Microstrip Propagation Times
Slower Than We Think
Most of us have been using incorrect values for the propagation speed of our microstrip traces! The correction
factor for εr 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 0.9835
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 0.085 ns/in.
In an earlier article on this site1 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 εr 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 εr that has been traditionally used for microstrip environments. It apparently derives
from work done in 19672 and is as follows:
ε 'r = 0.475ε r + .67
Equation 1
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
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
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Microstrip Propagation Times
Slower Than We Think
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).
(a)
(b)
(c)
Figure 1
Field concentration strength depends on several factors
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 εr).
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:
2
Microstrip Propagation Times
Slower Than We Think
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 εr
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:
εr
1
+
C
AveragePropagationTime= C
2
Equation 4
where C = the speed of light.
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:
StriplinePropagationTime=
er
C
Equation 5
Plugging Equations 4 and 5 into Equation 3, we derive that:
Fraction=
.5
+.5
er
Equation 6
3
Microstrip Propagation Times
Slower Than We Think
1
1
See Fig. 2(b)
Approx. range of FR4
0.95
0.95
0.9
W=10
W=100
0.85
Fraction
Fraction
0.9
W=10
W=100
0.85
0.8
0.8
0.75
0.75
0
20
40
60
80
100
Er
(a)
120
0
2
4
6
8
10
Er
(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 εr. 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 εr 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 εr 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.
Figure 3
As we have suggested above, the
Simple
transmission
line model in HyperLynx
fraction in Equation 3 depends on
trace Width and Height above the
plane. But this analysis shows that, all
other things equal, the fraction also goes down as εr goes up. Figure 2 illustrates this relationship.
4
Microstrip Propagation Times
Slower Than We Think
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
Microstrip Propagation Times
Slower Than We Think
355
345
335
H=5
325
H=15
3 15
Equation 1
See Figure 6
305
Stripline
295
285
275
0
200
400
600
800
1000
1200
(a)
3 55
345
335
H=5
325
H=15
3 15
Equation 1
Stripline
305
295
285
2 75
0
20
40
60
80
10 0
(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
Microstrip Propagation Times
Slower Than We Think
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 εr 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 εr. 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
PropagationTime(microstrip)=fraction*
er
ns/in
Equation 7
11.8
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 = 0.8566 + 0.0294*Ln(W) - 0.00239*H - 0.0101*εr
Where: W = Trace Width (mils)
H = Trace Height above the reference plane (mils)
Ln = natural logarithm
Equation 8
with an R2 value for this relationship of .96.4 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
Microstrip Propagation Times
Slower Than We Think
1.05
Relative Propagation Time
1
0.95
Fraction
Predicted Fraction
0.9
Equation 1
0.85
0.8
0.75
96 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
εr = 4.3: (times in ps):
H(mil) W(mil) Prop.Time (Hy)
5
5
292.5
5
15
309.2
5
50
327.3
10
5
288.7
10
15
300.5
10
50
317.6
Prop.Time (Est)
298.2
309.6
322.0
294.0
305.4
317.8
Prop.Time (Eq. 1)
354.1
354.1
354.1
354.1
354.1
354.1
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
Microstrip Propagation Times
Slower Than We Think
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 (εr) 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
www.mentor.com/pcb/tech_papers.cfm
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 1967.
3. 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, 2002. His presentation is reportedly available for download at www.gigatest.com.
4. R2 is a measure of “goodness of fit.” It is loosely related to the concept of correlation. An R2 of 1.0 would imply a
perfect fit of the equation to the data. Here, an R2 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 1992.
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
http://www.ultracad.com and e-mail him at doug@ultracad.com.
9
Microstrip Propagation Times
Slower Than We Think
Appendix
Regression Statistics
Multiple R
0.980693347
R Square
0.961759441
Adjusted R Square
0.960512467
Standard Error
0.011619276
Observations
96
ANOVA
df
Regression
Residual
Total
Intercept
Height
Er
Ln(W)
3
92
95
Coefficients
0.856636206
-0.002385078
-0.010093541
0.029414146
SS
MS
F
Significance F
0.312383561 0.10412785 771.2742 4.73348E-65
0.012420696 0.00013501
0.324804257
Standard Error
t Stat
P-value
Lower 95% Upper 95%
0.006692809 127.993521 1.9E-105 0.843343709 0.8699287
0.000323726 -7.3675923 7.28E-11 -0.00302802 -0.0017421
0.00141078 -7.1545818 1.98E-10 -0.01289547 -0.0072916
0.000627581 46.8691096 5.26E-66 0.028167716 0.03066057
Regression output for the evaluation
10