RMS and average value definitions: Do the terms “RMS” (root-mean-square) and “average value” apply only to sinusoidal waveforms?

Introduction / Context:
Root-mean-square (RMS) and average (mean) values are general mathematical constructs used to quantify the magnitude of time-varying signals. In electrical engineering, they are crucial for relating AC signals to equivalent DC heating or computing offsets and energy. This question addresses a common misconception that these metrics are defined only for sinusoidal waveforms.

Given Data / Assumptions:

• No special assumptions on waveform: it may be sinusoidal, non-sinusoidal periodic, or even aperiodic (provided the relevant integrals converge).
• Standard time averaging over one period for periodic signals or over an interval for arbitrary signals.

Concept / Approach:

The RMS value of a signal x(t) over a period T is defined as x_rms = sqrt((1/T) * ∫_0^T x(t)^2 dt). The average (mean) over a period is x_avg = (1/T) * ∫_0^T x(t) dt. Neither definition requires x(t) to be sinusoidal; they apply universally to any signal for which these integrals exist. For non-sinusoidal periodic waveforms such as square, triangular, or PWM waveforms, RMS and average values are routinely computed and used for power and control design. Even for random signals, RMS is used (e.g., noise RMS).

Step-by-Step Solution:

Verification / Alternative check:

Standards for power quality and instrumentation specify RMS measurement of arbitrary waveforms (true-RMS meters) precisely because non-sinusoidal loads are common. Average values are similarly computed for any waveform to determine offsets or rectified averages.

Why Other Options Are Wrong:

• Any option asserting exclusivity to sine waves contradicts the general mathematical definitions.
• “Apply to DC only” is incorrect; DC is a trivial special case with RMS equal to the absolute value and average equal to the DC level.

Common Pitfalls:

• Assuming instrument displays that are calibrated for sine waves define the mathematics; they are practical approximations, not restrictions of the definitions.

Final Answer:

False