Power Quality Problems – Harmonics & Fourier Analysis

Harmonics in Electrical Power Plants

• Any frequency present in the power system other than the fundamental ($f_1=50\,\text{Hz}$ or $60\,\text{Hz}$) is called a harmonic.

  • Visibly appear as distortion of the desired sinusoidal voltage or current waveform.
  • Graphical example in transcript:
  • Fundamental $50\,\text{Hz}$ component.
  • 3rd harmonic $150\,\text{Hz}$ component superimposed.

• Typical utility supply is nearly sinusoidal; distortion is introduced mainly by non-linear loads.

Typical Sources of Harmonics

• AC and DC adjustable-speed drives.
• Lighting ballasts (magnetic or electronic).
• Uninterruptible Power Supplies (UPSs).
• Any other device whose V–I characteristic is non-linear with respect to the source voltage.

Classification of Harmonics

• Integer harmonics: frequencies that are integer multiples of the fundamental ($nf_1$ where $n=1,2,3,\dots$).
• Interharmonics: frequencies that lie between integer multiples (e.g., $1.2f<em>1,\;2.7f</em>1$). Not commensurate with the fundamental.
• Subharmonics: components whose frequencies are below the fundamental (f<f_1).

Mechanism of Harmonic Injection

• Non-linear load draws a non-sinusoidal current even when applied voltage is ideally sinusoidal.
  • Non-sinusoidal current contains harmonic components $I_n$.
• Line impedance $Z<em>L$ causes a voltage drop $\Delta V = I</em>{\text{dist}}\,Z_L$, so current distortion is mirrored as voltage distortion.

Negative Effects of Harmonics

• Mal-operation or mis-triggering of sensitive electrical equipment.
• Increased copper, core, and stray losses (I$^2$R, eddy-current, skin effect, etc.).
• Localized overheating of transformers and rotating machines.
• False tripping of protective relays.
• De-rating of cables and generators.

Mitigation Strategies

• Install harmonic filters (passive LC, active filters, hybrid).
• Design or retrofit loads to behave as closely linear as possible (e.g., 12-pulse drives, PWM converters with high switching frequency, power-factor-correction front ends).

Non-Sinusoidal Waves and Fourier Analysis

• Distorted periodic waveforms are called non-sinusoidal (nonsinusoidal) waves.
• Jean-Baptiste Fourier proved any periodic function can be expressed as a sum of sinusoids.

Continuous Fourier Transform (CFT)

• Forward transform of a function $f(t)$:
  $F(\omega) = \frac{1}{2\pi}\int_{-\infty}^{\infty} f(t)\,e^{-j\omega t}\,dt$
• Inverse transform:
  $f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)\,e^{j\omega t}\,d\omega$
• Pair enables conversion between time domain and frequency domain.

Fourier Series of a Periodic Function (Trigonometric Form)

For a function $f(x)$ of period $2\pi$ (electrical texts often use $x=\omega t$):

1. General form:
   $f(x)= A<em>0 + \sum</em>{n=1}^{\infty}\left(A<em>n\sin nx + B</em>n\cos nx\right)$
2. Alternate sine-only and cosine-only forms supplied in transcript:
   \begin{aligned}
   f(x) &= A0 + \sum{n=1}^{\infty}Cn\sin(n x+\varphin)\
   Cn &= \sqrt{An^2+Bn^2}\ \varphin &= \tan^{-1}!\left(\dfrac{Bn}{An}\right)
   \end{aligned}

• Coefficient formulas (assuming period $2\pi$):
  $A<em>n=\frac{1}{\pi}\int</em>{-\pi}^{\pi}f(x)\sin(nx)\,dx,\;\;B<em>n=\frac{1}{\pi}\int</em>{-\pi}^{\pi}f(x)\cos(nx)\,dx$

Worked Example – Three-Term Current Wave

Given current:
$i(t)=105\sin \alpha-84\cos\alpha-38\sin2\alpha +19\cos3\alpha$
Convert to sine-only form.

• First harmonic magnitude:
  $C<em>1=\sqrt{105^2+(-84)^2}=134.47\,\text{A}$$\varphi</em>1 = \tan^{-1}!\left(\frac{-84}{105}\right)=-38.66^{\circ}$
• Second harmonic:
  $C<em>2=\sqrt{(-38)^2+19^2}=42.49\,\text{A},\;\varphi</em>2=116.565^{\circ}$
• Third harmonic:
  $C<em>3=\sqrt{10^2+7^2}=12.20\,\text{A},\;\varphi</em>3=34.99^{\circ}$
  Final sine-only waveform:
  $i(t)=134.47\sin(\alpha-38.66^{\circ})+42.49\sin(2\alpha+116.565^{\circ})+12.20\sin(3\alpha+34.99^{\circ})$

Example – Half-Wave Rectified Sine

• Waveform: $f(\omega t)=V_m\sin(\omega t)$ for 0<\omega t<\pi, zero elsewhere within 0\le\omega t<2\pi.
• Period $T=2\pi/\omega$, has half-wave symmetry – only odd harmonics, no DC term.
• Derived coefficients (selected):
  • $A<em>0=\dfrac{V</em>m}{\pi}$ (in transcript constant eliminated due to half-wave condition).
  • Cosine coefficients $B<em>n$ non-zero only for even $n$; transcript shows $B</em>n=-\dfrac{2V_m}{n\pi}$ for even $n$.
• Resulting Fourier Series (simplified):
  $f(\omega t)=\frac{V<em>m}{\pi}+\frac{2V</em>m}{\pi}\sum_{k=1}^{\infty}\frac{\cos(2k\omega t)}{2k-1}$ (exact expression depends on chosen form; transcript provides step-wise derivation pages 5–6).

Exponential (Complex) Form of Fourier Series

• Euler relations:
  $\sin nx=\frac{e^{jnx}-e^{-jnx}}{2j},\;\cos nx=\frac{e^{jnx}+e^{-jnx}}{2}$
• Series becomes:
  $f(x)=A<em>0+\sum</em>{n=1}^{\infty}\left(D<em>n e^{jnx}+D</em>{-n}e^{-jnx}\right)$
  where $D<em>n=\dfrac{1}{2\pi}\int</em>{-\pi}^{\pi}f(x)e^{-jnx}\,dx$ and $D<em>{-n}=D</em>n^{!*}$ (complex conjugate).
• Conversion between trigonometric and exponential coefficients:
  \begin{cases}
  Bn=Dn+D{-n}\ An=j\,(Dn-D{-n})\
  |Cn|=2|Dn|=\sqrt{An^2+Bn^2}
  \end{cases}
• Transcript example reorganises half-wave rectified series into $\sum D_n e^{jnx}$ form.

Symmetry Properties and Their Impact on Fourier Coefficients

Knowing symmetry significantly reduces computation:

Even Function Symmetry

• Condition: $f(x)=f(-x)$.
• Graph symmetric about vertical axis.
• Series contains only cosine terms (and possibly $A_0$).
• If additionally $f(x)=f(\pi-x)$, only even harmonics appear.

Odd Function Symmetry

• Condition: $f(x)=-f(-x)$.
• Series contains only sine terms; $A_0=0$.
• If $f(x)=-f(\pi-x)$, only odd harmonics remain.

Half-Wave (and Quarter-Wave) Symmetry

• Condition: $f(x+\pi)=-f(x)$.
• Implies zero DC component, only odd harmonics.
• Quarter-wave symmetry (special case) allows integration over one quarter period then multiply by 4.

Example – Quarter-Wave Symmetric Voltage (Fig. 34)

• Piecewise definition: $v(\omega t)=V$ from $\omega t=0$ to $\pi/2$, $0$ elsewhere within 0<\omega t<\pi (then anti-symmetry).
• Symmetry conclusions: no DC term, cosine-only, odd harmonic indices.
• Coefficient by quarter-wave method (transcript p. 10–11):
  $B<em>n=\frac{4V}{n\pi}\sin\left(\frac{n\pi}{2}\right)\quad\text{(odd }n)$ Numerical: $B</em>1=\frac{4V}{\pi},\;B<em>3=-\frac{4V}{3\pi},\;B</em>5=\frac{4V}{5\pi},\dots$
• Series expression:
  $v(\omega t)=\sum_{m=0}^{\infty}\frac{4V}{(2m+1)\pi}(-1)^m\cos\bigl[(2m+1)\omega t\bigr]$

Example – 3-Phase Rectifier Phase Current

• Given DC output current $I_d=10\,\text{A}$; single-phase waveform resembles quasi-square pulses (Fig. 35, page 11–13).
• Wave has half-wave symmetry ⇒ no DC term in AC component, only odd harmonics.
• Sine coefficients derived:
  $A<em>n=\frac{4I</em>d}{n\pi}\cos\left(\frac{n\pi}{6}\right)\quad\text{for odd }n$
  $A_n=0\quad\text{for even }n$
• Time-domain current:
  $i(t)=\sum<em>{k=0}^{\infty}\frac{4I</em>d}{(2k+1)\pi}\cos\left(\frac{(2k+1)\pi}{6}\right)\sin\bigl[(2k+1)\omega t\bigr]$
• RMS values (data from transcript table):
  • Fundamental $I_1=7.8\,\text{A}$
  • 5th $I_5=1.56\,\text{A}$
  • 7th $I_7=1.11\,\text{A}$
  • 11th $I_{11}=0.70\,\text{A}$, etc.
• Total Harmonic Distortion (THD) could be approximated: $\text{THD}=\sqrt{\sum<em>{n>1}I</em>n^2}/I_1 \approx 70\%$ (qualitative; exact requires full series).

Analysis Notes & Practical Relevance

• Symmetric non-sinusoidal waves consisting only of odd harmonics are common in power electronics (e.g., 6-pulse or 12-pulse rectifier currents).
• Phase alignment of harmonics (e.g., 30° delay of 3rd harmonic) influences instantaneous waveform but not necessarily RMS or THD.
• Engineering practice uses: IEEE 519 limits, filter design, transformer K-factor rating.
• Ethical/Environmental: Mitigating harmonics reduces energy waste, CO₂ emissions, and prevents premature equipment failure—aligns with sustainability goals.