Zin of a Transmission Line Given ZL, Zo, l and f

[timeforplanb]

Homework Statement

What is the input impedance in polar coordinates of 222.85 cm long 50 ohms line with a load of 20-j40 ohms at 300MHz?

Homework Equations

lambda=v/f
Z_norm=Z_L/Z_O

The Attempt at a Solution

Normalizing the load impedance: Z_L/Z_O=(20-j40)/50=0.4-j0.8
Plotting that in the SC:

Calculate wavelength: lambda=velocity/frequency=3x10⁸/300x10⁶
I know that using c as my velocity is wrong, but I'm completely lost so I did that in my quiz anyways. The answer we get from the above equation is 1, which is completely useless if we use that to rotate towards the generator because we'll arrive at the same point.

So what I need help with is knowing what the transmission line length (in cm) is for, and how we will find the wavelength given the data.

 

[KataKoniK]

I would first clarify the units being used in this problem. Is the length of the line given in centimeters or meters? This is important because the wavelength will be calculated using the velocity of light in meters per second, so the length of the line should also be in meters for consistent units.

Assuming the length of the line is actually given in meters, we can proceed with the solution. The first step is to calculate the wavelength using the formula lambda = v/f, where v is the velocity of light (3x10^8 m/s) and f is the frequency (300 MHz). This gives us a wavelength of 1 meter.

Next, we can calculate the normalized load impedance using the formula Znorm = ZL/ZO, where ZL is the load impedance (20-j40 ohms) and ZO is the characteristic impedance of the line (50 ohms). This gives us a normalized load impedance of 0.4-j0.8.

To find the input impedance in polar coordinates, we can use the Smith chart. The transmission line length (in this case, 222.85 cm) is used to find the position on the Smith chart. We can use the normalized load impedance to find the position on the chart, and from there we can rotate towards the generator to find the input impedance in polar coordinates.

In summary, to find the input impedance in polar coordinates, we need to:
1. Calculate the wavelength using lambda = v/f
2. Calculate the normalized load impedance using Znorm = ZL/ZO
3. Use the transmission line length to find the position on the Smith chart
4. Use the normalized load impedance to find the position on the chart
5. Rotate towards the generator to find the input impedance in polar coordinates.