Microstrip Propagation Times Slower Than We Think

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1 Most of us have been using incorrect values for the propagation speed of our microstrip traces! The correction factor for ε r we have been using all this time is based on an incorrect premise. This article explains why and develops a superior model for estimating propagation speeds and propagation delays for microstrip configurations. Signal Propagation Speeds Electrical signals on wires and traces travel at the speed of light: 186,280 miles/second! That works out to ft/nanosecond, or 11.8 in/nanosecond, if you do the arithmetic. The speed of light slows down in any other medium by the square root of the relative dielectric coefficient of the medium. Signal propagation time is the inverse of this figure, or ns/in. In an earlier article on this site 1 I posed the question of 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 wire under water would be about 1/9th that of the wire in the air! Same signal, same copper, same electrons. But only one-ninth the propagation speed (9 times the propagation time). You see, moving electrons (current) create an electromagnetic field around the wire (or trace). The issue is not how fast the electrons can travel through the wire, the issue is how fast the electromagnetic field can travel through the medium it travels through. In the example above the medium the electromagnetic wave travels through is water. On our circuit boards the medium is the board material, usually (but not always) FR4. So, for example, a stripline trace in FR4 with an ε r of 4.0 would travel at the speed of light divided by the square root of 4 (which is 2) or about 6 /ns. Most of us are pretty comfortable with this figure. Microstrip Environment The propagation speed for a microstrip trace poses the problem that the trace is in a mixed environment. The medium underneath the trace is the board dielectric. 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 ε r that has been traditionally used for microstrip environments. It apparently derives from work done in and is as follows: But there is a problem with this. This correction factor 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. 3 Figure 1 suggests why. Electromagnetic Fields ε ' = ε +.67 Equation 1 r r Figure 1 case (a) illustrates how the electric field lines might be concentrated under a microstrip trace. The 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 upwards into the air before they curve back down to the plane. Copyright 2002 by UltraCAD Design, Inc. and Mentor Graphics Corporation 1

2 Case (b) illustrates the same trace, but with a thinner spacing between the trace and plane. Since the spacing between the trace and the plane is closer in case (b), the field intensity will be stronger than in case (a), and the field lines will drop more quickly to the plane. We can think of this situation as case (b) having a higher percentage of its field lines internal to the dielectric, and a lower percentage of its field lines in the air. Since the propagation speed is slower in the dielectric, we can speculate that the signal propagation speed for case (b) will be slower than case (a). Now consider case (c). Here we have a very wide trace. Most of the field lines will be in the dielectric between the trace and the plane, with only a small percentage of them above the trace in the air. Therefore, we might speculate that the speed of this trace will be slower yet. Let s imagine the trace width taken to its limit infinitely wide. If the trace is infinitely wide, then ALL the field lines will be within the dielectric. In fact, there is little conceptual difference in propagation speed between an infinitely wide microstrip trace and a stripline trace. BOTH have the electromagnetic field lines fully contained within the dielectric. The issue becomes looking at the concentration of field lines under the trace. If the (percentage) concentration of field lines increases under the trace, the propagation speed will slow down. Two things contribute to an increase in the concentration of field lines underneath the trace: 1. Bringing the trace closer to the plane. 2. Increasing the t race width. (Note: Increasing the trace thickness has a minor effect on propagation speed, but the effect is much smaller than with the other variables and will be ignored in this paper.) Each of these will cause the propagation speed to slow down. Therefore the typical propagation speed adjustment we have been using for microstrip (Equation 1) cannot be sufficient since it is simply a constant (it only depends on ε r ). Alternative Approach I propose this as an alternative approach. Note that, in the limit, the propagation speed for a microstrip trace is the same as for a stripline trace. The limit is reached with an infinitely wide trace or a zero separation between trace and plane. Under any other conditions, the propagation speed increases. Therefore, we should think of the microstrip propagation speed as some factor of the propagation speed for the same trace in a stripline environment surrounded by a material with the same dielectric coefficient. This latter figure is easy to calculate, it is simply the speed of light, 11.8 in/ns, divided by the square root of the relative dielectric coefficient: (a) Figure 1 Field concentration strength depends on several factors (b) (c) 2

3 PropagationSpeed= 11.8/ ε r in/ns Equation 2 (Note: Up to this point we have talked about propagation speed. From now on we are going to talk about propagation time, which is the inverse of propagation speed. Propagation time is expressed in units of time per unit length, or, when multiplied by length, simply in units of time.) 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.) If we know it no other way, we can at least calculate it based on Equation 2. Now assume we have a microstrip trace with the same dielectric material between it and the reference plane and 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: Propagation Time (Microstrip) = fraction * (Propagation Time (Stripline)) Equation 3 This fraction cannot be greater than 1.0 (equating to the stripline propagation time), and there will be some lower limit that we probably don t need to be concerned with (approximately 0.5 for FR4). From the discussion above we know that this fraction will be a function of W (width of the trace) and H (Height above the plane). Role of e r The relative dielectric coefficient also plays a role in this relationship. Consider the situation where there is no underlying plane. Let air extend for an infinite distance above the trace and dielectric material extend for an infinite distance below the trace. The propagation time (the inverse of propagation speed) (expressed per unit length) would be the average of the propagation times above and below the trace, or: AveragePropagationTime= where C = the speed of light. 1 C ε r + C 2 Equation 4 The propagation time for the trace if it were completely surrounded by the dielectric (which is the same as the stripline case) is given by the expression: e r StriplinePropagationTime= C Equation 5 Plugging Equations 4 and 5 into Equation 3, we derive that:.5 Fraction= +.5 e r Equation 6 3

4 See Fig. 2(b) Approx. range of FR4 Fraction W=10 W=100 Fraction W=10 W= Er Er (a) (b) Figure 2 Ratio of the propagation time of a microstrip trace compared to a stripline trace in the same relative dielectric environment, as a function of ε r. Figure 2(b) expands the horizontal axis of Figure 2(a). The two curves represent a 10 mil and 100 mil wide traces spaced 10 mils above the underlying plane. The data is derived from HyperLynx, as described later in the article. Now, if ε r is 1 (i.e. the traces are all surrounded by air), then the fraction is 1.0. That is, the microstrip case equals the stripline case (equals a trace in air). But if ε r approaches infinity, then the fraction goes to 0.5. That is, with very high dielectric coefficient, the propagation time for a wire or trace with air on one side and dielectric on the other would be half that of the propagation time for a wire or trace with dielectric completely surrounding it. This actually makes intuitive sense. As we have suggested above, the Figure 3 fraction in Equation 3 depends on Simple transmission line model in HyperLynx trace Width and Height above the plane. But this analysis shows that, all other things equal, the fraction also goes down as ε r goes up. Figure 2 illustrates this relationship. 4

5 Figure 4 The HyperLynx Edit Stackup menu window. Change stackup parameters here. Figure 5 The HyperLynx Edit Transmission Line Window. Propagation delay can be read directly from this screen. 5

6 See Figure 6 H=5 H=15 Equation 1 Stripline (a) H=5 H=15 Equation 1 Stripline (b) Figure 6. Propagation time varies with trace width. Note how it approaches the stripline value for very wide traces. Equation 1 provides a very poor estimate. Figure 6(b) expands the horizontal axis of Figure 6(a). 6

7 Microstrip Calculations Mentor s HyperLynx simulator is a convenient tool for investigating this relationship. Figure 3 illustrates a very simple transmission line model. Figure 4 illustrates the edit stackup screen where changes to dielectric coefficient, layer thicknesses, etc. can be made. Right-clicking the transmission line element opens the Edit Transmission Line Values window (Figure 5) where trace width and length can be varied and where the trace propagation delay can be read directly. For example, some data derived from the HyperLynx model are shown in Figure 6. The curves show how the propagation time for a two-inch trace changes with trace width. In this example the value for ε r is held constant at 4.3. The horizontal axis reflects increasing trace width along a scale from 1.0 mil to 1,000 mils. The two curves represent two different heights, H, above the plane, 5.0 mils and 15.0 mils respectively. Note how the two traces converge near the stripline propagation time for very wide traces. They also tend to converge at the faster end of the range at very narrow trace widths, where the lowest percentage of the electric field travels through the dielectric. If we used the traditional equation for calculating propagation time in microstrip, based on Equation 1, the estimate would be a constant at 279 ns. regardless of height or trace width! In Search of a Better Formula Regression analysis was used to estimate a better relationship between propagation time and the other variables, Width, Height, and ε r. The procedure used was to generate 96 data points using HyperLynx, as described above. These data points were expressed as a fraction of what the propagation delay would be for a stripline trace in the same dielectric environment. (A fractional delay means a faster propagation speed for the same length trace.) The intent is to generate a calculation model of the form: Propagation time (Microstrip) = fraction * (propagation time in stripline),or e r PropagationTime(microstrip)=fraction* 11.8 ns/in Equation 7 where the fraction variable is the variable we are trying to determine. These 96 data points were entered into a regression model using an Excel spreadsheet. The output from the regression model is shown in the Appendix to this article. The formula for the fraction term works out to be: Fraction = *Ln(W) *H *ε r Equation 8 Where: W = Trace Width (mils) H = Trace Height above the reference plane (mils) Ln = natural logarithm with an R 2 value for this relationship of The average error that can be calculated from this fit is about 1.1 percent of the actual data calculated by the HyperLynx model, and almost all data points are within +/- 2 percent of actual (as determined by the HyperLynx field solver). 7

8 Relative Propagation Time Fraction Predicted Fraction Equation Data Observations --> Figure 7 The fraction determined from Equation 4 (dotted red line) very closely fits the actual data (black line). Actual data is 96 data points derived from HyperLynx. The traditional estimate, based on Equation 1 (blue) isn t even close! 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. This graph is shown in Figure 7. As is obvious from the graph, the results determined from this procedure are far better than the results one would obtain from simply applying the traditional formula, Equation 1. For example, consider the following propagation time estimates for a 2.0 in. long microstrip trace with FR4 with ε r = 4.3: (times in ps): H(mil) W(mil) Prop.Time (Hy) Prop.Time (Est) Prop.Time (Eq. 1) where Hy refers to an estimate from HyperLynx, Est refers to an estimate from the model, and Eq.1 refers to an estimate based on the historical approach. 8

9 Summary Electrical signals radiate electromagnetic waves. The propagation speed 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 speeds for stripline traces depend solely on the relative dielectric constant of the dielectric material surrounding the trace (assuming homogeneous material.) But propagation speeds for microstrip traces are more complicated. That is 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 (ε r ) all interact to effect how the field divides between the dielectric and the air. The propagation time for a signal traveling along a microstrip trace can be best 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. Footnotes: 1. See Propagation Times and Critical Length; How They Interrelate; available at 2. There are numerous places where this equation can be found. See for example, IPC-D-317A, Design Guidelines for Electronic Packaging Utilizing High-Speed Techniques p.18. The reference given for this formula is H.R Kaupp, Characteristics of Microstrip Transmission Lines, IEEE Trans., Vol EC-16, No 2 April Eric Bogatin, GigaTest Labs, triggered my thinking about this when he showed slide 20 in his presentation Characterization by Measurement: The Value of High Bandwidth Measurements at the PCB East 2002 Conference, October 16, His presentation is reportedly available for download at 4. R 2 is a measure of goodness of fit. It is loosely related to the concept of correlation. An R 2 of 1.0 would imply a perfect fit of the equation to the data. Here, an R 2 of.96 should be considered as quite good. About the author: Douglas Brooks has a B.S and an M.S in Electrical Engineering from Stanford University and a PhD from the University of Washington. During his career has held positions in engineering, marketing, and general management with such companies as Hughes Aircraft, Texas Instruments and ELDEC. Brooks has owned his own manufacturing company and he formed UltraCAD Design Inc. in UltraCAD is a printed circuit board design service bureau in Bellevue, WA, that specializes in large, complex, high density, high speed designs, primarily in the video and data processing industries. Brooks has written numerous articles through the years, including articles and a column for Printed Circuit Design magazine, and has been a frequent seminar leader at PCB Design Conferences. His primary objective in his speaking and writing has been to make complex issues easily understandable to those without a technical background. You can visit his web page at and him at doug@ultracad.com. 9

10 Appendix Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 96 ANOVA df SS MS F Significance F Regression E-65 Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept E Height E Er E Ln(W) E Regression output for the evaluation 10