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Electrical Harmonics

A detailed understanding of harmonics in electrical power systems

What are harmonics?

Harmonics are a distortion of the electrical voltage (or current) waveform, generally due to nonlinear loads such as: switched power converters, TVs, computers, and battery chargers.

A harmonic is defined as a sinusoidal component of a periodic wave or quantity having a frequency that is an integral multiple of the fundamental frequency (IEEE 141-1993).

To clarify this definition, consider the following example of a distorted current waveform:

distorted current illustrative example

This periodic signal corresponds to a sum of sine waves with different frequencies and amplitudes:

Sine wave 1:

The first sine function frequency is equal to 50Hz, which corresponds to the fundamental frequency. It is called 1st harmonic!

Distorted Current first harmonic

The above sine wave function is defined by the following expression:

Equation Harmonic 1

Sine wave 2:

The second sine function frequency is equal to 250Hz, which corresponds to a 5 multiple of the fundamental frequency. It is called the 5th harmonic!

Distorted Current fifth harmonic

The 5th harmonic is defined by the following expression of a sine wave function:

Equation Harmonic 5

Sine wave 3:

The third sine function frequency is equal to 350Hz, which corresponds to a 7 multiple of the fundamental frequency. It is called the 7th harmonic!

Distorted Current 7th harmonic

The 7th harmonic with 7*50Hz frequency and 6 Amps amplitude is defined by:

Equation Harmonic 7

Sine wave 4:

The fourth sine function frequency is equal to 550Hz, which corresponds to an 11 multiple of the fundamental frequency. It is called the 11th harmonic!

Distorted Current 11th harmonic

In this case, we have a sine wave with 11*50=550 Hz frequency and smaller amplitude (5 Amps). This harmonic is defined by the following expression:

Equation Harmonic 11

Taking the sum of the above 4 harmonics, we get an approximated version of the distorted current, check the following image:

Distorted current compared to the sum of its four harmonics

In summary and in simple terms:

A harmonic of a voltage waveform (or current) is a sinusoidal wave whose frequency is integer multiple of the fundamental frequency.

From above, you may ask how I can get the harmonics of a distorted signal?

Short answer: by applying Fast Fourier Transform (FFT) to the distorted signal.

Let’s talk about this later and now we move to the origin of electrical harmonics!

What causes electrical harmonics

As we mentioned above, Harmonics are mainly caused by non-linear loads. But what do we mean by non-linear loads? what is the difference between linear and non-linear ones?

1. Linear loads

For this class of loads, If we apply a sinusoidal voltage, it draws a sinusoidal current proportional to the applied voltage. Examples of linear loads include:

  • Resistive loads such as heaters,
  • Capacitive loads,
  • Inductive loads such as inductive motors and transformers,

or any combination thereof.

2. Non linear loads

Contrary to linear loads, when we apply a sinusoidal voltage to a non-linear load, the current it draws does not have the same waveform as the supply voltage. It draws currents in abrupt short pulses which distort the current waveforms. Examples of non-linear loads include:

  • Switched power converters,
  • Variable frequency drives (VFDs),
  • Arcing devices
  • High-voltage dc transmission stations
  • Computers, TVs, etc

Harmonics effects

In this part, we will highlight the effect of harmonics on electrical equipment and power network in general.

Electrical systemMain Effects
1. Power system\
– Losses due to heating
– Voltage distortions
/
2. Motors and generators\
– Increased heating due to iron and copper losses
– Higher audible noise
– Mechanical oscillations
– Temperature rise on the stator and in the rotor
/
3. Power cables\
– Voltage stress and corona
– Insulation failure
– Additional heating
/
4. Transformers\
– Increase of copper, Iron, and stray flux losses
– Higher audible noise
– Parasitic heating
/
5. Telephone interference\
– Noises
/

Harmonics analysis

Harmonic percentage can be calculated with respect to the 1st harmonic (corresponding to the fundamental frequency) or with respect to all harmonics.

1. Single harmonic percentage

Let’s consider the above example:

  • The distorted current has the amplitude 65.5A, 1st harmonic has 60A amplitude, 5th harmonic has 12A amplitude, 7th harmonic has 6A amplitude, and 11th harmonic has 5A amplitude.
  • With respect to all harmonics, we divide each harmonic’s amplitude by the amplitude of the distorted current, 65.5A in this case.
  • With respect to the 1st harmonic, we divide each harmonic’s amplitude by 60A

The following table reports different harmonics percentages with respect to total harmonics (column 2) as well as with respect to 1st harmonic (column 1)

HarmonicsTotal harmonics1st harmonics
1st harmonic91.6%100%
5th harmonic18.32%20%
7th harmonic09.16%10%
11th harmonic07.63%8.33%



Harmonics spectrum
  • It should be noted that we can also use RMS in the above calculations instead of amplitude. This also applies to distorted voltage signals.

2. Total Harmonic Distortion (THD)

Total harmonic distortion can be calculated with respect the 1st harmonic by taking the square root of the sum of all squared single harmonic percentages.

From the above example, assume the distorted current has only 1+3 harmonics, we have:

This image has an empty alt attribute; its file name is Total-distortion-harmonic-1024x377.webp

Total harmonic distortion (THD) can be used to troubleshoot symptoms of harmonics.

Further reading: Harmonics Analysis in Matlab: Full guide

Power factor

The power factor, cos(Φ), is defined as the ratio of real power to apparent power. It is an indicator about how effectively power is used by the load.

In the case of harmonics presence, this parameter is no longer equal to cos(Φ).

1. What is relation between power factor and harmonics?

Assuming that the most distorted signal is current, and a nearly sinusoidal voltage at the fundamental frequency. Hence, the power factor is given by the following approximated expression:

Power factor harmonics

where: cos (Φ1) corresponds to the power factor of the 1st harmonic current and voltage.

2. Does harmonics affect power factor?

The answer to this question is YES, from the previous equation, High THD yields to a low power factor which means that harmonics affects negatively the power factor of electrical installations.

On the other side, THD =0 means the power factor is the same as cos (Φ1), the power factor of the 1st harmonic current and voltage.

How can we reduce harmonics in electrical system?

Harmonics in electrical power systems can be reduced using different methods, such as:

These solutions will be considered with more details in the coming tutorials!

Conclusion

This tutorial is an introduction to harmonics and it is followed by a set of tutorials that address each part with more details!

What You Need to Know About Third Harmonic

Definition, causes, and effects!

Harmonics are an important concept in power systems, which significantly impacts power quality. In this post, we will address only third harmonics and for a general case, check this article.

Let’s get into the details!

What is 3rd harmonic and what is its frequency?

In AC three-phase systems, each phase has a periodic current/voltage wave with a specific fundamental frequency with 120 degrees phase angle difference between phases. There are two common fundamental frequencies in power systems, including 50Hz and 60Hz. For simplification, 50Hz frequency is considered as the fundamental frequency, but all subjects can be extended to other frequencies.

So what is 3rd harmonic frequency? As the harmonics are the integral multiple of the fundamental frequency, the third harmonic frequency is three times greater than the fundamental frequency which means that the third harmonic frequency would be 150Hz (3×50 Hz), check the above illustration. It is worth noting that the frequency of the third harmonic in the 60 Hz system is 180 Hz.

What causes 3rd harmonics?

In symmetrical three-phase systems, there are no harmonics, and all waves are purely sinusoidal with the fundamental frequency. However, some types of electric loads create disturbances in electric systems, and waves with non-fundamental frequencies are added to the original one. Therefore, the system is no longer pure sinusoidal.

Electric Vehicle (EV) Charger: Photo by dcbel on Unsplash

In fact, non-linear loads are the main cause of distortion and produce harmful third harmonics in energy systems. A wide range of non-linear loads is available in each system due to advances in power electronics. Nowadays, there are many power electronics devices such as:

  • power converters,
  • switching power supplies,
  • inverters, and transistors in the systems,

known as non-linear loads.

Non linear load example

Apart from power electronic devices, electric motors and non-ideal transformers can be the main sources of the third harmonics.

In distribution systems, having a balanced power system is challenging because there are many consumers who use different types of single-phase electric loads. Therefore, the distribution may be unbalanced, and, in this situation, the third harmonics are shown up in voltages and currents.

Asymmetrical short-circuit faults like phase to ground faults act as a huge non-linear load and make the system unbalanced. Thus, third harmonics would also be available in fault conditions.

Why 3rd harmonic is dangerous?

In power systems, the presence of odd harmonics is more probable than even ones. Moreover, higher-order harmonic magnitudes are relatively low in general, and in most cases, they are filtered in the system. It means that low-order odd harmonics can be more harmful because their magnitudes are considerable.

Generally, as the harmonic magnitudes decrease in higher-order harmonics, the third harmonic in the system would be the dominant harmonic component, which distorts the voltage or current waveform.

In the third harmonic, all phases have the same phase angle, and the instantaneous sum of the currents in the three phases taken at any moment will not be zero (phases cannot cancel out each other).

The current harmonics can flow in the entire system and devices, which may reduce their efficiencies and performance significantly.

Moreover, the third harmonic currents in the three-phase system are not eliminated in the neutral conductor, and in this situation, the neutral conductor will draw a cumulative third harmonic current which is too dangerous because it may lead to maloperation of electric apparatus and protection systems.

The problem gets more concerning once the neutral conductor/cable size is lower than phase one because when it becomes overloaded by third-order harmonic, substantial losses (heat) are produced in the neutral. As the overheating results from overloading losses, the neutral cable insulation can deteriorate, or its conductor may be deformed due to the high temperature.

Moreover, the neutral point can be loosened if its connection is not well. All these consequences are probable when a neutral path is available, and designers must be aware of the mentioned issues when designing neutral conductors.

However, the third harmonic can be a problem when the system does not have a ground path for the current third harmonic. If the current third harmonic cannot find a way to the ground, the current will be sinusoidal, but the peak voltage will go up significantly, which is undesirable and may lead to damage.

Effect of 3rd harmonics (power system, transformer, generator, induction motor)

To be more specific, the third harmonic has detrimental effects on generators, transformers, and induction motors which are available in each electric system. These effects can be somehow similar. For instance;

  • The third harmonic in a grounded generator with a resistor may create continuous heating (without fault) of the earthing resistor and malfunction the protection system.
  • About induction motors, the winding connection configuration is important because, in delta connection, the current third harmonic circulates in the delta and does not reflect at the source side. This harmonic increases the current RMS value, and more copper and core losses will be generated. Owing to an increase in losses, the motor output de-rating should be considered to prevent any damages. It means that the output will be decreased to follow the allowable temperature rise, and efficiency will be reduced due to losses. 
  • These impacts are also in transformers as third harmonic can increase core and copper losses, and core saturation may decrease the transformer efficiency and reduce the transformer operating power. In star-star connection, the third harmonic current is either towards the neutral point or away from it, and this harmonic will flow through the line. However, in the star-delta connection, the third harmonic will be trapped in the delta, and there is no path for flowing the harmonic.

In most cases, the third harmonic is dominant among Triplen harmonics (third, ninth, etc.), and other harmonics can be neglected. Thus, as mentioned before, the current third harmonic is accumulated in the neutral conductor, and it can be measured easily. The measured neutral current would be three times greater than the phase third harmonic currents and I3N=3×I3p.

Conclusion

To sum up this article, it is worth noting that the solution for the third harmonic in the power system is using a delta connection in the transformer. In this way, the current third harmonic will be rotated in the delta connection, and it never shows up in line currents.

Apart from that, to reduce the risk of the third harmonic in grounded star connection, the neutral conductor size should be three times one phase conductor’s size to decrease the risk of overloading and overheating.