Rate of change versus frequency: A 15 kHz sine wave changes faster (has larger dv/dt) than which of the following sine-wave frequencies?
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A25 kHz
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B12 kHz
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C18 kHz
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D1.3 MHz
Answer
Correct Answer: 12 kHz
Explanation
Introduction / Context:The instantaneous rate of change of a sine wave (slope or dv/dt) scales with angular frequency. Higher frequency signals swing through their cycles faster and therefore exhibit greater maximum slope. This concept underpins bandwidth limits, slew-rate considerations, and filter behavior.
Given Data / Assumptions:
- Reference signal: f_ref = 15 kHz.
- Candidates: 25 kHz, 12 kHz, 18 kHz, 1.3 MHz.
- Equal amplitudes assumed for fair comparison.
Concept / Approach:For v(t) = Vp sin(2π f t), the maximum slope is dv/dt|max = 2π f Vp. With fixed amplitude, slope increases linearly with f. Therefore, any frequency lower than 15 kHz has a lower maximum rate of change.
Step-by-Step Solution:
Identify frequencies lower than 15 kHz among options.Only 12 kHz is less than 15 kHz; the others are higher.Hence, the 15 kHz sine changes faster than a 12 kHz sine.Verification / Alternative check:Compute the ratio of slopes: dv/dt|max(15k)/dv/dt|max(12k) = 15/12 = 1.25 > 1, confirming higher rate at 15 kHz for equal amplitudes.
Why Other Options Are Wrong:
- 25 kHz, 18 kHz, 1.3 MHz: All exceed 15 kHz and thus have equal or greater rate of change.
Common Pitfalls:
- Comparing periods without recognizing inverse relationship to frequency.
- Ignoring amplitude; here we assume equal Vp for a fair slope comparison.
Final Answer:12 kHz