DO EVEN HARMONICS ALWAYS CANCEL OUT?

Even harmonics are those whose frequencies are even integer multiples of fundamental frequency of a waveform. For example, if fundamental frequency is 50 Hz, even harmonics would be 100 Hz (2nd harmonic), 200 Hz (4th harmonic), and so on. Even harmonics can arise in power systems due to non-linear loads, particularly under conditions with half-wave asymmetry in waveform. This can occur in systems with firing timing irregularities in rectifiers or during phase imbalances. The explanatory diagram below shows odd and even harmonic components in a system with multiples of fundamental frequency.

Most electrical machines almost always have symmetrical positive and negative half cycles. Hence, they do not produce even harmonics. Whenever the waveform is symmetrical about the time axis (or equivalently, the first half cycle is negative of the second half cycle (which is known as half-wave symmetry), all even harmonics are zero. Output of a half wave rectifier is an example of asymmetric waveform, where negative and positive half cycles are not symmetric around the time axis.

Generally, even harmonics are smaller in amplitude compared to odd harmonics. Most non-linear loads predominantly generate odd harmonics (e.g., 3rd, 5th, 7th) due to their nature of drawing uneven current during both the positive and negative halves of the waveform cycle. Though less common, even harmonics can have their own detrimental effects, like increased heating in transformers and motors. They may also contribute to resonance issues when combined with other harmonic frequencies.

Even harmonics typically cancel out in symmetrical power systems due to the nature of their waveforms. Understanding even harmonics is crucial for power quality analysis and mitigation strategies. They can affect system performance and reliability when considering harmonic filters or other corrective measures.  

Common Causes of Even Harmonics

• Firing Time Irregularities: In rectifier systems, irregularities in the timing of firing pulses can lead to generation of even harmonics. This occurs when there is half-wave asymmetry, resulting in poor cancellation of odd harmonics, and production of even harmonic currents.
• Asymmetrical Loads: Uneven current between the positive and negative halves of a cycle can generate even harmonics. This is often seen in systems with unbalanced loads or when equipment operates outside its designed parameters.
• Phase Unbalance: While phase unbalance primarily generates odd harmonics, it can also contribute to the creation of even harmonics under certain conditions, e.g. a single-phase rectifier.
• Group Unbalance: When certain phases are displaced relative to others, this can lead to generation of even harmonics as well. For example, if pulses from one group of phases are consistently out of sync with another group, it can result in multiples of 3±1 harmonics.
• Non-linear Loads: Although most non-linear loads predominantly produce odd harmonics, certain configurations and operating conditions can lead to the generation of even harmonics. Examples include specific types of electronic equipment and power converters that do not operate symmetrically.
• Transformer Phase Angle Tolerances: Imperfections in transformer winding phase angles can also contribute to creation of even harmonics within a power system. This is particularly relevant in systems where transformers are used to balance loads.

Understanding these causes is crucial for effectively managing and mitigating the impact of even harmonics in power systems, ensuring improved power quality and system reliability.

WHY DO EVEN HARMONICS CANCEL OUT IN SYMMETRICAL SYSTEMS?

Third Law of Bullard Laws of Harmonics says, “Even harmonics do not appear in symmetrically distorted spectra because even harmonics on the positive side cancel the even harmonics on the negative side” Even harmonics cancel out in symmetrical systems due to the inherent properties of waveform symmetry. Here are the key reasons:

Symmetry in Waveforms: In symmetrical waveforms, such as those found in balanced three-phase systems, positive and negative halves of the waveform are mirror images of each other. This symmetry means that any even harmonic component (which is mathematically defined as having a frequency that is an even integer multiple of fundamental frequency) will have equal magnitude but opposite phase in the positive and negative halves. As a result, they effectively cancel each other out over a complete cycle.

In a balanced three-phase system, positive and negative halves of the waveform are symmetrical. This symmetry leads to the cancellation of even harmonics, as their contributions from one half of the cycle negate those from the other half.

Mathematical Explanation: Cancellation occurs because, mathematically, when a waveform is symmetric about the horizontal axis (like a sine wave), all even harmonics will sum to zero over a complete cycle. This is particularly true for waveforms exhibiting half-wave symmetry.

Odd vs. Even Harmonics: Odd harmonics (such as the 3rd, 5th, and 7th) do not exhibit this cancellation effect because they do not possess this symmetry. They tend to add constructively in a system, which is why they are more commonly observed in non-linear loads. Most non-linear loads, such as rectifiers, predominantly generate odd harmonics (e.g., 3rd, 5th). Even harmonics are generally much smaller and often negligible because they do not have a significant presence in these systems.

Cancellation of even harmonics in symmetrical systems is thus primarily due to the symmetry of waveform, leading to equal and opposite contributions from these harmonics across positive and negative cycles. In ideal symmetrical systems, even harmonics tend to cancel out due to their inherent properties, while odd harmonics remain more prominent and problematic.

In conclusion, while even harmonics are less common and often cancel out, they can still appear in cases of waveform distortion caused by timing irregularities or imbalances in the system. Such scenarios can lead to unexpected even harmonic components that may need to be addressed.