The power factor and the upper harmonics

Upper harmonics that occur in electrical installations due to the use of semiconductor components increases the angle of phase difference between the voltage and the current of a phase. Following this, the power factor decreases, which necessitate, then, installations to compensate for it - to reduce or cancel the upper harmonics. In this work we present a case study on how upper harmonics influence the power factor. This work has been supported by Project CNFIS-FDI-2018-0282.


Introduction
In single-phased circuits with a sinusoidal voltage and current variations, the power factor is defined as the cosine of the angle of phase difference between these two entities [1]. For installations that operate under alternating, non-sinusoidal current, we define the following power types: -  (1) where U ν and I ν are the real values of the phase voltage and the phase current, corresponding to the ν level harmonics, while φ ν is the phase displacement between the current I ν and the phase voltage U ν . This power is translated to mechanical work or other active type of energy.  (2) which is the average oscillatory power value that is exchanged between the consumer and the source as the energy of the windings' electromagnetic field or as the energy of the capacitors' electric field.
-Distorting power, D: For circuits where upper harmonics occur, the power factor, λ, is defined as the ratio between the active and aparent output [2], [3]: Equations (1), (2) and (3) The distorting power, D, in Figure 1 leads to a large phase displacement between the terminal voltage and the current, which causes a small power factor (below the neutral one of 0.9) [4]. The power factor is characterising the electricity consumer. A small power factor value leads to needless load of the synchronous generators as well as of the electric transportation and distribution lines, with negative impacts on the energy bill which contains the used reactive energy. For this reason an adjustment of the user's power factor is necessary.
The use of capacitator batteries to adjust the power factor is significantly expensive, and does not reduce nor eliminate the upper harmonics. At the same time, using such batteries, resonance phenomena may happen [5]. Therefore, to reduce the upper harmonics and the reactive and distorting powers it is indicated that resonance circuits L-C are used, circuits which reduce the phase shift between the phase voltage and phase current, for each phase, and increases the power factor above the neutral value [6][7][8].
The installations to adjust power factors use reactive coils to protect the capacitator against harmonic distortions in the network introduced by equipment with integrated high power electronics.
To correct the wave form and to adjust the power factor, industrial appliances use automated electronic controllers which exactly establish the necessary capacity depending on the values of the active, reactive, and distorting power [9][10][11][12].

Case study
The installation we use to testify for the statements in the previous section use the UniTrain test bench and an experimenting module to raise the power factor by reducing the harmonics content.
The experimenting module is connected to the "Experimenter" adaptor in the UniTrain bench. This adaptor has two d.c. power sources (5 V and 15 V) as well as a three-phase voltage source [13].
The experimenting module ( Figure 2) to compensate the power factor consists of: two 330 Ω/8 W loading resistances, a power correction circuit, and a circuit with a simple rectifier. The power correction circuit is done at 15 V. Similar systems that have a correcting controller and which measures the voltage and current values are used also for real-time power factor adjustments [14].  Figure 4 also shows a power factor correction circuit (PFC) whose role is to reduce or cancel the upper voltage and current harmonics, reducing the distorting power and increasing the power factor.

Experimental findings
We experimented with the two circuits shown in Figures 3 and 4. Realizing the schema in Figure 3 allows us to determine the a.c. power voltage variation and the corresponding voltage harmonics variation, in the absence of load (no-load run), see  When we connect a load resistance, the power voltage wave shape is distorted (Figure 7) and low amplitude harmonics of level 3, 5, and 7 occur in both load voltage and load current (Figures 8 and 9, respectively). The c.c. voltage provided by the rectifier is almost constant around 22.9 V (Figure 7). The a.c. variation diverts quite a lot compared to the sinusoid variation, causing even order harmonics (levels 2, 4, and 10) and odd level harmonics (levels 3, 5, 7, 9, 1, 13, 17, and 19). We note that the amplitude of the level 3 harmonic is almost equal to the amplitude of the fundamental harmonics, contributing considerably to the distorted power. Note, also, that the level 5 harmonics has a amplitude of approximately 64% of the fundamental harmonics. Thus, the power factor will be quite reduced (Figure 8). For this power factor value, the reactive energy cost is comparable with the cost of the active energy. In this case we must correct for the Power factor. Using now the wiring in Figure 4, to dampen the harmonics and adjust the power factor, we observe that for a no-load circuit operation the wave form of the power voltage is sinusoidal  Figure 11) and has no harmonics ( Figure 12). The output voltage is lower by 1 V, being almost linear. The current absorbed from the source, during no-load operation, has a lower value, but higher than during the non-PFC operation. Figure 13 shows: a continuous component of approximately 0.025 A, even components harmonics (levels 2 and 4) with dampened amplitudes, and odd harmonics (levels 3,5,7,9,11,13). We note that the amplitude of the level 3 components is of approximately 0.03 A, that is almost 3 times higher than the amplitude of the fundamental.   Figure 13. A.c. harmonics for the no-load operation, with PFC Connecting a load we note a very slight alteration of the a.c. power voltage wave form (Figure 14), the occurrence of a level 3 voltage harmonics ( Figure 15) whose amplitude is approximately 0.0185 of the amplitude of the fundamental harmonic, while we can neglect the level 5 and 7 harmonics.
The absorbed current measured at the power source has a variation shape much more closer to a sinusoidal shape, being in the same phase with the power voltage. The level 3 harmonics have a approximately 13% amplitude of the fundamental harmonic amplitude (Figure 16). We note the presence of level 7 and 9 harmonics, with an amplitude of approximately 4.6% of the fundamental harmonic amplitude. Among the even order harmonics we note the presence of the level 4 harmonic whose amplitude is of about 5.6% of the fundamental harmonic amplitude. For this power factor value, the reactive energy cost is comparable to the cost of the active energy. In this case a correction of the power factor is needed.   Figure 17. Active, reactive, and distorted power as well as the apparent output for load operation, with PFC

Conclusions
Analysing the two circuits using the same load resistance has shown that for the wiring using an PFC the active power sent to the load has an almost double value compared to the non-PFC wiring. The reactive power in the PFC wiring is reduced to 40% of the reactive power, corresponding to the first case we analysed, while the distorted power is reduced to 36.8%.
Using the PFC, the power factor grows from 0.58 to 0.966, with the latter value considerably higher than the neutral factor.
The two analysed wirings show the necessity of using controllers to compensate the power factor. Thus, when the electric circuits are non-linear, to compensate the power factor the capacitator plays the most important role, intercalating lower or higher capacitor values must be commanded electronically.