The Per-Unit System

An impedance quoted in ohms changes every time you cross a transformer — by the square of the turns ratio. Express it as a fraction of a chosen base and it doesn't: pick one system MVA base, set voltage bases from the transformer ratios, and the ratios drop out of the network entirely. Convert between Ω, pu, and %, move a nameplate impedance onto your study base, and drag the secondary voltage in the demo to see why this works.

System Base

S_base (three-phase)

MVA

V_base (line-line)

Common study convention: 100 MVA system-wide, V_base set zone-by-zone from transformer ratios.

Convert a Value

Value

Given as

Impedance, converted on the bases above.

Change of Base

Z on old base

Old S_base

MVA

Old V_base

The everyday move: a transformer nameplate %Z is on its own rating and must land on the study base (the system base above) before combining.

Invariance Demo

Secondary voltage V₂

V₂ (exact)

Z on secondary side

Why Per-Unit Exists — Transformer Invariance

Drag the secondary voltage — the impedance in ohms reflected to the primary changes with the square of the ratio, while the per-unit value (on bases that follow the transformer ratio) does not move.

Readout

Method — your values substituted, follow with a calculator

The Equations

Base relations. One base pair (V, S) fixes everything else:

Change of base. Nameplate impedances are given on the equipment's own rating; move them to the study base before combining:

Conventions. Pick one S_base system-wide (often 100 MVA). Set V_base per zone from the transformer ratios — then transformer per-unit impedance is the same from either side, the ratio disappears from the network, and √3 factors vanish: S_pu = V_pu·I_pu*. Per-unit machine impedances also fall in predictable ranges, which makes data errors visible.

Related: AC Power & Waveforms — the quantities being normalized.

Model scope. This page is exact arithmetic — no estimated values. Bases are three-phase / line-line throughout: Z_base = V_LL²/S_3φ, numerically identical to the single-phase / line-neutral form. It is a single-zone converter — it does not build a multi-zone base table for a whole system (set V_base per zone and convert zone by zone). The invariance demo assumes voltage bases follow the transformer turns ratio; off-nominal taps and phase-shifting transformers are not modeled.

Validation — cases re-computed live by this page at load

Case	Expected	Computed	Result

References

J. D. Glover, T. J. Overbye, and M. S. Sarma, Power System Analysis & Design, 6th ed., Cengage, 2017, ch. 3 (per-unit system).
J. L. Blackburn and T. J. Domin, Protective Relaying: Principles and Applications, 4th ed., CRC Press, 2014, ch. 2.