Questions concerning Smith Charts

[ethslan]

Have some problems concerning these questions relating to the Smith Chart:

A 300 ohm transmission line is terminated with an antenna with a (450Ω - j600Ω) load impedance.
(1) What is the normalized load admittance?
(2) How far from the load should a short circuit stub be placed to match the load?
(3) How long should the stub be?
(4) What is the VSWR before matching at the load?
(5) What is the VSWR after matching?

Attempt at Solution:

(1) (450Ω - j600Ω) / 300 = 1-5 - j2

The answers to questions 2 to 5 require plotting on the Smith Chart. I got the following answers:

(2) 0.25λ, (3) 0.334λ (4) 1.7 (5) 1

But I am not sure if these answers are correct.

 

[KataKoniK]

Can you please provide some guidance?

I would first clarify the context of the problem and make sure I understand the definitions and concepts involved. The Smith Chart is a graphical tool used in radio frequency engineering to solve transmission line problems, and it is based on the concept of impedance and admittance. In this problem, we are dealing with a 300 ohm transmission line terminated with an antenna that has a load impedance of (450Ω - j600Ω).

To answer the questions, we need to convert the load impedance to admittance by taking its reciprocal. This gives us (1/450Ω + j1/600Ω) = 0.0022 + j0.0017. This is the normalized load admittance.

To find the distance from the load to the short circuit stub, we need to plot the normalized load admittance on the Smith Chart. The Smith Chart is a graphical representation of the complex reflection coefficient, which is defined as the ratio of the reflected voltage to the incident voltage at a given point on the transmission line. The center of the Smith Chart represents a perfect match (VSWR=1) and the outer edge represents a complete mismatch (VSWR=infinity). The distance from the center of the chart to a point on the chart represents the magnitude of the reflection coefficient, and the direction from the center represents the phase of the reflection coefficient.

Using the Smith Chart, we can find that the normalized load admittance of 0.0022 + j0.0017 is located at a distance of 0.25λ from the center and in the positive reactance direction. This means that the short circuit stub should be placed at a distance of 0.25λ from the load, in the direction of the load.

To find the length of the stub, we need to determine the distance between the normalized load admittance and the short circuit stub on the Smith Chart. This distance represents the electrical length of the stub. In this case, it is 0.25λ. To find the physical length of the stub, we need to know the operating frequency and the velocity factor of the transmission line.

To find the VSWR before and after matching, we need to use the reflection coefficient formula:

Γ = (ZL - Z0) / (ZL + Z0)

Where Γ is the reflection coefficient, ZL is the load impedance, and Z0 is the characteristic