# How to Use a Transmission Line Reflection Coefficient Correctly

February 23, 2021

### Key Takeaways

• All electromagnetic waves experience some reflection when they reach the interface between two media that have refractive index contrast.

• The reflection of a plane wave can be perfectly described using a reflection coefficient, but this is not the whole story in a complex structure like a printed circuit board.

• Designers need to use input impedance and S-parameters to describe reflections in transmission lines.

Plane waves reflecting off of water are described with a reflection coefficient.

New designers often refer to the reflection coefficient to describe reflections off the load end of a transmission line. Unfortunately, most designers who are not versed in signal integrity analysis may not know that the reflection coefficient is not a complete metric for describing reflection from the load on a transmission line. In a channel with finite size and definite geometry, signals will not propagate as plane waves and their reflection cannot be described using the transmission line reflection coefficient. Instead, we need S-parameters and input impedance to properly describe signal behavior at an impedance discontinuity along a transmission line.

## Transmission Line Reflection Coefficient vs. S-parameters and Input Impedance

All transmission lines are media used to direct propagation of an electromagnetic pulse or wave. It doesn’t matter whether we are dealing with digital pulses or harmonic AC waves, an incoming wavefront of an electromagnetic wave can reflect off of the interface between two materials. In optics, we say this occurs due to refractive index contrast. In electromagnetics, we say that this is due to a mismatch between the dielectric constants of the two media. In electronics, this is due to a mismatch in impedances (note that all of these quantities are related!).

The impedances involved in a transmission line connected to a load impedance Zin, source impedance ZS, and with input impedance Zin are shown below:

Transmission line schematic with input, source, and load impedances.

Deriving the reflection coefficient for a plane wave is a standard homework problem given in every electromagnetics class. The general definition for the transmission line reflection coefficient is:

Definition of transmission line reflection coefficient at the load.

Here, ZL is the load impedance and Z0 is the transmission line’s characteristic impedance. This quantity describes the voltage reflected off the load of a transmission line due to an impedance mismatch. Normally, this equation is derived while assuming the electromagnetic wave is a plane wave, and most treatments only consider what happens between the transmission line and the load component.

Although there is a reflection coefficient at the load end of a transmission line, there is also a reflection between the source and the input impedance of the transmission line:

Definition of transmission line reflection coefficient at the source.

Here, we need to understand the input impedance of the transmission line, which is also a function of the transmission line reflection coefficient as measured at the load.

### Input Impedance

The input impedance of a transmission line section is a function of the transmission line reflection coefficient. The input impedance is the impedance of the line looking into the source end. In other words, it is the impedance seen by the source due to the presence of the load and the transmission line’s characteristic impedance. We generally consider the load impedance to be composed of the true input impedance and any termination specified at the load. The input impedance of the transmission line is the value used for impedance matching at the source and is defined as:

Input impedance for a transmission line section of length l in terms of the transmission line reflection coefficient at the load end.

If there were perfect matching at the load, we would have the transmission line reflection coefficient at the load being equal to zero; the same applies at the source end. From the above equation, we can see that if the transmission line reflection coefficient is zero (perfect impedance matching), the input impedance is just the line’s characteristic impedance, regardless of the length of the line. There can still be a reflection if the source impedance and transmission line characteristic impedance do not match! The reflection just happens off of the source end, not at the load end.

In reality, the reflection coefficient is never equal to zero at all frequencies, which is why we use S-parameters, specifically S11 or return loss, to describe reflections over a broad range of frequencies.

### S11 (Return Loss)

S11 and return loss are closely related; the two quantities are reciprocals of each other. Most RF designers are familiar with the formula for return loss:

Definition of return loss in terms of transmission line reflection coefficient.

The input impedance and S11 (return loss) are both related to the transmission line reflection coefficient. Real S-parameters are complex functions of frequency and can have a complicated set of resonances/antiresonances; an example for a transmission line connected to a 1 pF load capacitance terminated to 50 Ohms is shown below.

Comparison of S11 and reflection coefficient at the input to a load component with 1 pF input capacitance.

Here, we see that the transmission line acts like a typical resonator cavity and has a definite resonance structure when the line is very short. Eventually, as the line gets longer, losses start to dominate, and the resonances in the S11 spectrum start to disappear.

From the above graph, it should be obvious that if we lengthen the line out to infinity, we’ll find that the input impedance at each port reduces to one of the standard reflection coefficient equations shown above. For real transmission lines operating at practical frequencies, you must describe signal behavior in terms of the input impedance and S-parameters, particularly when the line is short.

## Examine Reflection Using S-parameters and Input Impedance

As we’ve seen above, the S-parameters and input impedance of a transmission line are the correct tools for describing signal reflection at the load end of a transmission line. The reflection coefficient is only part of the story. Because input impedance only depends on the reflection coefficient and propagation constant, it can be approximated as long as you can approximate the propagation constant of the transmission line.

S-parameters are the standard tool for describing reflection and losses in a propagating signal as it travels through a channel. If you want to extract all the information needed to understand your interconnects, you can use the following process to determine S-parameters and input impedance of your transmission lines:

1. Use PCB layout software with an integrated field solver to extract the S-parameters of your channel.

2. Use the standard S-parameter to ABCD parameter conversion to determine the broadband propagation constant of your channel.

3. Calculate the reflection coefficient at the load end using the load impedance spectrum (including load capacitance!) and the transmission line’s impedance spectrum.

4. Calculate the input impedance using the results from Step 3.

With this process, you’ll have the input impedance and S-parameters for transmission lines in your PCB layout without the need to gather measurements.

When you need to analyze signal attenuation and reflection in your transmission lines, you need to use the best PCB design and analysis software. Allegro PCB Editor from Cadence integrates with the S-parameter extraction features in Sigrity Extraction and includes a complete set of PCB design and layout features. You’ll have a complete set of design features for building and optimizing your next high speed/high frequency system. You can also use InspectAR to accurately assess and improve PCBs using augmented reality and intuitive interaction. Inspecting, debugging, reworking, and assembling PCBs has never been faster or easier.