Battery Internal Resistance Formulas

Battery internal resistance formulas: the terminal voltage under load, the resistance from two readings, the heat loss and the maximum current and power.

The formulas

Voc is the open-circuit voltage in volts, I the current in amperes and Ri the internal resistance in ohms, which is the pack value (the cell value times the series count and divided by the parallel count). The resistance is not fixed: it is higher when the battery is cold, low in charge or old, so measure it under the same conditions that you design for. This is a simple model with a constant resistance and no change of the open-circuit voltage with charge.

Worked example: measure the resistance

A pack reads 50 V with no load and 47.6 V at 40 A.

1. Ri = (50 − 47.6) / 40 = 0.06 Ω

Worked example: a 100 A load

The same pack, Voc = 50 V and Ri = 0.06 Ω, with a minimum voltage of 42 V.

1. V = 50 − 100 × 0.06 = 44 V, so the load gets 4,400 W
2. Ploss = 100² × 0.06 = 600 W, so the efficiency is 44 / 50 = 88%
3. Imax = (50 − 42) / 0.06 = 133.3 A
4. Pmax = 50² / (4 × 0.06) = 10.42 kW

Terminal voltage and loss by current (Voc = 50 V, Ri = 0.06 Ω)

Frequently asked questions

Why does the voltage fall when I draw current?

The current flows through the internal resistance of the cells and the connections, and that costs a voltage of I × Ri before the load sees it.

Is the maximum power point usable?

Not in practice. At that point half of the power is lost as heat in the battery, so the efficiency is only 50%, and the voltage is half of the open-circuit value.

What raises the internal resistance?

Low temperature, a low state of charge, age and poor connections. The value on the data sheet is for a fresh cell at room temperature.

Does this change the runtime?

Yes. The loss in Ri is energy that the load does not get, and the lower voltage can reach the cut-off voltage early at a high current.

Related

Battery Pack FormulasBattery FormulasBattery Charge Time FormulasBattery bank calculatorsThree-Phase Power FormulasPower Factor and the Power TriangleAC Circuit Formulas: Reactance and ImpedanceVoltage Drop Formulas for CablesTransformer Formulas: Current and Fault LevelInduction Motor FormulasLighting Formulas: Lux, Lumens and CandelaSolar 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 FactorShort-Circuit Formulas: Fault Level by MVAUnbalance 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 PowerPer-Unit Formulas: Base Values and ImpedanceLED 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 TemperatureMotor 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 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 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