Base Impedance Calculator
Per-unit system: Zbase = Vbase² / Sbase. Solve for the base impedance, voltage or power, and read the base current, base admittance, per-unit conversion and base change.
Inputs
Choose what to solve for, then fill in the other two. Scientific notation such as 1e8 is accepted. Units convert automatically.
Results
Enter values and press Calculate. Your answer will appear here.
Base quantity relationships
The four base quantities are linked by the two base definitions. Every other base quantity follows from these.
Base impedance vs. base power (log–log)
Each dashed line is a different base voltage. On log–log axes they are straight lines of slope −1: doubling Sbase halves Zbase. The user's point is highlighted.
What is being calculated
Typical base impedances
| Level | Vbase | Sbase | Zbase | Ibase |
|---|
Computed by the same engine. Illustrative values, not manufacturer data.
Standard voltage levels and typical base power
| Application | Vbase (LL) | Typical Sbase | Zbase |
|---|---|---|---|
| Low-voltage distribution | 480 V | 0.5 MVA | 0.4608 Ω |
| Low-voltage distribution | 480 V | 1 MVA | 0.2304 Ω |
| Medium-voltage distribution | 4.16 kV | 2 MVA | 8.65 Ω |
| Medium-voltage distribution | 13.8 kV | 10 MVA | 19.04 Ω |
| Subtransmission / substation | 69 kV | 100 MVA | 47.6 Ω |
| Subtransmission / substation | 138 kV | 100 MVA | 190.4 Ω |
| Transmission | 230 kV | 100 MVA | 529.0 Ω |
| Extra-high-voltage transmission | 345 kV | 100 MVA | 1190.3 Ω |
| Large generator | 20 kV | 500 MVA | 0.800 Ω |
| Large generator | 24 kV | 1000 MVA | 0.576 Ω |
These are standard IEEE / IEC voltage levels. The rule of thumb is 100 MVA for transmission studies and 10 MVA for distribution studies.
Key relations and unit conversions
| Quantity | Relation |
|---|---|
| Base impedance | Zbase = Vbase² / Sbase |
| Base current | Ibase = Sbase / Vbase = Vbase / Zbase |
| Base admittance | Ybase = 1 / Zbase = Sbase / Vbase² |
| Base voltage from Z and S | Vbase = √(Zbase · Sbase) |
| Base power from Z and V | Sbase = Vbase² / Zbase |
| Per-unit impedance | Zpu = Zactual / Zbase |
| Actual impedance | Zactual = Zpu · Zbase |
| Per-unit current | Ipu = Iactual / Ibase |
| Per-unit power | Spu = Sactual / Sbase |
| Base change | Zpu,new = Zpu,old · (Vold/Vnew)² · (Snew/Sold) |
| Voltage conversions | 1 V = 10⁻³ kV = 10⁻⁶ MV · 1 kV = 1000 V |
| Power conversions | 1 VA = 10⁻³ kVA = 10⁻⁶ MVA = 10⁻⁹ GVA |
| Impedance conversions | 1 Ω = 10³ mΩ = 10⁻³ kΩ = 10⁻⁶ MΩ |
| Current conversions | 1 A = 10³ mA = 10⁻³ kA · 1 kA = 1000 A |
Formula
- Zbase: base impedance at the chosen voltage level (Ω)
- Vbase: line-to-line base voltage (V)
- Sbase: three-phase base power (VA)
- For single-phase work, Vbase is the phase voltage and Sbase is the phase power — the formula is unchanged.
Example — 13.8 kV, 100 MVA. Vbase = 13 800 V and Sbase = 100 × 10⁶ VA give Zbase = 13 800² / 100 × 10⁶ = 1.9044 Ω. The base current is Ibase = 100 × 10⁶ / 13 800 ≈ 7246 A, and the base admittance is Ybase = 1 / 1.9044 ≈ 0.5251 S. A cable with 0.5 Ω per phase would be 0.5 / 1.9044 ≈ 0.263 pu.
The base quantities are arbitrary by construction: you choose a base voltage and a base power, and every other base quantity follows from them. What matters is that once a base is chosen, the per-unit impedances of all equipment on that voltage level are consistent. The two standard conventions are to pick a single system-wide Sbase (100 MVA is typical for transmission studies, 10 MVA for distribution) and a Vbase at each voltage level equal to the nominal voltage or the transformer's rated voltage on that side.
When you change base, the actual impedance does not change — only its per-unit value does. The base-change formula Zpu,new = Zpu,old · (Vold/Vnew)² · (Snew/Sold) is the workhorse of per-unit calculations, because equipment datasheets almost never quote per-unit values on the base you happen to be using. Notice the voltage term is squared, so a 2× change in voltage base changes Zpu by 4×. The power term is linear and reciprocal: raising Sbase lowers Zpu proportionally.
Transformer per-unit impedances are the same on both sides when the base voltages are in the turns ratio. That is the whole point of the per-unit system: it removes the transformer ratio from the equivalent circuit, so a multi-voltage network becomes a single connected diagram with no ideal transformers in it. The base impedance at each level is different, but the per-unit values across a transformer are identical.
Advanced mode adds an actual impedance input to convert between ohms and per-unit, and an old-base input to perform the base-change calculation on the fly. Presets cover the standard levels from 480 V distribution to 345 kV transmission.
References
- Wikipedia, Per-unit system: base quantities, base change and the transformer-ratio elimination
- Wikipedia, Electrical impedance: definition and units
- Wikipedia, Three-phase electric power: line-to-line voltage and three-phase power conventions
- NIST SP 811, Guide for the Use of the SI: unit conversion factors
How to Use This Tool
- Choose what to solve for: Zbase, Vbase or Sbase.
- Pick a preset, or type the two known values and choose their units.
- In Advanced, add an actual impedance to see its per-unit value, or an old base to convert per-unit values between bases.
- Click Calculate.
- Read the highlighted answer, then check the base current, base admittance and the log–log chart.
What This Tool Offers
- Solves Zbase = Vbase² / Sbase for any of the three variables
- Automatic unit conversion for voltage, power, impedance and current
- Base current Ibase and base admittance Ybase
- Actual impedance to per-unit conversion (and back)
- Per-unit base change from an old voltage and power base
- Relationship diagram and a log–log chart with slope −1 reference lines
- Standard voltage level table and ready-made presets
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| Z_base | Base impedance | Ω |
| V_base | Base voltage (line-to-line RMS) | kV |
| S_base | Base apparent power (three-phase) | MVA |
What it means
In the per‑unit system, quantities are expressed as fractions of base values. Base impedance is defined as Z_base = V_base² / S_base, where V_base is the base voltage and S_base is the base power. This allows normalisation of impedances and simplifies calculations in power systems with multiple voltage levels. The per‑unit system eliminates the need for transformer ratios and makes it easier to compare equipment ratings. It is widely used in power system analysis, fault calculations, and protective relay coordination. The base values are chosen arbitrarily but should reflect the system’s nominal voltage and power. Example: For a system with V_base = 11 kV and S_base = 100 MVA, Z_base = (11,000)² / 100,000,000 = 121,000,000 / 100,000,000 = 1.21 Ω. An actual impedance of 2.42Ω would be 2.42/1.21 = 2.0 per unit.
Worked example
Base Impedance – Practical Example
Real‑World| Parameter | Value |
|---|---|
| Vbase | 11 kV |
| Sbase | 100 MVA |
| Formula | Zbase = Vbase² / Sbase |
Common mistakes
- V_base: The base voltage (line‑to‑line for three‑phase systems).
- S_base: The base apparent power (usually MVA or kVA).
- Units: Ensure V_base and S_base use consistent units (e.g., kV and MVA).
- Per‑unit system: Normalises quantities for easier analysis.
- Transformer: Base impedance changes when crossing transformer boundaries.
Applications
Base impedance in the per‑unit system is Z_base = V_base²/S_base, providing a reference for normalising parameters. The per‑unit system simplifies the analysis of power systems by eliminating transformer ratios and allowing comparison of equipment of different ratings. Engineers use it for load flow, fault analysis, and stability studies. By using per‑unit values, they can avoid mistakes and focus on the relative values. Understanding base impedance is essential for working with the per‑unit system, which is standard in power system analysis. This formula is a key part of system modelling.
- Power system modelling using per‑unit system
- Load flow analysis and fault studies
- Transformer and generator data representation
- Standardised comparison of equipment ratings
- Educational foundation of per‑unit calculations
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