Definition
We could informally define an stub as a transmission line that is not in the signal path. Keep this in mind. Some examples of could-be stubs are:
Traces in Y shape
Test points
Passive terminations
Vias in PCB
I say could-be stubs because its effect can be completely neglected depending of signal bandwidth. For example a side-to-side via could completely ignored for most applications but it is a concern in a PCIe Gen3 (8 Gbps) system.
The effect of an stub
Let us have a look to next figure. We have two transmission lines but at a certain point we have another one that it is not in the signal path and it is not terminated. [If the termination would be perfect there would be reflections and the story would be other].
In this systems we have two signals:
The direct one (in blue color),
The one that travels across the stub, reflects back and mixes with the direct one (drawn in red color).
Beware that the picture is a simplification, as there will be reflections in every discontinuity and these reflections propagate in both directions. However, it is very helpful to gain comprehension on what's happening.
Today, we are going to make an analysis in the frequency domain in order to gain some insight on how this work.
Let us imagine that stub is completely open at its termination. There will be no current to ground and thus the incident signal will reflect back with the same waveform and will appear in the joint point with the same amplitude and a delay. At certain frequency, the stub round trip delay will be a half cycle and the delayed signal signal collide with the incident wave, producing a null signal: a notch in frequency response. The same will happen when the delay is at one period and a half. We will have a periodic notch behavior.
If the stub is not completely open (and a capacitive load is very common in practice) the reflection will not be perfect and the frequency response of the circuit will be not a pure notch (complete attenuation) but more or less severe. The same will happen if the losses at the stub transmission line are significant: there will be some ripple, extra attenuation at these frequencies.
Let us use LTspice to make some insightful simulations. In the following system we will see:
A voltage generator configured to sweep frequency
A voltage divider. We could use transmission lines but would complicate much the understanding of how the stub operates.
A transmission line terminated in three possible capacitor values because capacity termination (parasitic) of an stub is quite common.
The stub is described as a transmission line with a arbitrary delay value of 5 ns. The notch frequency will be that in which the round trip delay in the stub is equal to a semi-period: the reflected signal will be in antiphase of the incident wave. Said in other words: the notch will be at the frequency in which the stub delay is one quarter of the signal period. This is the reason why we sometimes refer to as the quarter wave stub frequency. The notch takes place at quarter wave frequency and all odd multiples of it.
In the simulation we have included three possible termination capacitor values:
0.1 pF which is very similar to a pure open circuit: we see that the stub produces a very deep notch at 50 MHz.
1 pF, which is a very light load which produces an small shift in notch frequency and an attenuation that is deep but not so much,
10 pF, which produces a small but not negligible change in the line delay (as we will see in a future post) and thus reduces the resonant frequency. Less profound notch.
The effect of a stub is ripple in the frequency response of the system.
As a particular case, an unterminated stub produces notches in the frequency response.
As a rule of thumb, the effect of a stub can be neglected for frequencies under half the first notch frequency.
It worth looking at the magnitudes of the notch. A delay of 5 ns is the one we obtain of 1 m of RG58 cable! This means stubs are relevant only for very high frequencies. But we use very high frequencies in some of our circuitry!
The first notch frequency takes place when the one way transmission line delay of the stub is a quarter of the period.
We can neglect the stub effect for frequencies under one half of the notch frequency.
Today, we have done a frequency domain approach, that gives some vision of the magnitudes involved. However, does not lead to a direct application, for which a time domain analysis will be more satisfactory, but this is something that we will cover in next week post.
Just to end remember that
There are stubs that cannot be avoided. There are others that are simple bad design practices. Try to tame the former ones while avoiding the latter.