---
title: "Fourier series of triangle train, Vo and T"
subject: "Electrical Circuit Theory"
chapter: 6
key: "semester_3/electrical_circuit_theory:fourier-train-of-isosceles-triangles-vo"
source: https://www.bctengineer.com/q/semester_3/electrical_circuit_theory/fourier-train-of-isosceles-triangles-vo
site: BCT Engineering
---

# Fourier series of triangle train, Vo and T

Electrical Circuit Theory · Chapter 6: Fourier Series and transform

- Asked in: 2068 Baisakh, 8 marks
- Video: [Fourier series expansion of a periodic triangular wave](https://www.youtube.com/watch?v=60FTmRIruR4) — Seema Prasad
- 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-train-of-isosceles-triangles-vo
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_circuit_theory/chapter-6

## Question

The given figure shows a voltage waveform in the form of a train of isosceles triangles. Determine the Fourier series and plot the line spectrum.
[Figure: v(t) rises linearly from 0 at t = 0 to Vo at t = T/2, falls linearly to 0 at t = T, and repeats with peaks at 3T/2 etc. (period T)]

## Short answer

The train of isosceles triangles has period T, average Vo/2, and only odd cosine harmonics. Its series is v(t) = Vo/2 − (4Vo/π²)[cos(2πt/T) + (1/9) cos(6πt/T) + (1/25) cos(10πt/T) + …].

## Answer

**Reading the figure.** v(t) rises linearly from 0 at t = 0 to Vo at t = T/2 and falls to 0 at t = T; period T, fundamental ω0 = 2π/T. Let θ = ω0 t.

The wave is symmetric about θ = π (v(2π − θ) = v(θ)), so it has no sine terms: bn = 0, and the average is Vo/2.

#### Coefficients (θ = ω0 t over 0 to 2π)

```text
v(θ) = Voθ/π            0 ≤ θ ≤ π
v(θ) = Vo(2π − θ)/π     π ≤ θ ≤ 2π

a0 = (area)/(period) = (Vo·T/2)/T = Vo/2
an = (1/π)∫₀^2π v cos nθ dθ = (2/π)∫₀^π (Voθ/π) cos nθ dθ     (symmetry about π)
   = (2Vo/π²)·(cos nπ − 1)/n²
   = 2Vo[(−1)ⁿ − 1] / (π² n²)
an = −4Vo/(π² n²) for odd n,   an = 0 for even n
bn = 0
```

#### Fourier series

```text
v(t) = Vo/2 − (4Vo/π²) [ cos ω0t + (1/9) cos 3ω0t + (1/25) cos 5ω0t + ... ]
     with ω0 = 2π/T
     = 0.5Vo − 0.405Vo cos ω0t − 0.0450Vo cos 3ω0t − 0.0162Vo cos 5ω0t − ...
```

(Check at t = 0: 0.5 − 0.405 − 0.045 − 0.0162 − ... → 0, as it must since v(0) = 0.)

#### Line spectrum (amplitudes, in units of Vo)

| n | 0 | 1 | 3 | 5 | 7 |
|---|---|---|---|---|---|
| cn | 0.5 | 0.4053 | 0.0450 | 0.0162 | 0.0083 |
| phase (cos form) | – | 180° | 180° | 180° | 180° |

```text
  n=0 |##########################| 0.5
  n=1 |#####################     | 0.4053
  n=3 |##                        | 0.04503
  n=5 |#                         | 0.01621
  n=7 |#                         | 0.008271
```

Lines are at 0, ω0, 3ω0, 5ω0, ... = 0, 2π/T, 6π/T, ...

**Answer: v(t) = Vo/2 − (4Vo/π²) Σ (1/n²) cos(2πnt/T), n = 1, 3, 5, ...**

## Check yourself

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

1. Why is bn = 0?
   A. The wave is symmetric about θ = π, even about the centre
   B. The wave is odd
   C. a0 = 0
   D. The wave has half-wave symmetry
   Answer: A. v(2π - θ) = v(θ) gives only cosines. (Key point: Sine terms)

2. What is a0?
   A. 0
   B. Vo
   C. Vo/4
   D. Vo/2
   Answer: D. Area (Vo T/2)/T = Vo/2. (Key point: DC term)

3. What is an?
   A. 4Vo/(π^2 n^2) for odd n
   B. -4Vo/(π n) for all n
   C. -4Vo/(π^2 n^2) for odd n, 0 for even n
   D. -2Vo/(π^2 n^2) for all n
   Answer: C. Compute (2Vo/π^2)(cos nπ - 1)/n^2. (Key point: Cosine coefficient)

4. Why does v(0) = 0 confirm the series?
   A. The d.c. term alone is zero
   B. 0.5 - 0.405 - 0.045 - ... sums to zero
   C. The sine terms sum to Vo
   D. All cosines equal 0
   Answer: B. At t = 0 all cosines are 1 so terms sum to v(0) = 0. (Key point: Check)

## Close variants set in other exams

- [Fourier series of triangle train, peak 10](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Afourier-train-of-isosceles-triangles-10v) — Electrical Circuit Theory, chapter 6, asked 2×

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- [Fourier series of odd sawtooth, ±Vo](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Afourier-sawtooth-vo-odd) — Electrical Circuit Theory, chapter 6
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- [Fourier series of triangular wave, peak 20](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Afourier-triangular-wave-peak-20-minus) — Electrical Circuit Theory, chapter 6

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