I don't understand how amplitude of incident and reflected voltage of wave equation comes out with
\$e^{j2\beta z}\$ next to reflection coefficient. What is the meaning of it? Why is it part of the amplitude? How can I derive it?
Electrical – Voltage Standing Wave Ratio Derivation
microwavewaveform
Related Solutions
If the transmission line is perfectly matched (or its length is infinite), there is no reflection.
Therefore the voltage at any point x, and any time t is (assuming losesless line): $$V(x,t) =A\cdot \sin{(\phi + 2\cdot\ \pi\ \cdot ( f\cdot t - \frac{x}{\lambda}))} $$ where λ is the wavelength, and Φ is the phase.
If you choose any x point, you can rewrite the voltage as: $$V(t) =A\cdot \sin{(\theta + 2\cdot\ \pi\ \cdot f\cdot t )} $$
As you can see, the amplitude of that sinewave is A, regardless the particular x.
In a two port network (e.g. your quarter-wave transmission line) \$S_{21}\$ is the transmission coefficient only if port 2 is matched. This is an important distinction that trips up a lot of people.
Let's define the reference impedance of port 1 and port 2 of the transmission line as \$\sqrt{2}\Omega\$, the same impedance as the transmission line. Then the \$S_{21}\$ of the two port network (ie the quarter-wave transmission line with characteristic impedance \$\sqrt2\$) is \$-j\$ like you calculated. Now say you connect a \$\sqrt{2}\Omega\$ source to port 1 and a \$\sqrt{2}\Omega\$ load to port 2. The fact that we matched port 2 to its reference impedance now implies that $$T_{load}=S_{21}=-j$$
But now suppose we replace the \$\sqrt{2}\Omega\$ load at port 2 with a \$2\Omega\$ load and drive port 1 with a \$1\Omega\$ source. Port 2 (and 1) is no longer matched! This means that we can no longer equate \$T_{load}\$ and \$S_{21}\$. Note that the \$S_{21}\$ of the network hasn't changed though, only the transmission coefficient. So in other words, we previously defined the reference impedance of port 2 as \$\sqrt{2}\Omega\$, so port 2 "expects" to see \$\sqrt{2}\Omega\$. Instead it sees \$2\Omega\$, meaning the reflection coefficient at port 2 has changed from the matched condition. (Note that matched does not mean zero reflection coefficient in this case, only that the load impedance is equal to \$\sqrt{2}\Omega\$).
So how can we solve this problem? One way is to redefine the reference impedances for port 1 and 2 as \$1\Omega\$ and \$2\Omega\$, respectively. Then you would have to recalculate \$S_{21}\$, essentially how you've done using your second (correct) method. In this case, \$S_{21}\$ would correspond to the transmission coefficient.
Another way of looking at this problem is in terms of power conservation. The transmission line is assumed to be lossless, thus the only potential spots for power dissipation are in the source impedance and in the load impedance. But we already know that the source sees no reflected wave, so we can ignore any loss in the source. Similar to standard electromagnetic waves incident on a boundary, the power of an incident voltage wave must equal the total power of the transmitted and reflected voltage waves.
So let's consider port 1: there is an incident voltage wave, \$V_0\$. There is no reflected wave, as you have calculated. The transmitted wave's power must all be dissipated in the \$2\Omega\$ load. The incident power is $$ P_i=\frac{|V_0|^2}{Z_0}=\frac{|V_0|^2}{1\Omega}=|V_0|^2 $$ The reflected power is $$ P_r=0 $$ The transmitted power is $$ P_t=\frac{|V_{load}|^2}{2Z_0}=\frac{|V_{load}|^2}{2\Omega} $$ Now performing the power conservation step, $$ P_i=P_r+P_t => |V_0|^2 = \frac{|V_{load}|^2}{2\Omega} $$ From which we find $$ |V_{load}|=\sqrt{2}|V_0| $$ like you found in your correct derivation. Now of course, this method won't give you phase information, but power conservation is a good way to verify a result found from a different method.
A way of interpreting this result is that voltage amplitude will change to whatever it needs to be in order to maintain power conservation, similar to the electric field incident on a boundary. The amount it changes depends on the square root of the ratio between the load and the source (in this case \$\sqrt{2}\$), which follows from the ratio for power.
Best Answer
A transmission line introduces two primary factors that account for the application specific behavior of the transmission line: attenuation and phase delay. These factors dominate what differentiate a transmission line from normal lumped circuit analysis. While they are typically present in a lumped circuit, their effect is so minimal that it can be ignored. However, once the interconnections become longer than approximately 1/10 wavelength, the transmission line effects will become measurable and therefore should be factored into the analysis.
The attenuation constant and phase constant of a transmission line are expressed as α and β respectively. Both of these constants are frequency dependent. Combined, they form the complex propagation constant γ given as γ=α+jβ. Both attenuation and phase delay increase as the length of the transmission line increases. Skipping the lengthy proof, we can say that this effect is expressed as the complex term eγl where l is the length of the line. Given the voltage or current of the signal generator, it is now possible to predict the amplitude and phase of the voltage and current at any length l of a transmission line terminated in its characteristic impedance ZO or more generally, the amplitude and phase of the voltage and current of the incident (generator inducted) wave.
When a transmission line is not terminated in its characteristic impedance, a reflection of part or all of the incident energy occurs. The characteristics of the reflection include an amplitude and phase component. This complex term is given as Γ.
The reflected energy must also travel along the length of the line back toward the generator. As a result, the reflected wave once again encounters the eγl effect of the complex propagation constant but traveling in the reverse direction. But recall that what caused the reflection may also have introduced some attenuation (the returned energy compared to the incident energy) and phase change. The combined effect is therefore Γeγl. But this expresses the combined effect at the junction of the transmission line and the terminating impedance. To allow this effect to be determined at any point z along the line, as measured from the terminating impedance, we must account for the effect of the complex propagation constant as the incident energy travels from point z toward the terminating impedance and again as the reflected energy returns to point z. Since actual distance traveled is 2z, we arrive at the formula Γeγ2z to express the magnitude of the reflected wave at point z. By convention, reflected current is expressed as a negative term to indicate its direction of travel.
Therefore to determine the voltage or current at any point z along the transmission line, one must simply add the incident and reflected terms at position z along the line to arrive at the desired result. This is also defined as the amplitude of the standing wave at position z. It is also possible then to calculate the impedance at point z since the relative or absolute amplitude and phase of the voltage and current are easily calculated.