Question E4B03:
Which S parameter is equivalent to forward gain?  

Choices:  
A: S11  
B: S12  
C: S21  
D: S22    

Correct Answer is  C: S21  

Correct answer: C — S21

Why S21 is forward gain:
- In a 2-port network, $S_{21}$ is the forward transmission coefficient from port 1 (input) to port 2 (output) with the output port matched.
- Definition: $$S_{21} = \frac{b_2}{a_1}\Big|_{a_2=0}$$
  where $a_1$ is the incident wave at port 1 and $b_2$ is the outgoing wave at port 2. This directly represents forward gain (or insertion gain) when both ports are matched.

Why the others are incorrect:
- S11: Input reflection coefficient at port 1, not gain.
  - $$S_{11} = \frac{b_1}{a_1}\Big|_{a_2=0}$$
- S12: Reverse transmission (isolation) from port 2 back to port 1, not forward gain.
  - $$S_{12} = \frac{b_1}{a_2}\Big|_{a_1=0}$$
- S22: Output reflection coefficient at port 2, not gain.
  - $$S_{22} = \frac{b_2}{a_2}\Big|_{a_1=0}$$

Useful formulas and rules of thumb:
- Forward power gain (matched ports): $$G = |S_{21}|^2$$
- Gain in dB: $$G_{\text{dB}} = 10\log_{10}(|S_{21}|^2) = 20\log_{10}|S_{21}|$$
- Return loss (input): $$RL_1 = -20\log_{10}|S_{11}|$$
- VSWR from $S_{11}$: $$\text{VSWR} = \frac{1+|S_{11}|}{1-|S_{11}|}$$
- Practical cues:
  - Good amplifier: large $|S_{21}|$ (positive dB), small $|S_{12}|$ (good isolation), small $|S_{11}|$ and $|S_{22}|$ (good matches).