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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!