Instant Per-Unit to Ohms & Amps Converter — Convert Generator/Transformer Values to Real Units

This article provides precise conversion methods for per-unit, ohms, and amperes in power systems analysis

Detailed transformer and generator examples convert per-unit values into real units for engineering calculations accurately

Per-unit to Ohms and Amperes Converter for Generator and Transformer Bases

Per-unit impedance Z (p.u.)

Per-unit current I (p.u.)

System type Three-phase (typical) Single-phase

Base apparent power S_base (MVA)

Base voltage V_base (kV)

Advanced options

Custom base impedance Z_base (Ω)

Custom base current I_base (A)

System frequency (Hz)

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Enter per-unit impedance and/or current with base power and voltage to obtain actual ohms and amperes.

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Formulas used

All per-unit quantities are referred to the specified base apparent power S_base and base voltage V_base.

• Base impedance (ohms):
  For any system if S_base is in MVA and V_base in kV (rms):
  Z_base (Ω) = V_base² (kV²) / S_base (MVA)
• Base current (amperes):
  Three-phase system (V_base is line-to-line):
  I_base (A) = S_base (MVA) × 10³ / (√3 × V_base (kV))
  Single-phase system (V_base is phase-to-neutral or two-wire):
  I_base (A) = S_base (MVA) × 10³ / V_base (kV)
• Actual impedance from per-unit:
  Z_actual (Ω) = Z_pu × Z_base (Ω)
• Actual current from per-unit:
  I_actual (A) = I_pu × I_base (A)
• If custom bases are provided:
  • If a custom Z_base is entered, it replaces the calculated Z_base in Z_actual = Z_pu × Z_base.
  • If a custom I_base is entered, it replaces the calculated I_base in I_actual = I_pu × I_base.

Typical base values and resulting bases

System type	S_base (MVA)	V_base (kV)	Z_base (Ω)	I_base (A)
Three-phase	100	13.8	1.90	4,183
Three-phase	100	33	10.89	1,749
Three-phase	50	11	2.42	2,624
Single-phase	10	11	12.10	909

Technical FAQ

How should I choose S_base and V_base?
For per-unit studies, S_base is often a common system base such as 50, 100 or 500 MVA, and V_base is the nominal bus voltage (line-to-line for three-phase). Equipment per-unit data should be converted to this common base before using the converter.

Can I enter only impedance or only current in per-unit?
Yes. If you enter Z_pu and leave I_pu empty, the calculator will output only the actual impedance in ohms. If you enter I_pu and leave Z_pu empty, it will output only the actual current in amperes. If you provide both, both quantities are converted.

Does the system frequency affect the ohmic impedance result?
The ohmic impedance calculated from per-unit values depends only on the selected bases S_base and V_base. Frequency is included as an advanced field for documentation, but it does not modify the ohmic result in this converter.

When should I use custom Z_base or I_base?
Custom base impedance and current are useful if your per-unit quantities are defined on a non-standard base that is already expressed in ohms or amperes, for example when working from relay settings or detailed manufacturer data. In that case, enter the known base directly and the converter will scale using that value.

Fundamental conversion principles for per-unit, ohms, and amperes

Per-unit (pu) representation normalizes impedances, voltages, and currents against chosen base values. This normalization simplifies multi-voltage network calculations, allowing direct comparison and algebraic combination of impedances referenced to a common base. The per-unit system relies on consistent choice of base power (S_base) and base voltage (V_base). Common quality control checks include verifying that transformer turns ratios and impedances are converted to the same base before network assembly. Instant conversion between per-unit and SI units requires three basic base definitions:

• S_base — base apparent power (usually MVA).
• V_base — base line-to-line voltage for three-phase systems (kV).
• I_base and Z_base derived from S_base and V_base.

Once S_base and V_base are selected, the following conversions provide deterministic results for all components (generators, transformers, transmission lines, motors).

Core formulas and variable definitions

Primary base formulas (three-phase)

Z_base = (V_base²) / S_base
I_base = S_base / (sqrt(3) × V_base)
Z_actual = Z_pu × Z_base
Z_pu = Z_actual / Z_base
I_actual = I_pu × I_base
I_fault (three-phase) ≈ V_line / (sqrt(3) × Z_actual)

Variable explanations and typical engineering units

• V_base: line-to-line voltage, in volts or kilovolts (V or kV). For single-phase use line-neutral voltage.
• S_base: apparent power base, in VA or commonly MVA.
• Z_base: base impedance in ohms (Ω). If V_base in kV and S_base in MVA, Z_base = (kV²)/MVA yields Ω.
• I_base: base current in amperes (A). For three-phase networks: I_base = S_base (VA) / (√3 × V_base (V)).
• Z_actual: actual impedance in ohms (Ω) as measured or referred to a terminal.
• Z_pu: impedance in per-unit (dimensionless) on the chosen base.
• I_pu: current in per-unit (dimensionless).

Conversion relationships between different bases

When you must convert a per-unit impedance from one base set (S_base_old, V_base_old) to another (S_base_new, V_base_new) use: Z_pu_new = Z_pu_old × (S_base_new / S_base_old) × (V_base_old / V_base_new)² Derivation outline:

1. Z_actual = Z_pu_old × Z_base_old.
2. Z_base_old = V_base_old² / S_base_old.
3. Z_pu_new = Z_actual / Z_base_new = Z_pu_old × (Z_base_old / Z_base_new).
4. Replace Z_base expressions and simplify to the formula above.

For current conversion between bases: I_pu_new = I_pu_old × (S_base_old / S_base_new) × (V_base_new / V_base_old) (but typically currents are converted by computing actual current using I_actual = I_pu × I_base.)

Typical base values and extensive lookup tables

Step-by-step procedures for common conversions

Procedure: per-unit to ohms (generator or transformer)

1. Select the correct S_base and V_base for the equipment terminal where Z_pu is specified. If the device uses its own MVA rating as base, use that or convert using the base conversion formula.
2. Compute Z_base = V_base² / S_base (ensure units consistent: kV²/MVA yields Ω).
3. Compute Z_actual = Z_pu × Z_base.
4. Use Z_actual to compute currents or fault levels: I = V_line / (sqrt(3) × Z_actual).

Procedure: ohms to per-unit

1. Determine S_base and V_base used in system model.
2. Compute Z_base.
3. Z_pu = Z_actual / Z_base.

Two real-case worked examples with full development

Case 1 — Synchronous generator: compute actual reactance and fault current

Problem statement:

• Generator nameplate: 50 MVA, 13.8 kV (line-to-line).
• Subtransient reactance X" = 0.20 pu (given on generator base 50 MVA).
• Compute X" in ohms and three-phase bolted fault initial symmetrical current at generator terminals (fault bolted to ground on the generator terminal bus, neglect system external sources).

Step 1: Choose base values

• S_base = 50 MVA (use generator base, since Z_pu given on machine base).
• V_base = 13.8 kV.

Step 2: Compute Z_base Z_base = V_base² / S_base = (13.8²) / 50 = 190.44 / 50 = 3.8088 Ω. Step 3: Compute actual reactance X"_actual = X"_pu × Z_base = 0.20 × 3.8088 = 0.76176 Ω. Step 4: Compute fault current (three-phase bolted) I_fault = V_line / (sqrt(3) × Z_actual) = 13.8e3 / (1.732 × 0.76176) A. Compute numeric:

• Denominator = 1.732 × 0.76176 = 1.3186.
• I_fault = 13800 / 1.3186 ≈ 10470 A.

Alternate per-unit check: I_pu_fault = 1 / X"_pu = 1 / 0.20 = 5.0 pu. I_base = S_base / (sqrt(3) × V_base) = 50e6 / (1.732 × 13.8e3) ≈ 2091 A. I_actual = I_pu_fault × I_base = 5.0 × 2091 ≈ 10455 A. Results match within rounding: three-phase fault current ≈ 10.45 kA. Engineering notes:

• Use transient/subtransient reactance appropriate for the time frame of the fault current (subtransient for initial peak).
• Include system external impedances and transformer leakage when connected to an infinite bus — the above assumes isolated generator fault.

Case 2 — Power transformer: percent impedance to ohms and to per-unit on alternate base

Problem statement:

• Transformer rating: 100 MVA, 230/13.8 kV, nameplate Z% = 10.0% (on 100 MVA rating).
• Objective A: Compute equivalent series impedance on HV and LV terminals in ohms (referred to HV and LV respectively).
• Objective B: Convert transformer impedance to per-unit on a system base of 200 MVA and 13.8 kV low-voltage base (for network study).

Step A1: Transformer pu impedance on its own base Z_pu_nameplate = Z%/100 = 10/100 = 0.10 pu. Step A2: Compute Z_base on HV side (use HV V_base = 230 kV, S_base = 100 MVA) Z_base_HV = (230²) / 100 = 52900 / 100 = 529.00 Ω. Z_actual_referred_HV = Z_pu_nameplate × Z_base_HV = 0.10 × 529.00 = 52.900 Ω. Step A3: Compute Z_base on LV side (LV V_base = 13.8 kV) Z_base_LV = (13.8²) / 100 = 190.44 / 100 = 1.9044 Ω. Z_actual_referred_LV = Z_pu_nameplate × Z_base_LV = 0.10 × 1.9044 = 0.19044 Ω. Note: The same transformer physical impedance referred either to HV or LV should satisfy the turns ratio relationship; computed Z_actual values above are consistent when changing reference with squared turns ratio. Step B1: Convert Z_pu to a new base set

• Original base: S_base_old = 100 MVA, V_base_old (LV) = 13.8 kV.
• New system base: S_base_new = 200 MVA, V_base_new (LV system) = 13.8 kV (same voltage, double power base).

Step B2: Use conversion formula Z_pu_new = Z_pu_old × (S_base_new / S_base_old) × (V_base_old / V_base_new)² Since V_base_old = V_base_new, the voltage ratio term cancels to 1: Z_pu_new = 0.10 × (200 / 100) = 0.10 × 2 = 0.20 pu on 200 MVA base referenced to 13.8 kV. Step B3: Validate via Z_actual

• Z_actual from nameplate = 0.10 × Z_base_LV_old = 0.19044 Ω.
• New Z_base_LV (200 MVA) = (13.8²) / 200 = 190.44 / 200 = 0.9522 Ω.
• Z_pu_new = Z_actual / Z_base_new = 0.19044 / 0.9522 = 0.20 pu. Validation complete.

Engineering notes:

• Transformer percent impedance on nameplate is referenced to rated MVA; conversion to system base is mandatory before network combination.
• If voltage bases differ, include squared voltage ratio in conversion formula.

Advanced conversions: single-phase, neutral grounding and asymmetrical faults

Single-phase base choices

For single-phase equipment or phase-to-ground calculations, use single-phase bases: Z_base_single = (V_base_phase²) / S_base_single where V_base_phase is the phase voltage and S_base_single is the single-phase base VA. Common practice: keep S_base three-phase and use phase voltage derived from line-to-line (V_phase = V_line / sqrt(3)) and then compute single-phase Z_base accordingly.

Neutral grounding and zero-sequence conversion

Zero-sequence impedances require specific base selection because their reference is often different (ground path). When converting Z0 between bases:

• Use same procedure but ensure V_base corresponds to phase voltage for zero-sequence calculations.
• Be careful when referring transformer vector groups — zero-sequence transformation depends on winding connection (delta blocks zero-sequence between windings, wye provides path).

Practical tips and best practices for accuracy

• Always record the base MVA and base voltage associated with any per-unit data on equipment nameplates or datasheets.
• Prefer using the equipment rating base for transformer and machine per-unit values, then convert to the study base consistently.
• Keep units consistent: kV and MVA pair gives Z in ohms without additional scaling; convert to volts and VA if using SI base units.
• Document if impedances are given as percent (Z%) — convert to pu by dividing by 100 and noting base.
• For protection studies include transient reactances (X', X") appropriate to time frame; steady-state fault currents use synchronous reactance for sustained conditions.
• When assembling networks with multiple voltage levels, select a single global S_base and keep V_base per voltage level consistent with bus nominal voltages.

Additional conversion examples and edge cases

Example: converting transmission line series impedance to per-unit on 100 MVA base

Given:

• Line impedance per km: Z_line = 0.05 + j0.40 Ω/km.
• Line length: 100 km.
• System study base: 100 MVA, 230 kV.

Step:

• Z_total_actual = Z_line × length = (0.05 + j0.40) × 100 = 5 + j40 Ω.
• Z_base (230 kV, 100 MVA) = 529 Ω (see table).
• Z_pu = Z_total_actual / Z_base = (5 + j40) / 529 ≈ 0.00946 + j0.07564 pu.

Engineering note: Keep complex arithmetic for stability and modal analysis. Line charging and shunt admittances convert similarly using susceptance base B_base = 1/Z_base.

Standards, normative references and further authoritative reading

Key standards and references:

• IEEE Std 141 (Red Book) — Electrical Power Distribution for Industrial Plants: general power system conversion practices. https://standards.ieee.org/
• IEEE Std C37.010 — Application Guide for AC High-Voltage Circuit Breakers. https://standards.ieee.org/
• IEC 60076 series — Power transformers — design, testing and impedance definitions. https://www.iec.ch/
• NIST Reference on units and quantities for electrical engineering. https://www.nist.gov/
• CIGRE technical brochures on transformer impedance and short-circuit calculation methods. https://www.cigre.org/

These sources provide normative definitions for impedance representation, measurement methods, and recommended practice for short-circuit calculation and transformer testing. Use them for compliance and detailed modelling guidance.

Common pitfalls and verification checks

1. Unit mismatch: mixing kV with V or MVA with kVA leads to errors of orders of magnitude; always normalize units before calculation.
2. Incorrect voltage base selection: using HV base for LV impedance without appropriate conversion results in erroneous pu values.
3. Neglecting transformer vector group constraints for zero-sequence paths when computing ground fault currents.
4. For fault studies, ensure use of appropriate reactance type (subtransient for initial peak, transient for intermediate times, synchronous for steady-state).
5. When combining impedances in pu, ensure all are on the same base — otherwise convert using the base conversion formula.

SEO optimized summary of conversion workflow (quick reference)

• Select S_base and V_base per voltage level.
• Compute Z_base = V_base² / S_base and I_base = S_base / (√3 × V_base).
• Convert Z_pu to ohms: Z_actual = Z_pu × Z_base.
• Convert ohms to pu: Z_pu = Z_actual / Z_base.
• Convert between pu bases: Z_pu_new = Z_pu_old × (S_new / S_old) × (V_old / V_new)².
• Use per-unit currents: I_actual = I_pu × I_base or directly compute I_actual = V_line / (√3 × Z_actual).

References for practical implementation and software tools

• PSS®E user manuals and conversion examples for system base handling. https://new.siemens.com/global/en/products/energy.html
• ETAP technical notes on per-unit conversions and protective relay settings. https://etap.com/
• Open-source libraries and academic references demonstrating per-unit standardization and conversion routines for power system simulation (e.g., MATPOWER documentation). http://www.pserc.cornell.edu/matpower/

This article equips power engineers with deterministic formulas, worked calculations, and conversion best practices to implement an instant per-unit to ohms and amps converter in studies involving generators, transformers, and network components. Use the provided tables, conversion formulas, and worked examples as templates for automation in spreadsheets, scripts, or specialized power-system analysis software.