Introduction / Context:
The rms value of a periodic signal represents the DC-equivalent heating value in a resistive load. For sinusoidal waveforms, there is a fixed relationship between the peak value and the rms value that is used extensively in AC circuit analysis and instrumentation.
Given Data / Assumptions:
- Signal is a pure sine: v(t) = Vm * sin(ωt).
- We compare Vrms to Vm (peak amplitude).
- No distortion or DC offset.
Concept / Approach:
- Definition: Vrms = sqrt( (1/T) * ∫ v(t)^2 dt over one period ).
- For a sine: Vrms = Vm / √2 ≈ 0.707 * Vm.
Step-by-Step Solution:
Start with Vrms = Vm / √2.Compute numeric factor: 1 / √2 ≈ 0.707.Conclude: Vrms is about 70.7% of the peak amplitude.
Verification / Alternative check:
Given Vp-p = 2 * Vm, one can convert between rms and peak using Vrms = Vm / √2 and Vm = Vrms * √2 ≈ 1.414 * Vrms.
Why Other Options Are Wrong:
- Number of cycles in one second: That is frequency (Hz), not rms.
- Time for one cycle: That is period T, not rms.
- 1.41 times peak: Inverse relation; peak ≈ 1.414 * Vrms, not rms ≈ 1.41 * peak.
- None of the above: Incorrect because 0.707 * peak is correct.
Common Pitfalls:
- Confusing peak, peak-to-peak, and rms values.
- Applying the sine relation to non-sinusoidal waveforms where the factor differs.
Final Answer:
equal to 0.707 times the peak amplitude
Discussion & Comments