Introduction:
The quality factor (Q-factor) is a measure of how "good" or efficient a resonant system is. It is a dimensionless parameter that describes the bandwidth of a resonant circuit relative to its center frequency. Q-factor is given by the ratio of the center frequency to the bandwidth.

Explanation:
When the bandwidth of a resonant circuit increases, it means that the range of frequencies over which the circuit can resonate also increases. This leads to a decrease in the Q-factor. Let's understand this in detail:

1. Definition of Quality Factor:
The Q-factor of a resonant circuit is defined as the ratio of the center frequency to the bandwidth. It is mathematically represented as follows:

Q = f_center / Δf

Where:
Q = Quality factor
f_center = Center frequency
Δf = Bandwidth

2. Relationship between Bandwidth and Quality Factor:
As per the definition, Q-factor is inversely proportional to the bandwidth. This means that as the bandwidth increases, the Q-factor decreases. The relationship can be expressed as:

Q ∝ 1 / Δf

3. Interpretation:
The Q-factor represents the sharpness or selectivity of a resonant circuit. A higher Q-factor indicates a narrower bandwidth, meaning the circuit can respond to a smaller range of frequencies around its center frequency. Conversely, a lower Q-factor indicates a wider bandwidth, meaning the circuit can resonate over a larger range of frequencies.

4. Example:
Consider a resonant circuit with a center frequency of 1 kHz and a bandwidth of 100 Hz. The Q-factor of this circuit would be:

Q = 1000 Hz / 100 Hz
Q = 10

Now, if the bandwidth increases to 200 Hz while keeping the center frequency constant, the Q-factor would become:

Q = 1000 Hz / 200 Hz
Q = 5

As observed, the Q-factor decreases when the bandwidth increases.

Conclusion:
In summary, when the bandwidth of a resonant circuit increases, the Q-factor decreases. This is because a wider bandwidth allows the circuit to resonate over a larger range of frequencies, resulting in a lower selectivity or sharpness of the circuit. Conversely, a narrower bandwidth leads to a higher Q-factor, indicating a more efficient and selective resonant system.