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This is the continuation of Part 1 where we established the theory behind this subject and also proved by using calculations and by plotting the sine wave curves of fundamental and 3rd harmonic currents.
Let us then try to investigate why 3rd harmonic currents seems to be the only that adds up in the neutral conductor. How about the 2nd, 4th, 5th, 7th, etc.?
In Table-1 below, I worked out some calculations for the harmonic orders 1 to 15, showing the instantaneous current values for Phases A, B, C.
Table-1: Instantaneous Current Values for Harmonic Orders 1-15
Harmonic Order Frequency (Hz) Time (Sec) Phase A Current @ Time T- sin(n(wt+0)) Phase B Current @ Time T- sin(n(wt-120)) Phase C Current @ Time T- sin(n(wt+120)) Sum in Neutral
1 60 1/720 0.5 -1 0.5 0
2 120 1/720 0.87 0 -0.87 0
3 180 1/720 1 1 1 3
4 240 1/720 0.87 0 -0.87 0
5 300 1/720 0.5 -1 0.5 0
6 360 1/720 0 0 0 0
7 420 1/720 -0.5 1 -0.5 0
8 480 1/720 -0.87 0 0.87 0
9 540 1/720 -1 -1 -1 -3
10 600 1/720 -0.87 0 0.87 0
11 660 1/720 -0.5 1 -0.5 0
12 720 1/2880 1 1 1 3
13 780 1/720 0.5 -1 0.5 0
14 840 1/720 0.87 0 -0.87 0
15 900 1/720 1 1 1 3
As shown in Table-1 above, the currents for Phases A, B & C would cancel out for all harmonic orders except those in the Triplen Harmonic Orders (3, 6, 9, 15). I have selected an instantaneous time of 1/720 Seconds and 1/2880 Seconds to demonstrate that at these time values Triplen Harmonic orders have the same maximum values of 1p.u. or minimum values of -1p.u. for all three phases, which means that all three phases are in-phase with each other and thus will add up in the neutral conductor.
On the other hand, if you look at the non-triplen harmonic orders, the currents in three phases will always cancel out and sum-up to zero whatever instantaneous time value we use.
Relationships Between Harmonics and Symmetrical Components
In balanced 3-phase circuits where the currents are equal and are in 120° apart between phases, the harmonics can be expressed as a sequence components. For example, the 2nd harmonic has 240° (at 60Hz base) between it’s phasers, the 3rd harmonics has 360° between it’s phasers, etc. In Table-2 below, I listed the harmonic orders and their respective sequence.
Table-2: Harmonic sequences in a balanced three-phase system
It is interesting to note that based on Table-2 above, all the Triplen Harmonics fall on the Zero Sequence column. This further supports our findings from Table-1 calculations which basically tells us that Triplen Harmonics are indeed have it’s three phases in-phase with each other, meaning that for a Triplen Harmonic, the Magnitude and Angle of harmonic currents in Phase A, B & C are the same just as in a Zero Sequence network. Refer to below Figure-1 for a quick review of symmetrical network and note that for Zero Sequence network, all the three phases are also have the same Magnitude and Direction.
Sources of Harmonic Currents
IEEE Std 141-1993 gives us a good list of electrical equipment that are sources of harmonic currents, such as
- Arc furnaces and other arc-discharge devices, such as fluorescent lamps
- Resistance welders (impedance of the joint between dissimilar metals is different for the flow of positive vs. negative current)
- Magnetic cores, such as transformer and rotating machines that require third harmonic current to excite the iron
- Synchronous machines (winding pitch produces 5th and seventh harmonics)
- Adjustable speed drives used in fans, blowers, pumps, and process drives
- Solid-state switches that modulate the current-to-control heating, light intensity, etc.
- Switched-mode power supplies, used in instrumentation, PCs, televisions, etc.
- High-voltage dc transmission stations (rectification of ac to dc, and dc to ac invertors)
- Photovoltaic invertors converting dc to ac
What The National Electrical Code Says About Harmonic Currents
The National Electrical Code or NFPA 70 2017, on Article 368.258 states the following…
…”Neutral Conductor – Neutral bus, where required, shall be sized to carry all neutral load current, including harmonic currents, and shall have adequate momentary and short-circuit rating consistent with system requirements.”
It is therefore necessary that as a design engineer, we take harmonic currents into consideration when sizing the neutral conductors.
Odd Versus Even Harmonic Orders (Addendum)
Note that only odd numbered harmonic orders (such as 3, 9, 15, etc) are responsible for adding in the neutral conductors. The fundamental and even numbered harmonic orders would cancel out in the neutral conductors. Therefore what we refer to as Triplen Harmonics are the odd number multiples of three.
- National Electrical Code, NFPA 70 2017
- IEEE Std 519-1992 Recommended Practices and Requirements for Harmonic Control in Electrical Power Systems
- IEEE Std 141-1993 Recommended Practice for Electric Power Distribution for Industrial Plants
- Electrical Machines, Drives and Power Systems Sixth Edition by Theodore Wildi