Electrical – Transmission Line open and short analysis

Background: The equations derived in transmission line analysis are based off of standing waves. The idea is that the various impedance in the line and load will create reflections that will add and subtract until steady state is achieved. Therefore the equation you have used is based on wave reflection and standing wave theory. The "z" component which is often lost in these simplified forms is the distance from the load (and becomes negative) towards the source. This component is often set to 0 in order to find the voltage at the load.

Problem: The equation you have suggested comes with various assumptions you must consider. The main assumptions is that the forward and reverse propagating waves exhibit the same reflection and thus can both be interchanged via a coefficient. This allows us to use a multiplier to simply calcuate the reverse wave as a fraction of the forward (this is the same theory we use for optic reflections). This factor/coefficent is called the reflection coefficient and it is found using the Zo and Zl of the system.

$$\Gamma = {{Zl - Zo} \over {Zl + Zo}}$$

Where \Gamma is the reflection coefficient

Solution: So yes you could go back to the original derivation and use length and location to show that it can be used for what I could call zero-length-transmission-lines but we can actually use what you have to get to it.

The point is to remove the idea of "reflections" from the equation you have. The way we can remove it while still holding true to the original derivations is simply to state that Zo = Zl and thus no reflection will occur at the junction. If you substitute Zl = Zo into the reflection coefficient you will see it becomes zero and thus there is no reflection. Your equation stated above will also simplify to VL = Vs.

Considerations: I would recommend looking into the theory behind the derivation of these equations to better understand how they should/shouldn't be used. Also, often we use different models for what seem to be the same thing simply because certain models provide assumptions others don't. This can be seen in the consideration of capacitance in medium-length transmission lines that is not found in short-length transmission lines.

It's ideal when the trace width ends up being the same width as your pad because the pad simply works as an extension of the transmission line. However, in most cases this is not possible, but don't panic because this is not the end of the world. When the pad is much wider than the 50Ω trace it's a good idea to have a void on the ground plane(s) immediately under the pad to reduce the parasitic capacitance represented by the pad and the GND underneath. The module and the antenna manufacturer (I'm assuming you're using a chip antenna) should provide guidance on what to do.

Having said that, simply use a 50Ω microstrip transmission line with the width you calculated based on the dielectric height and dielectric constant you have. I would use a clearance to GND on the top layer of at least two times the trace width. It's very critical that you place GND vias underneath the GND pads next to your module antenna feed pad. The same for the GND pads of your antenna (I'm assuming you're using a chip antenna).

I don't recommend adding a matching network for the purpose of compensating for the presence of a non-50Ω transmission line. The problem is that the matching will be narrowband and you need it to work properly over a wide frequency range. Having said that, it's always a good idea to have provision for antenna impedance matching (instead of transmission line impedance compensation). I would keep the pi network right next to the antenna you already have. I would keep the shunt components a little closer to the series component though. I can't tell what component size you're using - I would use 0201s (so you have a wider range of component values below the self-resonance frequency you can use).