Triplen harmonics in the neutral
The key idea
A balanced three-phase load returns no current through its neutral. That cancellation is one of the main reasons three-phase systems exist. At the 3rd harmonic, it stops being true. The 3rd, 9th and 15th arrive in step on all three phases and add instead of canceling, so a perfectly balanced group of electronic loads can run its neutral hotter than any of its phases.
The idea
Every phase carries the same current waveform, each starting a third of a cycle after the one before. That delay is what makes the three currents sum to zero at the star point, so a balanced load needs no return conductor. Harmonics do not change the delay; phase B is still phase A a third of a cycle later. What they change is what that delay means.
A third of a cycle is 120° of the fundamental. But the 3rd harmonic completes three cycles in the time the fundamental completes one, so the same delay is 360° of the 3rd — a full turn, right back to the start. The 3rd harmonic of phase B therefore sits exactly on top of the 3rd harmonic of phase A. All three phases carry the same 3rd-harmonic wave.
Now add the three currents in the neutral. The fundamentals are 120° apart and cancel, as always. The three 3rd harmonics are identical, so they add to three times one of them: a clean 150 Hz current on a 50 Hz system (180 Hz on a 60 Hz one), flowing in a conductor the designer sized for a current near zero.
The same arithmetic gives the general rule. The h-th harmonic of phase B is displaced by h × 120°. Orders 1, 4, 7, 10 (remainder 1 after dividing by three) put the phases 120° apart in ABC order: a positive-sequence set. Orders 2, 5, 8, 11 (remainder 2) put them 120° apart in the reverse order: negative sequence. Both are three equal phasors at even angles, and both cancel in the neutral. Only the multiples of three add. These are the triplen harmonics, and they form a zero-sequence set: three phasors pointing the same way.
This would be a curiosity if nothing produced much 3rd harmonic. Unfortunately, single-phase rectifier loads produce a lot of it. The switch-mode supply in every computer, LED driver and charger draws its current in a short pulse near the voltage peak, and the 3rd is the largest harmonic in that pulse, commonly 30% to 70% of the fundamental. A building full of these loads is a building full of triplen sources, all in phase with each other.
Try it
Increase the 3rd harmonic and watch the neutral waveform grow. Then push the 5th harmonic to its maximum — the neutral does not move at all.
neutral rms = 90% · fundamental + 5th cancel · 3rd adds ×3
| Order | In each phase | Sequence | In the neutral |
|---|---|---|---|
| fundamental | 100% | positive | cancels: the three sum to zero |
| 3th | 30% | zero | adds ×3: 90% of the fundamental |
| 5th | 20% | negative | cancels: the three sum to zero |
The three phases here are perfectly balanced. They have the same load and the same spectrum, and only the third-of-a-cycle delay separates them. The fundamental and the 5th cancel in the neutral at any size. The 3rd is identical in all three phases. The neutral therefore carries a clean 150 Hz wave three times the size of the 3rd harmonic of one phase. Real loads are never exactly balanced, and that unbalance adds its own fundamental-frequency current to the neutral.
Why it matters
- The neutral can be the most heavily loaded conductor in the cable. The old guideline that a neutral may be smaller than the phases comes from an era of linear loads. With a 33% 3rd harmonic, the neutral already carries about 90% of the fundamental phase current, and in most installations it has no overcurrent device protecting it.
- Balancing the load does not remove this current. An equal load on each phase fixes every other neutral problem, but not this one: triplen harmonics add in phase whether the phases are balanced or not. Real load unbalance then adds its own fundamental-frequency current on top.
- Transformers see triplen current as zero sequence. A delta winding is a closed loop, and the triplen current circulates inside it rather than passing upstream, which is how a delta–wye transformer protects the network above it. The same circulating current also heats the winding that traps it.
- Triplen current distorts the voltage too. Flowing through the earthing impedance, it produces a 3rd-harmonic voltage that moves the star point and adds to the total distortion every load on the bus receives.
The math, if you want itOptional — the page reads completely without it
Phase B carries the waveform of phase A delayed by a third of a cycle, so its h-th harmonic is displaced by h × 120°. At the 3rd harmonic the displacement vanishes:
a third of a cycle, seen by the 3rd harmonic
sin( 3(θ − 120°) ) = sin( 3θ − 360° ) = sin( 3θ )
Only the remainder of h ÷ 3 decides how a set of three behaves, so every order belongs to one of three families:
sequence of the h-th harmonic
h mod 3 = 1 → positive · = 2 → negative · = 0 → zero (triplen)
Positive- and negative-sequence sets are three equal phasors 120° apart; they cancel at the star point at any size. A zero-sequence set is three identical phasors, so it adds to three times one of them. With balanced phases, the neutral carries the triplen harmonics and nothing else. The orders are orthogonal, so their RMS values add in squares:
neutral rms, balanced phases
IN = √( Σh = 3, 9, 15… (3·Îh)²2 )
where Îh is the peak of the h-th harmonic in one phase. Two assumptions sit behind this: the phases are balanced, and they share the same spectrum. Both are close to true for a group of identical rectifier loads, which are also the loads with the largest 3rd harmonic. The widget models orders 1, 3 and 5 only; the 9th and 15th would raise the neutral current further, never lower it.
See it in Phasor
A harmonic penetration study in Phasor carries each order through the network with its own sequence: triplen current follows the zero-sequence path, down earthed star points and around delta windings, while the 5th and 7th take a different route. Neutral and earthing conductors are part of the model, so the study reports the current the neutral actually carries. You never have to assume it is empty.