The quality factor Q of a series resonant circuit is defined as:

Explanation:

Quality Factor (Q) of a Series Resonant Circuit

Definition: The quality factor (Q) of a series resonant circuit is a parameter that measures the sharpness or selectivity of the resonance in the circuit. It is defined as the ratio of the reactive power stored in the inductor or capacitor to the average power dissipated in the resistor at resonance. This parameter is significant in determining the efficiency and performance of resonant circuits used in applications such as filters, oscillators, and communication systems.

Correct Option Explanation:

The correct option is:

Option 4: The ratio of the reactive power of either the inductor or the capacitor to the average power of the resistor at resonance.

This definition accurately captures the essence of the quality factor (Q). In a series resonant circuit, resonance occurs when the inductive reactance (XL) equals the capacitive reactance (XC), resulting in a purely resistive impedance. At this condition, the circuit achieves maximum current flow, and the reactive power in the inductor or capacitor is at its peak. The quality factor (Q) is then calculated using the formula:

Q = (Reactive Power of Inductor or Capacitor) / (Average Power of Resistor)

Let us break this down:

• Reactive Power: Reactive power is the energy alternately stored and released by the inductor or capacitor during each cycle of AC oscillation. It is given by:
• Reactive Power of Inductor: \( Q_L = I^2 × X_L \)
• Reactive Power of Capacitor: \( Q_C = I^2 × X_C \)
• Average Power: Average power is the energy dissipated in the resistor during each cycle. It is given by:
• \( P_R = I^2 × R \)

At resonance, the inductive reactance (XL) equals the capacitive reactance (XC), and the current (I) in the circuit is maximized. Therefore, the quality factor (Q) can be expressed as:

\( Q = \frac{X_L}{R} \) or \( Q = \frac{X_C}{R} \)

This ratio indicates the sharpness of the resonance. Higher values of Q signify narrower bandwidth and greater selectivity, making the circuit more effective in applications requiring precise frequency selection.

Analysis of Other Options:

To further understand why the other options are incorrect, let us evaluate them:

Option 1: The sum of the reactive power of either the inductor or the capacitor and the average power of the resistor at resonance.

This option is incorrect because the quality factor (Q) is defined as a ratio, not a sum. The reactive power and average power represent entirely different types of energy in the circuit, and their sum does not provide any meaningful measure of resonance sharpness or selectivity.

Option 2: The ratio of the average power of the resistor to the reactive power of either the inductor or the capacitor at resonance.

This option is incorrect because it reverses the definition of the quality factor (Q). The correct definition involves the reactive power being divided by the average power, not the other way around. Such a reversal would yield a value that does not correspond to the intended characteristic of resonance sharpness.

Option 3: The product of the reactive power of either the inductor or the capacitor and the average power of the resistor at resonance.

This option is incorrect because the quality factor (Q) is not defined as a product. Multiplying the reactive power and average power would result in a value that does not have any physical relevance to the concept of resonance or selectivity.

Option 5: No additional information provided in the context of the problem.

This option is invalid as no specific details or explanation are given. It does not align with the standard definition or formula for the quality factor (Q).

Conclusion:

The quality factor (Q) is a crucial parameter in resonant circuits, serving as a measure of their sharpness and efficiency at resonance. It is defined as the ratio of the reactive power in the inductor or capacitor to the average power dissipated in the resistor. Among the given options, only Option 4 correctly describes this relationship, making it the correct answer. Understanding the Q factor is essential for designing and analyzing resonant circuits in various electrical and electronic applications.