---
title: "Average and RMS of sawtooth waveform"
subject: "Fundamental of Electrical and Electronics Engineering"
chapter: 2
key: "semester_1/fundamental_of_electrical_and_electronics_engineering:sawtooth-average-rms-value"
source: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/sawtooth-average-rms-value
site: BCT Engineering
---

# Average and RMS of sawtooth waveform

Fundamental of Electrical and Electronics Engineering · Chapter 2: Average and RMS Values

- Asked in: 2081 Chaitra (new course), 2+3 marks
- Video: [Sawtooth Waveform Average & RMS Value||With & Without Integration||Shortcut/Direct Method||Trick](https://www.youtube.com/watch?v=U5ho_OlkAPU) — Tutor - Praveen Singh
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/sawtooth-average-rms-value
- Whole chapter: https://www.bctengineer.com/md/semester_1/fundamental_of_electrical_and_electronics_engineering/chapter-2

## Question

For the periodic signal given below, find the average and rms value. Assume T = 10 ms. [Figure: sawtooth waveform y(t): it rises linearly from y = 10 at t = 0 (point A) to y = 20 at t = T (point B), then drops vertically back to y = 10 (point C) at t = T, and repeats with period T (rising again to 20 at 2T, etc.). The dashed line through A, E, C marks the level y = 10; D is a point on the ramp at time t with height y above the time axis.]

## Short answer

The waveform rises linearly from 10 to 20 over period T. Its average value is (10+20)/2 = 15. The RMS value is √(1/T ∫₀ᵀ (10+10t/T)² dt) = 15.28 (independent of T).

## Answer

#### Equation of the waveform

In one period (0 ≤ t < T) the wave rises linearly from 10 to 20:

```text
y(t) = 10 + (20 − 10) t/T = 10 + 10t/T
T = 10 ms
```

#### Average value

```text
Y_avg = (1/T) ∫₀ᵀ (10 + 10t/T) dt
      = (1/T) [10t + 5t²/T]₀ᵀ
      = (1/T) (10T + 5T)
      = 15
```

This is also clear from the shape: a straight ramp from 10 to 20 has mean (10 + 20)/2 = 15.

#### RMS value

Put x = t/T (x goes from 0 to 1, dt = T dx):

```text
Y_rms² = (1/T) ∫₀ᵀ (10 + 10t/T)² dt
       = 100 ∫₀¹ (1 + x)² dx
       = 100 [(1 + x)³/3]₀¹
       = 100 × (8 − 1)/3
       = 233.33
Y_rms  = √233.33 = 15.275
```

**Answer: Average value = 15, RMS value ≈ 15.28** (in the units of y, e.g. volts).

The value T = 10 ms does not change the result: the average and RMS of a periodic wave depend only on its shape over one period, not on how long the period is. RMS (15.28) is slightly larger than the average (15) because squaring gives more weight to the larger values near 20.

## Check yourself

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

1. The sawtooth rises from 10 to 20 over one period T. Which is y(t) for 0 <= t < T?
   A. 10 + 10t/T
   B. 20 - 10t/T
   C. 10t/T
   D. 20 + 10t/T
   Answer: A. Starting at 10 with total rise of 10 over T gives y = 10 + 10t/T. (Key point: Waveform equation)

2. What is the average value of this ramp?
   A. 10
   B. 15
   C. 20
   D. 7.5
   Answer: B. A straight ramp from 10 to 20 has mean (10+20)/2 = 15. (Key point: Average value)

3. Which expression gives the mean square for RMS?
   A. 100 times integral of (1+x) dx = 150
   B. (15)^2 = 225 exactly
   C. 100 times integral of (1+x)^2 dx from 0 to 1 = 233.3
   D. 100 times integral of x^2 dx = 33.3
   Answer: C. Substituting x = t/T gives Y_rms^2 = 100 x 7/3 = 233.3, so Y_rms = 15.28. (Key point: RMS calculation)

4. Why does T = 10 ms not change the average or RMS?
   A. Because 10 ms is too short to matter
   B. Because RMS always equals average
   C. Because T cancels only for sine waves
   D. They depend only on the shape over one period, not its length
   Answer: D. Both are averages over a period, so scaling the time axis leaves them unchanged. (Key point: Effect of T on results)

## Related in Fundamental of Electrical and Electronics Engineering

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