Question E5A09:
How is the Q of an RLC parallel resonant circuit calculated?  

Choices:  
A: Reactance of either the inductance or capacitance divided by the resistance  
B: Reactance of either the inductance or capacitance multiplied by the resistance  
C: Resistance divided by the reactance of either the inductance or capacitance  
D: Reactance of the inductance multiplied by the reactance of the capacitance    

Correct Answer is  C: Resistance divided by the reactance of either the inductance or capacitance  

To calculate the Q of a parallel RLC resonant circuit, use the shunt (parallel) resistance divided by the magnitude of the reactance of either the L or the C at resonance.

Correct answer (C): Resistance divided by the reactance of either the inductance or capacitance

1) Why this is correct
- For a parallel resonant circuit, the unloaded quality factor is:
  $$Q_p = \frac{R_p}{X}$$
  where:
  - $R_p$ = equivalent parallel (shunt) resistance representing losses
  - $X$ = magnitude of the reactance of either $L$ or $C$ at resonance ($X_L = X_C$)
- At resonance, $X_L = \omega_0 L$ and $X_C = \frac{1}{\omega_0 C}$, and either can be used:
  $$Q_p = \frac{R_p}{X_L} = \frac{R_p}{X_C}$$

2) Why the other choices are incorrect
- A: Reactance divided by resistance ($X/R$) — this is the series-resonant formula, not parallel. For parallel, it’s $R/X$.
- B: Reactance multiplied by resistance ($X \times R$) — has units of ohm-squared and cannot be Q (which is dimensionless).
- D: Reactance of L multiplied by reactance of C ($X_L \times X_C$) — also has units of ohm-squared and doesn’t relate to losses or bandwidth.

3) Useful formulas and rules of thumb
- Parallel resonant Q:
  $$Q_p = \frac{R_p}{X} \quad \text{with} \quad X = \omega_0 L = \frac{1}{\omega_0 C}$$
- Series resonant Q (for comparison):
  $$Q_s = \frac{X}{R_s}$$
- Bandwidth relationship (for simple resonators):
  $$Q \approx \frac{f_0}{\text{BW}_{3\,\text{dB}}}$$
- Resonant frequency:
  $$\omega_0 = 2\pi f_0 = \frac{1}{\sqrt{LC}}$$

Key point: Series uses $X/R$; parallel uses $R/X$.