Bandwidth and Selectivity
How to Use
- Q factor slider — increase Q to sharpen the peak and narrow the bandwidth (1–100).
- f₀ slider — shift the resonant center frequency (100 Hz – 5 kHz).
- Multi-Q Curves button — overlay five curves at Q = 2, 5, 10, 25, 50 to compare selectivity.
- dB Scale button — toggle between linear and decibel (dB) vertical axis.
What to Observe
- The blue shaded region is the passband — frequencies passed with |H(f)| > 0.707 (−3 dB).
- The −3 dB line marks the cutoff points f₁ and f₂; the bandwidth BW = f₂ − f₁.
- Higher Q → narrower bandwidth → the circuit is more selective (passes fewer frequencies).
- Lower Q → wider bandwidth → the circuit passes a broad range of frequencies.
- With Multi-Q mode, notice that all curves peak at 1.0 at f₀, but higher-Q curves fall off more steeply.
- In dB mode, the −3 dB cutoff appears as a horizontal line at −3 dB.
- Real-world radios use high-Q resonant circuits to select one station while rejecting adjacent channels.
Key Equations
\[\text{Bandwidth:} \quad BW = \frac{f_0}{Q}\]
\[\text{Cutoff frequencies:} \quad f_{1,2} = f_0\sqrt{1 + \frac{1}{4Q^2}} \mp \frac{f_0}{2Q}\]
\[\text{Frequency response:} \quad |H(f)| = \frac{1}{\sqrt{1 + Q^2\!\left(\dfrac{f}{f_0} - \dfrac{f_0}{f}\right)^2}}\]
\[\text{At } f_1, f_2: \quad |H| = \frac{1}{\sqrt{2}} \approx 0.707 \quad (\text{half power, } {-3}\text{ dB})\]
| Q | BW (% of f₀) | Selectivity |
|---|---|---|
| 1 | 100% | Poor |
| 5 | 20% | Fair |
| 10 | 10% | Moderate |
| 25 | 4% | Good |
| 100 | 1% | Excellent |
Key Concepts
- Bandwidth BW: Frequency range between the −3 dB points; BW = f₀/Q
- Selectivity: Ability to distinguish between closely spaced frequencies; higher Q = better
- Passband: Frequencies passed with less than 3 dB attenuation
- Stopband: Frequencies attenuated significantly below the −3 dB level
- −3 dB point: Frequency where power drops to half its peak value (voltage to 70.7%)
- Q factor: Quality factor; ratio of center frequency to bandwidth; measures sharpness