Zig-Zag Transformers: Theory, Harmonic Mitigation and Grounding Applications

Zig-zag transformers: theory, harmonic mitigation and power-quality applications

As power electronics have advanced, the load profile of industrial and commercial distribution systems has changed fundamentally. Non-linear loads — variable frequency drives (VFDs), uninterruptible power supplies (UPS) and switch-mode power supplies of every kind — have greatly improved conversion efficiency and process control, but they also inject large harmonic currents into the network ¹. Those harmonics distort the voltage waveform and cause a chain of power-quality problems: overloaded neutral conductors, overheated transformers, nuisance relay operation and communication interference ³. Against this background the zig-zag transformer, also known as the interconnected-star transformer, has become a key device for system grounding and harmonic mitigation thanks to its distinctive winding arrangement and electromagnetic behaviour.

This report examines the zig-zag transformer in depth. It starts from the underlying electromagnetic theory, analysing the vector composition and impedance characteristics, then explains how the transformer removes zero-sequence harmonics (such as the 3rd) and how phase-shifting suppresses positive- and negative-sequence harmonics (such as the 5th and 7th). It also compares the zig-zag transformer with K-factor transformers, delta-wye isolation transformers and active power filters (APF), with selection guidance for engineering design.

2. Winding Theory and Impedance Characteristics

The value of the zig-zag transformer lies in the way its windings are interconnected, which gives it very different impedances to the different sequence components of current.

2.1 Winding Structure and Vector Diagram

In a conventional wye or delta connection, the whole winding of each phase is wound on a single core limb. A zig-zag connection instead splits each phase winding into two equal sections wound on different limbs ⁵.

2.1.1 The Interconnection

Each phase voltage of a zig-zag transformer is the vector sum of the voltages of two winding sections on two different limbs.

• Outer coil (zig): connected to the supply terminal.

• Inner coil (zag): connected to the neutral point.

• Phase-shifted interconnection: the A-phase output terminal, for example, is formed by the outer coil on limb A in series with the inner coil on limb B (or limb C, depending on the vector group). Because the flux in limbs A and B differs by $120^{\circ}$, the voltages induced in the two sections are also $120^{\circ}$ apart in space ⁶.

2.1.2 Vector Composition and Voltage Ratio

If each winding section produces a voltage of magnitude $V_{w}$, the two sections are connected in series with opposing polarity (head to tail), so that the electrical angle between them is $60^{\circ}$. By the cosine rule the resulting phase voltage $V_{ph}$ is:

$$V_{ph} = \sqrt{V_{w}^2 + V_{w}^2 + 2 V_{w} V_{w} \cos(60^{\circ})} = \sqrt{3} V_{w}$$

This contrasts with a conventional transformer, where two sections on the same limb simply add ($V_{ph} = 2 V_{w}$). To reach the same output voltage, a zig-zag transformer therefore needs $2 / \sqrt{3} \approx 1.155$ times as many turns ⁶. That extra 15.5% is known as the copper penalty — the price paid for the transformer’s special behaviour.

2.2 Impedance: Positive Sequence vs. Zero Sequence

The most striking engineering characteristic of the zig-zag transformer is how asymmetric its sequence impedances are. This is the physical basis for its use both as a grounding transformer and as a harmonic filter.

2.2.1 Positive- and Negative-sequence Impedance ($Z_1, Z_2$)

When balanced positive-sequence current flows into a zig-zag transformer, the flux in the limbs is $120^{\circ}$ apart. Because of the way the windings are interconnected, this flux circulates through the core exactly as in a normal transformer and the magnetic circuit presents its high magnetising impedance. To positive-sequence voltage or current the transformer therefore looks close to an open circuit, drawing only a small magnetising current ⁵. In normal operation a zig-zag grounding transformer adds virtually no load to the system.

2.2.2 Zero-sequence Impedance ($Z_0$)

Zero-sequence current ($I_0$) is equal in magnitude and in phase in all three phases. When it flows into terminals A, B and C:

1. The current flows simultaneously through the outer coils on limbs A, B and C.

2. It then flows through the inner coils on limbs B, C and A.

By Ampère’s circuital law, on any one limb (say limb A) the zero-sequence current in the outer coil flows in the opposite direction to that in the inner coil — or the polarities are arranged so that the magnetomotive forces oppose. With equal turns, the two MMFs cancel:

$$MMF_{net} = N I_0 – N I_0 \approx 0$$

With the MMFs cancelled, almost no zero-sequence flux is produced in the core, so the inductance ($L \propto \Phi / I$) is very low. The transformer therefore presents a very low leakage impedance to zero-sequence current ⁹. That low-impedance path makes it an ideal sink for zero-sequence current, whether earth-fault current or third-harmonic current.

2.3 Leakage Reactance and Simulation Models

Calculating the leakage reactance ($X_{zigzag}$) of a zig-zag transformer is not straightforward and normally involves the Rogowski method or finite element analysis (FEA). The literature ¹¹ gives an approximation derived from the short-circuit reactances between windings:

$$X_{zigzag} \approx \frac{1}{2} (X_{HV-zig} + X_{HV-zag}) – \frac{1}{6} X_{zig-zag}$$

The expression reflects the magnetic coupling between the winding sections. In simulation tools such as PSCAD or MATLAB/Simulink, three single-phase transformer models — or a dedicated zig-zag grounding transformer block — are normally required to reproduce this zero-sequence behaviour accurately ¹⁰.

3. Comparison with Other Transformer and Filtering Technologies

To place the zig-zag transformer in context, this section compares it with K-factor transformers, delta-wye isolation transformers and active power filters.

3.1 Zig-zag vs. K-factor Transformers

The two are often confused, but their design philosophies are opposites: one endures harmonics, the other removes them.

Criterion	K-factor transformer	Zig-zag (harmonic mitigation) transformer
Design philosophy	Survival: built to withstand the extra heat harmonics produce without failing.	Mitigation: removes harmonics through flux cancellation and phase shifting.
How harmonics are handled	Harmonic current still flows; heavier conductors, reinforced insulation and a double-size neutral prevent overheating.	Winding interconnection cancels zero-sequence harmonics and stops them reaching the upstream network.
Effect upstream	No improvement in upstream current distortion (THDi); harmonics still pollute the network.	Markedly lower upstream harmonic content, better power factor and voltage waveform.
Efficiency and losses	Lower. Harmonics create additional eddy-current and hysteresis loss (heat) inside the transformer.	Higher. With harmonics cancelled, reactive circulation and thermal losses in the system fall.
Cost	Moderate ($1.1 \sim 1.3$ times a standard transformer).	Higher ($1.5 \sim 2.0$ times), because of the more complex winding work and greater copper content.
Where it fits	Where only the life of the transformer itself needs protecting and network quality is not a concern.	Where IEEE 519 has to be met and several non-linear loads can be filtered at one point.

Conclusion: a K-factor transformer is a soldier in armour — it can absorb the attack but cannot stop it. A zig-zag transformer is a shield: it neutralises the attack. If the only aim is to keep the transformer from burning out, a K-factor unit is enough; if the aim is cleaner power, energy savings and protection of upstream equipment, the zig-zag transformer is the better choice ¹.

3.2 Zig-zag vs. Delta-Wye Isolation Transformers

Delta-wye is the most common distribution transformer arrangement and does offer some harmonic mitigation, but by a different mechanism.

• Zero-sequence harmonics: a delta-wye transformer traps third-harmonic current circulating inside the primary delta so that it does not reach the source. Those circulating currents still generate heat in the delta winding. A zig-zag transformer instead cancels the flux on the same limb, so in theory the elimination happens inside the core and the thermal stress is distributed more evenly ².

• As a grounding device:

  • Delta-wye: bulky and expensive if used purely for grounding, because a full secondary winding is required.

  • Zig-zag: for grounding duty alone (no secondary load), only a single winding structure is needed (autotransformer form), so volume and weight are roughly 33% lower than a delta-wye unit of the same rating, with lower losses ⁵.

• Phase shift: delta-wye always provides $30^{\circ}$. By adjusting the turns ratio between the zig and zag sections, a zig-zag transformer can be designed for any angle (for example $15^{\circ}$ or $20^{\circ}$), which is essential for multi-pulse rectifier systems ¹⁷.

3.3 Zig-zag vs. Active Power Filters (APF)

An APF represents the state of the art in power-electronic harmonic treatment, in contrast to the passive zig-zag transformer.

Criterion	Active power filter (APF)	Zig-zag transformer (passive)
Principle	Measures the harmonic current and injects an equal and opposite current (current injection).	Electromagnetic cancellation and phase-shift superposition.
Filtering capability	Excellent. Handles the 2nd to 50th harmonic simultaneously and is unaffected by load imbalance.	Good. Targets specific orders (zero sequence, or the 5th/7th when combined with phase shifting); performance depends on load balance.
Interaction with system impedance	No resonance, although high-frequency switching can introduce very high frequency noise.	No resonance risk; it inherently adds to the short-circuit impedance of the system.
Reliability and life	Lower. Contains capacitors, fans and IGBTs, so maintenance requirements are significant.	Very high. Copper and steel only; design life of 20–30 years and essentially maintenance-free.
Capital cost (CAPEX)	High — typically several times a zig-zag solution 19.	Moderate to high. Lower initial investment than an APF, with no firmware upgrades or ageing components to replace.

Conclusion: an APF suits sites with rapidly varying loads, complex harmonic spectra and the highest power-quality requirements, such as semiconductor fabs and data centres. The zig-zag transformer suits high-power industrial sites with relatively steady loads where long-term reliability and freedom from maintenance matter most — heavy-industry rectifiers or offshore platforms, for example ¹⁹.

4. Harmonic Filtering Benefits: the 3rd, 5th and 7th Harmonics

Zig-zag transformers are applied to harmonics in two distinct ways: using the low zero-sequence impedance to remove triplen harmonics, and using phase shifting to cancel characteristic harmonics.

4.1 Suppressing Triplen Harmonics

The third harmonic and its multiples (9th, 15th, 21st…) are zero sequence. In a three-phase four-wire system they are in phase on all three line conductors, so they add arithmetically in the neutral, where the current can reach 1.73 times the phase current or more ².

• How it filters: the zig-zag transformer is connected in parallel with the load, for example at the distribution board nearest the non-linear load. Because it offers very low impedance to zero-sequence current ($Z_0$) but high impedance to positive-sequence supply voltage, the third-harmonic current produced by the load flows preferentially into the neutral point of the zig-zag transformer and is cancelled inside its windings, as described in section 2.2.2.

• Practical benefits:

  1. Much lower neutral current: third-harmonic current no longer returns to the upstream transformer, significantly unloading the upstream neutral.

  2. No upstream overheating: eddy-current and skin-effect losses in the main transformer are reduced.

  3. Less voltage distortion: less harmonic current on long feeders means less voltage drop and a cleaner waveform at the load ²¹.

4.2 Suppressing the 5th and 7th Harmonics: Phase Shifting

The 5th (negative sequence) and 7th (positive sequence) harmonics cannot be cancelled inside the zig-zag windings the way triplen harmonics are, because they are not zero sequence. Suppressing them relies on phase-shift cancellation ¹.

4.2.1 Dual-unit Phase Shift

Where there are two identical non-linear loads (two 6-pulse VFDs, say), a phase difference can be created by changing the connection of the supply transformers.

• Principle: if two harmonic sources differ by $\alpha$ at fundamental frequency, their $h$-th harmonic currents differ by $h \times \alpha$. When $h \times \alpha = 180^{\circ}$, that harmonic cancels at the common busbar.

  • Phase difference needed to cancel the 5th and 7th: $\alpha = 30^{\circ}$.

  • Check: the 5th harmonic is shifted by $5 \times 30^{\circ} = 150^{\circ}$ (close to anti-phase) and the 7th by $7 \times 30^{\circ} = 210^{\circ}$ (also close to anti-phase). In a 12-pulse system a 30-degree shift cancels the 5th and 7th almost completely.

4.2.2 Zig-zag Transformers in Multi-pulse Rectifiers

A conventional delta-wye transformer only provides a fixed $30^{\circ}$ shift. Higher-order harmonic mitigation — 18-pulse or 24-pulse rectification — needs finer angles:

• 12-pulse systems: need a $30^{\circ}$ difference, which can be obtained from one delta-delta ($0^{\circ}$) plus one delta-wye ($30^{\circ}$) unit. Where the secondary must be earthed, or a particular voltage ratio is required, a zig-zag connection provides the adjustment ²³.

• 24-pulse systems: need a $15^{\circ}$ difference, and here the zig-zag transformer is irreplaceable. Choosing the right turns ratio between the zig and zag sections produces $+15^{\circ}$ and $-15^{\circ}$ phase-shifting transformers.

  • Arrangement: one group of loads is fed from a $+15^{\circ}$ zig-zag unit and another from a $-15^{\circ}$ unit; the $30^{\circ}$ difference between them gives 12-pulse behaviour. Adding a $0^{\circ}$ transformer and others extends this to 24 pulses, further eliminating the 11th, 13th, 23rd and 25th harmonics ²⁴.

Benefit: this makes the zig-zag transformer indispensable as front-end equipment for high-power industrial rectification — aluminium smelting or large rolling mills, for example — where it can bring total current harmonic distortion (THDi) below 5% and satisfy the strictest reading of IEEE 519 ²⁶.

5. Neutral Current Reduction and Grounding Applications

5.1 Case One: System Grounding for Ungrounded Networks

Many older industrial installations and wind farms are supplied from a delta system (three-phase three-wire). Such systems have no neutral point, so a single line-to-earth fault raises the voltage to earth on the healthy phases by a factor of $\sqrt{3}$, which readily breaks down insulation and develops into a phase-to-phase fault.

• Zig-zag solution: install a zig-zag grounding transformer (grounding bank).

• How it works:

  • Normally: it presents a high magnetising impedance and draws only microamp-level current, so system operation is unaffected.

  • During a fault: it provides a low-impedance zero-sequence path, allowing earth-fault current to return to the neutral point and drive the earth-fault protection relays (51N/50N) to trip ¹².

• Cost advantage: compared with using a standard two-winding delta-wye transformer for grounding, a zig-zag unit needs no secondary winding and only carries high current for a short time during a fault (10 seconds, typically), so its rating is usually only 10–20% of the system capacity — smaller, lighter and cheaper ⁵.

5.2 Case Two: Neutral Overload in Commercial Buildings and Data Centres

A modern office building is full of single-phase non-linear loads such as computers and LED lighting.

• The problem: the third harmonics these devices produce add arithmetically in the neutral. Traditional designs size the neutral the same as the line conductors, so it frequently overheats, ages the insulation and can even start a fire.

• Zig-zag solution: install a zero-sequence filter (a zig-zag reactor) in parallel near each floor’s power distribution unit (PDU).

• Benefits:

  1. Local diversion: third-harmonic current circulates inside the local zig-zag winding instead of returning to the main transformer.

  2. Energy saving: $I^2 R$ losses are reduced over the long cable run from the floor back to the main transformer.

  3. Voltage stability: eliminating the zero-sequence voltage drop caused by neutral current keeps phase voltages at the equipment more balanced ²¹.

6. Specification Comparison

The table below summarises the key differences between the zig-zag transformer and the alternatives, as a selection aid ¹.

Characteristic	Zig-zag transformer	K-factor transformer	Delta-wye isolation transformer	Active power filter (APF)
Primary function	Grounding / zero-sequence harmonic elimination / phase shift	Withstanding harmonic heating	Voltage transformation / creating a neutral	Full-spectrum harmonic elimination
Zero-sequence impedance ($Z_0$)	Very low (the key advantage)	Medium to high	High (unless the secondary is shorted)	N/A
Third harmonic	Cancelled internally (trap)	Withstood passively	Circulated in the primary delta (trap)	Cancelled by active injection
5th / 7th harmonic	Requires paired phase-shifted units (12-pulse)	Withstood passively	Requires paired phase-shifted units (12-pulse)	Eliminated directly
Energy saving	High (reduces system losses)	Low (high losses of its own)	Medium	Medium (switching losses of its own)
Size and weight	Small (grounding duty) / medium (filtering duty)	Large (oversized core and conductors)	Medium	Small (high power density)
Maintenance	Minimal (no moving parts)	Low	Low	High (fan and capacitor life)
Relative cost	Medium	Low	Low to medium	Very high

7. Conclusion

The zig-zag transformer is more than a way of creating an artificial neutral; it plays an indispensable role in modern power-quality engineering. The analysis above shows that its interconnected winding structure effectively decouples zero-sequence impedance from positive-sequence impedance, which makes it the most economical and effective zero-sequence harmonic filter available.

Compared with a K-factor transformer it takes the active route — eliminating harmonics rather than enduring them — and genuinely improves network quality. Compared with an expensive active filter it offers a passive solution with very high reliability and no maintenance, which suits harsh environments and steady industrial loads.

In practice, from earth-fault protection on wind farms to harmonic mitigation in heavy-industry multi-pulse rectifiers, the zig-zag transformer delivers engineering benefits no other device matches. Its winding complexity makes it somewhat more expensive than a standard transformer and it carries the 1.155 turns penalty, but weighed against the energy savings, equipment protection and system stability it brings, it remains one of the key technologies for solving modern power-quality problems.