Transmission Lines & Impedance Matching

[AngelofMusic]

Hi all, I'm currently studying up on transmission lines for my next midterm, and I stumble onto this problem.
Given information: A 800 MHz transmission line that is lossless
- Vmax, Vmin, V_L for a given load impedance R_L.
- Characteristic impedance of a line
- length of a line & $$\lambda$$
Is it possible to find the amplitude of the voltage (Vmatched) assuming that instead of the original load impedance, the line is now matched?
So far, I've tried to use the voltage equation of a transmission line
$$V(z) = \frac{Z_0 V_g}{Z_0 + Z_g} e^{-j\beta z} [1+\gamma e^{-j2\beta (l-z)}]$$
I set V(z=l) = V_L, which is given. And I solve for the value of the big fraction in front. Then I use that equation again, except set $$\gamma = 0$$ for the matched case.
Is this the right approach to solving this problem?
I do get an answer out of this one, but it's not one that seems obvious or intuitive. Is there a better approach to this? Or an incredibly simple answer to this problem?
Any help would be appreciated!

 

[WalterContrata]

Standing Wave

Perhaps I am wrong, but I think there is a simple way.
Are Vmax and Vmin given? They result from adding or subtracting a wave going towards the load and a reflected wave from the load. Can you solve for the amplitudes of those two waves?
If the load were macthed, what is the amplitude of the incoming wave? The reflected wave?
Hope this helps.

 

[quicknote]

I would say that your approach to solving this problem is correct. The voltage equation of a transmission line is a fundamental equation that describes the voltage along the line at any given point. By setting the voltage at the end of the line (V(z=l)) equal to the given load impedance (V_L), you are effectively matching the line to the load. This is the goal of impedance matching – to minimize reflections and maximize power transfer.

However, I understand your concern about the seemingly non-intuitive answer you obtained. This could be due to the complexity of the equation, which involves the characteristic impedance, propagation constant, and length of the line. It may be helpful to plot the voltage equation for both the matched and unmatched cases to get a better visual understanding of the differences.

Another approach to this problem could be to use the Smith chart, which is a graphical tool commonly used for impedance matching in transmission lines. The Smith chart allows you to easily determine the required impedance for matching at a given frequency and line length. Using the chart, you can find the matched voltage amplitude without having to solve the complex voltage equation.

In conclusion, your approach to solving this problem is valid, but it may be helpful to also consider using the Smith chart or visualizing the voltage equation to gain a better understanding of the solution. Keep up the good work in your studies!