Single-Tuned Harmonic Filter Formulas
Single-tuned harmonic filter formulas: the reactor from the capacitor and the tuning order, the tuned frequency, the fundamental kVAR and the capacitor voltage.
The formulas
| To find | Formula |
|---|---|
| Capacitor reactance | Xc = V² / (Qc × 1000), with Qc in kVAR and V in volts |
| Reactor reactance | XL = Xc / h² |
| Reactor inductance | L = XL / (2π × f1) |
| Tuned frequency | ftune = h × f1 |
| Fundamental kVAR of the filter | Qf = V² / (Xc − XL) / 1000 = Qc × h² / (h² − 1) |
| Capacitor voltage | Vc = V × h² / (h² − 1) |
V is the line voltage, Qc the rating of the capacitor at that voltage, f1 the supply frequency and h the order at which the filter is tuned. A filter is tuned a little below the harmonic it must absorb, for example 4.7 for the 5th and 6.7 for the 7th, so that a drift of the parts does not move the resonance onto the harmonic. At the fundamental the capacitor and reactor together are capacitive, so the filter supplies more kVAR than the capacitor alone and the capacitor sees a higher voltage. The filter also needs a rating for the harmonic current that it carries, which comes from a harmonic study, and the maker confirms the parts. The values here are examples.
Worked example: a 100 kVAR, 400 V capacitor tuned to h = 4.7 on a 50 Hz supply
- Xc = 400² / 100,000 = 1.6 Ω
- XL = 1.6 / 4.7² = 0.07243 Ω
- L = 0.07243 / (2π × 50) = 0.2306 mH
- ftune = 4.7 × 50 = 235 Hz
- Qf = 400² / (1.6 − 0.07243) / 1000 = 104.7 kVAR
- Vc = 400 × 4.7² / (4.7² − 1) = 419 V
Tuning order, reactor share and kVAR factor
| Order h | Tuned frequency at 50 Hz | XL as a share of Xc | Qf / Qc |
|---|---|---|---|
| 3.8 | 190 Hz | 6.925% | 1.074 |
| 4.7 | 235 Hz | 4.527% | 1.047 |
| 6.7 | 335 Hz | 2.228% | 1.023 |
| 10.7 | 535 Hz | 0.8734% | 1.009 |
Frequently asked questions
Why tune below the harmonic and not on it?
Capacitor tolerance, ageing and temperature move the tuned frequency. A filter tuned exactly on the 5th could drift above it and cause a parallel resonance with the supply.
How does a filter differ from a detuning reactor?
A detuning reactor is tuned well below the lowest harmonic, so the bank never absorbs it. A filter is tuned close to a harmonic on purpose, so it takes that current out of the supply.
Why is the capacitor voltage higher than the line voltage?
The reactor and the capacitor form a voltage divider at the fundamental, with the reactor in opposition, so the capacitor voltage rises by the factor h² / (h² − 1).
What else decides the size?
The harmonic current to absorb, the short-circuit level of the supply and the kVAR that the plant needs. A harmonic study gives the final design.
Related
Detuning Reactor FormulasHarmonics FormulasCapacitor FormulasPower systems 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 CandelaBattery Formulas: Capacity and RuntimeSolar PV Formulas: Array Size and StringsEarthing 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 BatteryShort-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 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 LimitTransformer 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 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