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Frequently Asked Question
Topic:
What is the cos φ of an inverter
Author
Yaskawa
Product:
All Drives
Date
28-Feb-2011
Keywords:
Cos φ, Power factor, real power, apparent power,
reactive power, harmonics, THD
Pages
1 of 3
Distribution:
Internal Use only
Customer
1 General Basics
Very often the question comes up; what is the power factor or cos φ of an inverter? The inverter operates
an AC inductance motor that has a load dependent often quite poor power factor. So the wide spread
opinion is, that the inverter that supplies the motor has the same poor power factor.
The question is interesting, because reactive power is measured as well as real power and has to be paid
or compensated. This means additional cost anyway.
Physically the inverter is for the mains supply a 6 pulse rectifier (3ph supply) with a DC capacitor battery.
The no-load current or magnetizing current of the inductance motor is the reason for the poor power factor when it is operated directly on the mains supply. When the inductance motor is operated by an inverter, the magnetizing current is supplied by the DC link capacitor bank and has no bad influence on the
mains power factor.
If the input and output current of an inverter is measured and compared (running a motor at no-load), you
will see, that the input current is almost zero, and the output current is around 30…40% of the motor
nominal current.
Unfortunately an inverter does not take sinusoidal current from the mains. As it is a so called non-linear
load the inverter mains current is a mixture of fundamental current and harmonic currents.
Talking about cos φ, power factor and harmonics with frequency inverters requires to have a basis of
common under-standing about some terms and expressions.
1.1 Harmonics
In real live currents or voltage signals are very often non-sinusoidal. But even those periodic signals can
be described as a sum of sine and cosine terms with different frequencies, represented by a Fourier series. The Fourier series consists of one sine waveform with the base frequency (the fundamental) of the
periodic signal and of sine and/or cosine with an integer multiple frequency of the fundamental (the harmonics).
In Fig. 1 an example for a non-sinusoidal signal can be seen, in this case it could be the input current of a
standard inverter (more explanations regarding this waveform will follow later).
Fig. 1 Non-sinusoidal Waveform (Inverter Input Current)
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28.02.2011
The sine waveforms needed to get the waveform in Fig. 2 are the following:
Fundamental:
Harmonics:
5
th
7
th
Amplitude Î1: 10A
Frequency f1: 50Hz
RMS value: I1 = 7.1A
Amplitude Î5: 5.3A
Frequency f5: 250Hz
RMS value: I5 = 3.7A
Amplitude Î7: 3.0A
Frequency f7: 350Hz
RMS value: I7 = 2.1A
11
th
Amplitude Î11: 0.23A
Frequency f11: 550Hz
RMS value: I11 = 0.16A
13
th
Amplitude Î13: 0.49A
Frequency f13: 650Hz
RMS value: I13 = 0.35A
If all the above oscillations are added (10A*sin(2π*50Hz*t) + 5.3A*sin(2π*250Hz*t) +
3.0A*sin(2π*350Hz*t) + 0.23A*sin(2π*550Hz*t) + 0.49A*sin(2π*650Hz*t) ) the resulting waveform will
exactly look like the one in Fig. 1.
1.2 Total Harmonic Current Distortion (THDi)
The Total Harmonic Current Distortion is a measure for the amount of harmonic oscillations in the current
drawn from the power supply. It is defined as the ratio of the RMS value of all harmonics to the RMS
value of the fundamental, expressed in %:
THDi =
IRMS, harm
I1
The RMS value of the harmonics (i.e. all oscillations with a frequency above the fundamental) is the
square root of the sum of the squared current RMS values:
IRMS, harm = I2 + I3 + I4 + I5 + .... =
2
2
2
2
∞
∑I
k
2
k =2
In the above example this means:
IRMS, harm = 3.7A 2 + 2.1A 2 + 0.16A 2 + 0.35A 2 = 4.27A
The RMS current value of the signal in Fig. 1 is 4.27 A RMS
Further, for the THDi this means
THDi =
4.27A
× 100% = 60.1%
7.1A
The same measure, of course, exists for every signal that can be described with a Fourier series.
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1.3 Power Factor
Generally the power factor is defined as the ratio of working (active) power P (in W or kW) to apparent
power S (VA or kVA); or it is defined as the ratio of the current drawn that produces real work to the total
current drawn from the power supply. This is also called the True Power Factor and can be expressed
through following formula:
PFtrue =
P
S
Beside the True Power Factor a Displacement Power Factor is defined. It is a measure for the displacement of voltage and current and is defined as the cosine of the phase angle difference between voltage
and current. It can be expressed through following formula:
PFdisp = cos (AngleV – AngleI) = cosϕ
In a power system with sinusoidal voltages and currents only the True Power Factor and the Displacement Power Factor are equal. This would be the case for an induction motor driven directly on the power
supply lines.
Voltage
Current
Fig. 1
Angle
Sinusoidal
and Voltage
Phase
between
Current
Phase Angle
between Current
and Voltage
Fig. 1
example of the phase shift between current and voltage of the phase angle ϕ
shows an
Last but not least a third power factor is also defined, the Distortion Power Factor. For power supplies in
which the voltage is only slightly distorted (which is the case for most of the strong, European power supplies) the Distortion Power Factor is described as:
PFdist =
I1
IRMS
=
1
1
=
= 0.86
60.1% 2
THDI 2
1+ (
)
1+ (
)
100%
100%
2 Conclusion
All power factors can be used to express the quality of the power consumers.
In case of frequency inverters the True Power Factor as well as the Displacement Power Factor can
be regarded to be 1 as no reactive power is drawn from the power supply.
The Distortion Power Factor is the actual relevant one. Its value depends on the frequency inverter type
used and on the additional equipment connected (like AC reactors or DC chokes). In the example above
it can be calculated to 0.85, which is a usual value for a drive without any countermeasures.
What is the power factor of an inverter Rev. 1.doc