Power Quality Problems – Harmonics & Fourier Analysis

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.

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.

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., 1.2f1,\;2.7f1). Not commensurate with the fundamental.
Subharmonics: components whose frequencies are below the fundamental (f<f_1).

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 ZL causes a voltage drop \Delta V = I{\text{dist}}\,Z_L, so current distortion is mirrored as voltage distortion.

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.

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).

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

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.

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

General form:
f(x)= A0 + \sum{n=1}^{\infty}\left(An\sin nx + Bn\cos nx\right)
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):
An=\frac{1}{\pi}\int{-\pi}^{\pi}f(x)\sin(nx)\,dx,\;\;Bn=\frac{1}{\pi}\int{-\pi}^{\pi}f(x)\cos(nx)\,dx

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

First harmonic magnitude:
C1=\sqrt{105^2+(-84)^2}=134.47\,\text{A} \varphi1 = \tan^{-1}!\left(\frac{-84}{105}\right)=-38.66^{\circ}
Second harmonic:
C2=\sqrt{(-38)^2+19^2}=42.49\,\text{A},\;\varphi2=116.565^{\circ}
Third harmonic:
C3=\sqrt{10^2+7^2}=12.20\,\text{A},\;\varphi3=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})

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):
- A0=\dfrac{Vm}{\pi} (in transcript constant eliminated due to half-wave condition).
- Cosine coefficients Bn non-zero only for even n; transcript shows Bn=-\dfrac{2V_m}{n\pi} for even n.
Resulting Fourier Series (simplified):
f(\omega t)=\frac{Vm}{\pi}+\frac{2Vm}{\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).

Euler relations:
\sin nx=\frac{e^{jnx}-e^{-jnx}}{2j},\;\cos nx=\frac{e^{jnx}+e^{-jnx}}{2}
Series becomes:
f(x)=A0+\sum{n=1}^{\infty}\left(Dn e^{jnx}+D{-n}e^{-jnx}\right)
where Dn=\dfrac{1}{2\pi}\int{-\pi}^{\pi}f(x)e^{-jnx}\,dx and D{-n}=Dn^{!*} (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.

Knowing symmetry significantly reduces computation:

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.

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):
Bn=\frac{4V}{n\pi}\sin\left(\frac{n\pi}{2}\right)\quad\text{(odd }n) Numerical: B1=\frac{4V}{\pi},\;B3=-\frac{4V}{3\pi},\;B5=\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]

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:
An=\frac{4Id}{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{k=0}^{\infty}\frac{4Id}{(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{n>1}In^2}/I_1 \approx 70\% (qualitative; exact requires full series).

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.