Three-Phase Transformer Analysis
Theory: three phases sharing one core
Overview
In a no-load test on a three-phase transformer, only the middle phase showed a smaller current. Is it faulty?
Probably normal. In the common 3-limb core-type core, the three phases share one core. Three-phase fluxes sum to zero, so each limb's flux returns through the others. The middle limb returns to its neighbours by a short path, but outer limbs must travel further through the yokes (the horizontal top and bottom sections). The higher reluctance means outer phases need more magnetizing current. Three-phase transformer analysis starts from this magnetic coupling between phases and the voltage–current relations set by the connections.
Magnetic circuit equations
$F$ is each phase's MMF (turns × magnetizing current), $\phi$ the limb flux, $R_l$ one limb's reluctance and $R_y$ the reluctance of the yoke between adjacent limbs (top and bottom combined). With a delta or an ungrounded star, the three magnetizing currents sum to zero.
Rated and fault current
The birth of three-phase power and transformers
Three-phase transmission was shown practical in 1891 by a demonstration linking Lauffen and Frankfurt in Germany, about 175 km apart. The three-phase transformers used were designed by the Russian-born engineer Dolivo-Dobrovolsky, who devised combining three limbs in one core. It uses less iron than three single-phase units, making it smaller and cheaper. Many of today's distribution transformers still build on this 3-limb core.
Worked examples
Example 1: magnetizing imbalance
With equal flux magnitudes in all three phases, the MMF each phase needs was computed from the magnetic circuit equations (linear, no saturation):
| Yoke / limb reluctance | Outer phases (A, C) | Middle phase (B) | Outer / middle | Outer phase angle shift |
|---|---|---|---|---|
| 0 | 1.00 | 1.00 | 1.00 | 0° |
| 0.6 | 1.51 | 1.20 | 1.26 | 6.6° |
| 1.2 | 2.03 | 1.40 | 1.45 | 9.8° |
MMF is relative to the value with zero yoke reluctance. Outer phases need more current than the middle, and their phases shift away from the 120° spacing.
If the outer phases need 1.26 to 1.45 times as much, of course the middle phase is smaller.
Exactly. Core joints and saturation add their own effects, so the ratio depends on design, but a smaller middle phase is characteristic of 3-limb cores — which is why no-load current is evaluated as the three-phase average. Conversely, if the two outer phases differ greatly, or the middle phase is largest, suspect bad joints or winding faults. With a 5-limb core or three single-phase units, this imbalance is much smaller.
Example 2: 1,000 kVA distribution transformer (Dyn11)
| Item | Value |
|---|---|
| Rated current (HV 6.6 kV) | 87.5 A |
| Delta winding phase current | 50.5 A |
| Rated current (LV 420 V) | 1,375 A |
| Turns ratio (HV/LV) | 27.2 |
| Phase shift | LV leads by 30° |
| Fault current (%Z 5%) | LV 27.5 kA / HV 1.75 kA |
| Fault current (%Z 6%) | LV 22.9 kA / HV 1.46 kA |
The turns ratio is not the line-voltage ratio (15.7) but 6.6 kV across the delta winding over the 242 V phase voltage of the star winding. Raising %Z lowers fault current but increases voltage drop under load.
Analysis workflow
- Confirm core form (3-limb, 5-limb, shell) and connection (e.g. Dyn11).
- Compute each phase's magnetizing current and core loss by magnetic circuit or electromagnetic FEM.
- Find short-circuit impedance and leakage flux, and evaluate electromagnetic forces on windings during faults.
- Check tank and structural heating from eddy currents under unbalanced or zero-sequence currents.
- Compare with no-load and short-circuit test results.
“The tank runs locally hot”
A 3-limb transformer connected star–star, at a site with many single-phase loads, developed an abnormally hot patch on its tank. Unbalanced load drives zero-sequence current, giving all three limbs a flux component in the same direction. A 3-limb core has no iron return path for it, so the flux passes through air and oil into the tank and causes eddy-current heating there. Connections with a delta winding (such as Dyn11) cancel zero-sequence MMF, so this rarely happens. Consider connection and core form together.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Treating small middle-phase current as a fault | Unnecessary inspection | Recognise the 3-limb trait |
| Turns ratio from line voltages | Wrong with delta/star | Use winding voltages |
| Ignoring zero-sequence flux path | Tank heating missed | Model the tank |
| Paralleling without matching phase shift | Large circulating current | Same vector group only |
| Over-reducing fault current with source impedance | Undersized breakers | Size for the upper bound |
I'd like to try related calculations.
The transformer design calculator, transformer efficiency calculator and three-phase power calculator cover design, efficiency and three-phase power. For detail see transformer equivalent circuit, core loss, leakage inductance and winding loss.
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