User manual
PQM-702, PQM-703 Operating Manual
76
When the sign is positive (+), then the character is inductive, and when it is negative (-), it is capac-
itive.
Passive source current may be reduced to zero when the following condition is met for each
harmonic
2
:
where:
B
h
– receiver susceptance for the h-th harmonic,
B
kh
– parallel compensator susceptance for the h-th harmonic.
As the compensator complexity grows proportionally to the number of harmonics subjected to
compensation, usually only the fundamental component is compensated and maximum a few higher
harmonics with the largest values. However, the compensation of the fundamental component may
considerably improve the power factor and may be sufficient.
5.4.3 Harmonics in three-phase systems
In three-phase systems, harmonics of given orders have a particular feature which is shown in
the table below:
Order
1
2
3
4
5
6
7
8
9
Frequency [Hz]
50
100
150
200
250
300
350
400
450
Sequence
(+ positive,
– negative,
0 zero)
+
–
0
+
–
0
+
–
0
"Sequence" line refers to the symmetrical components method which allows to decompose any
of the three vectors into three sets of vectors: positive, negative and zero sequence (more in section
related to unbalance).
For example: Let’s assume that a three-phase motor is supplied from a balanced, 4-wire mains
(RMS phase-to-neutral voltage values are equal, and angles between the individual fundamental
components are 120 each).
"+" sign in the line specifying the sequence for the 1st harmonics means the normal direction of
the motor shaft rotation. The voltage harmonics, for which the sign is also "+" cause the torque
corresponding with the direction of the fundamental component. The harmonics of the 2nd, 5th, 8th
and 11th order are the opposite sequence harmonics, meaning that they generate the torque which
counteracts normal motor direction of rotation, which can cause heating, unnecessary energy
losses, and reduced efficiency. The last group are the zero sequence components, such as the 3rd,
6th and 9th, which do not generate torque but flowing through the motor winding cause additional
heating.
Basing on the data from the table, it is easy to note that the series +, –, 0 is repeated for all
successive harmonic orders. The formula which links the sequence with order is very simple, and
for 'k' being any integer:
Sequence
Harmonic order
positive "+"
3k +1
negative "–"
3k – 1
zero "0"
3k
The even order harmonics do not appear when a given waveform is symmetrical in relation to
its average value, and this is the case in majority of power supply systems. In a typical situation, the
2
L.S. Czarnecki: "„Power values in electrical circuits with non-sinusoidal voltage and current wave-
forms", Publishing House of Warsaw Technical University, Warsaw 2005, p.109