Triplen harmonics

What it is (precise)

The triplen harmonics are the odd multiples of the third — 3rd, 9th, 15th, 21st. Their special status comes from one line of arithmetic on the phase displacement: delay a waveform by the fundamental's 120° and its h-th harmonic shifts by h × 120°. For the 5th that is 600°, which lands at 240°; for the 7th it is 840°, landing at 120° — still displaced, still balanced sets. For any multiple of three the shift is a whole number of full turns, so where the three phases carry the same distorted waveform a third of a cycle apart, their triplen components come out identical in magnitude and in phase.

Three equal phasors with no displacement between them is exactly the zero-sequence component, which is why triplens obey the zero-sequence path rules — a neutral or earth return required, addition where balanced current cancels, confinement by a delta. The zero-sequence entry owns that classification and the full pass/block/trap behavior; this page follows the harmonic story.

The same arithmetic sorts the rest of the spectrum. Orders of the form 3n+1 — the fundamental, 4th, 7th, 13th — keep the normal displacement and behave as positive sequence; orders of the form 3n+2 — the 5th, 11th — reverse it and behave as negative sequence, which is why a 5th harmonic drives a counter-rotating field in a machine while the 7th drives a forward one; the 3n family is the co-phasal one.

The mapping is exact only under balance. Unbalanced distortion — unequal single-phase loads, transformer inrush — produces third-harmonic sets whose three phasors differ in magnitude and angle, and such a set decomposes, still at 150 or 180 Hz, into positive-, negative- and zero-sequence parts. Only the zero-sequence part follows the blocking rules; the remainder travels wherever balanced current can.

The neutral arithmetic

In a four-wire wye the balanced fundamental cancels in the neutral, and so does every balanced non-triplen harmonic — the 5th, 7th, 11th and 13th keep their displacement and sum to zero the way the fundamental does. Triplens add arithmetically: the neutral carries three times one phase's triplen content, at 150 Hz on a 50 Hz system or 180 Hz on 60 Hz, with no fundamental in it at all while the load balances.

The factor of three applies to the per-phase triplen component, and the worked cases show how fast it bites: 20% third-harmonic content per phase puts a neutral current of 60% of the phase fundamental in the fourth wire; 30% per phase puts 90% there; and beyond roughly 35% — the exact crossover is 1/(2√2) — the neutral carries more RMS current than any phase conductor.

The extremes are worth knowing because real load mixes approach them. Where the load is single-phase rectifier electronics whose current pulses do not overlap between phases, neutral RMS reaches √3 ≈ 1.73 times phase RMS, and the theoretical ceiling as the current becomes pure triplen is three times.

That arithmetic is the engineering basis for 100%- to 200%-rated neutrals on panelboards serving nonlinear load, and for treating the neutral as a current-carrying conductor when ampacity is worked out — a survey that clamps only the three phases misses the most loaded conductor on the board. The neutral-current entry follows those amps into conductor sizing, the protection blind spot and the broken-neutral failure mode.

Delta circulation, and which connections stop them

At the fundamental, the three winding EMFs of a delta sum to zero around the closed mesh — the reason a delta can be closed at all. At a triplen frequency the three co-phasal EMFs add, so the loop sees a driving voltage of three times one winding's triplen EMF, and a current circulates around the triangle limited only by the winding impedances at that frequency.

The circulating current never reaches the line terminals, because each line current is the difference of two winding currents and identical co-phasal components subtract out exactly — yet the copper works for it: winding heating follows the quadrature sum of load current and circulating current, spending thermal margin without delivering load power.

That heating is the sizing basis for the stabilizing and tertiary delta windings of large transformers, commonly rated around a third of the unit's throughput, and one reason K-rated or oversized transformers are specified for triplen-rich load.

Harmonic amperes also multiply stray loss out of proportion: winding eddy loss scales roughly with the square of order times amplitude, so a 20% third harmonic adds about 36% of the fundamental's eddy loss — the sum the K-factor packages. The circulation gives something back too: it supplies the third-harmonic magnetizing component the core needs to keep its flux, and hence the voltage waveform, sinusoidal.

The winding letters sort out who passes what, and the directions matter. A grounded wye passes triplen currents between its lines and windings, the neutral connection carrying the threefold sum. An ungrounded wye blocks them outright — no return path exists.

A delta keeps them off its own line side in both directions while carrying the circulating set internally, which is how a Dyn transformer confines triplens generated on its grounded-wye side: they flow in that side's phases and neutral, the delta windings carry the reflected co-phasal current around the loop, and the delta-side lines and the network above see none — a standing reason the Dyn arrangement is so common for LV distribution and auxiliary transformers.

The connection that lets triplens pass straight through between systems is the grounded-wye/grounded-wye unit with no delta anywhere on it, which is why YNyn transformers are often fitted with a stabilizing delta tertiary. The zero-sequence entry carries the full path framework, ground faults included.

Three-wire systems suppress balanced triplen line currents structurally: with no neutral, Kirchhoff's current law forces the three line currents to sum to zero at every instant, and a co-phasal set would sum to three times itself, so it cannot flow. The blocked demand re-emerges as third-harmonic voltage distortion of the phase-to-neutral quantities — the floating star point oscillates at triplen frequency against ground — while the line-to-line voltages stay clean, because the subtraction removes identical co-phasal terms exactly.

That last cancellation has a measurement consequence: metering and protection supplied only from phase-to-phase VTs are structurally blind to triplen voltage distortion, and assessing it takes phase-to-ground measurement — wye-connected VTs or a broken-delta secondary, whose output is the residual 3V0. The suppression applies only to the zero-sequence part: an unbalanced third-harmonic set has positive- and negative-sequence content that flows in three-wire lines regardless.

Where they come from in a BESS plant

The steady source is transformer magnetizing current. With sinusoidal voltage applied the core flux is sinusoidal, and the saturating magnetization curve maps that flux into a peaked magnetizing current dominated by the third harmonic — often 30-50% of the magnetizing fundamental, rising when high grid voltage pushes the core toward saturation.

Because the distortion is identical in all three phases the demand is co-phasal, and it is path-dependent: a delta winding or a grounded neutral supplies it and the voltage stays sinusoidal; block every path and the flux waveform distorts instead, putting third-harmonic voltage on the phase-to-neutral quantities.

Energization inrush is the violent cousin, and it breaks the balanced-triplen rules: the three phases close at different points on the wave with different residual flux, so the triplen content is unbalanced, genuine zero-sequence current appears in the lines and in the residual during energization, and sensitive ground elements can pick up — the reason ground and differential protection carries harmonic restraint or a time delay.

The converter's contribution is smaller than intuition suggests. A balanced three-phase voltage-source converter on a three-wire connection injects no triplen line current — the Kirchhoff argument above holds at every harmonic order, whatever the modulation strategy — and its characteristic emission sits at the non-triplen odd orders, 5th, 7th, 11th and 13th, plus switching-frequency sidebands, which is where the harmonics entry picks up the spectrum.

Deliberate third-harmonic injection, or the space-vector modulation that implies it, adds a common-mode triplen to the phase-leg voltages that cancels exactly line-to-line: its purpose is to stretch the linear modulation range by 2/√3 ≈ 1.155 — about 15% more AC voltage from the same DC bus — and the common-mode voltage drives only parasitic capacitive currents through stray capacitance, an EMC matter rather than a line-current one.

Triplen current from a PCS arrives through imperfection: phase unbalance, asymmetric dead-time distortion, or a grounded neutral path someone provided. The dependable producers on site are the single-phase electronics — switched-mode supplies, UPS front ends, LED drivers and control gear on the auxiliary board, each drawing the rectifier current spectrum that is rich in exactly the third harmonic.

Where a grid-scale site genuinely meets them

The main power path of a utility-scale BESS is close to triplen-free by construction: the PCS is a balanced three-wire converter, and the skid transformer's delta winding — whichever side of the unit the design places it, the transformer entry has the letters — keeps zero-sequence content from propagating across it.

Harmonic studies and IEEE 519 compliance work at the point of interconnection therefore center on the 5th, 7th, 11th and 13th orders and the switching-frequency range, with resonance between collector-cable capacitance and filters as the live risk; the compliance metric and its limits are the total-harmonic-distortion entry's subject.

Inside the fence the exposure is concrete, and it comes in five forms. The LV auxiliary distribution is a four-wire system feeding single-phase electronics — a commercial-building load profile in miniature — so neutral sizing and the auxiliary transformer's rating for harmonic load follow the arithmetic above, with a Dyn auxiliary transformer confining the triplens to the LV side.

Energization and any DC-bias event, such as geomagnetically induced current entering through grounded neutrals, put genuinely unbalanced triplen and even-harmonic content into the collection network for as long as they last.

Every delta winding doing the confining pays for the service in circulating-current copper loss the thermal design must carry, even though the grid never sees the current.

Any deliberately grounded zero-sequence path at MV — a grounding transformer or a grounded-wye main-transformer winding — sinks whatever background zero-sequence harmonic voltage exists on the bus by design, drawing continuous triplen current that heats the grounding bank and can push standby ground elements toward nuisance pickup unless the relays filter to fundamental. And third-harmonic-based protection schemes inherit the plant's ambient triplen level as their noise floor for the same reason.