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This diagram is currently used for variable speeddrives, uninterruptible power supplies andinduction heating systems. The appendixcontains Fourier coefficient calculations fordetermining the magnitudes of harmonics in thecurrent drawn by an ideal 3-phase rectifier.1.4 Single-phase loadsRemember that symmetrical loads do notgenerate even order harmonics (seesection 1.2). As the spectrum is generallydecreasing, the third harmonic is therefore thedominant harmonic for single-phase loads. Also,for very common loads such as a single-phasediode rectifier with capacitive filtering (see fig. 8),the third harmonic can be as much as 80% of thefundamental. The waveform of the current whichthese loads draw and its harmonic spectrum arerepresented by figures 9 and 10.Numerous appliances, in all spheres of activity,contain a circuit of this type (see fig. 11). Theseare the main generators of third harmonics.1009080(%) 7060504030201001 3 5 7 9 11 13 15 17 19 21 23 25Harmonic orderFig. 8: Single-phase rectifier with capacitive filteringFig. 10: Harmonic spectrum of the current drawn bythe diagram in figure 8151050-5-10-15Network voltageLine current (A)t(s)0 0.020.04Sphere of activityDomesticCommercialIndustrialAppliancesTV, hi-fi, video, microwaveovens, fluorescent lampswith electronic ballast, etc.Micro-computers, printers,photocopiers, faxmachines, etc.Switch mode powersupplies, variable speeddrivesFig. 9: Appearance of the current drawn by thediagram in figure 8Fig. 11: Some examples of appliances containing asingle-phase rectifier with capacitive filteringCahier Technique Schneider Electric no. 202 / p.6
2 Overload on the neutral conductor2.1 Third harmonics and multiples of 3Consider a simplified system consisting of abalanced 3-phase source and three identicalsingle-phase loads, connected between phaseand neutral (see fig. 12).irisLoadLoadIn this simplified example, the third orderharmonic currents in all 3 phases aretherefore identical.Since the current in the neutral is equal to thesum of the currents in the phases, the componentof order 3 of the neutral current is equal to thesum of the components of order 3, ie:in 3 = 3ir 3As a general rule, for balanced loads, harmoniccurrents of orders which are a multiple of 3 are inphase and are added up arithmetically in theneutral conductor, while the fundamentalcomponents and harmonics of orders which arenot multiples of 3 cancel one another out.Third order harmonic currents are thereforezero-sequence currents, circulating in phasein all three phases.itinLoadReasoning based on graphic representationc Superimposition of third harmonicsFigure 13 shows three 3-phase sinusoidalcurrents at 50 Hz and three sinusoidal currentsat 150 Hz, each in phase with one of thecurrents at 50 Hz. These three currents areequal and are therefore superimposed.Fig. 12: Single-phase loadsIf the loads are linear, the currents constitute abalanced 3-phase system. The sum of the phasecurrents is therefore zero, as is the neutralcurrent.in = ∑i i = 0If the loads are non-linear, the phase currentsare non-sinusoidal and therefore containharmonics, particularly of orders which aremultiples of 3.Since all three-phase currents are equal, thethird order harmonic currents, for example, havethe same magnitude and can be written as:i I 3 sin 3 ωt⎛ 2π⎞i s3 = I3 sin 3 ⎜ωt− ⎟ = I sin ( 3 t − 2 ) = i⎝r33 ⎠ 3 ω π⎛ 4π⎞i t3 = I3 sin 3 ⎜ωt− ⎟ = I sin ( 3 t − 4 ) = i⎝r33 ⎠3 ω πr3 = ( )050Hz150Hz0 s 0.02 s0.04 sFig. 13: 3-phase currents at 50 Hz and 150 Hz drawnCahier Technique Schneider Electric no. 202 / p.7
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