A first-order system with a time constant η and unity gain receives a sinusoidal input at frequency w = 1/η. What is its amplitude ratio______________?
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Answer: C. 1/√2
Explanation
The frequency response amplitude ratio formula for first-order systems contains a denominator square root term. Substituting the specific input frequency value equal to the inverse time constant simplifies this equation. Evaluating the resulting mathematical expression yields exactly one divided by the square root of two. First-order systems are characterized by a single time constant representing dynamic sluggishness. Sinusoidal input testing reveals how system output magnitude and phase change with frequency. The amplitude ratio quantifies the attenuation of output sine wave magnitude relative to input.
Frequency response analysis helps engineers determine bandwidth and filtering characteristics of physical systems. The frequency where amplitude drops to this specific ratio defines the system cutoff frequency. Control system designers use these metrics to evaluate disturbance rejection and stability margins. An amplitude ratio of one occurs only at very low frequencies near steady state. A value of zero point five represents a different magnitude attenuation not matching this frequency. A value of zero point two five corresponds to higher attenuation ratios found in other systems.
- Subject
- Engineering
- Category
- Chemical Engineering
- Topic
- —
- Difficulty
- medium
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