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

# Fourier series of half-period ramp pulses

Electrical Circuit Theory · Chapter 6: Fourier Series and transform

- Asked in: 2069 Chaitra, 8 marks
- Video: [How to Compute a FOURIER SERIES // Formulas & Full Example](https://www.youtube.com/watch?v=ijQaTAT3kOg) — Dr. Trefor Bazett
- 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-half-period-vo
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-6

## Question

Find the trigonometric Fourier series for the given waveform shown and also sketch the line spectrum.
[Figure: v(t) rises linearly from 0 at ωt = 0 to Vo at ωt = π, drops to 0, stays at 0 until 2π, then rises to Vo at 3π and drops to 0 again (period 2π)]

## Short answer

The voltage ramps from 0 to Vo over π, then stays 0. Its series has dc, odd cosine, and all sine terms: v(t) = Vo/4 − (2Vo/π²)[cos ωt + (1/9) cos 3ωt + …] + (Vo/π)[sin ωt − (1/2) sin 2ωt + (1/3) sin 3ωt − …].

## Answer

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

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

#### Coefficients

```text
a0 = (1/2π)∫₀^π (Voθ/π) dθ = (1/2π)(Vo π/2) = Vo/4
an = (1/π)∫₀^π (Voθ/π) cos nθ dθ = (Vo/π²)[ (cos nπ − 1)/n² ]
   = Vo[(−1)ⁿ − 1]/(π² n²)
   → an = −2Vo/(π² n²) for odd n, an = 0 for even n
bn = (1/π)∫₀^π (Voθ/π) sin nθ dθ = (Vo/π²)(−π cos nπ/n)
   = Vo(−1)^(n+1)/(nπ)
```

(using ∫₀^π θ cos nθ dθ = (cos nπ − 1)/n² and ∫₀^π θ sin nθ dθ = −π cos nπ/n).

#### Fourier series

```text
v(t) = Vo/4 − (2Vo/π²)[cos ωt + (1/9) cos 3ωt + ...]
       + (Vo/π)[ sin ωt − (1/2) sin 2ωt + (1/3) sin 3ωt − ... ]
```

#### Line spectrum (cn = √(an² + bn²), Vo = 1)

| n | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| an | 0.25 | −0.2026 | 0 | −0.0225 | 0 | −0.0081 |
| bn | – | 0.3183 | −0.1592 | 0.1061 | −0.0796 | 0.0637 |
| cn | 0.25 | 0.3773 | 0.1592 | 0.1085 | 0.0796 | 0.0642 |

```text
  n=0 |#################         | 0.25
  n=1 |##########################| 0.3773
  n=2 |###########               | 0.1592
  n=3 |#######                   | 0.1085
  n=4 |#####                     | 0.0796
  n=5 |####                      | 0.0642
```

The wave has no even, odd or half-wave symmetry, so the d.c. term, cosine terms and sine terms are all present, and lines appear at every harmonic. The phase of each line is ψn = arctan(−bn/an), taken in the correct quadrant, e.g. ψ1 = −122.5° (in the form cn cos(nθ + ψn)).

**Answer: v(t) = Vo/4 + Σ [ an cos nωt + bn sin nωt ], with an = [(−1)ⁿ − 1]Vo/(π²n²) and bn = (−1)^(n+1) Vo/(nπ).**

## Check yourself

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

1. What is a0?
   A. Vo/4
   B. Vo/2
   C. Vo
   D. 0
   Answer: A. Area Vo π/2 divided by 2π is Vo/4. (Key point: DC term)

2. Which terms appear?
   A. Only sines
   B. Only odd cosines
   C. Only d.c. and cosines
   D. d.c., sine and cosine terms, all harmonics
   Answer: D. The wave lacks even, odd and half-wave symmetry. (Key point: Symmetry)

3. What is an?
   A. Vo/(nπ) for all n
   B. 0 for all n
   C. -2Vo/(π^2 n^2) for odd n and 0 for even n
   D. -2Vo/(π n) for odd n
   Answer: C. (cos nπ - 1)/n^2 vanishes for even n. (Key point: Cosine coefficient)

4. What is bn?
   A. -Vo/(nπ)
   B. Vo(-1)^(n+1)/(nπ)
   C. Vo/(n^2 π^2)
   D. 0
   Answer: B. Sign alternates with n. (Key point: Sine coefficient)

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