Question E9G05:
Which of the following is a common use for a Smith chart?  

Choices:  
A: Determine the length and position of an impedance matching stub  
B: Determine the impedance of a transmission line, given the physical dimensions  
C: Determine the gain of an antenna given the physical and electrical parameters  
D: Determine the loss/100 feet of a transmission line, given the velocity factor and conductor materials    

Correct Answer is  A: Determine the length and position of an impedance matching stub  

Correct answer: A — Determine the length and position of an impedance matching stub

Why A is correct:
- The Smith chart graphically shows how impedances transform along a transmission line and lets you:
  - Move from the load toward the generator by an electrical length to find where to place a shunt or series stub.
  - Read the required stub length (open or short) to cancel the reactive part and achieve a match.
- Workflow:
  1) Normalize the load: $z_L = Z_L/Z_0$.
  2) Move toward the generator along the constant-VSWR circle to a convenient point (often where the normalized conductance is 1 for shunt-stub solutions).
  3) Switch to admittance ($y = 1/z$) if using a shunt stub.
  4) Read the needed stub susceptance/reactance and its length from the wavelength scales.

Why the others are incorrect:
- B: Determine the impedance of a transmission line, given the physical dimensions
  - The Smith chart needs $Z_0$ as an input; it does not compute $Z_0$ from geometry.
  - $Z_0$ comes from transmission-line geometry and dielectric constants, e.g., for coax:
    $$Z_0 \approx \frac{138}{\sqrt{\epsilon_r}} \log_{10}\!\left(\frac{D}{d}\right)$$
    where $D$ is shield I.D., $d$ is conductor O.D., and $\epsilon_r$ is dielectric constant.
- C: Determine the gain of an antenna
  - Antenna gain is from radiation patterns and aperture/efficiency, not from impedance matching plots. Example relation:
    $$G \approx \eta \frac{4\pi A_e}{\lambda^2}$$
- D: Determine the loss/100 feet of a transmission line
  - Line loss per length depends on conductor and dielectric losses from manufacturer data; the Smith chart can show mismatch effects but not intrinsic loss from velocity factor and materials alone.

Useful formulas and rules of thumb:
- Reflection coefficient:
  $$\Gamma = \frac{Z_L - Z_0}{Z_L + Z_0}, \quad \text{VSWR} = \frac{1+|\Gamma|}{1-|\Gamma|}$$
- Input impedance at distance $l$ from the load:
  $$Z_{\text{in}} = Z_0 \frac{Z_L + j Z_0 \tan(\beta l)}{Z_0 + j Z_L \tan(\beta l)}, \quad \beta = \frac{2\pi}{\lambda}$$
- Quarter-wave transformer:
  $$Z_{\text{in}}(l=\lambda/4) = \frac{Z_0^2}{Z_L}$$
- Stub reactance (lossless):
  - Shorted stub: $X_{\text{in}} = Z_0 \tan(\beta l)$
  - Open stub: $X_{\text{in}} = -Z_0 \cot(\beta l)$
- Smith chart rotation:
  - One full revolution corresponds to $0.5\lambda$ along the line.
  - Moving toward the generator is clockwise; toward the load is counterclockwise.