First-order filters: A basic single-pole RC or RL filter exhibits a magnitude slope (roll-off rate) of approximately 20 dB per decade beyond its cutoff frequency. State whether this is true or false.
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ATrue
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BFalse
Answer
Correct Answer: True
Explanation
Introduction / Context:Single-pole filters are the simplest building blocks in analog electronics. Understanding their asymptotic roll-off rate is essential for quick Bode plot sketches, noise estimation, and cascade design for higher-order responses.
Given Data / Assumptions:
- Filter is first-order (one reactive element and one resistive element).
- Type is RC or RL (either low-pass or high-pass topology).
- Cutoff (break) frequency is defined where magnitude is 1/√2 of passband (−3 dB point).
Concept / Approach:
A first-order transfer function has the form |H(jω)| = 1 / √(1 + (ω/ωc)^2) for a low-pass (or its dual for high-pass). On a log-magnitude (dB) plot, the asymptotes differ by 20 dB per decade between successive decades of frequency beyond the break point, yielding the well-known slope of 20 dB/dec (≈6 dB/octave).
Step-by-Step Solution:
Write transfer magnitude in dB: 20log10|H(jω)|.For ω ≫ ωc in a low-pass, |H| ≈ ωc/ω ⇒ 20log10(ωc/ω) = −20*log10(ω/ωc).Increase ω by a factor of 10 (one decade): magnitude changes by −20 dB.For a high-pass, the same 20 dB/dec applies below cutoff (increasing as frequency rises).Verification / Alternative check:
Construct Bode asymptotes: horizontal in the passband, then a straight line with slope ±20 dB/dec starting one decade away from the break, intersecting at the corner. SPICE simulations confirm within a small margin near the corner where the exact response deviates by up to 3 dB.
Why Other Options Are Wrong:
- “False” would imply a different slope for first-order filters; slopes like 40 dB/dec or 60 dB/dec correspond to second- or third-order networks (additional poles).
Common Pitfalls:
Confusing per-octave and per-decade slopes (6 dB/oct vs 20 dB/dec); mixing passband ripple (active filters) with simple RC/RL responses; misidentifying the −3 dB point as the slope rather than the breakpoint.
Final Answer:
True