Motor Start Voltage Dip in the Cable Formulas

Motor starting voltage drop formulas: the starting current, the drop in the supply cable at the starting power factor and the longest cable for a dip limit.

The formulas

k is the ratio of starting to full-load current, In the full-load current, L the cable length in km, r and x the resistance and reactance of the cable in Ω/km, cos φs the power factor while starting (low, often 0.2 to 0.4) and V the line voltage. The formula is for a three-phase cable with the motor at its end. It gives the drop in the cable alone, and the supply and the transformer add to it. The motor needs enough voltage at its terminals to accelerate the load, and the maker or the standard gives the lowest value. The cable data, the ratio and the limit here are examples.

Worked example: a 95 A motor with a ratio of 6, on 150 m of 70 mm² copper at 415 V

1. Ist = 6 × 95 = 570 A
2. r = 1.72 × 10⁻⁸ / (70 × 10⁻⁶) × 1000 = 0.2457 Ω/km, and with x = 0.08 Ω/km and cos φs = 0.3, sin φs = 0.9539
3. ΔV = √3 × 570 × 0.15 × (0.2457 × 0.3 + 0.08 × 0.9539) = 22.22 V
4. ΔV% = 22.22 / 415 × 100 = 5.354%
5. For a limit of 10%, Lmax = 280.2 m

Starting drop by cable length for the same motor

Frequently asked questions

Why is the starting power factor so low?

A stalled or slow rotor is mostly an inductive load, so the starting current lags the voltage by a large angle. The reactive part drives the drop through the reactance of the supply.

Why does a dip matter?

The torque of an induction motor falls with the square of the voltage, so a deep dip can stop the motor from accelerating the load and can upset other equipment on the supply.

What helps most?

A larger cable, a shorter run, a stronger supply or a softer start such as a soft starter or a drive, which cut the starting current.

Is this the same as the running voltage drop?

No. The running current is much lower and its power factor is high, so the running drop is a different and smaller figure. Check both.

Related

Motor Starting FormulasVoltage Drop FormulasSoft Starter and VFD Sizing FormulasMotor starting calculatorsThree-Phase Power FormulasPower Factor and the Power TriangleAC Circuit Formulas: Reactance and ImpedanceTransformer Formulas: Current and Fault LevelInduction 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 FactorShort-Circuit Formulas: Fault Level by MVAUnbalance Formulas: Voltage and Neutral CurrentResistor Formulas: Series, Parallel, DividerCircuit 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 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 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 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