Suppose I have a transmission line of length \$L\$ with characteristic impedance \$Z_0\$ and load voltage \$Z_L\$. If it is connected to a ideal sinusoidal voltage source \$V_s\$, then the voltage across the load \$V_L=V_s \frac{2Z_L}{Z_L+Z_0}\$. However, I think that when L becomes very small (compared to wavelength of the source) , the line essentially acts as a short and I should get \$V_L=V_s\$ across the load. I do not understand how this approximation can be obtained by reducing the length since \$V_L\$ got from transmission line model seems to be independent of the length (and of source frequency).
Electrical – lum – How does short length of transmission line approximate to a wire
impedance-matchinglumpedreflectiontransmission line
Best Answer
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.