Instant Ohms ↔ Per-Unit Converter — Two-Way Base kV & MVA Calculator

This article provides an instant ohms-per-unit converter for two-way Kv and MVA base system calculations.

Engineers can convert impedances between bases, compute Zbase, and validate protection settings quickly with accuracy.

Instant Ohms ↔ Per-Unit Impedance Converter with Two-Way Base kV/MVA

Conversion direction Ohms to per unit (same base) Per unit to ohms (same base)

Impedance in ohms (Ω)

Impedance in per unit (p.u.)

Base voltage (kV, line-to-line)

Base voltage presets (kV) Custom 4.16 kV 6.6 kV 11 kV 13.8 kV 33 kV 69 kV 132 kV 230 kV 400 kV

Base power (MVA, three-phase)

Advanced options

From base voltage (kV, line-to-line)

From base power (MVA)

To base voltage (kV, line-to-line)

To base power (MVA)

Per-unit impedance on 'from' base (p.u.)

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Enter base kV, base MVA and one impedance value (ohms or per unit) to obtain the converted result.

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Formulas used (three-phase systems, per-unit on kV/MVA base):

• Base impedance in ohms:
  Z_base (Ω) = (V_base,kV)² / S_base,MVA
  where V_base,kV is line-to-line base voltage in kV and S_base,MVA is three-phase base power in MVA.
• Per-unit impedance from ohms:
  Z_pu = Z_actual,Ω / Z_base,Ω
• Ohmic impedance from per-unit:
  Z_actual,Ω = Z_pu × Z_base,Ω
• Per-unit impedance change of base:
  Z_pu,new = Z_pu,old × (Z_base,old / Z_base,new)
  = Z_pu,old × [(V_old,kV² / S_old,MVA) / (V_new,kV² / S_new,MVA)]

Parameter	Typical values / notes
Distribution base voltage	4.16, 6.6, 11, 13.8, 33 kV (line-to-line)
Transmission base voltage	69, 132, 230, 400 kV (line-to-line)
System study base power	Common choices: 10, 50, 100, 500, 1000 MVA (three-phase)
Transformer leakage reactance	0.05–0.2 p.u. on the transformer own MVA base
Short overhead line reactance	0.05–0.3 p.u. depending on length and voltage level

Technical FAQ about the Ohms / Per-Unit Converter

Does this calculator assume three-phase systems?

Yes. The base impedance is computed using the standard three-phase per-unit convention: line-to-line kV and three-phase MVA. For single-phase systems you can still use it if you define kV and MVA consistently with how you define base impedance in your study.

Which impedance value should I enter: R, X, or |Z|?

The calculator works with the magnitude of impedance. In most short-circuit and load-flow studies, the dominant component is the series reactance X, so users typically enter |Z| ≈ |X|. If you have R and X separately, you can compute the magnitude as sqrt(R² + X²) and then convert that value.

When should I use the advanced change-of-base fields?

Use the advanced section when you have an impedance already expressed in per unit on one base (kV, MVA) and you need the equivalent per-unit value on another base. This is common when combining manufacturer transformer data with a system-wide study base.

Why is the base impedance formula Z_base = kV² / MVA so simple?

The constants cancel because kV and MVA are scaled units of volts and volt-amperes. Starting from Z = V² / S and substituting V = kV × 10³ and S = MVA × 10⁶ yields Z = (kV² × 10⁶) / (MVA × 10⁶) = kV² / MVA in ohms.

Overview of the Per-Unit and Ohm Relationship

The per-unit (p.u.) system normalizes impedances and simplifies multi-voltage power system calculations. Using base quantities of voltage and apparent power (kV and MVA) yields direct formulas that are compact and unit-consistent. An instant converter that handles two-way base transformation must implement both the ohms-to-pu and pu-to-ohms relationships plus a robust base-change routine.

Why two-way Kv–MVA base conversion matters

Power-system components (transformers, generators, lines) are often specified on different bases. Protection settings and short-circuit studies require all impedances expressed on a common base. Two-way conversion means:

• Converting an impedance measured in ohms into p.u. on a chosen base (kV, MVA).
• Converting a p.u. impedance from one base to another without first converting to ohms explicitly.

Fundamental formulas and derivations

All formulas below assume Vbase in kilovolts (kV) and Sbase in MVA. Using these units simplifies the base impedance expression.

Base impedance (Zbase) in ohms

Formula:

Zbase (Ω) = Vbase² / Sbase

Explanation of variables and typical values:

• Vbase — line-to-line base voltage in kilovolts (kV). Typical values: 11 kV, 33 kV, 69 kV, 132 kV, 230 kV.
• Sbase — apparent power base in MVA. Typical values: 1 MVA, 10 MVA, 100 MVA, 500 MVA.
• Units: with the chosen units (kV and MVA) the numerical result is in ohms directly because (kV)² / (MVA) simplifies to (10^3 V)^2 / (10^6 VA) = ohm.

Conversion between ohms and per-unit

Formulas:

Zpu = Zohm / Zbase

Zohm = Zpu × Zbase

Variable explanation and typical values:

• Zohm — actual impedance value in ohms measured at the terminals of the device.
• Zpu — per-unit impedance (dimensionless). Typical generator synchronous impedance: 0.1–1.0 p.u.; distribution transformer: 0.04–0.1 p.u.; transmission lines: small values depending on length.

Two-way base conversion (pu on base1 → pu on base2)

Derivation: Let Zpu1 be on base (V1,S1) and Zpu2 on base (V2,S2). Starting from Zohm = Zpu1 × Zbase1 and Zpu2 = Zohm / Zbase2 gives:

Zpu2 = Zpu1 × (Zbase1 / Zbase2)

Substituting Zbase = V²/S yields:

Zpu2 = Zpu1 × (V1²/S1) / (V2²/S2)

Which simplifies to:

Zpu2 = Zpu1 × (S2 / S1) × (V1 / V2)²

Variable meaning:

• Zpu1 — known per-unit impedance on base1.
• Zpu2 — desired per-unit impedance on base2.
• V1, S1 — old base voltage (kV) and power (MVA).
• V2, S2 — new base voltage (kV) and power (MVA).

Algorithm for an instant two-way converter

1. Accept inputs: either (Zohm and Vbase, Sbase) or (Zpu, Vbase, Sbase) and the target base (Vtarget, Starget).
2. If input is Zohm and target is pu: compute Zbase using Zbase = Vbase² / Sbase then Zpu = Zohm / Zbase.
3. If input is Zpu and target is ohm: compute Zbase = Vbase² / Sbase then Zohm = Zpu × Zbase.
4. If input is Zpu on base1 and target is pu on base2: compute Zpu2 = Zpu1 × (S2 / S1) × (V1 / V2)².
5. Include validation checks: non-zero and positive base values, consistent units, and rounding tolerances.

Common reference tables for quick lookup

Detailed worked examples (real cases)

Example 1 — Convert line impedance 0.5 Ω to per-unit on 33 kV, 10 MVA base

Problem statement: A transmission line segment has measured series impedance Z = 0.5 Ω at 33 kV. Convert this impedance to per-unit on a local study base of Vbase = 33 kV and Sbase = 10 MVA.

Step 1: Compute base impedance:

Zbase = Vbase² / Sbase = 33² / 10

Step 2: Compute per-unit impedance:

Interpretation: On a 33 kV, 10 MVA base the line impedance is extremely small in p.u. This small value indicates that for short-circuit or load-flow studies the line contributes modest series impedance relative to generator/transformer impedances.

Example 2 — Convert generator impedance from one base to another across a transformer

Problem statement: A generator nameplate gives Xgen = 0.18 p.u. on its own base of Vg = 13.8 kV and Sg = 50 MVA. The system study base is Vsys = 138 kV and Ssys = 100 MVA. The generator is connected to the 138 kV bus through a 13.8/138 kV transformer with turns ratio a = 13.8/138 = 0.1. Transfer the generator impedance to the system base (p.u.).

Two alternative ways: (A) Convert to ohms then to new p.u.; (B) Use the two-way base conversion formula directly. Both are shown for validation.

Method A — via ohms:

1. Compute generator base impedance Zbase_g = Vg² / Sg = 13.8² / 50 = 190.44 / 50 = 3.8088 Ω
2. Compute generator actual impedance in ohms: Zgen_ohm = Zpu_g × Zbase_g = 0.18 × 3.8088 = 0.685584 Ω
3. Refer Zgen_ohm to system base: Zbase_sys = Vsys² / Ssys = 138² / 100 = 19044 / 100 = 190.44 Ω
4. Compute Zpu_sys = Zgen_ohm / Zbase_sys = 0.685584 / 190.44 = 0.00360 p.u.

Method B — direct base conversion formula:

Zpu_sys = Zpu_g × (Ssys / Sg) × (Vg / Vsys)²

Zpu_sys = 0.18 × (100 / 50) × (13.8 / 138)²

Zpu_sys = 0.18 × 2 × (0.1)² = 0.18 × 2 × 0.01 = 0.0036 p.u.

Same result. Note on transformer turns ratio: If the generator impedance had been specified on transformer HV or LV terminal base, one must account for the impedance referred through the transformer using the squared turns ratio. The direct conversion formula already captures the voltage base ratio (Vg/Vsys), which implicitly includes transformer ratio if you use the machine terminal voltage for Vg.

Example 3 — Recalculate transformer impedance when changing MVA base

Problem statement: A transformer is specified Xtr = 0.08 p.u. on its own base Vtr = 66 kV, Str = 20 MVA. For a fault study, you need Xtr on base Vstudy = 66 kV but Sstudy = 100 MVA. Compute the new p.u. value.

Zpu2 = Zpu1 × (S2 / S1) × (V1 / V2)²

Since V1 = V2, (V1/V2)² = 1, so:

Zpu2 = 0.08 × (100 / 20) = 0.08 × 5 = 0.4 p.u.

Interpretation: Changing only the power base inflates the p.u. impedance by the ratio S2/S1. This is why equipment specified on small MVA bases becomes comparatively large in p.u. when the system base is much larger.

Practical considerations for an instant converter

• Unit consistency: Always confirm kV and MVA are the units used. Mismatch in units is the primary source of errors.
• Floating-point precision: Provide sensible rounding (e.g., 4–6 significant digits) but allow full precision for internal calculations.
• Transformer connections and delta-wye shifts: When converting impedances across transformer windings, include turns ratio squared and consider grounding if converting sequence impedances.
• Per-unit on different voltage levels: For sequence components, ensure the correct base voltages: line-to-line for three-phase impedances, line-to-neutral where required for single-phase or sequence conversions depending on convention.

Verification tests and validation examples

Recommended validation checks for any converter implementation:

1. Round-trip test: Convert a known ohm value to p.u. and back; difference should be numerical rounding only.
2. Base-change commutativity: Converting a p.u. from base A to base B and back to A should return the original p.u. within computational tolerance.
3. Cross-check with measured short-circuit currents: Use converted impedances to compute expected fault currents and compare with bench measurements or detailed electromagnetic transient simulations.

Implementation details for an online instant calculator

Minimum input fields:

• Input type selector: (Ohm → p.u., p.u. → Ohm, p.u. → p.u.).
• Impedance value (numeric).
• Original base: Vbase_original (kV), Sbase_original (MVA).
• Target base: Vbase_target (kV), Sbase_target (MVA) — optional for ohm↔pu conversions if only one base is needed.
• Unit confirmations and tooltips for each field.

Output and user feedback:

• Display intermediate Zbase values to aid transparency.
• Show step-by-step arithmetic results to support auditability.
• Provide warnings for extreme or unrealistic inputs (e.g., negative base, zero).

Standards, normative references and further reading

Authoritative standards and guidance relevant to per-unit and impedance calculations include:

• IEC 60076 — Power transformers. Provides testing and impedance measurement standards. See https://www.iec.ch
• IEEE Std 141 (Green Book) — Electric Power Distribution for Industrial Plants. Guidance on power system analysis and bases. See https://standards.ieee.org/standard/141-1993.html
• IEEE Std C37.010 — Application of Current Transformers. Useful for protection scaling and impedance-based relay settings. See https://standards.ieee.org
• IEC 60909 — Short-circuit currents in three-phase AC systems. Contains standard methodologies referencing p.u. practices. See https://www.iec.ch/standards
• IEEE Std C57.12 — Transformer standards covering impedance declarations and conversions. See https://standards.ieee.org

Additional technical resources and tutorials:

• CIGRE technical brochures on impedance and transformer modeling: https://www.cigre.org
• Textbook reference: P. Kundur, Power System Stability and Control — chapters on per-unit systems and modeling.

Best practices and pitfalls

1. Always state the base when publishing p.u. values — p.u. without base is meaningless.
2. For multi-network computations, select a consistent system base early and convert all device impedances to that base to avoid confusion.
3. Be cautious with low-voltage systems and single-phase conversions: use correct voltage references (line-to-line versus line-to-neutral).
4. Document rounding rules and significant digits for engineering reports. Small rounding errors can cascade in large network analyses.

Extended examples and checks for protection engineers

Protection settings often rely on per-unit impedances to determine pickup levels and margin. Below is a design check example.

Example 4 — Fault current check using converted p.u. impedance

Given: A generator with Zgen_pu = 0.15 on 20 MVA, 13.8 kV base. The generator is connected to a 69 kV bus through a 13.8/69 kV transformer and a line with series impedance Zline = 1.2 Ω. The system base is 69 kV and 100 MVA. Compute approximate three-phase fault current at the 69 kV bus assuming the system behind the bus is stiff (negligible impedance).

Step 1 — Convert generator to system base using formula:

Zpu_gen_sys = 0.15 × (100 / 20) × (13.8 / 69)²

Zpu_gen_sys = 0.15 × 5 × (0.2)² = 0.15 × 5 × 0.04 = 0.03 p.u.

Step 2 — Convert line impedance to p.u. on 69 kV, 100 MVA base:

Zbase_sys = Vbase² / Sbase = 69² / 100 = 4761 / 100 = 47.61 Ω

Zline_pu = Zline / Zbase_sys = 1.2 / 47.61 = 0.0252 p.u.

Step 3 — Total equivalent p.u. impedance to fault (approx): Ztotal_pu ≈ Zpu_gen_sys + Zline_pu = 0.03 + 0.0252 = 0.0552 p.u.

Step 4 — Fault current magnitude (three-phase) in per-unit: If system nominal line-to-line voltage is 69 kV and base current Ibase = Sbase / (sqrt(3) × Vbase), then Ifault_pu = 1 / Ztotal_pu. So:

Ifault_pu = 1 / 0.0552 = 18.12 p.u.

Now convert to amperes:

Ibase = Sbase (MVA) × 10^6 / (sqrt(3) × Vbase (kV) × 10^3) = (100 × 10^6) / (1.732 × 69 × 10^3) ≈ 837.5 A

This quick calculation provides an estimate for relay coordination and breaker selection. Compare with detailed short-circuit software for exact results (including subtransient generator reactance and network contributions).

Summary of formulas (quick reference)

• Zbase (Ω) = Vbase² / Sbase
• Zpu = Zohm / Zbase
• Zohm = Zpu × Zbase
• Zpu_new = Zpu_old × (Snew / Sold) × (Vold / Vnew)²
• Impedance referral across transformer: Zref = Z × a² where a = Vprimary / Vsecondary (when moving impedance between windings)

Authoritative links for implementation and verification

• IEC Standards portal: https://www.iec.ch/ — for transformer and short-circuit standards (IEC 60076, IEC 60909).
• IEEE Xplore: https://ieeexplore.ieee.org/ — look up IEEE Std 141 and IEEE transformer/relay standards for detailed methodologies.
• CIGRE: https://www.cigre.org/ — technical brochures on modeling and impedance practices.
• NERC reliability guidelines: https://www.nerc.com/ — operational planning and studies guidance.

Implementing an instant two-way ohms-per-unit converter requires attention to unit conventions (kV, MVA), precise arithmetic for base impedances, and provision for transformer referrals and sequence impedance considerations. Following the formulas and best practices above ensures accurate, auditable conversions for design, protection, and studies.