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Harmonics in Polyphase Power SystemsharmonicsIn the chapter on mixedfrequency signals, we explored the concept of harmonics in AC systems: frequencies that are integer multiples of the fundamental source frequency. With AC power systems where the source voltage waveform coming from an AC generator (alternator) is supposed to be a singlefrequency sine wave, undistorted, there should be no harmonic content . . . ideally. This would be true were it not for nonlinear components. Nonlinear components draw current disproportionately with respect to the source voltage, causing nonsinusoidal current waveforms. Examples of nonlinear components include gasdischarge lamps, semiconductor powercontrol devices (diodes, transistors, SCRs, TRIACs), transformers (primary winding magnetization current is usually nonsinusoidal due to the B/H saturation curve of the core), and electric motors (again, when magnetic fields within the motor's core operate near saturation levels). Even incandescent lamps generate slightly nonsinusoidal currents, as the filament resistance changes throughout the cycle due to rapid fluctuations in temperature. As we learned in the mixedfrequency chapter, any distortion of an otherwise sinewave shaped waveform constitutes the presence of harmonic frequencies. waveform symmetry and harmonicsWhen the nonsinusoidal waveform in question is symmetrical above and below its average centerline, the harmonic frequencies will be odd integer multiples of the fundamental source frequency only, with no even integer multiples. Most nonlinear loads produce current waveforms like this, and so evennumbered harmonics (2nd, 4th, 6th, 8th, 10th, 12th, etc.) are absent or only minimally present in most AC power systems. Examples of symmetrical waveforms  odd harmonics only:
Examples of nonsymmetrical waveforms  even harmonics present: Even though half of the possible harmonic frequencies are eliminated by the typically symmetrical distortion of nonlinear loads, the odd harmonics can still cause problems. Some of these problems are general to all power systems, singlephase or otherwise. Transformer overheating due to eddy current losses, for example, can occur in any AC power system where there is significant harmonic content. However, there are some problems caused by harmonic currents that are specific to polyphase power systems, and it is these problems to which this section is specifically devoted. It is helpful to be able to simulate nonlinear loads in SPICE so as to avoid a lot of complex mathematics and obtain a more intuitive understanding of harmonic effects. First, we'll begin our simulation with a very simple AC circuit: a single sinewave voltage source with a purely linear load and all associated resistances: The R_{source} and R_{line} resistances in this circuit do more than just mimic the real world: they also provide convenient shunt resistances for measuring currents in the SPICE simulation: by reading voltage across a 1 Ω resistance, you obtain a direct indication of current through it, since E = IR. A SPICE simulation of this circuit with Fourier analysis on the voltage measured across R_{line} should show us the harmonic content of this circuit's line current. Being completely linear in nature, we should expect no harmonics other than the 1st (fundamental) of 60 Hz, assuming a 60 Hz source:
linear load simulation vsource 1 0 sin(0 120 60 0 0) rsource 1 2 1 rline 2 3 1 rload 3 0 1k .options itl5=0 .tran 0.5m 30m 0 1u .plot tran v(2,3) .four 60 v(2,3) .end fourier components of transient response v(2,3) dc component = 4.028E12 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.198E01 1.000000 72.000 0.000 2 1.200E+02 5.793E12 0.000000 51.122 123.122 3 1.800E+02 7.407E12 0.000000 34.624 37.376 4 2.400E+02 9.056E12 0.000000 4.267 76.267 5 3.000E+02 1.651E11 0.000000 83.461 11.461 6 3.600E+02 3.931E11 0.000000 36.399 108.399 7 4.200E+02 2.338E11 0.000000 41.343 30.657 8 4.800E+02 4.716E11 0.000000 53.324 125.324 9 5.400E+02 3.453E11 0.000000 21.691 93.691 total harmonic distortion = 0.000000 percent
A No discrete Fourier transform is perfect, and so we see very small harmonic currents indicated (in the picoamp range!) for all frequencies up to the 9th harmonic (in the table ), which is as far as SPICE goes in performing Fourier analysis. We show 0.1198 amps (1.198E01) for the "fourier component" of the 1st harmonic, or the fundamental frequency, which is our expected load current: about 120 mA, given a source voltage of 120 volts and a load resistance of 1 kΩ. Next, I'd like to simulate a nonlinear load so as to generate harmonic currents. This can be done in two fundamentally different ways. One way is to design a load using nonlinear components such as diodes or other semiconductor devices which as easy to simulate with SPICE. Another is to add some AC current sources in parallel with the load resistor. The latter method is often preferred by engineers for simulating harmonics, since current sources of known value lend themselves better to mathematical network analysis than components with highly complex response characteristics. Since we're letting SPICE do all the math work, the complexity of a semiconductor component would cause no trouble for us, but since current sources can be finetuned to produce any arbitrary amount of current (a convenient feature), I'll choose the latter approach:
Nonlinear load simulation vsource 1 0 sin(0 120 60 0 0) rsource 1 2 1 rline 2 3 1 rload 3 0 1k i3har 3 0 sin(0 50m 180 0 0) .options itl5=0 .tran 0.5m 30m 0 1u .plot tran v(2,3) .four 60 v(2,3) .end In this circuit, we have a current source of 50 mA magnitude and a frequency of 180 Hz, which is three times the source frequency of 60 Hz. Connected in parallel with the 1 kΩ load resistor, its current will add with the resistor's to make a nonsinusoidal total line current. I'll show the waveform plot here just so you can see the effects of this 3rdharmonic current on the total current, which would ordinarily be a plain sine wave:
fourier components of transient response v(2,3) dc component = 1.349E11 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.198E01 1.000000 72.000 0.000 2 1.200E+02 1.609E11 0.000000 67.570 139.570 3 1.800E+02 4.990E02 0.416667 144.000 216.000 4 2.400E+02 1.074E10 0.000000 169.546 97.546 5 3.000E+02 3.871E11 0.000000 169.582 241.582 6 3.600E+02 5.736E11 0.000000 140.845 212.845 7 4.200E+02 8.407E11 0.000000 177.071 249.071 8 4.800E+02 1.329E10 0.000000 156.772 228.772 9 5.400E+02 2.619E10 0.000000 160.498 232.498 total harmonic distortion = 41.666663 percent
In the Fourier analysis, the mixed frequencies are unmixed and presented separately. Here we see the same 0.1198 amps of 60 Hz (fundamental) current as we did in the first simulation, but appearing in the 3rd harmonic row we see 49.9 mA: our 50 mA, 180 Hz current source at work. Why don't we see the entire 50 mA through the line? Because that current source is connected across the 1 kΩ load resistor, so some of its current is shunted through the load and never goes through the line back to the source. It's an inevitable consequence of this type of simulation, where one part of the load is "normal" (a resistor) and the other part is imitated by a current source. If we were to add more current sources to the "load," we would see further distortion of the line current waveform from the ideal sinewave shape, and each of those harmonic currents would appear in the Fourier analysis breakdown:
Nonlinear load simulation vsource 1 0 sin(0 120 60 0 0) rsource 1 2 1 rline 2 3 1 rload 3 0 1k i3har 3 0 sin(0 50m 180 0 0) i5har 3 0 sin(0 50m 300 0 0) i7har 3 0 sin(0 50m 420 0 0) i9har 3 0 sin(0 50m 540 0 0) .options itl5=0 .tran 0.5m 30m 0 1u .plot tran v(2,3) .four 60 v(2,3) .end fourier components of transient response v(2,3) dc component = 6.299E11 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.198E01 1.000000 72.000 0.000 2 1.200E+02 1.900E09 0.000000 93.908 21.908 3 1.800E+02 4.990E02 0.416667 144.000 216.000 4 2.400E+02 5.469E09 0.000000 116.873 44.873 5 3.000E+02 4.990E02 0.416667 0.000 72.000 6 3.600E+02 6.271E09 0.000000 85.062 157.062 7 4.200E+02 4.990E02 0.416666 144.000 72.000 8 4.800E+02 2.742E09 0.000000 38.781 33.219 9 5.400E+02 4.990E02 0.416666 72.000 144.000 total harmonic distortion = 83.333296 percent
As you can see from the Fourier analysis, every harmonic current source is equally represented in the line current, at 49.9 mA each. So far, this is just a singlephase power system simulation. Things get more interesting when we make it a threephase simulation. Two Fourier analyses will be performed: one for the voltage across a line resistor, and one for the voltage across the neutral resistor. As before, reading voltages across fixed resistances of 1 Ω each gives direct indications of current through those resistors:
YY source/load 4wire system with harmonics * * phase1 voltage source and r (120 v /_ 0 deg) vsource1 1 0 sin(0 120 60 0 0) rsource1 1 2 1 * * phase2 voltage source and r (120 v /_ 120 deg) vsource2 3 0 sin(0 120 60 5.55555m 0) rsource2 3 4 1 * * phase3 voltage source and r (120 v /_ 240 deg) vsource3 5 0 sin(0 120 60 11.1111m 0) rsource3 5 6 1 * * line and neutral wire resistances rline1 2 8 1 rline2 4 9 1 rline3 6 10 1 rneutral 0 7 1 * * phase 1 of load rload1 8 7 1k i3har1 8 7 sin(0 50m 180 0 0) i5har1 8 7 sin(0 50m 300 0 0) i7har1 8 7 sin(0 50m 420 0 0) i9har1 8 7 sin(0 50m 540 0 0) * * phase 2 of load rload2 9 7 1k i3har2 9 7 sin(0 50m 180 5.55555m 0) i5har2 9 7 sin(0 50m 300 5.55555m 0) i7har2 9 7 sin(0 50m 420 5.55555m 0) i9har2 9 7 sin(0 50m 540 5.55555m 0) * * phase 3 of load rload3 10 7 1k i3har3 10 7 sin(0 50m 180 11.1111m 0) i5har3 10 7 sin(0 50m 300 11.1111m 0) i7har3 10 7 sin(0 50m 420 11.1111m 0) i9har3 10 7 sin(0 50m 540 11.1111m 0) * * analysis stuff .options itl5=0 .tran 0.5m 100m 12m 1u .plot tran v(2,8) .four 60 v(2,8) .plot tran v(0,7) .four 60 v(0,7) .end Fourier analysis of line current: fourier components of transient response v(2,8) dc component = 6.404E12 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.198E01 1.000000 0.000 0.000 2 1.200E+02 2.218E10 0.000000 172.985 172.985 3 1.800E+02 4.975E02 0.415423 0.000 0.000 4 2.400E+02 4.236E10 0.000000 166.990 166.990 5 3.000E+02 4.990E02 0.416667 0.000 0.000 6 3.600E+02 1.877E10 0.000000 147.146 147.146 7 4.200E+02 4.990E02 0.416666 0.000 0.000 8 4.800E+02 2.784E10 0.000000 148.811 148.811 9 5.400E+02 4.975E02 0.415422 0.000 0.000 total harmonic distortion = 83.209009 percent
Fourier analysis of neutral current: fourier components of transient response v(0,7) dc component = 1.819E10 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 4.337E07 1.000000 60.018 0.000 2 1.200E+02 1.869E10 0.000431 91.206 31.188 3 1.800E+02 1.493E01 344147.7638 180.000 240.018 4 2.400E+02 1.257E09 0.002898 21.103 81.121 5 3.000E+02 9.023E07 2.080596 119.981 59.963 6 3.600E+02 3.396E10 0.000783 15.882 44.136 7 4.200E+02 1.264E06 2.913955 59.993 0.025 8 4.800E+02 5.975E10 0.001378 35.584 24.434 9 5.400E+02 1.493E01 344147.4889 179.999 240.017
This is a balanced YY power system, each phase identical to the singlephase AC system simulated earlier. Consequently, it should come as no surprise that the Fourier analysis for line current in one phase of the 3phase system is nearly identical to the Fourier analysis for line current in the singlephase system: a fundamental (60 Hz) line current of 0.1198 amps, and odd harmonic currents of approximately 50 mA each. What should be surprising here is the analysis for the neutral conductor's current, as determined by the voltage drop across the R_{neutral} resistor between SPICE nodes 0 and 7. In a balanced 3phase Y load, we would expect the neutral current to be zero. Each phase current  which by itself would go through the neutral wire back to the supplying phase on the source Y  should cancel each other in regard to the neutral conductor because they're all the same magnitude and all shifted 120^{o} apart. In a system with no harmonic currents, this is what happens, leaving zero current through the neutral conductor. However, we cannot say the same for harmonic currents in the same system. Note that the fundamental frequency (60 Hz, or the 1st harmonic) current is virtually absent from the neutral conductor. Our Fourier analysis shows only 0.4337 μA of 1st harmonic when reading voltage across R_{neutral}. The same may be said about the 5th and 7th harmonics, both of those currents having negligible magnitude. In contrast, the 3rd and 9th harmonics are strongly represented within the neutral conductor, with 149.3 mA (1.493E01 volts across 1 Ω) each! This is very nearly 150 mA, or three times the current sources' values, individually. With three sources per harmonic frequency in the load, it appears our 3rd and 9th harmonic currents in each phase are adding to form the neutral current. This is exactly what's happening, though it might not be apparent why this is so. The key to understanding this is made clear in a timedomain graph of phase currents. Examine this plot of balanced phase currents over time, with a phase sequence of 123: With the three fundamental waveforms equally shifted across the time axis of the graph, it is easy to see how they would cancel each other to give a resultant current of zero in the neutral conductor. Let's consider, though, what a 3rd harmonic waveform for phase 1 would look like superimposed on the graph: Observe how this harmonic waveform has the same phase relationship to the 2nd and 3rd fundamental waveforms as it does with the 1st: in each positive halfcycle of any of the fundamental waveforms, you will find exactly two positive halfcycles and one negative halfcycle of the harmonic waveform. What this means is that the 3rdharmonic waveforms of three 120^{o} phaseshifted fundamentalfrequency waveforms are actually in phase with each other. The phase shift figure of 120^{o} generally assumed in threephase AC systems applies only to the fundamental frequencies, not to their harmonic multiples! If we were to plot all three 3rdharmonic waveforms on the same graph, we would see them precisely overlap and appear as a single, unified waveform (shown here in bold): For the more mathematically inclined, this principle may be expressed symbolically. Suppose that A represents one waveform and B another, both at the same frequency, but shifted 120^{o} from each other in terms of phase. Let's call the 3rd harmonic of each waveform A' and B', respectively. The phase shift between A' and B' is not 120^{o} (that is the phase shift between A and B), but 3 times that, because the A' and B' waveforms alternate three times as fast as A and B. The shift between waveforms is only accurately expressed in terms of phase angle when the same angular velocity is assumed. When relating waveforms of different frequency, the most accurate way to represent phase shift is in terms of time; and the timeshift between A' and B' is equivalent to 120^{o} at a frequency three times lower, or 360^{o} at the frequency of A' and B'. A phase shift of 360^{o} is the same as a phase shift of 0^{o}, which is to say no phase shift at all. Thus, A' and B' must be in phase with each other: This characteristic of the 3rd harmonic in a threephase system also holds true for any integer multiples of the 3rd harmonic. So, not only are the 3rd harmonic waveforms of each fundamental waveform in phase with each other, but so are the 6th harmonics, the 9th harmonics, the 12th harmonics, the 15th harmonics, the 18th harmonics, the 21st harmonics, and so on. Since only odd harmonics appear in systems where waveform distortion is symmetrical about the centerline  and most nonlinear loads create symmetrical distortion  evennumbered multiples of the 3rd harmonic (6th, 12th, 18th, etc.) are generally not significant, leaving only the oddnumbered multiples (3rd, 9th, 21st, etc.) to significantly contribute to neutral currents. In polyphase power systems with some number of phases other than three, this effect occurs with harmonics of the same multiple. For instance, the harmonic currents that add in the neutral conductor of a starconnected 4phase system where the phase shift between fundamental waveforms is 90^{o} would be the 4th, 8th, 12th, 16th, 20th, and so on. Triplen harmonicsDue to their abundance and significance in threephase power systems, the 3rd harmonic and its multiples have their own special name: triplen harmonics. All triplen harmonics add with each other in the neutral conductor of a 4wire Yconnected load. In power systems containing substantial nonlinear loading, the triplen harmonic currents may be of great enough magnitude to cause neutral conductors to overheat. This is very problematic, as other safety concerns prohibit neutral conductors from having overcurrent protection, and thus there is no provision for automatic interruption of these high currents. angular velocityThe following illustration shows how triplen harmonic currents created at the load add within the neutral conductor. The symbol "Ω" is used to represent angular velocity, and is mathematically equivalent to 2πf. So, "Ω" represents the fundamental frequency, "3Ω " represents the 3rd harmonic, "5Ω" represents the 5th harmonic, and so on: In an effort to mitigate these additive triplen currents, one might be tempted to remove the neutral wire entirely. If there is no neutral wire in which triplen currents can flow together, then they won't, right? Unfortunately, doing so just causes a different problem: the load's "Y" centerpoint will no longer be at the same potential as the source's, meaning that each phase of the load will receive a different voltage than what is produced by the source. We'll rerun the last SPICE simulation without the 1 Ω R_{neutral} resistor and see what happens:
YY source/load (no neutral) with harmonics * * phase1 voltage source and r (120 v /_ 0 deg) vsource1 1 0 sin(0 120 60 0 0) rsource1 1 2 1 * * phase2 voltage source and r (120 v /_ 120 deg) vsource2 3 0 sin(0 120 60 5.55555m 0) rsource2 3 4 1 * * phase3 voltage source and r (120 v /_ 240 deg) vsource3 5 0 sin(0 120 60 11.1111m 0) rsource3 5 6 1 * * line resistances rline1 2 8 1 rline2 4 9 1 rline3 6 10 1 * * phase 1 of load rload1 8 7 1k i3har1 8 7 sin(0 50m 180 0 0) i5har1 8 7 sin(0 50m 300 0 0) i7har1 8 7 sin(0 50m 420 0 0) i9har1 8 7 sin(0 50m 540 0 0) * * phase 2 of load rload2 9 7 1k i3har2 9 7 sin(0 50m 180 5.55555m 0) i5har2 9 7 sin(0 50m 300 5.55555m 0) i7har2 9 7 sin(0 50m 420 5.55555m 0) i9har2 9 7 sin(0 50m 540 5.55555m 0) * * phase 3 of load rload3 10 7 1k i3har3 10 7 sin(0 50m 180 11.1111m 0) i5har3 10 7 sin(0 50m 300 11.1111m 0) i7har3 10 7 sin(0 50m 420 11.1111m 0) i9har3 10 7 sin(0 50m 540 11.1111m 0) * * analysis stuff .options itl5=0 .tran 0.5m 100m 12m 1u .plot tran v(2,8) .four 60 v(2,8) .plot tran v(0,7) .four 60 v(0,7) .plot tran v(8,7) .four 60 v(8,7) .end Fourier analysis of line current: fourier components of transient response v(2,8) dc component = 5.423E11 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.198E01 1.000000 0.000 0.000 2 1.200E+02 2.388E10 0.000000 158.016 158.016 3 1.800E+02 3.136E07 0.000003 90.009 90.009 4 2.400E+02 5.963E11 0.000000 111.510 111.510 5 3.000E+02 4.990E02 0.416665 0.000 0.000 6 3.600E+02 8.606E11 0.000000 124.565 124.565 7 4.200E+02 4.990E02 0.416668 0.000 0.000 8 4.800E+02 8.126E11 0.000000 159.638 159.638 9 5.400E+02 9.406E07 0.000008 90.005 90.005 total harmonic distortion = 58.925539 percent
Fourier analysis of voltage between the two "Y" centerpoints: fourier components of transient response v(0,7) dc component = 6.093E08 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.453E04 1.000000 60.018 0.000 2 1.200E+02 6.263E08 0.000431 91.206 31.188 3 1.800E+02 5.000E+01 344147.7879 180.000 240.018 4 2.400E+02 4.210E07 0.002898 21.103 81.121 5 3.000E+02 3.023E04 2.080596 119.981 59.963 6 3.600E+02 1.138E07 0.000783 15.882 44.136 7 4.200E+02 4.234E04 2.913955 59.993 0.025 8 4.800E+02 2.001E07 0.001378 35.584 24.434 9 5.400E+02 5.000E+01 344147.4728 179.999 240.017 total harmonic distortion = ************ percent
Fourier analysis of load phase voltage: fourier components of transient response v(8,7) dc component = 6.070E08 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.198E+02 1.000000 0.000 0.000 2 1.200E+02 6.231E08 0.000000 90.473 90.473 3 1.800E+02 5.000E+01 0.417500 180.000 180.000 4 2.400E+02 4.278E07 0.000000 19.747 19.747 5 3.000E+02 9.995E02 0.000835 179.850 179.850 6 3.600E+02 1.023E07 0.000000 13.485 13.485 7 4.200E+02 9.959E02 0.000832 179.790 179.789 8 4.800E+02 1.991E07 0.000000 35.462 35.462 9 5.400E+02 5.000E+01 0.417499 179.999 179.999 total harmonic distortion = 59.043467 percent
Strange things are happening, indeed. First, we see that the triplen harmonic currents (3rd and 9th) all but disappear in the lines connecting load to source. The 5th and 7th harmonic currents are present at their normal levels (approximately 50 mA), but the 3rd and 9th harmonic currents are of negligible magnitude. Second, we see that there is substantial harmonic voltage between the two "Y" centerpoints, between which the neutral conductor used to connect. According to SPICE, there is 50 volts of both 3rd and 9th harmonic frequency between these two points, which is definitely not normal in a linear (no harmonics), balanced Y system. Finally, the voltage as measured across one of the load's phases (between nodes 8 and 7 in the SPICE analysis) likewise shows strong triplen harmonic voltages of 50 volts each. The following illustration is a graphical summary of the aforementioned effects: In summary, removal of the neutral conductor leads to a "hot" centerpoint on the load "Y", and also to harmonic load phase voltages of equal magnitude, all comprised of triplen frequencies. In the previous simulation where we had a 4wire, Yconnected system, the undesirable effect from harmonics was excessive neutral current, but at least each phase of the load received voltage nearly free of harmonics. Since removing the neutral wire didn't seem to work in eliminating the problems caused by harmonics, perhaps switching to a Δ configuration will. Let's try a Δ source instead of a Y, keeping the load in its present Y configuration, and see what happens. The measured parameters will be line current (voltage across R_{line}, nodes 0 and 8), load phase voltage (nodes 8 and 7), and source phase current (voltage across R_{source}, nodes 1 and 2):
DeltaY source/load with harmonics * * phase1 voltage source and r (120 v /_ 0 deg) vsource1 1 0 sin(0 207.846 60 0 0) rsource1 1 2 1 * * phase2 voltage source and r (120 v /_ 120 deg) vsource2 3 2 sin(0 207.846 60 5.55555m 0) rsource2 3 4 1 * * phase3 voltage source and r (120 v /_ 240 deg) vsource3 5 4 sin(0 207.846 60 11.1111m 0) rsource3 5 0 1 * * line resistances rline1 0 8 1 rline2 2 9 1 rline3 4 10 1 * * phase 1 of load rload1 8 7 1k i3har1 8 7 sin(0 50m 180 9.72222m 0) i5har1 8 7 sin(0 50m 300 9.72222m 0) i7har1 8 7 sin(0 50m 420 9.72222m 0) i9har1 8 7 sin(0 50m 540 9.72222m 0) * * phase 2 of load rload2 9 7 1k i3har2 9 7 sin(0 50m 180 15.2777m 0) i5har2 9 7 sin(0 50m 300 15.2777m 0) i7har2 9 7 sin(0 50m 420 15.2777m 0) i9har2 9 7 sin(0 50m 540 15.2777m 0) * * phase 3 of load rload3 10 7 1k i3har3 10 7 sin(0 50m 180 4.16666m 0) i5har3 10 7 sin(0 50m 300 4.16666m 0) i7har3 10 7 sin(0 50m 420 4.16666m 0) i9har3 10 7 sin(0 50m 540 4.16666m 0) * * analysis stuff .options itl5=0 .tran 0.5m 100m 16m 1u .plot tran v(0,8) v(8,7) v(1,2) .four 60 v(0,8) v(8,7) v(1,2) .end
Note: the following paragraph is for those curious readers who follow every detail of my SPICE netlists. If you just want to find out what happens in the circuit, skip this paragraph! When simulating circuits having AC sources of differing frequency and differing phase, the only way to do it in SPICE is to set up the sources with a delay time or phase offset specified in seconds. Thus, the 0^{o} source has these five specifying figures: "(0 207.846 60 0 0)", which means 0 volts DC offset, 207.846 volts peak amplitude (120 times the square root of three, to ensure the load phase voltages remain at 120 volts each), 60 Hz, 0 time delay, and 0 damping factor. The 120^{o} phaseshifted source has these figures: "(0 207.846 60 5.55555m 0)", all the same as the first except for the time delay factor of 5.55555 milliseconds, or 1/3 of the full period of 16.6667 milliseconds for a 60 Hz waveform. The 240^{o} source must be timedelayed twice that amount, equivalent to a fraction of 240/360 of 16.6667 milliseconds, or 11.1111 milliseconds. This is for the Δconnected source. The Yconnected load, on the other hand, requires a different set of timedelay figures for its harmonic current sources, because the phase voltages in a Y load are not in phase with the phase voltages of a Δ source. If Δ source voltages V_{AC}, V_{BA}, and V_{CB} are referenced at 0^{o}, 120^{o}, and 240^{o}, respectively, then "Y" load voltages V_{A}, V_{B}, and V_{C} will have phase angles of 30^{o}, 90^{o}, and 210^{o}, respectively. This is an intrinsic property of all ΔY circuits and not a quirk of SPICE. Therefore, when I specified the delay times for the harmonic sources, I had to set them at 15.2777 milliseconds (30^{o}, or +330^{o}), 4.16666 milliseconds (90^{o}), and 9.72222 milliseconds (210^{o}). One final note: when delaying AC sources in SPICE, they don't "turn on" until their delay time has elapsed, which means any mathematical analysis up to that point in time will be in error. Consequently, I set the The result of this analysis is almost as disappointing as the last. Line currents remain unchanged (the only substantial harmonic content being the 5th and 7th harmonics), and load phase voltages remain unchanged as well, with a full 50 volts of triplen harmonic (3rd and 9th) frequencies across each load component. Source phase current is a fraction of the line current, which should come as no surprise. Both 5th and 7th harmonics are represented there, with negligible triplen harmonics:
Fourier analysis of line current: fourier components of transient response v(0,8) dc component = 6.850E11 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.198E01 1.000000 150.000 0.000 2 1.200E+02 2.491E11 0.000000 159.723 9.722 3 1.800E+02 1.506E06 0.000013 0.005 149.996 4 2.400E+02 2.033E11 0.000000 52.772 97.228 5 3.000E+02 4.994E02 0.416682 30.002 119.998 6 3.600E+02 1.234E11 0.000000 57.802 92.198 7 4.200E+02 4.993E02 0.416644 29.998 179.998 8 4.800E+02 8.024E11 0.000000 174.200 324.200 9 5.400E+02 4.518E06 0.000038 179.995 329.995 total harmonic distortion = 58.925038 percent
Fourier analysis of load phase voltage: fourier components of transient response v(8,7) dc component = 1.259E08 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.198E+02 1.000000 150.000 0.000 2 1.200E+02 1.941E07 0.000000 49.693 100.307 3 1.800E+02 5.000E+01 0.417222 89.998 239.998 4 2.400E+02 1.519E07 0.000000 66.397 83.603 5 3.000E+02 6.466E02 0.000540 151.112 301.112 6 3.600E+02 2.433E07 0.000000 68.162 81.838 7 4.200E+02 6.931E02 0.000578 148.548 1.453 8 4.800E+02 2.398E07 0.000000 174.897 324.897 9 5.400E+02 5.000E+01 0.417221 90.006 59.995 total harmonic distortion = 59.004109 percent
Fourier analysis of source phase current: fourier components of transient response v(1,2) dc component = 3.564E11 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 6.906E02 1.000000 0.181 0.000 2 1.200E+02 1.525E11 0.000000 156.674 156.493 3 1.800E+02 1.422E06 0.000021 179.996 179.815 4 2.400E+02 2.949E11 0.000000 110.570 110.390 5 3.000E+02 2.883E02 0.417440 179.996 179.815 6 3.600E+02 2.324E11 0.000000 91.926 91.745 7 4.200E+02 2.883E02 0.417398 179.994 179.813 8 4.800E+02 4.140E11 0.000000 39.875 39.694 9 5.400E+02 4.267E06 0.000062 0.006 0.186 total harmonic distortion = 59.031969 percent
Really, the only advantage of the ΔY configuration from the standpoint of harmonics is that there is no longer a centerpoint at the load posing a shock hazard. Otherwise, the load components receive the same harmonicallyrich voltages and the lines see the same currents as in a threewire Y system. If we were to reconfigure the system into a ΔΔ arrangement, that should guarantee that each load component receives nonharmonic voltage, since each load phase would be directly connected in parallel with each source phase. The complete lack of any neutral wires or "center points" in a ΔΔ system prevents strange voltages or additive currents from occurring. It would seem to be the ideal solution. Let's simulate and observe, analyzing line current, load phase voltage, and source phase current:
DeltaDelta source/load with harmonics * * phase1 voltage source and r (120 v /_ 0 deg) vsource1 1 0 sin(0 120 60 0 0) rsource1 1 2 1 * * phase2 voltage source and r (120 v /_ 120 deg) vsource2 3 2 sin(0 120 60 5.55555m 0) rsource2 3 4 1 * * phase3 voltage source and r (120 v /_ 240 deg) vsource3 5 4 sin(0 120 60 11.1111m 0) rsource3 5 0 1 * * line resistances rline1 0 6 1 rline2 2 7 1 rline3 4 8 1 * * phase 1 of load rload1 7 6 1k i3har1 7 6 sin(0 50m 180 0 0) i5har1 7 6 sin(0 50m 300 0 0) i7har1 7 6 sin(0 50m 420 0 0) i9har1 7 6 sin(0 50m 540 0 0) * * phase 2 of load rload2 8 7 1k i3har2 8 7 sin(0 50m 180 5.55555m 0) i5har2 8 7 sin(0 50m 300 5.55555m 0) i7har2 8 7 sin(0 50m 420 5.55555m 0) i9har2 8 7 sin(0 50m 540 5.55555m 0) * * phase 3 of load rload3 6 8 1k i3har3 6 8 sin(0 50m 180 11.1111m 0) i5har3 6 8 sin(0 50m 300 11.1111m 0) i7har3 6 8 sin(0 50m 420 11.1111m 0) i9har3 6 8 sin(0 50m 540 11.1111m 0) * * analysis stuff .options itl5=0 .tran 0.5m 100m 16m 1u .plot tran v(0,6) v(7,6) v(2,1) i(3har1) .four 60 v(0,6) v(7,6) v(2,1) .end
Fourier analysis of line current: fourier components of transient response v(0,6) dc component = 6.007E11 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 2.070E01 1.000000 150.000 0.000 2 1.200E+02 5.480E11 0.000000 156.666 6.666 3 1.800E+02 6.257E07 0.000003 89.990 60.010 4 2.400E+02 4.911E11 0.000000 8.187 141.813 5 3.000E+02 8.626E02 0.416664 149.999 300.000 6 3.600E+02 1.089E10 0.000000 31.997 181.997 7 4.200E+02 8.626E02 0.416669 150.001 0.001 8 4.800E+02 1.578E10 0.000000 63.940 213.940 9 5.400E+02 1.877E06 0.000009 89.987 60.013 total harmonic distortion = 58.925538 percent
Fourier analysis of load phase voltage: fourier components of transient response v(7,6) dc component = 5.680E10 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.195E+02 1.000000 0.000 0.000 2 1.200E+02 1.039E09 0.000000 144.749 144.749 3 1.800E+02 1.251E06 0.000000 89.974 89.974 4 2.400E+02 4.215E10 0.000000 36.127 36.127 5 3.000E+02 1.992E01 0.001667 180.000 180.000 6 3.600E+02 2.499E09 0.000000 4.760 4.760 7 4.200E+02 1.992E01 0.001667 180.000 180.000 8 4.800E+02 2.951E09 0.000000 151.385 151.385 9 5.400E+02 3.752E06 0.000000 89.905 89.905 total harmonic distortion = 0.235702 percent
Fourier analysis of source phase current: fourier components of transient response v(2,1) dc component = 1.923E12 harmonic frequency fourier normalized phase normalized no (hz) component component (deg) phase (deg) 1 6.000E+01 1.194E01 1.000000 179.940 0.000 2 1.200E+02 2.569E11 0.000000 133.491 46.449 3 1.800E+02 3.129E07 0.000003 89.985 89.955 4 2.400E+02 2.657E11 0.000000 23.368 156.571 5 3.000E+02 4.980E02 0.416918 180.000 359.939 6 3.600E+02 4.595E11 0.000000 22.475 202.415 7 4.200E+02 4.980E02 0.416921 180.000 359.939 8 4.800E+02 7.385E11 0.000000 63.759 243.699 9 5.400E+02 9.385E07 0.000008 89.991 89.949 total harmonic distortion = 58.961298 percent
As predicted earlier, the load phase voltage is almost a pure sinewave, with negligible harmonic content, thanks to the direct connection with the source phases in a ΔΔ system. But what happened to the triplen harmonics? The 3rd and 9th harmonic frequencies don't appear in any substantial amount in the line current, nor in the load phase voltage, nor in the source phase current! We know that triplen currents exist, because the 3rd and 9th harmonic current sources are intentionally placed in the phases of the load, but where did those currents go? Remember that the triplen harmonics of 120^{o} phaseshifted fundamental frequencies are in phase with each other. Note the directions that the arrows of the current sources within the load phases are pointing, and think about what would happen if the 3rd and 9th harmonic sources were DC sources instead. What we would have is current circulating within the loop formed by the Δconnected phases. This is where the triplen harmonic currents have gone: they stay within the Δ of the load, never reaching the line conductors or the windings of the source. These results may be graphically summarized as such: This is a major benefit of the ΔΔ system configuration: triplen harmonic currents remain confined in whatever set of components create them, and do not "spread" to other parts of the system.


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