---
title: "Define RMS and average value"
subject: "Basic Electrical and Electronics Engineering"
chapter: 1
key: "semester_1/basic_electrical_and_electronics_engineering:define-rms-average"
source: https://www.bctengineer.com/q/semester_1/basic_electrical_and_electronics_engineering/define-rms-average
site: BCT Engineering
---

# Define RMS and average value

Basic Electrical and Electronics Engineering · Chapter 1: Fundamentals of Electrical and Electronics Circuits

- Asked in: 2081 Baishakh (new course), 2 marks
- Video: [RMS & Average Values of Standard Waveforms](https://www.youtube.com/watch?v=KygtYb2EsQE) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/basic_electrical_and_electronics_engineering/define-rms-average
- Whole chapter: https://www.bctengineer.com/md/semester_1/basic_electrical_and_electronics_engineering/chapter-1

## Question

Define RMS and Average value of AC quantities.

## Short answer

Average value is the mean of instantaneous values over a half cycle (0.637 times peak for a sine wave); full-cycle average is zero. RMS (effective) value is the equivalent DC that produces the same heating, equal to peak divided by √2 (0.707 peak).

## Answer

**Average value** is the mean of all instantaneous values of an alternating quantity over one complete cycle:

$$
I_{av}=\frac{1}{T}\int_0^T i\,dt
$$

For a symmetrical wave, such as a sine wave, this full-cycle mean is zero, so the average value is taken over a half cycle:

$$
I_{av}=\frac{2}{T}\int_0^{T/2} i\,dt
$$

For a sine wave: $I_{av}=\frac{2}{\pi}I_m = 0.637\,I_m$.

**RMS (effective) value** is the value of direct current that produces the same heating effect in a resistor as the alternating current does in the same time. It is the square root of the mean of the squares of the instantaneous values over one cycle.

$$
I_{rms}=\sqrt{\frac{1}{T}\int_0^T i^2\,dt}
$$

For a sine wave: $I_{rms}=\frac{I_m}{\sqrt2}=0.707\,I_m$.

The ratio $I_{rms}/I_{av}=1.11$ is called the form factor of a sine wave. Ammeters and voltmeters on AC supply read RMS values (for example, a 230 V supply has a peak of $230\sqrt2 = 325$ V).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the RMS value of an alternating current represent?
   A. The mean of the instantaneous values over a half cycle
   B. The DC value that produces the same heating effect in a resistor as the AC
   C. The maximum value reached in a cycle
   D. The mean of the instantaneous values over a full cycle
   Answer: B. RMS (effective) value is the steady DC that gives the same heat in the same resistor over the same time. (Key point: RMS definition)

2. Why is the average value of a sine wave normally taken over a half cycle?
   A. The sine wave is only defined for half a cycle
   B. Heating effect depends only on the positive half
   C. The full-cycle mean of a symmetrical wave is zero
   D. Half-cycle averaging gives the RMS value
   Answer: C. Positive and negative halves cancel, so the full-cycle mean is zero; the half-cycle mean gives a useful value, 0.637 Im. (Key point: Why half-cycle average)

3. For a sine wave, what is the form factor Irms/Iav, and how is a 230 V supply's peak found?
   A. 1.57; peak = 230/sqrt(2)
   B. 1.11; peak = 230 x sqrt(2) = 325 V
   C. 1.11; peak = 230/sqrt(2) = 163 V
   D. 1.57; peak = 230 x 2 = 460 V
   Answer: B. Irms/Iav = 0.707/0.637 = 1.11, and meters read RMS so peak = 230 x sqrt(2) = 325 V. (Key point: Sine wave factors)

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