Per-Unit Formulas: Base Values and Impedance
Per-unit system formulas: base current and base impedance from MVA and kV, converting ohms and percent impedance to per unit, and changing the base.
The formulas
| To find | Formula |
|---|---|
| Base current | Ibase (A) = Sbase (MVA) × 10⁶ / (√3 × Vbase (kV) × 1000) |
| Base impedance | Zbase (Ω) = Vbase² (kV) / Sbase (MVA) |
| Impedance in per unit | Z(pu) = Z (Ω) / Zbase |
| Percent to per unit | Z(pu) = Z% / 100 |
| Change of base | Z(pu, new) = Z(pu, old) × (Snew / Sold) × (Vold / Vnew)² |
Sbase is the three-phase power base in MVA and Vbase the line-to-line voltage base in kV. Pick one Sbase for the whole network and a Vbase for each voltage level so that the bases follow the transformer ratios.
Worked example: base values for 10 MVA and 11 kV
- Ibase = 10 × 10⁶ / (√3 × 11 × 1000) = 524.9 A
- Zbase = 11² / 10 = 12.1 Ω
- A cable of 0.5 Ω is 0.5 / 12.1 = 0.04132 pu
Worked example: change of base
A 1 MVA transformer with an impedance of 5% (0.05 pu) is put on a 10 MVA base at the same voltage.
- Z(pu) = 0.05 × (10 / 1) × 1 = 0.5 pu, which is 50%
Base values at 10 MVA
| Voltage base | Base impedance | Base current |
|---|---|---|
| 0.415 kV | 0.01722 Ω | 13,910 A |
| 6.6 kV | 4.356 Ω | 874.8 A |
| 11 kV | 12.1 Ω | 524.9 A |
| 33 kV | 108.9 Ω | 175 A |
Frequently asked questions
Why use per unit?
It removes the transformer ratios. All impedances are on one common base, so the fault level is simply 1 divided by the total per-unit impedance.
How do I get the fault level in per unit?
MVAsc = Sbase / Z(pu), and the fault current is Ibase / Z(pu).
Is a percent impedance the same as per unit?
It is the per-unit value times 100, on the rating of the equipment. Convert it to the network base before adding impedances.
Does the voltage base have to be the nominal voltage?
It is usually the nominal system voltage at each level, chosen so the bases follow the transformer turns ratios.
Related
Short-Circuit FormulasTransformer FormulasTransformer Fault Current CalculatorPower systems calculatorsThree-Phase Power FormulasPower Factor and the Power TriangleAC Circuit Formulas: Reactance and ImpedanceVoltage Drop Formulas for CablesInduction Motor FormulasLighting Formulas: Lux, Lumens and CandelaBattery Formulas: Capacity and RuntimeSolar PV Formulas: Array Size and StringsCapacitor Formulas: Energy and RC Time ConstantEarthing Formulas: Earth Rod and Fault LoopCable Size Formulas: Current and Short-CircuitEnergy and Cost Formulas: kWh and PaybackGenerator Formulas: kVA, Current and FuelUPS Formulas: kVA, DC Current and BatteryHarmonics Formulas: THD and True Power FactorUnbalance Formulas: Voltage and Neutral CurrentResistor Formulas: Series, Parallel, DividerMotor Starting Formulas: DOL, Star-DeltaCircuit Breaker Formulas: Rating and TripCooling Formulas: Airflow and Heat LoadHeating Formulas: Energy and Heater PowerLED Formulas: Resistor Value and PowerConduit Fill Formulas: Percent Fill and SizeLightning Protection Formulas: Rolling SphereTransformer Efficiency and Regulation FormulasPump Formulas: Hydraulic, Shaft and Motor PowerFan Formulas: Air Power and Affinity LawsInductor Formulas: Energy and L/R Time ConstantRMS, Peak and Average Value FormulasResonance Formulas: Frequency, Q, BandwidthMotor Efficiency Formulas: Input and SavingsDemand, Load and Diversity Factor FormulasCable Tray Fill Formulas: Area and Cable CountConductor Resistance Formulas and TemperatureTransmission Line Formulas: Drop and LossesStar-Delta Conversion Formulas for Resistors4-20 mA Loop Formulas: Scaling and ResistorCable Loss Formulas: Power and Annual CostOp-Amp Formulas: Inverting and Non-Inverting555 Timer Formulas: Frequency and Duty CycleThévenin and Norton FormulasWheatstone Bridge Formulas: Balance and OutputConductor Force Formulas: Short-Circuit ForcesCT Burden Formulas: Lead Resistance and VAGear and Belt Drive Formulas: Speed and TorqueSkin Effect Formulas: Skin DepthRC Filter Formulas: Cutoff Frequency and GainRectifier Ripple Formulas: Capacitor FilterBuck and Boost Converter Formulas: Duty CycleHeat Sink Formulas: Junction TemperatureBattery Pack Formulas: Series and ParallelMotor Acceleration Time FormulasZener Regulator Formulas: Series ResistorBJT Bias Formulas: Voltage Divider BiasPt100 RTD Formulas: Resistance and TemperatureNTC Thermistor Formulas: Beta EquationFusing Current Formulas: Wire Fuse EstimateFlywheel Formulas: Stored EnergyLinear Regulator Formulas: Power and EfficiencyMOSFET Loss Formulas: Conduction and SwitchingRelay Coil Formulas: Current and ResistorTransformer Inrush Current FormulasBattery Charge Time FormulasWind Turbine Power Formulas: Betz LimitDetuning Reactor Formulas: Capacitor BanksTransformer Tap Changer FormulasCable Pulling Tension FormulasInsulation Resistance Formulas: PI and DARPower Transfer and Power Angle FormulasSolar Panel Temperature Derating FormulasVoltage Change and Flicker FormulasSoft Starter and VFD Sizing FormulasEarth Fault Loop and RCD FormulasBattery Internal Resistance FormulasBusbar Size and Short-Circuit FormulasADC Resolution and Sampling FormulasCapacitor Switching Inrush FormulasCable Charging Current FormulasSolar DC to AC Ratio FormulasMeasurement Uncertainty FormulasParallel Transformer Load Sharing FormulasPeukert Law and Battery Runtime FormulasCapacitor Discharge Time FormulasHydraulic Power and Cylinder Force FormulasThermocouple Voltage FormulasTransformer Vector Group FormulasPower Supply Hold-Up Time FormulasHoist and Lift Power FormulasCompressor Power FormulasRectifier Bridge Sizing FormulasPCB Trace Width and Current FormulasHeat Pump COP FormulasSound Level and dB Addition FormulasDuty Cycle and RMS Load FormulasSupercapacitor Energy FormulasStep and Touch Voltage FormulasContactor Sizing FormulasInverter-Fed Motor Cable FormulasPneumatic Cylinder Force and Air UseBusbar Temperature Rise FormulasMotor Temperature Rise FormulasBattery Storage Round-Trip EfficiencyThermal Time Constant FormulasSolar Charge Controller Sizing FormulasPoint Illuminance FormulasGenerator Droop and Load Sharing FormulasTwo-Wattmeter Method FormulasVFD Braking Resistor FormulasLoad Bank Test FormulasTransformer Cooling Class Rating FormulasEnergy Meter Multiplier FormulasOvercurrent Relay Curve FormulasSingle-Tuned Harmonic Filter FormulasCable Sheath Circulating Current FormulasDifferential Protection Slope FormulasEarth Electrode Spacing FormulasSolar Array Row Spacing FormulasDC Lighting Voltage Drop FormulasTransformer Loss of Life FormulasMotor Start Voltage Dip in the Cable FormulasNeutral Earthing Resistor FormulasCable I²t Withstand Check FormulasMotor Locked-Rotor Code Letter FormulasSynchronous Speed and Pole FormulasBattery C-Rate and Cycle Life FormulasEV Charging Time FormulasConveyor Belt Power FormulasParallel Plate Capacitor FormulasSolenoid and Coil Inductance FormulasPower Factor Correction Payback FormulasTransformer Through-Fault Current FormulasTransformer EMF Equation FormulasDelta-Star Resistor Conversion FormulasCurrent Shunt Resistor FormulasSolar Energy Yield and Degradation FormulasElectrolysis and Faraday’s Law FormulasAGL Series Circuit Load and CCR Sizing FormulasAGL Maximum Fittings on a CCR FormulasAGL Isolating Transformer FormulasAGL CCR Input Power and Energy FormulasAGL Brightness Step FormulasAGL Lamp Life FormulasComplex Number Formulas: Polar and RectangularPhasor Addition Formulas: Sum and DifferencePercentage Error Formulas for MeasurementsTrigonometry Formulas for Electrical EngineersMean and Standard Deviation FormulasQuadratic, Logarithm and Power FormulasPercentage, Ratio and Rounding FormulasProbability, Combinations and Binomial FormulasConversion Centre