Definition of Harmonics and Their Origin

Definition of Harmonics and Their Origin
Definition of Harmonics and Their Origin

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:

  1. A sinusoidal term at the fundamental frequency,
  2. Definition of Harmonics and Their Origin

  3. Sinusoidal terms (harmonics) whose frequencies are whole multiples of the fundamental frequency,
  4. A DC component, where applicable.
The nth order harmonic (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:

Equation for the harmonic expansion


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.

Example of a current containing harmonics and expansion of the overall current
Figure 1 – example of a current containing harmonics and expansion of the overall current into its harmonic orders 1 (fundamental), 3, 5, 7 and 9

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.

Spectrum of a signal comprising a 50 Hz fundamental and harmonic orders
Figure 2 – spectrum of a signal comprising a 50 Hz fundamental and harmonic orders 3 (150 Hz), 5 (250 Hz), 7 (350 Hz) and 9 (450 Hz)

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).

Single-line diagram showing the impedance of the supply circuit for h-order harmonic
Figure 3 – single-line diagram showing the impedance of the supply circuit for h-order harmonic

Note that the impedance of a conductor increases as a function of the frequency of the current flowing through it. For each h-order harmonic current, there is therefore an impedance Zh in the supply circuit.

The h-order harmonic current creates via impedance Zh a harmonic voltage Uh, where Uh = Zh x Ih, i.e. a simple application of Ohm’s law. The voltage at 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 4b show an installation confronted with harmonic disturbances. Figure 4a shows the flow of the fundamental 50 Hz current, whereas in 4b, the h-order harmonic current is presented.

Diagram of an installation supplying a non-linear load
Figure 4a – diagram of an installation supplying a non-linear load, showing only the fundamental 50 Hz current

Diagram of the same installation, showing only the phenomena related to the h-order harmonic
Figure 4b – diagram of the same installation, showing only the phenomena related to the h-order harmonic

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 Ih (shown in figure 4b) corresponding to each harmonic (order h).

Resource: Harmonic Detection and Filtering – Schneider Electric

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Edvard Csanyi

Edvard - Electrical engineer, programmer and founder of EEP. Highly specialized for design of LV/MV switchgears and LV high power busbar trunking (<6300A) in power substations, commercial buildings and industry fascilities. Professional in AutoCAD programming. Present on


  1. Khalid Niazi
    May 07, 2016

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

      Dec 28, 2016

      We can minimize the harmonics, but not totally.
      The objective is to eliminate the harmonics of ranks 3,5,7

    • David Ogaye Adhola
      Jan 09, 2017

      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

    Apr 08, 2016

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

    Apr 08, 2016

    1. clear difination of what is Harmonics
    2. how it happen

  4. sljivan
    Feb 10, 2016

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

  5. Ram
    Apr 19, 2015

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

  6. Senthilkumar
    Dec 23, 2014

    Use full information

  7. Senthilkumar
    Dec 23, 2014


  8. Krunal Patel
    Dec 22, 2014

    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

  9. Vijai
    Mar 25, 2014

    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 ….

  10. marvin
    Mar 14, 2014

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

  11. kizitoVyam
    Aug 13, 2013

    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!!!


  12. chourishiraj
    May 31, 2013

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

  13. stendley
    Sep 26, 2012

    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
      Sep 26, 2012

      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:

      Best regards,

    • EB1000
      May 29, 2013

      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)

  14. mohammed
    Sep 19, 2012

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

    • EB1000
      May 29, 2013

      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…

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