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## 18 May 2020

 Figure 1. Quarter wave stubs exhibit resonance which impacts S21 as broad dips.  The resonant frequency can be computed based on the characteristics of the transmission lines and its length.
Besides ripples and monotonic drop offs, the third pattern commonly seen in plots of S-parameters are broad dips due to stub resonances. Figure 1 shows an interconnect which includes a stub.  The interconnect is composed of 50 ohm uniform transmission lines which incorporate a branch point with an open at the end.

Not surprisingly, the open stub is a discontinuity. What's going to happen? At the open, it's going to reflect and head back. At the junction, it's going to branch. Some of it's going to go back toward the source and some of it's going to go forward toward the receiver.

At the receiver, we have two waves. We have the incident wave, and we have the wave that went down and back the stub and then continued to the receiver. Well, in going down the stub and back, it increased its path length and increased its phase shift compared to the incident wave. We're going to get the interference of these two waves, the incident plus the reflection from the stub. When that phase length difference between them is shifted 180 degrees out of phase, we're going to have cancellation. This will result in a minimum in the insertion loss.

We can calculate, given the length of this path, that when the down and back corresponds to half a wavelength is when we have the minimum. That means the down corresponds to a quarter of wavelength. We call this frequency at which we have a minimum the quarter wavelength stub or the quarter wave stub resonance.

We can calculate the resonant frequency (f_res) where that happens based on the length of that stub and the dielectric constant, the speed of light in air, using another good Figure of Merit. It's roughly 1.5 over the length of  the stub in inches. (This is for the resonant frequency in gigahertz with FR4 dielectric.)

We can confirm our calculation with a measurement.  Figure 2 is an example of a quarter wave stub resonance. In blue is the analysis of an SMA barrel measured up to 20 gigahertz. The blue trace shows a nice, beautiful, wonderful, transparent interconnect. The return loss is below minus 15 dB, a very low return loss, although we get some reflection because it's not a perfect 50 ohm match.
 Figure 2. Comparing the S-parameter response between an SMA barrel and an SMA ‘tee’. The tee  adds a 0.35 inch stub to the interconnect, which adds a quarter wave stub resonance at 5.9 GHz.

Now, we add a little stub by replacing the barrel with an SMA ‘tee’. It's still a through path, all we've done is add a little stub to the interconnect. Well, that stub is about 0.35 inches. We'd estimate the quarter wave stub resonant frequency based on the dielectric constant of the PTFE to be about 5.9 gigahertz.

Let’s see how that compares to the green trace of the SMA tee. Sure enough, at low frequency, the insertion loss S21 is near 0 dB; the return loss is a large negative dB value. As we get close to that 5.9 gigahertz, S21 drops and we get nothing coming to us through the interconnect.
We have this relatively broad resonance here at the broad dip in the insertion loss, right at the frequency we expect based on the length of that stub. At the same time, we have that dip. We have a lot of the signal reflecting back. Note also that it repeats again at about 17.7 GHz, another frequency that results in a 180-degree phase shift of the reflected wave.

Out next post in this series will cover sharper dips due to resonant coupling structures. In the meantime, watch the on-demand webinar Reading S-Parameters Like a Book by Dr. Eric Bogatin, Signal Integrity Evangelist, for a more in-depth treatment of this topic.