Microstrip propagation time: the difference in determining stripline and microstrip propagation time is more than a fine line.

Electrical signals on wires and traces travel at the speed of light, 186,280 miles/sec. Do the math and that works out to 0.9835 ft/ns, or 11.8 in./ns. The speed of light slows down in any other medium by the square root of the relative dielectric coefficient of the medium. So the propagation speed of a signal in a stripline environment would be

(1) Speed = 11.8 / [square root of ([epsilon.sub.r])] in/nsec

This would lead to a propagation time for a signal in a stripline environment of

(2) Time = [square root of ([epsilon.sub.r])] / 11.8 nsec/in

I once (1) posed the following question: What happens if we string a wire across a lake and measure the propagation speed, and then lower the wire into the lake and measure the propagation speed when the wire is under water? I pointed out that the propagation speed through the underwater wire would be about 1/9 that of the wire in the air! Same signal, same copper, same electrons, but only 1/9 the propagation speed (that is, 9 times the propagation time).

The issue is not how fast the electrons can travel through the wire. Moving electrons (current) create an electromagnetic field around the wire (or trace). The issue is how fast the electromagnetic field can travel through its medium. In the example water is the medium that the electromagnetic wave is traveling through. On a circuit board the medium is the board material (usually but not always FR-4).

So, for example, a stripline trace in FR-4 with an [epsilon.sub.r] of 4.0 would travel at the speed of light divided by the square root of 4 (which is 2) or about 6 in./ns. Most of us are pretty comfortable with this figure.

The problem with the propagation speed for a microstrip trace is that a microstrip trace is in a mixed environment. The medium beneath the trace is the dielectric and the medium above the trace is air. So the electromagnetic wave travels through this mixed medium at a speed somewhere between that of the speed of light and the propagation speed in stripline--approximately one-half that of the speed of light.

There is a correction factor for [epsilon.sub.r] that has been traditionally used for microstrip environments. It simply replaces the standard value for [epsilon.sub.r] in (1) or (2). The correction factor apparently derives from work2 done in 1967 and is

(3) [epsilon'.sub.r] = 0.475[epsilon.sub.r] + 0.67

But there is a problem with this correction factor. It is a constant, yet we sometimes observe that different width microstrip traces may have different propagation speeds even though they are in otherwise identical environments. A microstrip trace is referenced to a plane. The return signal will be on the plane directly under the trace. So, the electric field lines will extend from the trace to the plane. Most of the field lines are under the trace, in the dielectric environment, but many extend upward into the air before they curve back down to the plane.

How the electric field lines are concentrated underneath the microstrip trace becomes the issue. If the concentration of field lines increases underneath the trace, then propagation speed will slow down. Thus we can ask: What actions (all other things being equal) lead to increased field concentration underneath the trace?

1. Bringing the trace closer to the plane.

2. Increasing the relative dielectric coefficient of the material underneath the trace.

3. Increasing the trace width. (If the trace were infinitely wide, then virtually all field lines would be under the trace between the trace and the plane!)

Note: Increasing the trace thickness has a minor effect on propagation speed, but the effect is much smaller than with the other variables.

Each of these actions will cause the propagation speed to decrease (or the propagation time to increase). Therefore the typical propagation speed adjustment we have been using for microstrip cannot be sufficient since it is simply a constant (it only depends on [[epsilon].sub.r]).

I propose this as a better approach. Note that in the limit, the propagation time for a microstrip trace is the same as for a stripline trace. The limit is reached with an infinitely wide trace, an infinitely high [[epsilon].sub.r], or a zero separation between trace and plane. Under any other conditions, the propagation time slows down. Therefore, we should think of the microstrip propagation time as some fraction of the propagation time for the same trace in a stripline environment with the same dielectric coefficient. This latter figure is easy to calculate, as seen in (2).

Let's assume we know the propagation time for a trace in a stripline environment (the time for the signal to propagate from one end of the trace to the other). Even if this is the only way we know the propagation time, we can still calculate it based solely on (2). Now assume we have a microstrip trace with the same dielectric material between it and the reference plane. We want to determine the propagation time for a signal traveling down the microstrip trace. We know two things: The time cannot be shorter than what would be the propagation time through the air, and it cannot be longer than the propagation time for the stripline trace.

We can express the propagation time as a fraction of the stripline propagation time. This fraction cannot be greater than 1.0 (as long as the stripline propagation time), and there will be some lower limit that we probably don't need to be concerned with now (approximately 0.5 for FR-4). From the discussion above we know that this fraction will be a function of W (width of the trace), H (height above the plane) and [[epsilon].sub.r] (relative dielectric coefficient of the material). Therefore, we could express it as

Propagation time = Br * (propagation time in stripline) or

(4) Propagation Time = Br [square root of ([[epsilon].sub.r])]/11.8 ns/in 11.8

where Br is the fraction to be determined.

Some simulators (such as Mentor Graphics' Hyperlynx) permit collection of a set of data for determining propagation time for otherwise identical traces in both a stripline and a microstrip environment while changing trace parameters. Of course the propagation time for a trace in a stripline environment does not depend on the trace parameters, only the relative dielectric coefficient of the material.

I used regression analysis to estimate a better relationship between propagation time and the other variables: width, height and [[epsilon].sub.r]. The procedure was to generate 96 data points using HyperLynx. These data points were expressed as a fraction of what the propagation delay would be for a stripline trace in the same dielectric environment. These 96 data points were entered into a regression model using an Excel spreadsheet. The formula for the Br term works out to be

(5) Br = 0.8566 + 0.0294*Ln(W) - 0.00239*H - 0.0101*[[epsilon].sub.r])

with an [R.sup.2] value for this relationship of [0.96.sup.3]. The "average" error that can be calculated from this fit is about 1.1% of the actual data calculated by the HyperLynx model, and almost all data points are within +/- 2% of the actual, as determined by the simulation tool.

Model validation: One way to validate the results of a model like this is to compare the calculated results from the model against the actual input data. FIGURE 1 graphs the fraction (Br) as determined by the simulation tool, the calculated value for Br from (5), and the fraction calculated using the traditional approach in (3). As is obvious from the graph, the results determined from the regression model are far better than the results one would obtain from simply applying the traditional formula in (3).

[FIGURE 1 OMITTED]

Remember that electrical signals radiate electromagnetic waves. The propagation time for an electrical signal depends on how fast these electromagnetic waves can travel through the medium surrounding the wire or trace the signal is traveling along. Propagation times for stripline traces depend solely on the relative dielectric constant of the dielectric material surrounding the trace (assuming homogeneous material).

But propagation times for microstrip traces are more complicated, because the electromagnetic field is divided between the dielectric below and the air above. The relationship between the trace height above the reference plane (H), the trace width (W), and the relative dielectric coefficient of the material between the trace and the plane ([[epsilon].sub.r]) all interact to affect how the field divides between the dielectric and the air. This relationship can be quantified and expressed as a fraction of what the propagation time would be for the same trace in a stripline environment surrounded with material with the same relative dielectric constant. PCD&M

REFERENCES

(1.) Douglas Brooks, "Propagation Times, Yet Another Brain Teaser," Printed Circuit Design, August 2000, pp, 30.

(2.) IPC-D-317A, "Design Guidelines for Electronic Packaging Utilizing High-Speed Techniques," pp. 18. The reference given for this formula is H.R Kaupp, "Characteristics of Microstrip Transmission Lines," IEEE Transcript, vol. EC-16, no. 2, April 1967.

(3.) Details (and the interpretation of the value [R.sup.2] from Douglas Brooks, "Microstrip Propagation Times, Slower Than We Think," mentor.com/pcb/tech_papers.cfm.

DOUGLAS BROOKS is president of UltraCAD Design Inc. (ultracad.com). He is a frequent speaker at the PCB Design Conferences, Brooks can be reached at [e-mail].