EMNG 3028 - Chapter 5: Periodic Functions & Fourier Series Notes

EMNG 3028 Power Quality & Distribution - Chapter 5: Periodic Functions & Fourier Series

Objectives

• Understand periodic functions and their characteristics.
• Understand Fourier Series Theory.
• Calculate Fourier constants.

Definition of a Periodic Function

• A function $f(x)$ is periodic if and only if $f(x + L) = f(x)$ is true for some value of $L$ and for all values of $x$.
• The smallest value of $L$ for which the equation $f(x + L) = f(x)$ holds true for every value of $x$ is called the period of the function.
• A graph of a periodic function $f(x)$ with period $L$ exhibits the same pattern every $L$ units along the x-axis.
• Knowing the function's behavior over one complete period allows us to sketch the graph over a wider interval containing multiple periods.
• Examples: $\sin x$ and $\cos x$ are periodic functions with a period of $2\pi$.

Fourier Series

• Periodic functions frequently occur in engineering problems and can be complicated.
• It is desirable to represent these functions in terms of simple periodic functions like sine and cosine.
• Joseph Fourier (1768-1830) studied the development of a periodic function into a series of sines and cosines.
• The series of sines and cosines is named after him: Fourier Series.
• If $f(x)$ is a periodic function with period $2\pi$ defined in $(c, c + 2\pi)$ and satisfies Dirichlet's conditions, it can be expanded as a Fourier series.

Dirichlet Conditions

• Dirichlet conditions (also known as the Dirichlet-Jordan test) provide sufficient conditions for a periodic function to be equal to the sum of its Fourier series.
• These conditions ensure the convergence of the Fourier series to the original function.

Three Dirichlet Conditions:

1. Absolute Integrability: The function $f(t)$ must be absolutely integrable over one period. This means that the integral of the absolute value of the function over a period is finite.
2. Bounded Variation: Within any finite interval, the function must have a finite number of maxima and minima. The function's variation over the interval is bounded.
3. Finite Discontinuities: The function can have a finite number of discontinuities in any finite interval, but each discontinuity must be finite (not infinite).

Fourier Series Coefficients

• The coefficients are calculated using the following formulas:
  • $a<em>0 = \frac{1}{\pi} \int</em>{c}^{c+2\pi} f(x) dx$
  • $a<em>n = \frac{1}{\pi} \int</em>{c}^{c+2\pi} f(x) \cos(nx) dx$
  • $b<em>n = \frac{1}{\pi} \int</em>{c}^{c+2\pi} f(x) \sin(nx) dx$

General Form of Fourier Series

• $f(t) = a<em>0 + \sum</em>{n=1}^{\infty} (a<em>n \cos(n \omega</em>0 t) + b<em>n \sin(n \omega</em>0 t))$
• $\omega_0 = \frac{2\pi}{T}$ is the fundamental frequency in radians per second.
• The Fourier series resolves the function into a DC component ($a_0$) and an AC component (the summation).
• The constants $a<em>0$, $a</em>n$, and $b_n$ are called the Fourier coefficients.

Periodic Functions - Cosine and Sine

• Cosine Function:
  • $\cos(n\pi) = \begin{cases} (-1)^n & \text{for all n} \ 1 & , n = 0, 4, 8, … \ -1 & , n = 2, 6, 10, … \end{cases}$
  • $\cos(n\frac{\pi}{2}) = \begin{cases} 0 & \text{, n odd} \ 1 & , n = 0, 4, 8, … \ -1 & , n = 2, 6, 10, … \end{cases}$
• Sine Function:
  • $\sin(n\pi) = 0 \text{ for all n}$
  • $\sin(n\frac{\pi}{2}) = \begin{cases} 0 & \text{, n even} \ 1 & , n = 1, 5, 9, … \ -1 & , n = 3, 7, 11, … \end{cases}$

Forms of Periodic Waves

• Modulation:
  • Amplitude varies in a repeating manner: Amplitude Modulation (AM).
  • Frequency varies in a repeating manner: Frequency Modulation (FM).
  • Waveform shape varies in a repeating manner: Non-sinusoidal periodic wave.

Fourier Series Theory

• A square-wave pattern can be approximated with a sum involving a fundamental sine-wave plus a combination of harmonics of this fundamental frequency.
• This sum is called a Fourier series.

Harmonics

• The sinusoid $\sin(n\omega<em>0 t)$ or $\cos(n\omega</em>0 t)$ is called the $n^{th}$ harmonic of $f(t)$.
  • If $n$ is odd, then it's called the odd harmonic.
  • If $n$ is even, then it's called the even harmonic.

Requirements for Fourier Series Representation

1. $f(t)$ must be single-valued everywhere.
2. It must have a finite number of finite discontinuities per period.
3. It must have a finite number of maxima and minima per period.

Integrals Useful for Evaluating Fourier Coefficients

• $\int \cos(at) dt = \frac{1}{a} \sin(at)$
• $\int \sin(at) dt = -\frac{1}{a} \cos(at)$
• $\int t \cos(at) dt = \frac{1}{a} \cos(at) + \frac{t}{a} \sin(at)$
• $\int t \sin(at) dt = \frac{1}{a} \sin(at) - \frac{t}{a} \cos(at)$

Formulas

• To find $a<em>0$: $a</em>0 = \frac{2}{T} \int_{0}^{T} f(t) dt$
• To find $a<em>n$: $a</em>n = \frac{2}{T} \int<em>{0}^{T} f(t) \cos(n \omega</em>0 t) dt$
• To find $b<em>n$: $b</em>n = \frac{2}{T} \int<em>{0}^{T} f(t) \sin(n \omega</em>0 t) dt$

Integrals Table

• $\int x^n dx = \frac{x^{n+1}}{n+1}$
• $\int \frac{1}{x} dx = \ln |x|$
• $\int e^x dx = e^x$
• $\int \sin x dx = -\cos x$
• $\int \cos x dx = \sin x$
• $\int [g(x)]^n g'(x) dx = \frac{[g(x)]^{n+1}}{n+1}$
• $\int \frac{g'(x)}{g(x)} dx = \ln |g(x)|$
• $\int a^x dx = \frac{a^x}{\ln a}$
• $\int \cosh x dx = \sinh x$
• $\int \sinh x dx = \cosh x$
• $\int \tanh x dx = \ln(\cosh x)$
• $\int \coth x dx = \ln(\sinh x)$
• $\int \tan x dx = -\ln(\cos x)$
• $\int \cot x dx = \ln(\sin x)$
• $\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \tan^{-1}(\frac{x}{a})$
• $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}(\frac{x}{a})$

Example: Fourier Series Calculation

• Function defined in $(0, 2\pi)$.
• $f(x) = \begin{cases} x, & 0 < x < \pi \ \pi, & \pi < x < 2\pi \end{cases}$

1. Calculate $a_0$:
   • $a<em>0 = \frac{1}{2\pi} \int</em>{0}^{2\pi} f(x) dx = \frac{1}{2\pi} \left[ \int<em>{0}^{\pi} x dx + \int</em>{\pi}^{2\pi} \pi dx \right]$
   • $a_0 = \frac{1}{2 \pi} \left[ \frac{\pi^2}{2} + \pi^2 \right] = \frac{3\pi}{4}$
2. Calculate $a_n$:
   • $a<em>n = \frac{1}{\pi} \int</em>{0}^{2\pi} f(x) \cos(nx) dx = \frac{1}{\pi} \left[ \int<em>{0}^{\pi} x \cos(nx) dx + \int</em>{\pi}^{2\pi} \pi \cos(nx) dx \right]$
   • $a_n = \frac{1}{\pi} \left[ \frac{\cos(n \pi) - 1}{n^2} + \frac{\sin(2n\pi) - \sin(n\pi)}{n} \right] = \frac{\cos(n \pi) - 1}{\pi n^2}$
   • $a_n = \begin{cases} 0 & \text{, n even} \ -\frac{2}{\pi n^2} & \text{, n odd} \end{cases}$
3. Calculate $b_n$:
   • $b<em>n = \frac{1}{\pi} \int</em>{0}^{2\pi} f(x) \sin(nx) dx = \frac{1}{\pi} \left[ \int<em>{0}^{\pi} x \sin(nx) dx + \int</em>{\pi}^{2\pi} \pi \sin(nx) dx \right]$
   • $b<em>n = \frac{1}{\pi} \left[ \frac{\pi \cos(n \pi)}{n} + \frac{\sin(n\pi)}{n^2} - \frac{(\cos(2n\pi) - \cos(n\pi))\pi}{n} \right]$$b</em>n = \frac{1 - (-1)^n}{n}$
4. Final Fourier Series:
   • $f(x) = \frac{3\pi}{4} + \sum<em>{n=1}^{\infty} \left[ a</em>n \cos(nx) + b_n \sin(nx) \right]$
   • Where $a<em>0 = \frac{3 \pi}{2}$, $a</em>n = 0$, (n even), $a<em>n = -\frac{2}{\pi n^2}$, (n odd), and $b</em>n = \frac{1 - (-1)^n}{n}$

Amplitude Modulation (AM)

• Amplitude varies with time.
• Includes a carrier frequency plus sidebands.

Frequency Modulation (FM)

• Frequency varies with time.
• Modulation determines the number of sidebands.

Non-sinusoidal Periodic Waves

• Examples: Sawtooth wave, Square wave, Triangular wave.
• Consist of harmonic series with varying amplitudes and frequencies.