Question E9G03:
Which of the following is often determined using a Smith chart?  

Choices:  
A: Beam headings and radiation patterns  
B: Satellite azimuth and elevation bearings  
C: Impedance and SWR values in transmission lines  
D: Point-to-point propagation reliability as a function of frequency    

Correct Answer is  C: Impedance and SWR values in transmission lines  

Correct answer: C — Impedance and SWR values in transmission lines

1) Why C is correct
- A Smith chart is a graphical tool based on the complex reflection coefficient. It is used to:
  - Plot and read complex impedances/admittances on transmission lines.
  - Move along a line (toward/away from the generator) to find input impedance.
  - Read the magnitude of the reflection coefficient and thus determine SWR.
- These are exactly “impedance and SWR values in transmission lines.”

2) Why the other choices are incorrect
- A: Beam headings and radiation patterns are antenna-pattern problems (azimuth/elevation plots), not Smith chart tasks.
- B: Satellite azimuth/elevation bearings come from orbital geometry and observer location, not transmission-line calculations.
- D: Point-to-point propagation reliability involves ionospheric models and statistics (e.g., VOACAP/ITU), not Smith charts.

3) Useful formulas and rules of thumb
- Normalize impedance (for plotting):
  $$z = \frac{Z_L}{Z_0}, \quad y = \frac{1}{z}$$
  The chart center is $z = 1 + j0$ (matched).
- Reflection coefficient at the load:
  $$\Gamma_L = \frac{Z_L - Z_0}{Z_L + Z_0}$$
  The Smith chart radius equals $|\Gamma|$ (constant-SWR circles).
- SWR from reflection coefficient:
  $$\text{SWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}$$
- Input impedance along a lossless line of length $\ell$:
  $$Z_{in} = Z_0 \,\frac{Z_L + j Z_0 \tan(\beta \ell)}{Z_0 + j Z_L \tan(\beta \ell)}, \quad \beta = \frac{2\pi}{\lambda}$$
  On the Smith chart, moving toward the generator is clockwise; one full revolution corresponds to $\lambda/2$ change in line length.
- Quarter-wave transformer (purely resistive case):
  $$Z_{in}(\ell = \lambda/4) = \frac{Z_0^2}{Z_L}, \quad Z_{0,\text{QWT}} = \sqrt{R_S R_L}$$
- Admittance from impedance: rotate 180° through the chart center (since $y = 1/z$).