with two capacitors: Take two caps (each C), one initially charged to V, the other to zero. Connect them at time zero. The final voltage will be V/2 by charge conservation, but then energy isn't conserved. Einitial = �*C*V^2, Efinal = 2*( �*C*(V/2)^2) = �*C*V^2 Where'd half the energy go? I don't buy "radiation" and I don't buy "it disappeared". The real problem is that the model is non-physical. You can't build a real circuit with two caps without introducing some inductance and some resistance (unless you use superconductors...). Once these real, but previously neglected, physical components are added, the conceptual problem disappears: Either half the energy is dissipated in the finite resistance, or (for the case of the superconductor) the tank circuit oscillates forever.

So, you're probably asking: is this guy foolish enough to say you can't model these effects? Absolutely not. I'm just warning you to be careful about how the models are constructed and interpreted. If you want to model radiation from a T-line, your network should consist of the T-line (with your favorite embedded model) plus a parasitic resistor that models the amount of the signal coupled into the surrounding media. The energy dissipated into this resistor is not lost as heat, but rather radiated into free space (i.e. from the antenna characteristics of the circuit). This "radiation" resistor is just a bookkeeping means of accounting for the energy loss from the circuit - i.e. it keeps the circuit simulator honest.

Accounting for the value of this resistor is a bit tricky. The best way that comes to mind would be to treat the T-line as a 3-port circuit (the third port being the radiated field) and calculate what the energy lost into the field really is from your port1 and port 2 measurements (energy into port1 minus energy rec'd at port2 = energy radiated into port3). This term would then appear in parallel to the standard input impedance of the T-line model. I'd fully expect that this term would vary with frequency and depend strongly on the surrounding structures. I think a similar measurement technique can be used to estimate the single port input impedance of an antenna when modeling the load on a transmitter. The real question at hand would then be if anyone has a theoretical model for the input impedance seen for an arbitrary antenna structure. I'm sure such models exist for dipoles and such, but general board layout isn't anywhere near as symmetrical or consistent.

This discussion thread seems to assume that since free space has an impedance of 377 ohms, the "radiation" resistor would also appear as 377 ohms. I disagree strongly. The value of the "radiation" resistor has to account for the all the various coupling efficiencies and the loading due to near field structures. Even a well designed half-wave dipole can be designed with an input impedance of 50 ohms (i.e. VSWR=1.0), and it still has to couple into free space at 377 ohms. A better way to conceptually look at an antenna is as an impedance transformer, not just a straight resistance.

Now I'll be the first to admit that I may be full of it since I haven't actually tried to correlate to real life with this technique, but it seems to be a much cleaner way of thinking about the problem.

John Spohnheimer

Vinu Arumugham wrote:
>
> If you were able to connect a transmitter to a receiver using a 377 ohm
> transmission line, this line would be in parallel to the "transmission
> line" between the two formed by free space. Therefore, one half the
> transmitted power would go through free space and the other half through
> the line. As the line impedance is lowered, more power would be
> transmitted through the line and less through space.
>
> What's wrong with this scenario?
>
> Thanks,
> Vinu

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