User manual
PQM-702, PQM-703 Operating Manual
66
where: U is RMS voltage, I is RMS current and
is the phase shift angle between voltage and
current.
The active power is calculated by the analyzer directly from the integral formula, using sampled
voltage and current waveforms:
where M is a number of samples in 10/12-period measuring window (2048) and U
i
and I
i
are suc-
cessive voltage and current samples.
5.3.2 Reactive power
The most known formula for reactive power is also correct only for one-phase circuits with si-
nusoidal voltage and current waveforms:
Interpretation of this power in such systems is as follows: it is the amplitude of AC component
of the instantaneous power on source terminals. Existence of a non-zero value of this power indi-
cates a bidirectional and oscillating energy flow between the source and the receiver.
Imagine a system with a single-phase sinusoidal voltage source, where the load is a RC circuit.
As under such conditions, these components behave linearly, the source current waveform will be
sinusoidal, but due to the properties of the capacitor it will be shifted in relation to the voltage source.
In such a circuit, reactive power Q is non-zero and may be interpreted as an amplitude of the energy
oscillation, which is alternately stored and returned by the capacitor. Active power of the capacitor
is zero.
However, it turns out the energy oscillation seems only an effect, and that it appears in particular
cases of circuits with sinusoidal current and voltage waveforms, and is not the cause of reactive
power. Research in this area has shown that reactive power occurs also in circuits without any
energy oscillation. This statement may surprise many engineers. In latest publications on power
theory, the only physical phenomenon mentioned which always accompanies appearance of reac-
tive power is phase shift between current and voltage.
The above mentioned formula for calculating the reactive power is valid only for single-phase
sinusoidal circuits. How then we should calculate the reactive power in non-sinusoidal systems?
For electrical engineers this question opens the 'Pandora’s box'. It turns out that the reactive power
definition in real systems (and not only those idealized) has been subject to controversy and now
(2013) we do not have one, generally accepted definition of reactive power in systems with non-
sinusoidal voltage and current waveforms, not to mention even unbalanced three-phase systems.
The IEEE (Institute of Electrical and Electronics Engineers) 1459-2000 standard (from 2000) does
not give a formula for total reactive power for non-sinusoidal three-phase systems – as three basic
types of power the standard mentions are active power, apparent power and – attention – non-
active power designated as N. Reactive power has been limited only to the fundamental component
and marked as Q
1
.
This standard is the last document of this type issued by recognized organization which was to
put the power definition issues in order. It was even more necessary as for many years specialists
in scientific circles reported that the power definitions used so far may give erroneous results. Con-
troversies concerned mainly the definition of reactive power and apparent power (and distortion
power – see below) in single- and three-phase circuits with non-sinusoidal voltages and currents.
In 1987, professor L.S. Czarnecki proved the widely used definition of reactive power defined
by Budeanu was wrong. This definition is still taught in some technical schools and it was presented
by prof. Budeanu in 1927. The formula is as follows: