Question E9G09:
What third family of circles is often added to a Smith chart during the process of designing impedance matching networks?  

Choices:  
A: Constant-SWR circles  
B: Transmission line length circles  
C: Coaxial-length circles  
D: Radiation-pattern circles    

Correct Answer is  A: Constant-SWR circles  

Correct Answer: Constant-SWR circles

Why this is correct:
- A standard Smith chart already shows two families: constant-resistance circles and constant-reactance arcs (for normalized impedance).
- During matching design, a third family that is commonly used/overlaid is constant-SWR (constant |Γ|) circles. These are circles centered at the chart center that represent all points with the same magnitude of reflection coefficient, hence the same SWR.

Key ideas and formulas:
- Reflection coefficient for normalized impedance $z = Z/Z_0$:
  $$\Gamma = \frac{z - 1}{z + 1}$$
- SWR and reflection coefficient magnitude:
  $$\text{SWR} = \frac{1 + |\Gamma|}{1 - |\Gamma|}, \quad |\Gamma| = \frac{\text{SWR} - 1}{\text{SWR} + 1}$$
- Moving along a lossless transmission line rotates the point around a constant-SWR circle (|Γ| is constant):
  $$\Gamma_{\text{in}} = \Gamma_L e^{-j2\beta \ell}$$
- On the Smith chart:
  - Center = match ($\Gamma = 0$, SWR = 1).
  - Edge = $|\Gamma| = 1$ (infinite SWR).
  - Constant-SWR circles are concentric with the center.

Why the other choices are incorrect:
- Transmission line length circles: Line length is indicated by rotation around the chart and by the wavelength scales on the rim, not by a distinct family of circles.
- Coaxial-length circles: Not a recognized Smith chart construct; “coaxial” doesn’t change the geometry used on the chart.
- Radiation-pattern circles: Radiation patterns are antenna far-field plots, not related to impedance plotting on a Smith chart.

Rule of thumb:
- For matching on a lossless line, your trajectory due to changing line length stays on a constant-SWR circle; reactive elements move you along resistance/reactance families to reach the chart center (the match).