Cos phi in the three-phase grid: formulas, calculation examples, typical motor values, power factor on the converter, measuring and compensating reactive power.
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Cos phi and power factor: definition and formulas
The power factor λ is the ratio of the magnitude of active power P to apparent power S and lies between 0 and 1. With sinusoidal AC quantities, this ratio is called the active factor and corresponds to cos φ, i.e. the cosine of the phase shift angle φ between current and voltage[1]. In practice, cos phi, power factor and active power factor are therefore often used synonymously. The symbol φ is the Greek letter phi; as the ratio of two powers, cos φ has no unit.
For a symmetrically loaded three-phase grid with the line-to-line voltage U (for example 400 V) and the line current I, these formulas apply:
- Apparent power: S = √3 · U · I in VA
- Active power: P = √3 · U · I · cos φ in W
- Reactive power: Q = √3 · U · I · sin φ in var
- Power triangle: S = √(P² + Q²) and cos φ = P / S
Reactive power oscillates periodically between inductances and capacitances and does no external work. With an inductive load, the current lags the voltage and Q is positive; with a capacitive load, it leads and Q is negative[2]. A cos φ of 1 means that no displacement reactive power flows; with a cos φ close to 0, the power is almost pure reactive power.
| Quantity | Symbol and unit | Meaning |
|---|---|---|
| Active power | P in W, kW | is converted into work or heat |
| Reactive power | Q in var, kvar | oscillates to build up magnetic and electric fields |
| Apparent power | S in VA, kVA | geometric sum of P and Q, determines the current load |
| Power factor | λ = P / S, without unit | equal to cos φ for sinusoidal quantities |
With line-to-line voltage and line current, the formulas apply equally to star and delta connection. The connection does, however, determine the voltage across each winding phase: with unchanged windings, the power consumed in star connection drops to one third of the power in delta connection, which is what star-delta starting makes use of[3]. In operation, the motor must be connected as the nameplate specifies for the available mains voltage.
You should not confuse the active power factor with the efficiency η: cos φ describes the ratio of active to apparent power at the grid, while efficiency describes the ratio of delivered mechanical power to active power consumed.
Calculating cos phi: examples in the three-phase grid
If active power, voltage and current are known, you calculate cos phi using cos φ = P / (√3 · U · I). An example: a motor draws a current of 30 A and an active power of 18 kW at 400 V. The apparent power is √3 · 400 V · 30 A ≈ 20.8 kVA, so cos φ is 18 / 20.8 ≈ 0.87. This gives φ ≈ 30° and a reactive power of around 10.4 kvar. Cos phi is converted into sin phi using sin φ = √(1 − cos² φ).
Conversely, you determine the apparent power from active power and cos φ: a drive with 4 kW of active power consumed and a cos φ of 0.85 requires S = 4 / 0.85 ≈ 4.7 kVA and Q ≈ 2.5 kvar. The line current at 400 V is about 6.8 A. How to determine the motor current from the power is shown in our article Converting kW to amps, and sizing via apparent power in the article Calculating apparent power.
Apparent power determines the sizing of the equipment, because the size of a transformer determines the maximum apparent power that can be drawn[4]. A 1,000 kVA transformer can only deliver 700 kW of active power at cos φ = 0.7, but 950 kW at cos φ = 0.95.
Typical cos phi of three-phase motors
Asynchronous motors always draw inductive reactive power from the grid to build up the rotating field. Because of the mechanically necessary air gap, the magnetizing current still amounts to around 30 % of the rated current; at no load, the machine draws only magnetizing current[5]. Because this reactive component remains almost the same as the load decreases, the power factor drops: below a utilization of about 50 %, it decreases sharply, and even earlier for smaller motors. In some industrial plants, poorly utilized motors are the main reason for a low power factor[6].
How pronounced the effect is can be seen from a test bench measurement on a two-pole 15 kW asynchronous motor connected directly to a 3 × 400 V supply[7]:
| Load (share of rated load) | Measured cos φ |
|---|---|
| 100 % | 0.90 |
| 75 % | 0.87 |
| 63 % | 0.84 |
| 50 % | 0.80 |
| 38 % | 0.73 |
| 25 % | 0.62 |
On a plastic injection moulding machine with very low motor utilization, the authors of the same study even measured values of 0.17 to 0.43[7]. You can find the cos φ of your motor at rated load on the nameplate. In practice, this means: an oversized motor worsens the cos φ of the system, while appropriate power sizing improves it. Our article Calculating electric motor power provides help with the sizing.
Power factor and displacement factor on the frequency converter
As soon as current or voltage are not sinusoidal, the power factor λ can no longer be stated as cos φ. In that case, λ = g · |cos φ1| applies: the displacement factor cos φ1 describes only the phase relationship of the fundamental components, while the fundamental content g = I1 / I captures the harmonics[1]. Non-sinusoidal currents are caused by non-linear loads such as power electronic converters; the displacement reactive power Q1 is then joined by the distortion reactive power Qd, and Qtot = √(Q1² + Qd²) applies[2].
On the frequency converter, the difference is clearly visible. With a diode rectifier, cos φ is practically 1; in very large systems, it can be a few percent lower because of commutation. However, the current of a 6-pulse rectifier is heavily distorted, with a THDI of over 30 %. With harmonics, the power factor is lower than cos φ; usually only cos φ is stated[8]. A calculation example: with cos φ1 = 0.95 and g = 0.9, λ ≈ 0.86. For assessing grid load and compensation in converter drives, including servo drives with servo converters, λ therefore counts, not cos φ alone.
Measuring reactive power in the three-phase grid
Measuring reactive power starts with the question of which reactive power you want to capture. A wattmeter forms the product of the instantaneous values of voltage and current and averages it; if the voltage in the voltage path is shifted by 90°, it measures reactive power. In an undistorted, symmetrical three-phase grid with a neutral conductor, such voltages rotated by 90° are directly available; with an unbalanced load in a four-wire system, Q = U1N · I1 · sin φ1 + U2N · I2 · sin φ2 + U3N · I3 · sin φ3 applies. In a three-wire system without a neutral conductor, two measuring elements in Aron connection are sufficient; care must be taken to connect them with the correct sign. With an unbalanced load, additional losses occur in the neutral conductor, which are superimposed on the total power as oscillations; this effect is described by the unbalance reactive power[2].
- Calculation from measured values: with sinusoidal quantities, Q follows from S and P via Q = √(S² − P²).
- Classic reactive power meters that work like active power meters only capture the fundamental reactive power Q1 when the voltage is sinusoidal.
- Electronic devices with sufficiently fast digital sampling also measure the total reactive power Qtot.
- The quantities are defined in DIN 40110-1; for the supply grid, further specifications in DIN 40108 and DIN 40110-2 must be observed[2].
Before you plan a compensation system, measurements of voltage quality should be carried out and the system perturbations on the planned system estimated; continuous monitoring during operation protects planners, plant operators and grid operators from surprises[9]. Measure over typical load cycles so that partial-load phases and load peaks are included in the result.
Grid requirements, reactive energy and losses
Utility companies often require their customers to maintain a power factor of at least 0.9; if it falls below this value, they bill the reactive energy drawn separately. Conversely, since 1 January 2012, photovoltaic systems in Germany have had to be able to operate at a cos φ of 0.9 or 0.95, underexcited to overexcited, depending on system size, in order to stabilize the grid voltage[1]. This mainly affects large industrial consumers; private and small consumers are often exempt from these costs[2]. The technical basis for connecting customer installations to the low-voltage grid in Germany is VDE-AR-N 4100[10]; which value applies to your system is governed by the grid operator’s connection conditions and your supply contract.
In addition to the costs for reactive energy, a low cos φ loads your own network and reduces system efficiency. Reactive current must be transported over the cables in addition to the active current, and the Joule line losses fall with the square of the current ratio[4]. If cos φ rises from 0.7 to 0.95 at the same active power, the current drops to around 74 % and the line losses fall by about 46 %. For an idea of scale: 100 kvar of reactive power corresponds to an additional reactive current of around 144 A at 400 V.
Designing reactive power compensation
Inductive reactive power is compensated mainly with capacitors connected in parallel to the load; alternatively, synchronous machines or power factor correction can be considered[2]. The required capacitor power results from Qc = P · (tan φ1 − tan φ2), where φ1 describes the actual state and φ2 the target value. Example: a system with 100 kW is to be raised from cos φ = 0.70 to 0.95. With tan φ1 ≈ 1.02 and tan φ2 ≈ 0.33, around 69 kvar is required. The capacitance of a capacitor follows from C = Q / (ω · U²), with its reactive power Q and the voltage U applied to it[4]; in delta connection, each of the three capacitors carries one third of Qc at the line-to-line voltage.
The literature gives 0.85 to 0.95 inductive as the target value for cos φ. Full compensation should generally be avoided, because overcompensation threatens when the load fluctuates, and asynchronous motors are at risk of self-excitation. Automatic systems continuously measure the reactive power demand and switch capacitor stages in as required. To keep reactive power costs low, compensation systems are installed as close as possible to the sources, right down to individual compensation directly at the motor; reactive power that is not compensated close to the source requires a central compensation system[4].
| Design | Principle | Application |
|---|---|---|
| Non-detuned compensation | Capacitors as a purely capacitive branch | common in the past; susceptible to harmonics |
| Detuned compensation | Capacitors with series reactor | state of the art today |
| Thyristor-switched compensation | contactless switching of the stages | rapid load changes |
| Passive and active filters | tuned filter circuits or power electronics | high harmonic content |
The table summarizes the solutions described by the ZVEI. Non-detuned capacitors are particularly sensitive to harmonics and can amplify harmonic currents. With the correct choice of detuning, the risk of overcurrent due to resonance can be largely ruled out; detuned compensation is state of the art today. Active filters, as power electronic systems, go further: they can cancel harmonics of different orders, compensate reactive power dynamically in both the capacitive and inductive range, and compensate voltage unbalance[9]. For the detuning factor p, the resonance frequency is calculated as fr = 50 Hz / √p: at 7 %, it is about 189 Hz, at 14 % about 134 Hz, i.e. below the 5th harmonic at 250 Hz.
However, the greatest leverage often lies with the drive itself: an appropriately sized motor that operates close to its rated load needs less compensation. As a supplier for the complete drive train, we at ATEK Drive Solutions design the motor, gearbox and brake to fit precisely from a single source. You can find an overview of our IEC standard motors in our product range; we will be happy to advise you personally on your application via our contact page.
Frequently asked questions
What does cos phi = 1 mean?
Current and voltage are in phase, no displacement reactive power flows and the entire apparent power is active power. This applies, for example, to purely ohmic loads. A cos phi close to 0, on the other hand, means almost pure reactive power, as with a motor at no load.
What cos phi is typical for a three-phase motor?
The rated value is shown on the nameplate. A measured 15 kW motor reached 0.90 at rated load, 0.80 at half load and 0.62 at a quarter load. At no load, the motor draws almost only reactive power.
How do I convert cos phi into sin phi?
Using sin φ = √(1 − cos² φ). For cos φ = 0.8, sin φ = 0.6; for cos φ = 0.9, around 0.44. With sin φ, you calculate the reactive power Q = √3 · U · I · sin φ.
What is the difference between active power factor and efficiency?
The active power factor (cos φ) is the ratio of active to apparent power and describes the load on the grid. Efficiency is the ratio of delivered useful power to active power consumed and describes the losses in the device. A device can have a cos φ close to 1 and still have poor efficiency.
What do inductive and capacitive cos phi mean?
With an inductive load such as a motor, the current lags the voltage; with a capacitive load, it leads. An oversized capacitor in a compensation system can make a system capacitive; a capacitive power factor can lead to overvoltages on insulation.
What do you use to measure cos phi and reactive power?
Power meters or network analysers that record voltage and current per phase. With converter loads, the device should display the total reactive power and the power factor λ, not just the cos φ of the fundamental.
At what cos phi do costs for reactive energy arise?
Often at a power factor below 0.9. The exact limit and the billing are set out in your electricity supply contract and the grid operator’s connection conditions.
Why is compensation not carried out to cos phi = 1?
With fluctuating load, full compensation would at times lead to overcompensation, and asynchronous motors are at risk of self-excitation. A target value of about 0.95 inductive is therefore usual.
Sources
- Wikipedia (DE): Leistungsfaktor (power factor)
- Wikipedia (DE): Blindleistung (reactive power)
- Wikipedia (DE): Dreieckschaltung (delta connection)
- Wikipedia (DE): Blindleistungskompensation (reactive power compensation)
- University of Rostock, Institute of Electrical Power Engineering (DE): Asynchronous motors (lecture notes, part 3)
- IEA 4E Electric Motor Systems (DE): Energy efficiency guide for electric motors (BMVIT, 2014)
- SwissEnergy / Swiss Federal Office of Energy (DE): Dynamic control of the power factor cos phi and torque, final report 2020
- Topmotors (SwissEnergy, DE): Fact sheet no. 25 Frequency converters, 2021
- ZVEI (DE): Use of new technologies for reactive power compensation in electrical networks
- VDE FNN (DE): TAR low voltage (VDE-AR-N 4100)
