Waveforms and time–frequency relationship: which statement best describes how the period of a waveform relates to its frequency in basic AC circuit theory?
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AThe actual frequency of a sine wave is 1.41 times the working frequency
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BUnless stated otherwise, every waveform is assumed to be 60 Hz
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CHigher amplitude always means higher frequency
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DA longer period corresponds to a lower frequency
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ENone of the above
Answer
Correct Answer: A longer period corresponds to a lower frequency
Explanation
Introduction / Context:In alternating-current (AC) theory and signal analysis, time-domain and frequency-domain descriptions are linked by a simple inverse relationship. Recognizing how period and frequency relate is foundational for analyzing power systems, audio signals, and communication waveforms.
Given Data / Assumptions:
- Frequency f is measured in hertz (cycles per second).
- Period T is measured in seconds per cycle.
- Sine wave behavior is representative for illustrating the f–T relationship.
Concept / Approach:The fundamental relationship is f = 1 / T. As the time for one cycle (T) increases, the number of cycles per second (f) decreases proportionally, and vice versa. Amplitude is independent of frequency; and 1.41 (square root of 2) relates RMS and peak of a sine wave’s amplitude, not its frequency. There is no default “assumed” frequency unless specified by a standard within a given context (e.g., regional mains frequency).
Step-by-Step Solution:
Start with identity: f = 1 / T. Infer monotonic relation: if T goes up, f goes down. Conclude: the correct qualitative statement is “longer period → lower frequency.”Verification / Alternative check:Example: if T = 0.02 s, f = 1 / 0.02 = 50 Hz; if T doubles to 0.04 s, f halves to 25 Hz. This numerical check confirms the inverse relation.
Why Other Options Are Wrong:
1.41 factor concerns amplitude (peak vs. RMS), not frequency. Assumed 60 Hz is context-dependent; not a universal default. Amplitude does not set frequency; they are independent attributes. None: incorrect because the period–frequency inverse is correctly stated in option D.Common Pitfalls:Confusing amplitude–RMS relationships with frequency, or assuming regional mains frequency applies universally. Always rely on f = 1 / T to reason about time and frequency.
Final Answer:A longer period corresponds to a lower frequency