## Distortion of a sinusoidal signal

The * Fourier theorem* states that all non-sinusoidal periodic functions can be represented as the sum of terms (

*i.e. a series*) made up of:

- A sinusoidal term at the fundamental frequency,
- Sinusoidal terms (harmonics) whose frequencies are whole multiples of the fundamental frequency,
- A DC component, where applicable.

Definition of Harmonics and Their Origin

*commonly referred to as simply the nth harmonic*) in a signal is the sinusoidal component with a frequency that is n times the fundamental frequency.

**The equation for the harmonic expansion of a periodic function is presented below:**

*where:*

**Yo** – value of the DC component, generally zero and considered as such hereinafter,

**Yn** – rms value of the nth harmonic,

**ω** – angular frequency of the fundamental frequency,

**ϕn** – displacement of the harmonic component at t = 0.

**Example of signals (current and voltage waves) on the French electrical distribution system:**

- The value of the fundamental frequency (or first order harmonic) is 50 Hertz (Hz),
- The second (order) harmonic has a frequency of 100 Hz,
- The third harmonic has a frequency of 150 Hz,
- The fourth harmonic has a frequency of 200 Hz, etc.

A distorted signal is the sum of a number of superimposed harmonics. * Figure 1* shows an example of a current wave affected by harmonic distortion.

### Representation of harmonics: the frequency spectrum

The frequency spectrum is a practical graphical means of representing the harmonics contained in a * periodic signal*.

The graph indicates the amplitude of each harmonic order. This type of representation is also referred to as spectral analysis. The frequency spectrum indicates which harmonics are present and their relative importance.

* Figure 2* shows the frequency spectrum of the signal presented in

*.*

**figure 1**## Origin of harmonics

Devices causing harmonics are present in all industrial, commercial and residential installations. Harmonics are caused by * non-linear loads*.

### Definition of non-linear loads

A load is said to be non-linear when the current it draws does not have the same wave form as the supply voltage.

### Examples of non-linear loads

Devices comprising power electronics circuits are * typical non-linear loads*. Such loads are increasingly frequent and their percentage in overall electrical consumption is growing steadily.

**Examples include:**- Industrial equipment (welding machines, arc furnaces, induction furnaces, rectifiers),
- Variable-speed drives for asynchronous and DC motors,
- Office equipment (PCs, photocopy machines, fax machines, etc.),
- Household appliances (television sets, microwave ovens, fluorescent lighting, etc.),
- UPSs.

Saturation of equipment (*essentially transformers*) may also cause non-linear currents.

### Disturbances caused by non-linear loads, i.e. current and voltage harmonics

The supply of power to non-linear loads causes the * flow of harmonic currents* in the distribution system.

Voltage harmonics are caused by the flow of harmonic currents through the impedances of the supply circuits (e.g. transformer and distribution system a whole in figure 3).

*harmonic current, there is therefore an impedance*

**h-order****Z**in the supply circuit.

_{h}The h-order harmonic current creates via impedance **Z _{h}** a harmonic voltage

**U**, where

_{h}**U**, i.e. a simple application of Ohm’s law. The voltage at

_{h}= Z_{h}x I_{h}**B**is therefore distorted and all devices supplied downstream of point

**B**will receive a distorted voltage.

Distortion increases in step with the level of the impedances in the distribution system, for a given harmonic current.

### Flow of harmonics in distribution systems

To better understand harmonic currents, it may be useful to imagine that the non-linear loads reinject harmonic currents upstream into the distribution system, in the direction of the source.

** Figures 4a** and

*show an installation confronted with harmonic disturbances.*

**4b***shows the flow of the fundamental 50 Hz current, whereas in 4b, the h-order harmonic current is presented.*

**Figure 4a**Supply of this non-linear load causes the flow in the distribution system of current I50Hz (*shown in figure 4a*) to which is added each of the harmonic currents

**I**(

_{h}*shown in*) corresponding to each harmonic (

**figure 4b***order h*).

**Resource:** Harmonic Detection and Filtering – Schneider Electric

Abdulrahman

Very good article

Izhal

Very good write up. Very useful for me to understand the fundamentals of Harmonics and its effects on the power grid. This is good material. Thank you

Dan Bullard

I finally figured out how harmonics are created, it’s documented in my book Distortion. It’s groundbreaking, you won’t find the answer anywhere else. Find it on Amazon.com

Khalid Niazi

is it possible, to totally finish the harmonics at out put. i.e. to save motor from effects of harmonics?

SALAH SLIMANI

We can minimize the harmonics, but not totally.

The objective is to eliminate the harmonics of ranks 3,5,7

David Ogaye Adhola

yes , suppose the highly sensitive to harmonic devices are introduced at the overall stage and then again at the zonal points of the installation, the total desired elimination of harmonics with individual compensation at terminals could save the motor from the effects

ARUN KUMAR M

im facing harmonics problem in HVAC Panel how can reduce the harmonics level by avoding cable heating rather than active filters

ARUN KUMAR M

1. clear difination of what is Harmonics

2. how it happen

sljivan

why are we considering only odd harmonics (3rd, 5th, 7th,…) and not even ones?

Ram

Thank you.So when harmonics increases peak value decreases right?

Senthilkumar

Use full information

Senthilkumar

Good

Krunal Patel

Dear Sir/mam,

I found 65% current distortion in my system, i measured it with fluke 1735 power logger. now i want solution for mitigation of this much distortion. any suggestion please

Vijai

Hi Guys,

Gone through all the remarks/i still have a fundamental doubt.

Suppose a half wave rectifier is connected to an ac supply, 50 HZ.The rectifier on/off can cause sudden load flow to the system & load cut off( upon diode switching).

How can we relate this with the harmonic concept ?. Is this a mere representation of the drop in the supply voltage when a sudden load is thrown in and out ??.

Is it so , why we are discussing on the production of harmonic components like we say 3,5,7,…. etc and all.

Please advice ….

marvin

how harmonics distortion affects voltage and current level in RLC circuit?

kizitoVyam

What a fabulous website!!!

Since I discovered EEP, it has became the site I spent several hours browsing, enriched with new knowledge every single day. Thank you guys for this ocean of information about electrical engineering…

Long life to EEP!!!

Cheers!

chourishiraj

How Can we calculate harmonic Losses in electrical system/ equipment like Cable, transformer?

stendley

Even though it is a basic electrical theory for harmonic, nonetheless it does help a lot to in the way the author explaining the theoretical side of harmonics, how harmonics produce within a system, what are the sources as well as the basic formulation. This is very helpful to whom it may concern which includes myself. Thanks! Actually, I would like to ask the author, Mr. Edvard with regards to the case study of this harmonics.

Let say, we want to create a harmonic in a system, how are we going to model the non load devices and what are the formulas that we should use for the calculation of injected h-harmonic value (let say current, Ih) within a system. Advise on this is very much appreciated.

Edvard

Harmonics are usually produced by devices that rectifies AC Voltage into a DC Voltage, such as the are variable speed drives (VFDs), lighting, battery chargers, UPS or computers. To create a DC voltage from an AC sine wave, a bridge rectifier circuit is used to maintain a DC charge on a capacitor.

You can study the formulas written in thi study:

http://www.icrepq.com/icrepq07/366-trovao.pdf

Best regards,

Edvard

EB1000

Almost all industrial loads produce some sort of harmonics because of current waveform distortion, even electrical motors (because of core saturation and the shape of winding slots). But the main cause of harmonics are loads that are connected to the grid through a switch type (solid state) power converted such as AC/DC and AC/AC. Current harmonics causes harmonics at voltage drops, which causes voltage harmonics at points of common coupling (pcc) therefore injecting voltage and current harmonics to other loads. The current harmonics increases the total RMS current, therefore increasing the apparent power S and reducing the power factor (This can be described in terms of new distortive reactive power D, where S^2=P^2+Q^2+D^2). Anyway, one simulation software that has all the tools to simulate non linear loads and analyze harmonics is Plexsim’s PLECS. To manually analyze harmonic content, you first need to know the exact shape of the non-linear load’s current. Then you can just analyze it using Fourier series (PLECS allows you to plot the spectrum of any voltage or current waveform very easily)

mohammed

Let me ask you one question.almost all electrical equipment works at 50Hz.then how come this 150,200Hz etc…explain please

EB1000

50Hz is the source basis frequency of origin… Once the 50Hz sine wave waveform gets distorted, it will be mathematically represented by a sum of many sine waves with higher frequencies (multiples of the basic 50Hz frequency) plus the basic 50Hz itself. This means that a distorted not perfect 50hz sine wave, has higher harmonics contents… hence the 150, 200Hz…