# Harmonics - Power Quality

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Legacy
Disciplines
• Mathematics
• Musicology

## Abstract

No Slide Title Harmonics How to deal with them ? Zbigniew Leonowicz, PhD Wroclaw University of Technology, Poland Based on ABB Power Quality Seminar What is the line current from this device? ? Current waveform Fundamental only Fundamental + fifth harmonic H1 + H5 + H7 H1 → H13 H1 → H25 Harmonics representation 0% 5% 10% 15% 20% 25% 5 7 11 13 17 19 23 25 Time domain Frequency domain Fourier series T • Any periodic function can be seen as FOURIER SERIES • A Fourier series is an infinite sum of sines with frequency which are equal to an integer multiple of the FUNDAMENTAL FREQUENCY • Each component of the series is called an HARMONIC Some mathematics ... T • Periodic function: f(t) = f(t+T) ∀ t • Representation in interval [-T/2,T/2]: f(t) = A0 + Σk [ Ak cos(kωt) + Bk sin(kωt) ] = A0 + Σk Ck cos(kωt-φk) • Fundamental pulsation: ω = 2π/T = 2π f1 • Harmonic magnitude: Ck = (Ak + Bk)1/2 • Harmonic phase: φk = atan(-Bk/Ak) • Sine/cosine components calculation: A T f t k t dtk T= +∫2 ( ).cos( ).ωθθ B T f t k t dtk T= +∫2 ( ).sin( ).ωθθ → Fourier’s theorem ... please forget it! f t A C t C k tk k( ) cos( ) cos( )= + − + − ∞∑0 1 1 1 ω φ ω φ DC Fundamental Harmonics RMS value = 1 1 2 2 0 2 2 1T f t dt A C T k k ( ).θ θ+ = ∞∫ ∑= + Example: No harmonics → RMS = C1/√2 Total Harmonic Distortion (THD) • Relative importance of harmonics regarding to fundamental • ( expressed in %) • THD(U): meaningful THD(I): ??? what is the reference ??? 1 2k 2 k C C THD ∑ == Harmonics dimensionless numbers • THDF ( Transformer HHarmonic Derating Factor) → kVAderated = THDF*kVA • KF (K-Factor) → Extra heat brought by harmonics • TIF-Factor (Telephone Interference Factor) → Describes interference of a power transmission line on a telephone line Example of distortion Peak 100% RMS 100% THD 0% 100% H1 Example of distortion Peak 100% 133

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