Consider the active filter shown (an operational-amplifier based network). Evaluate the following claims about the filter: (1) it is an active low-pass filter, (2) it is second order, and (3) its magnitude slope changes by 40 dB per decade beyond cutoff. Which statements are correct?
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A1, 2 and 3
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B1 and 2
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C1 and 3
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D2 and 3
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EOnly 1
Answer
Correct Answer: 1, 2 and 3
Explanation
Introduction / Context:Active filters employ operational amplifiers with resistors and capacitors to realize frequency-selective responses without using inductors. A classic low-pass topology (such as Sallen–Key or multiple-feedback) provides a second-order transfer function, giving a −40 dB/decade roll-off in the stopband. This question checks your ability to match qualitative statements to a standard active low-pass response.
Given Data / Assumptions:
- The depicted circuit is a common op-amp active filter topology.
- Component choices yield a low-pass response (DC gain finite, high-frequency attenuation).
- Order is determined by the highest power of s in the denominator (two reactive elements ⇒ second order).
Concept / Approach:A second-order low-pass filter has transfer function magnitude that is flat near DC (passband), transitions around the natural frequency or cutoff, and then attenuates with a slope of 40 dB/decade (12 dB/octave) as frequency increases past the corner. Whether implemented by Sallen–Key or multiple-feedback, two capacitors (or two independent energy storage elements) confirm the second order.
Step-by-Step Solution:
Identify passband behavior: unity or set gain at low frequency ⇒ low-pass.Count energy storage elements: two reactive components ⇒ second order.Apply asymptotic Bode slope: second order ⇒ −40 dB/decade beyond the cutoff.Verification / Alternative check:
Plot a Bode magnitude sketch: 0 dB slope in passband, then −40 dB/decade after ωc, confirming (1), (2), (3).Why Other Options Are Wrong:
Any option omitting one of the three statements contradicts the standard second-order low-pass characteristics.Common Pitfalls:
Confusing second-order low-pass (−40 dB/dec) with first-order (−20 dB/dec); misidentifying a high-pass due to capacitor placement without analyzing the transfer function.Final Answer:
1, 2 and 3