---
title: "Harmonic signal parameters and RMS value"
subject: "Instrumentation and Measurement"
chapter: 2
key: "semester_4/instrumentation_and_measurement:harmonic-signal-parameters-rms"
source: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/harmonic-signal-parameters-rms
site: BCT Engineering
---

# Harmonic signal parameters and RMS value

Instrumentation and Measurement · Chapter 2: Time Dependent Properties of Signal

- Asked in: year not given, 2+3 marks
- Video: [RMS Value of AC Circuits](https://www.youtube.com/watch?v=QW8ZXju1ruU) — The Organic Chemistry Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_and_measurement/harmonic-signal-parameters-rms
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_and_measurement/chapter-2

## Question

(a) A harmonic signal is $y(t) = 10\sin(100\pi t + \pi/6)$ volt. Find its frequency, period, amplitude, RMS value and the time by which it leads $10\sin(100\pi t)$.

(b) A signal is composed of two harmonics, $x(t) = 4\sin(2\pi \cdot 50\, t) + 2\sin(2\pi \cdot 150\, t)$ volt. Find its fundamental frequency and RMS value.

## Short answer

For y(t) = 10 sin(100πt + π/6) V: frequency 50 Hz, period 20 ms, amplitude 10 V, RMS 7.07 V, time lead 1.667 ms. For x(t) = 4 sin(100πt) + 2 sin(300πt) V: fundamental 50 Hz, RMS = √((4²+2²)/2) = √10 ≈ 3.16 V.

## Answer

#### (a) Single harmonic
Compare $y(t) = 10\sin(100\pi t + \pi/6)$ with $y = A\sin(\omega t + \phi)$.

- Amplitude $A = 10$ V
- Angular frequency $\omega = 100\pi$ rad/s, so $f = \omega/2\pi = 50$ Hz
- Period $T = 1/f = 0.02$ s = 20 ms
- RMS value $= A/\sqrt{2} = 10/\sqrt{2} = 7.07$ V
- Phase $\phi = \pi/6 = 30^\circ$. Time lead:

$$
t_d = \frac{\phi}{\omega} = \frac{\pi/6}{100\pi} = \frac{1}{600}\ \text{s} = 1.667\ \text{ms}
$$

**Answer: $f = 50$ Hz, $T = 20$ ms, $A = 10$ V, RMS $= 7.07$ V, lead $= 1.667$ ms.**

#### (b) Two harmonics
The components have frequencies 50 Hz and 150 Hz. The 150 Hz is the third harmonic of 50 Hz, so the fundamental frequency is 50 Hz (period 20 ms).

For components of different frequency, mean squares add:

$$
x_{rms} = \sqrt{\frac{A_1^2}{2} + \frac{A_2^2}{2}} = \sqrt{\frac{16}{2} + \frac{4}{2}} = \sqrt{10} = 3.16\ \text{V}
$$

**Answer: fundamental 50 Hz, RMS $= 3.16$ V.**

## Check yourself

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

1. For y(t) = 10 sin(100πt + π/6), what is the frequency?
   A. 100 Hz
   B. 50 Hz
   C. 25 Hz
   D. 314 Hz
   Answer: B. ω = 100π rad/s, so f = ω/2π = 50 Hz and T = 20 ms. (Key point: Frequency from omega)

2. What is the RMS value of a sine of amplitude 10 V?
   A. 10 V
   B. 5 V
   C. 6.37 V
   D. 7.07 V
   Answer: D. RMS = A/sqrt(2) = 10/1.414 = 7.07 V. (Key point: RMS of sine)

3. How is the time by which the signal leads 10 sin(100πt) found, and what is it?
   A. φ/ω = 1.667 ms
   B. φ×ω = 0.052 s
   C. φ/f = 10.5 ms
   D. T/6 × 2 = 6.67 ms
   Answer: A. Time lead = phase/ω = (π/6)/(100π) = 1/600 s = 1.667 ms. (Key point: Time lead)

4. For x(t) = 4 sin(2π50t) + 2 sin(2π150t), the RMS value is
   A. sqrt(10) = 3.16 V
   B. 6 V
   C. 4.24 V
   D. 2.83 V
   Answer: A. Mean squares of different-frequency components add: sqrt(16/2 + 4/2) = sqrt(10) = 3.16 V; the fundamental is 50 Hz. (Key point: RMS of two harmonics)

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