Questions: Quality Factor and Bandwidth Tradeoffs

Question 1

An AM radio designer wants to select one station from a band where stations are spaced 10 kHz apart, centered at 1 MHz. What minimum Q must the resonant circuit have to prevent adjacent stations from interfering?

• A. Q = 10, giving a 100 kHz bandwidth around 1 MHz
• B. Q = 100, giving a 10 kHz bandwidth that can just resolve adjacent stations
• C. Q = 1000, giving a 1 kHz bandwidth for maximum selectivity
• D. Q = 0.1, giving a broad 10 MHz bandwidth to capture all stations

Answer: (B)

The relationship Q = f₀/BW gives Q = 1 MHz / 10 kHz = 100. A bandwidth of 10 kHz centered at 1 MHz just resolves stations spaced 10 kHz apart. Lower Q gives a wider bandwidth, letting adjacent station energy leak through. Higher Q would be even more selective but might excessively attenuate the edges of the desired station's own signal (AM stations have sidebands spanning ±5 kHz). This is a real design constraint: you need exactly enough Q to separate stations, not more or less.

Question 2

A high-fidelity audio amplifier must pass signals from 20 Hz to 20 kHz uniformly. If a resonant bandpass stage is used with a center frequency around 1 kHz, what does the bandwidth requirement imply about Q?

• A. Q must be very high (>1000) to pass the full audio range without attenuation
• B. Q must be low (approximately 0.05) since the required bandwidth is comparable to and much larger than the center frequency
• C. Q must equal exactly 1, which is the only value that produces a flat response
• D. Q is irrelevant to audio frequency response — only component values matter

Answer: (B)

BW needed ≈ 20 kHz. Q = f₀/BW ≈ 1 kHz / 20 kHz = 0.05. This very low Q produces a very broad, flat resonance curve that passes the entire audio spectrum without significant roll-off within the band. High Q would give a narrow spike around 1 kHz, badly attenuating low bass and high treble. This example illustrates that low Q is desirable in wideband applications — the common student assumption that 'higher Q is always better' gets this backwards. The right Q is whatever the application demands, and for audio reproduction, that means low Q.

Question 3

For a fixed resonant frequency, a higher Q generally produces better circuit performance because it provides sharper, more precise frequency discrimination.

• T. True
• F. False

Answer: False

Higher Q is better for applications requiring narrow bandwidth — radio receivers selecting one channel, oscillators needing stable frequencies, notch filters eliminating one frequency. But high Q is harmful in applications requiring broad bandwidth — audio amplifiers, wideband communications, baseband signal processing. High Q narrows the passband, attenuating signal frequencies away from resonance. The statement confuses 'sharper' with 'better': selectivity is only a virtue if you need to select. The fundamental Q–bandwidth relationship Q = f₀/BW means that every gain in selectivity is paid for in bandwidth, and vice versa.

Question 4

The half-power bandwidth of a resonant circuit is defined as the frequency interval between the two points where the current magnitude falls to 1/√2 (approximately 0.707) of its peak value at resonance.

• T. True
• F. False

Answer: True

At the half-power points, the power dissipated in the circuit is half its peak value (since P ∝ I²), and I = I_peak/√2. The ratio 10·log₁₀(1/2) ≈ −3 dB, which is why these points are also called the −3 dB frequencies. The bandwidth between them is BW = f₀/Q. This is a rigorous definition, not a rule of thumb: the half-power frequencies are the exact points where the impedance magnitude equals √2 times the minimum impedance (the resistance R at resonance in a series circuit). All standard bandwidth measurements in filter design use this definition.

Question 5

Explain the fundamental tradeoff between Q and bandwidth, and give one example each of an application where high Q is desirable and one where low Q is desirable. What determines the 'right' Q?

Model answer: Q = f₀/BW, so Q and bandwidth are inversely proportional at fixed resonant frequency. High Q produces narrow bandwidth (sharp, selective resonance); low Q produces wide bandwidth (broad, flat response). High Q is desirable in AM/FM radio tuners (selecting one station while rejecting neighbors), crystal oscillators (stable single-frequency output), and bandpass filters needing high selectivity. Low Q is desirable in audio amplifiers (must pass 20 Hz–20 kHz uniformly), wideband communication receivers, and impedance matching networks where bandwidth matters more than rejection. The 'right' Q is determined by the application's bandwidth requirement: Q = f₀ / (required bandwidth). There is no universally correct Q — only the right tradeoff for the specific design.

The key insight is that Q is not a measure of quality in the everyday sense — it is a measure of selectivity. Selectivity is valuable when you need to distinguish signals at nearby frequencies (radio, oscillators) and harmful when you need to pass a wide range of frequencies equally (audio, broadband). Every circuit that uses resonance embodies a deliberate choice about where to sit on the Q–bandwidth curve, driven by the application's requirements.