Per-unit is the accounting system of power engineering. Once every quantity is normalised to a common base, transformer turns ratios disappear from the network model, impedances of machines of wildly different sizes become directly comparable, and a bus voltage of 0.96 means the same thing at 13.8 kV as it does at 500 kV. The arithmetic is elementary. The errors are not, and almost all of them come from one place: the bases.
This calculator does the three jobs that per-unit work actually consists of. It computes the base quantities from a chosen base power and base voltage and converts an impedance between ohms, per-unit and percent. It moves a device impedance from its own nameplate rating onto a system base. And it propagates voltage bases through a radial chain of transformers so you can read off the base impedance and base current in each zone.
One convention runs through everything below and through the tool itself: \(S_{base}\) is the three-phase apparent power in MVA and \(V_{base}\) is the line-to-line voltage in kV. The base impedance that follows is the per-phase, wye-equivalent impedance. Mixing three-phase and single-phase bases in the same study is the single most common per-unit error, and it produces answers wrong by a factor of three, which is close enough to plausible that it survives review.
Per-Unit Calculator
Base quantities, change of base, and voltage-base propagation across transformer zones
The base quantities
You choose two bases freely, conventionally the three-phase apparent power and the line-to-line voltage. Everything else follows:
\[ Z_{base}=\frac{V_{base}^{2}}{S_{base}}, \qquad I_{base}=\frac{S_{base}}{\sqrt3\,V_{base}}, \qquad Y_{base}=\frac{1}{Z_{base}}=\frac{S_{base}}{V_{base}^{2}} \]
With \(S_{base}\) in MVA and \(V_{base}\) in kV, \(Z_{base}\) comes out directly in ohms, which is the reason those units are the working pair. \(I_{base}\) in that mixed system is \(1000\,S_{base}/(\sqrt3\,V_{base})\) amperes. Any quantity is then converted by dividing by its base:
\[ Z_{pu}=\frac{Z_{\Omega}}{Z_{base}}, \qquad Z_{\%}=100\,Z_{pu} \]
Percent and per-unit are the same number scaled by 100. Manufacturers quote transformer impedance in percent; power flow and short-circuit programs work in per-unit. The only trap is forgetting which one you are holding.
The reason the wye-equivalent per-phase impedance falls out of a line-to-line voltage without a stray factor of three is worth seeing once. Per phase, \(Z_{base}=V_{LN}^{2}/S_{1\phi}\), with \(V_{LN}=V_{LL}/\sqrt3\) and \(S_{1\phi}=S_{3\phi}/3\). The two factors of three cancel exactly, leaving \(V_{LL}^{2}/S_{3\phi}\). A delta-connected impedance must be converted to its wye equivalent, dividing by three, before it is normalised.
Change of base
A device is delivered with its impedance expressed on its own rating. A 50 MVA transformer with 8% impedance means 8% on 50 MVA, not on whatever base your study happens to use. Because the physical ohms do not change, equating them on the two bases gives the change-of-base relation:
\[ Z_{pu,new}=Z_{pu,old}\cdot\frac{S_{new}}{S_{old}}\cdot\left(\frac{V_{old}}{V_{new}}\right)^{2} \]
Note the directions. Per-unit impedance scales up with base MVA and down with the square of base voltage. When the device nameplate voltage equals the zone base voltage, which is the usual case, the voltage term is exactly 1 and only the MVA ratio matters. When it does not, the squared voltage term is not optional: a 5% voltage discrepancy moves the impedance by about 10%.
Voltage bases across transformer zones
The whole point of per-unit in a network with transformers is that the ideal turns ratio vanishes from the model, but only if you choose the voltage bases to follow that same ratio. So a network is divided into zones separated by transformers. One base power \(S_{base}\) is used everywhere, and the voltage base is carried from zone to zone by the nameplate ratio:
\[ V_{base,k}=V_{base,k-1}\cdot n_{k}, \qquad n_{k}=\frac{V_{rated,k,\text{out}}}{V_{rated,k,\text{in}}} \]
Pick the base in one zone, propagate through every transformer, and the ideal transformers drop out of the impedance diagram entirely. Break that rule anywhere in the chain and the transformer reappears as a spurious per-unit turns ratio, which is exactly the bug that produces short-circuit currents off by an order of magnitude.
Worked example
Take a 100 MVA system base with a 230 kV transmission voltage. The base impedance is
\[ Z_{base}=\frac{(230\ \text{kV})^{2}}{100\ \text{MVA}}=529\ \Omega, \qquad I_{base}=\frac{100\ \text{MVA}}{\sqrt3\cdot 230\ \text{kV}}=251.0\ \text{A} \]
A line with 52.9 Ω of series reactance is therefore 0.10 pu, or 10%. Now take a transformer rated 50 MVA at 138 kV with 8% impedance and move it to a 100 MVA system base at the same 138 kV: the voltage ratio is 1, the MVA ratio is 2, and the impedance becomes 16%, or 0.16 pu. Doubling the base MVA doubles the per-unit impedance, which is worth internalising because it means a small machine looks stiff on its own base and weak on the system base.
Switch the tool to zone mode for the radial chain: a generator zone at 13.8 kV, a 13.8/230 kV step-up, a 230 kV line, a 230/34.5 kV step-down, and a 34.5 kV load zone, all on 100 MVA. The voltage bases propagate to 13.8 kV, 230 kV and 34.5 kV. The base impedances are 1.90 Ω, 529 Ω and 11.90 Ω, and the base currents are 4184 A, 251 A and 1673 A. Those three base currents differ by more than an order of magnitude while every per-unit current in the chain is the same number, which is the entire value of the method.
Reading the base-consistency check
Unlike most calculators here, per-unit has no good or bad axis: a base is a choice, not a performance. The chip therefore reports base consistency, meaning whether the device ratings you entered agree with the base of the zone they sit in.
- Green, bases consistent: every nameplate voltage matches its zone base. The voltage term in the change-of-base formula is 1 and impedances scale by the MVA ratio alone. This is the clean textbook case.
- Amber, nameplate offset: a rating differs from its zone base by a few percent. This is common and entirely legitimate. Generator windings are rated 13.8 kV against a transformer winding of 13.2 kV; a transformer sits on a fixed tap that shifts its effective ratio. Per-unit still works, but the squared voltage term is now doing real work and must not be dropped.
- Red, base missing or mismatched: a base is zero, negative or absent, so every derived quantity is undefined; or a rating differs from its zone base by more than 5%, which is usually a data-entry error or a genuinely different voltage class rather than a tap offset. Check which nameplate the impedance was quoted on before converting anything.
Keep the whole method on one page
The Per-Unit System Cheat Sheet collects the base formulas, the change-of-base relation, the three-phase and single-phase conventions side by side, and the zone-propagation rule on one printable page.
Frequently asked questions
Is the base voltage line-to-line or line-to-neutral?
Line-to-line, paired with three-phase MVA, throughout this tool and in standard practice. The two factors of three that appear when you go to a per-phase basis cancel exactly, so \(Z_{base}=V_{LL}^{2}/S_{3\phi}\) is the per-phase wye-equivalent impedance despite being written with a line-to-line voltage. If you insist on single-phase bases, use \(V_{LN}\) with \(S_{1\phi}\) consistently and never mix the two systems in one study.
Why is per-unit impedance not the same on both bases?
Because per-unit is a ratio, and only the numerator is a physical property. The ohms are fixed by the device; the denominator changes when you change base, so the ratio changes with it. An 8% transformer on 50 MVA is 16% on 100 MVA and 4% on 25 MVA, and all three describe the same piece of iron and copper.
Does the base power change from zone to zone?
No. One base power is used for the entire system, and only the voltage base changes across transformers. That is what makes per-unit powers directly comparable everywhere in the network and what allows a single-line impedance diagram to be assembled without turns ratios.
What about delta-connected impedances and off-nominal taps?
Convert a delta impedance to its wye equivalent, dividing by three, before normalising. Off-nominal taps break the assumption that the zone voltage bases follow the nameplate ratio: the standard treatment keeps the bases on the nominal ratio and models the residual ratio as an explicit off-nominal tap in the network, which is what a power flow program does internally. This tool models the nominal chain only.
Which base power should I pick?
It is arbitrary, so pick a round number and keep it. 100 MVA is the near-universal default for transmission studies. Industrial and distribution work often uses 10 MVA or the rating of the dominant source. The choice affects only the numbers on the page, never the physical answer.
Related reading
- All EE Power School calculators: the full set of interactive power system tools.
- Two-Bus Load Flow Calculator: takes per-unit impedances as its input, so it is the natural next step once the bases are set.
- SCR Calculator: short-circuit ratio, the per-unit measure of grid strength at a point of interconnection.
- Symmetrical Components Calculator: the companion decomposition for unbalanced conditions, also worked in per-unit.
References
- J. D. Glover, M. S. Sarma and T. J. Overbye, Power System Analysis and Design, Cengage, chapter 3 (the per-unit system).
- J. J. Grainger and W. D. Stevenson, Power System Analysis, McGraw-Hill, 1994, chapter 2.
- P. Kundur, Power System Stability and Control, McGraw-Hill, 1994, chapter 3 (per-unit representation of synchronous machines).
- IEEE Std C57.12.00, Standard for General Requirements for Liquid-Immersed Distribution, Power, and Regulating Transformers, for how nameplate impedance is defined and referenced to the transformer rating.