What are harmonics?
The harmonics allow to represent any periodic waveform. In fact, according to Fourier’s theorem, any periodic function of a period T may be represented as a summation of:
- A sinusoid with the same period T;
- Some sinusoids with the same frequency as whole multiples of the fundamental;
- A possible continuous component, if the function has an average value not null in the period.
The harmonic with frequency corresponding to the period of the original waveform is called fundamental and the harmonic with frequency equal to “n” times that of the fundamental is called harmonic component of order “n”.
A perfectly sinusoidal waveform complying with Fourier’s theorem does not present harmonic components of order different from the fundamental one.
Therefore, it is understandable how there are no harmonics in an electrical system when the waveforms of current and voltage are sinusoidal. On the contrary, the presence of harmonics in an electrical system is an index of the distortion of the voltage or current waveform and this implies such a distribution of the electric power that malfunctioning of equipment and protective devices can be caused.
Figure 1 below shows a graphical representation of this concept.
How harmonics are generated?
Harmonics are generated by nonlinear loads. When we apply a sinusoidal voltage to a load of this type, we shall obtain a current with non-sinusoidal waveform. The diagram of Figure 2 illustrates an example of nonsinusoidal current waveform due to a nonlinear load:
This nonsinusoidal waveform can be deconstructed into harmonics. If the network impedances are very low, the voltage distortion resulting from a harmonic current is low too and rarely it is above the pollution level already present in the network. As a consequence, the voltage can remain practically sinusoidal also in the presence of current harmonics.
The main equipment generating harmonics are:
- Personal computer
- Fluorescent lamps
- Static converters
- Continuity groups
- Variable speed drives
In general, waveform distortion is due to the presence of bridge rectifiers (inside of these equipment), whose semiconductor devices carry the current only for a fraction of the whole period, thus originating discontinuous curves with the consequent introduction of numerous harmonics.
Also transformers can be cause of harmonic pollution. In fact, by applying a perfectly sinusoidal voltage to a transformer, it results into a sinusoidal magnetizing flux, but, due to the phenomenon of the magnetic saturation of iron, the magnetizing current shall not be sinusoidal.
Figure 3 shows a graphic representation of this phenomenon:
The resultant waveform of the magnetizing current contains numerous harmonics, the greatest of which is the third one. However, it should be noted that the magnetizing current is generally a little percentage of the rated current of the transformer and the distortion effect becomes more and more negligible the most loaded the transformer results to be.
The main problems caused by harmonic currents are //
The main effects of the harmonics voltages are //
In a three phase symmetric and balanced system with neutral, the waveforms between the phases are shifted by a 120° phase angle so that, when the phases are equally loaded, the current in the neutral is zero.
The presence of unbalanced loads (phase-to-phase, phase-to-neutral etc.) allows the flowing of an unbalanced current in the neutral.
Figure 4 shows an unbalanced system of currents (phase 3 with a load 30% higher than the other two phases), and the current resultant in the neutral is highlighted in red. Under these circumstances, the Standards allow the neutral conductor to be dimensioned with a cross section smaller than the phase conductors.
In fact, although the currents at fundamental frequency in the three phases cancel each other out, the components of the third harmonic, having a period equal to a third of the fundamental, that is equal to the phase shift between the phases (see Figure 5 below), are reciprocally in phase and consequently they sum in the neutral conductor adding themselves to the normal unbalance currents.
The same is true also for the harmonics multiple of three (even and odd, although actually the odd ones are more common).
The effects of harmonics inside the transformers involve mainly three aspects //
- Increase of iron losses (or no-load losses)
- Increase of copper losses
- Presence of harmonics circulating in the windings
The iron losses are due to the hysteresis phenomenon and to the losses caused by eddy currents. The losses due to hysteresis are proportional to the frequency, whereas the losses due to eddy currents depend on the square of the frequency.
The copper losses correspond to the power dissipated by Joule effect in the transformer windings. As the frequency rises (starting from 350 Hz) the current tends to thicken on the surface of the conductors (skin effect). Under these circumstances, the conductors offer a smaller cross section to the current flow, since the losses by Joule effect increase.
These two first aspects affect the overheating which sometimes causes a derating of the transformer.
The third aspect is relevant to the effects of the triple-N harmonics (homopolar harmonics) on the transformer windings. In case of delta windings, the harmonics flow through the windings and do not propagate upstream towards the network since they are all in phase.
When the frequency rises, the current tends to flow on the outer surface of a conductor. This phenomenon is known as skin effect and is more pronounced at high frequencies.
At 50 Hz power supply frequency, skin effect is negligible, but above 350 Hz, which corresponds to the 7th harmonic, the cross section for the current flow reduces, thus increasing the resistance and causing additional losses and heating.
In the presence of high-order harmonics, it is necessary to take skin effect into account, because it affects the life of cables. In order to overcome this problem, it is possible to use multiple conductor cables or busbar systems formed by more elementary isolated conductors.
The distorted load current drawn by the nonlinear load causes a distorted voltage drop in the cable impedance. The resultant distorted voltage waveform is applied to all other loads connected to the same circuit, causing harmonic currents to flow in them, even if they are linear loads.
Harmonic voltage distortion causes increased eddy current losses in the motors, in the same way as seen for transformers. The additional losses are due to the generation of harmonic fields in the stator, each of which is trying to rotate the motor at a different speed, both forwards (1st, 4th, 7th, …) as well as backwards (2nd, 5th, 8th, …).
High frequency currents induced in the rotor further increase losses.
Reference // Electrical installation handbook Protection, control and electrical devices by ABB