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Home PCB Design S11 vs. Return Loss vs. Reflection Coefficient: When Are They
the Same?


S11 VS. RETURN LOSS VS. REFLECTION COEFFICIENT: WHEN ARE THEY THE SAME?


Zachariah Peterson
|  Created: November 4, 2020  |  Updated: February 11, 2021


S-parameters are an often misunderstood topic for the intrepid RF/SI/PI
engineer, and I even find myself questioning my understanding at times. One
reason I’ve found that this useful set of signal integrity metrics is often
misunderstood is that there are copious online resources with different
definitions and explanations, all of which are given for different systems. In
addition, S-parameters sometimes get used interchangeably with return loss,
insertion loss, and reflection coefficient, and often without context.

In particular, there seems to be the occasional confusion around the difference
between return loss vs. reflection coefficient, as well as how these relate to
S11. The important thread here is this: all of these quantities describe the
reflection of a propagating wave off of a load, whether it’s a terminated
transmission line or a circuit network. Let’s look at these different
definitions and show when they start to correspond to each other.


BREAKING DOWN REFLECTANCE FORMULAS

Since S11 is sometimes used interchangeably with return loss and reflection
coefficient, are they ever really the same? The answer is they are sometimes
negatives of each other, and sometimes the magnitudes of all three quantities
are equal in limited cases and in certain frequency ranges. The formulas shown
below define return loss in terms of the reflection coefficient

Return loss vs. reflection coefficient definition

When you look at a graph of return loss, the negative sign is often omitted and
is sometimes used interchangeably with S11. For transmission lines, and likely
due to the way the data are displayed on graphs, S11 is often set equal to the
reflection coefficient defined between the source/load and the transmission line
characteristic impedance, which is only correct for a specific situation of a
long transmission line. In general, we need the line's input impedance, which
might be equal to the load impedance in specific circuit networks. However, as
we’ll see below, circuits with propagating waves will have S11 that eventually
converges to the reflection coefficient.

If you want to get a formula for S11 for a particular circuit network, it’s best
to look at the ABCD parameters. There is a universal formula for converting
between ABCD parameters and S-parameters. Once you have the S-parameters for a
channel, you can determine return loss using the formula shown above.

ABCD to S-parameter conversion.

Here, Z is the reference impedance for the incoming port (column 1 --> port 1,
and column 2 --> port 2). See this article from Caspers (starting on page 87)
for ABCD parameters for some common 2-port networks, including transmission
lines. Here we have a simple definition of S11:

 * S11 is defined as the reflection coefficient between the port impedance and
   the network's input impedance (looking from the source end to the load end).

If we have different port impedances, we have:

S-parameters from ABCD parameters with different reference impedance at each
port.

Finally, with either of the above equations, we can calculate S11.

The important point here is that S11 is not always equal to the reflection
coefficient between a source impedance and the impedance of a single element.
Because we're dealing with an input impedance, we need to consider the impedance
of all other elements in the circuit network, not just the first element
encountered in the network. S11 still describes reflection, it just uses the
input impedance rather than characteristic impedance. To see one important
example, let’s look at S11, return loss, and the reflection coefficient of a
transmission line terminated at a known impedance. As we'll see, the value of
S11 converges to the typical reflection coefficient between the source impedance
and the characteristic impedance as we make the line longer.


S11 VS. RETURN LOSS VS. REFLECTION COEFFICIENT FOR TRANSMISSION LINES

You can see when this occurs by comparing these different values for different
line lengths. As an example, I’ve run a simple calculation using the method I’ve
outlined in my upcoming IEEE EPS paper (more details from this conference are
found here).

Here I’ve simulated S11 for three transmission lines with dispersion in the
dielectric substrate. All three lines are identical except for their length, and
the lines are terminated with a matched source and load (50 Ohms nominal
impedance) with 1 pF load capacitance. For comparison, I’ve included a
calculation of the reflection coefficient using the standard formula. The
results are shown below.

Comparison of S11 on a transmission line connected to a capacitive load.

The results for this calculation are quite interesting. First, we see that a
signal on the short line (25 cm) with a severe mismatch at the capacitive load
can experience strong resonances above ~500 MHz, leading to strong oscillations.
In other words, above ~500 MHz, the line acts as a resonant cavity at particular
frequencies. When the line is physically longer, we start to see when the line
becomes electrically long as the resonances begin to appear at higher
frequencies (see the 2.5 m line), In addition, the magnitude of the resonances
is lower.

Once we look at the 25 m line (an extremely long channel for a PCB that would
only appear through a cable or waveguide interface), it’s clear that S11 is
almost identical to the reflection coefficient. The reflection coefficient curve
almost entirely overlaps the S11 curve for the 25 m line (in grey). The only
exception is seen from ~18-20 GHz, where we see a set of S11 resonances. I’ve
zoomed in on this region in the graph below.

Zoomed in view showing S11 compared to reflection coefficient for three
transmission lines.

A few conclusions can be drawn here:

 1. The reflection coefficient for a transmission line can almost be seen as an
    “average” of S11 for a short transmission line.
 2. When the line is longer, we don’t have return loss resonance problems with
    high VSWR and radiation, but can we still have the same mismatch regardless.
 3. Although longer lines do not have the same return loss resonance problems as
    shorter lines, the channel is now dominated by S21 or insertion loss.


CONVERGENCE OF S11 TO THE REFLECTION COEFFICIENT

Why should we compare different line lengths at all? From the above equations,
it should be clear that the reflection coefficient does not depend on line
length, which suggests we may be able to derive a relationship between S11 and
the reflection coefficient if the length can be eliminated from S11. By taking
limits to zero and infinity, we can see where S11 converges to return loss and
reflection coefficient. Starting from S11 in terms of the line’s ABCD parameters
(see the above article from Caspers), we can take the limit of |S11| for a line
with characteristic impedance Z0 and input port impedance ZL:

Convergence of S11 to reflection coefficient for a transmission line.

We’re now back to the reflection coefficient and return loss! Note that this is
applied at the input port (port #1), but we can take the same limit at the
output port (port #2) and get the same result for S22 looking backwards into the
transmission line. In addition, we have S11 = S22 when the two sides of the line
are terminated to the exact same impedance, so we get the same result for both
limits. This works out well for us, and it nicely shows where S11 and S22
correspond with return loss vs. reflection coefficient at each port.

Finally, using the definition of reflection coefficient with the source
impedance and transmission line input impedance, we can get back to the above
result by taking the same limit. This occurs because the line's input impedance
converges to the characteristic impedance as the line length goes to infinity.

Conceptually, this means that the line acts like an isolated source impedance to
create some reflection off the load impedance whenever the line is extremely
long. Switching this around, the line acts like a very long load at the input
port, so S11 will simply reduce to its reflection coefficient when the line is
infinitely long. You can then use this result to interpret the numerical results
from the integrated simulation features in Altium Designer®, which allow you to
create accurate impedance profiles for your board and simulate time-domain
waveforms of your signals.

Once you’ve qualified your design, you can share your design data on the Altium
365® platform, giving you an easy way to work with your design team and manage
your design data. We have only scratched the surface of what is possible to do
with Altium Designer on Altium 365. You can check the product page for a more
in-depth feature description or one of the On-Demand Webinars.

 

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ABOUT AUTHOR


ABOUT AUTHOR

Zachariah Peterson has an extensive technical background in academia and
industry. He currently provides research, design, and marketing services to
companies in the electronics industry. Prior to working in the PCB industry, he
taught at Portland State University and conducted research on random laser
theory, materials, and stability. His background in scientific research spans
topics in nanoparticle lasers, electronic and optoelectronic semiconductor
devices, environmental sensors, and stochastics. His work has been published in
over a dozen peer-reviewed journals and conference proceedings, and he has
written 1000+ technical blogs on PCB design for a number of companies. He is a
member of IEEE Photonics Society, IEEE Electronics Packaging Society, American
Physical Society, and Printed Circuit Engineering Association (PCEA), and he
currently serves on the INCITS Quantum Computing Technical Advisory Committee.



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