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Per-unit base conversion

Calculate the base current and base impedance of a three-phase system and bring the nameplate impedance of a transformer, generator or line to the common study base, in %, pu or ohms.

Grainger and Stevensonper-unit system, ch. 2Glover, Sarma and Overbyechange of base, ch. 3

Example cases
Per-unit base conversion
E1Study base
MVA
kV
E2Equipment impedance
MVA
kV

The one on the side that is in the zone of the base voltage.

Use a decimal point. The calculation runs in your browser and updates as you type.

Impedance on the study base

— pu

The same, in percent

— %

In ohms, Vbase zone

— Ω

Base current

— A

Base impedance

— Ω

Three-phase fault behind this impedance

— kA

Step-by-step calculation

How to use the result

    Example: three elements on a common 100 MVA base

    13.8 kV in the generator zone and 115 kV in the line zone.

    ElementNameplate dataCalculationX (pu)
    Generator50 MVA, 13.2 kV, X″d = 18 %0.18 · (13.2 / 13.8)² · (100 / 50)0.3294
    Transformer60 MVA, 13.8/115 kV, X = 10 %0.10 · (100 / 60)0.1667
    Line115 kV, X = 25 Ω25 / (115² / 100)0.189

    What it is for

    To build the impedance diagram of a short-circuit, coordination or load-flow study: each piece of equipment comes with its impedance on its own base (the nameplate) and all of them must be brought to a common base before adding them. It also helps to check a nameplate value quickly, compare the impedance measured in a transformer test with its nameplate, or convert the ohms of a line to per unit.

    How it is calculated

    For a three-phase system, with the three-phase base power and the line-to-line base voltage (Grainger and Stevenson, Power System Analysis, ch. 2; Glover, Sarma and Overbye, Power System Analysis and Design, ch. 3):

    Base quantities

    Ibase = Sbase / (√3 · Vbase)Zbase = Vbase² / Sbase

    With Sbase in MVA and Vbase in kV, Ibase comes out in kA and Zbase in ohms. Per unit and percent: Zpu = ZΩ / Zbase and Z% = 100 · Zpu.

    Change of base

    Znew = Zold · (Vold / Vnew)² · (Snew / Sold)

    With both impedances in pu.

    Ohms to the study base

    Zpu = ZΩ · Sbase / Vzone²

    Vzone is the base voltage of the zone where the ohms are measured, obtained by carrying Vbase through the rated transformer ratios; it is not the rated voltage of the equipment. If that zone differs from the Vbase zone, the formula already refers the ohms with the ratio of the base voltages, so the two voltages do not need to match.

    Three-phase fault behind the impedance

    Isc = Ibase / Zpu

    It is the current with an infinite source behind and only this impedance in series, in the Vbase zone, with 1 pu pre-fault voltage. It is good for comparing orders of magnitude; the actual short circuit needs the source impedance and the IEC 60909 voltage factor.

    The same conversion applies separately to resistance and reactance, and to the positive-, negative- and zero-sequence impedances.

    Limits of the calculation

    • It assumes a balanced three-phase system and voltage bases that follow the rated transformer ratios.
    • It does not account for off-nominal taps, nameplate tolerances or standard-based generator impedance corrections (for example, the IEC 60909 KG factor).
    • In single-phase systems the base current is Sbase / Vbase.
    • A short-circuit or protection coordination study needs the complete network model.

    Frequently asked questions

    How is a transformer impedance changed to another base?

    Multiply the nameplate per-unit impedance by the square of the ratio between the nameplate voltage and the new base voltage, and by the ratio between the new base power and the nameplate power: Znew = Zrated · (Vrated / Vnew)² · (Snew / Srated). A 60 MVA transformer with 10 % becomes 0.1667 pu on a 100 MVA base at the same voltage.

    Why is the per-unit system used in power system studies?

    Because with voltage bases that follow the transformer ratios, the impedance of each element is the same seen from either side and the ideal transformers drop out of the circuit. Also, the impedances of similar equipment fall in known ranges, which helps spot wrong data.

    Which base voltage is used on each side of a transformer?

    A base voltage is chosen in one zone and carried to the others with the rated voltage ratio of the transformer. If the base is 115 kV on the HV side of a 115/13.8 kV transformer, the LV side base is 13.8 kV. The base power is the same throughout the system.

    We do your power system study

    We model the network with the nameplate data of your equipment, calculate the short circuit, coordinate the protection and deliver the calculation report. We do it remotely, with no travel.

    See design and studies

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