Per-unit analysis in transformers

Reference notes

Use Next → on the narrator above to step through per-unit analysis specifically as it applies to transformers — nameplate-as-base, %Z meaning, base conversion, the single-impedance per-unit equivalent, parallel-operation load sharing, auto-transformer kVA saving, and SLG fault calculation through a Δ-Y_n transformer.

Nameplate as base

A power transformer's nameplate IS the natural per-unit base for that machine:

• S_base = S_rated (MVA)
• V_base,HV = V_HV,LL (kV) · V_base,LV = V_LV,LL (kV)
• Z_base,HV = V_HV²/S_rated · Z_base,LV = V_LV²/S_rated (differ by N²)
• I_base,HV = S_rated / (√3 · V_HV,LL) · I_base,LV = S_rated / (√3 · V_LV,LL)

Nameplate %Z — what it means

The nameplate's %Z (sometimes called %X — for power transformers leakage reactance dominates over resistance) is just Z_pu × 100, on the transformer's own ratings as base. Practical implications:

• Bolted secondary short-circuit current: I_SC,pu = 1.0 / Z_pu → I_SC = (1/Z_pu) × I_rated. A 5 % Z transformer delivers ~20× rated current into a bolted short; a 10 % Z transformer delivers ~10×.
• Worst-case voltage drop at full load: ≈ %Z (for pure inductive load). A 6 % transformer drops up to 6 % under inductive full load.
• Inrush current magnitudes and harmonic content scale with %Z too — higher %Z limits inrush but raises voltage drop.

Typical ranges:

• Distribution transformers (10 kVA – 5 MVA): 2–6 % Z. Low Z is cheaper but raises fault current; utilities often specify minimum 4 %.
• Power transformers (5–500 MVA): 8–14 % Z.
• Generator step-up transformers: 13–16 % Z — high Z limits generator fault contribution.

Base conversion to system base

Real engineering uses one system-wide base (e.g., 100 MVA), not each transformer's nameplate. Convert with:

Equivalent form with the old/new naming used in the quiz: Z_pu,new = Z_pu,old · (S_new / S_old) · (V_old / V_new)². Same formula — just renamed for general base-change problems.

When the system base voltage matches the transformer's nameplate voltage at that bus (the usual case), the V ratio is 1 and only S matters:

Worked example: 50 MVA transformer with 10 % Z on its own base → 100 MVA system base → Z_pu = 0.10 × (100/50) = 0.20 = 20 %.

The per-unit equivalent circuit

The miracle of per-unit: the transformer's primary and secondary impedances combine into a single series Z_pu between two buses, and the ideal-transformer turns-ratio block becomes 1:1 and disappears. Reason: referring impedance from one side to the other multiplies by N² in actual ohms, and Z_base also scales by N², so Z_pu is identical on both sides. The magnetizing branch (typically ~0.001 pu shunt admittance) is usually negligible and dropped.

A two-winding transformer in a per-unit one-line is simply a series impedance between two buses. That's the most important practical reason to use per-unit.

Parallel transformer operation

Two or more transformers feeding the same bus from the same source share the load inversely with their per-unit impedances Z_1, Z_2 (all converted to a common base):

For just two transformers in parallel: S_1 / S_2 = Z_2 / Z_1 — the one with smaller Z_pu carries more load.

Worked example: T1 = 50 MVA @ 10 % Z, T2 = 100 MVA @ 8 % Z. Both serving the same bus with 100 MVA load.

1. Convert to common 100 MVA system base: T1 → 20 %, T2 → 8 %.
2. Inverse Z: 1/0.20 = 5, 1/0.08 = 12.5. Sum = 17.5.
3. T1 carries 100 · (5/17.5) ≈ 28.6 MVA (about 57 % of its rating).
4. T2 carries 100 · (12.5/17.5) ≈ 71.4 MVA (about 71 % of its rating).

If the two %Z values are mismatched, one transformer can be overloaded while the other has spare capacity. This is one of the four paralleling rules: matching %Z is required for proper load sharing.

Auto-transformer kVA saving

An auto-transformer uses a single tapped winding rather than two isolated windings. Part of the load current flows directly through the common conductive path; only the rest goes through transformer action across the core. With turns ratio a = V_HV / V_LV (a > 1):

Examples:

• a = 2.0 (2:1 step-down): two-winding equivalent is 50 % of S_auto. Significant saving in iron, copper, weight, cost.
• a = 1.1 (small ratio change): two-winding equivalent is only 9 % of S_auto. Dramatic saving — this is why auto-transformer starters and generator-step-up adjustments use auto-transformer construction.
• a = 1.0 (no ratio change): two-winding equivalent is 0 % — degenerate case.

Trade-off: no galvanic isolation. A short or transient on one side propagates electrically to the other side, so auto-transformers are not used where isolation between voltage levels matters (e.g., for hazard safety in low-voltage distribution to dwellings). Common where both sides are at similar voltage levels (e.g., 230 kV ↔ 138 kV tie transformers in transmission).

Three-winding transformer (briefly)

A three-winding transformer (primary, secondary, tertiary) is modeled in per-unit as a star (Y) equivalent with three impedances meeting at a common node. The three values Z_p, Z_s, Z_t come from three short-circuit tests (one winding short, one open at a time).

Note: individual Z values can be NEGATIVE in this model (they're a mathematical convenience, not physical impedances). Common tertiary use: a delta tertiary that provides a path for zero-sequence harmonics in a Y-Y main transformer.

SLG fault through a Δ-Y_n 3-φ transformer

Single line-to-ground (SLG) fault on the grounded-wye secondary of a delta-grounded-Y transformer. The system Thevenin impedance Z_th drives the fault loop:

• Positive sequence: source Thevenin + transformer Z_+ in series.
• Negative sequence: same as Z_+ for static equipment.
• Zero sequence: ONLY the transformer's Z_0 — the Δ primary winding blocks zero-sequence from propagating to the source. Z_0 terminates AT the transformer (looking from the fault).

Multiply by I_base on the secondary side to get fault current in amperes — the breaker-sizing value.

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