---
title: "Fourier series of ramp wave 0 to 20"
subject: "Electrical Circuit Theory"
chapter: 6
key: "semester_3/electrical_circuit_theory:fourier-ramp-0-to-20-period"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/fourier-ramp-0-to-20-period
site: BCT Engineering
---

# Fourier series of ramp wave 0 to 20

Electrical Circuit Theory · Chapter 6: Fourier Series and transform

- Asked in: 2071 Chaitra, 8 marks
- Video: [Trigonometric Fourier Series (Example 2)](https://www.youtube.com/watch?v=yVF_POXZBlw) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/fourier-ramp-0-to-20-period
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-6

## Question

Obtain trigonometric Fourier series of the waveform in figure below and sketch the line spectra.
[Figure: v(t) rises linearly from 0 at ωt = 0 to 20 at ωt = 2π, drops to 0 and rises again to 20 at 4π (period 2π)]

## Short answer

The voltage rises from 0 to 20 V over 2π, period 2π, with average 10 V. The Fourier series is v(t) = 10 − (20/π)[sin ωt + (1/2) sin 2ωt + (1/3) sin 3ωt + …] V.

## Answer

**Reading the figure.** v(t) rises linearly from 0 at θ = ωt = 0 to 20 at θ = 2π, drops to 0 and repeats. Period 2π.

```text
v(θ) = 20 θ/(2π) = 10θ/π       0 < θ < 2π
```

#### Coefficients

```text
a0 = (1/2π)∫₀^2π (10θ/π) dθ = 10                 (average value)
an = (1/π)∫₀^2π (10θ/π) cos nθ dθ = 0           (∫ θ cos nθ dθ = 0 over a period)
bn = (1/π)∫₀^2π (10θ/π) sin nθ dθ
   = (10/π²)·(−2π/n) = −20/(nπ)
```

(using ∫₀^2π θ sin nθ dθ = −2π/n).

#### Fourier series

```text
v(t) = 10 − (20/π) [ sin ωt + (1/2) sin 2ωt + (1/3) sin 3ωt + ... ]
     = 10 − 6.366 sin ωt − 3.183 sin 2ωt − 2.122 sin 3ωt − 1.592 sin 4ωt − ...
```

#### Line spectrum

| n | 0 (d.c.) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| cn | 10 | 6.366 | 3.183 | 2.122 | 1.592 | 1.273 |
| phase (cos form) | – | +90° | +90° | +90° | +90° | +90° |

```text
  n=0 |##########################| 10
  n=1 |#################         | 6.366
  n=2 |########                  | 3.183
  n=3 |######                    | 2.122
  n=4 |####                      | 1.592
  n=5 |###                       | 1.273
```

(−sin nθ = cos(nθ + 90°), so each harmonic has phase +90°.)

**Answer: v(t) = 10 − (20/π) Σ (1/n) sin nωt, n = 1, 2, 3, ...**

## Check yourself

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

1. What is a0 for the ramp 0 to 20?
   A. 10, the average
   B. 20
   C. 0
   D. 5
   Answer: A. Average of a ramp from 0 to 20 is 10. (Key point: DC term)

2. Why is an = 0?
   A. The wave is odd
   B. a0 cancels
   C. The wave is even
   D. ∫θ cos nθ over a full period is zero
   Answer: D. The integral over 0 to 2π vanishes. (Key point: Cosine terms)

3. What is bn?
   A. 20/(nπ)
   B. -10/(nπ)
   C. -20/(nπ)
   D. -40/(nπ)
   Answer: C. bn = (10/π^2)(-2π/n) = -20/(nπ). (Key point: Sine coefficient)

4. What are the line values for n = 0, 1, 2?
   A. 5, 6.366, 3.183
   B. 10, 6.366, 3.183
   C. 10, 3.183, 6.366
   D. 10, 6.366, 1.592
   Answer: B. cn = 20/(nπ) falling as 1/n. (Key point: Spectrum)

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