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Harmonics

Harmonics

Harmonics are unwanted higher frequencies which superimposed on the fundamental waveform creating a distorted wave pattern

Harmonics and harmonic distortion is the difference between the ideal sinusoidal waveform the supply voltage or the load current waveform should look like, and what it really is as a result of a non-linear load.

In an AC circuit, a resistance behaves in exactly the same way as it does in a DC circuit. That is, the current flowing through the resistance is proportional to the voltage across it. This is because a resistor is a linear device and if the voltage applied to it is a sine wave, the current flowing through it is also a sine wave so the phase difference between the two sinusoids is zero.

Generally when dealing with alternating voltages and currents in electrical circuits it is assumed that they are pure and sinusoidal in shape with only one frequency value, called the “fundamental frequency” being present, but this is not always the case.

In an electrical or electronic device or circuit that has a voltage-current characteristic which is not linear, that is, the current flowing through it is not proportional to the applied voltage. The alternating waveforms associated with the device will be different to a greater or lesser extent to those of an ideal sinusoidal waveform. These types of waveforms are commonly referred to as non-sinusoidal or complex waveforms.

Complex waveforms are generated by common electrical devices such as iron-cored inductors, switching transformers, electronic ballasts in fluorescent lights and other such heavily inductive loads as well as the output voltage and current waveforms of AC alternators, generators and other such electrical machines. The result is that the current waveform may not be sinusoidal even though the voltage waveform is.

Also most electronic power supply switching circuits such as rectifiers, silicon controlled rectifier (SCR’s), power transistors, power converters and other such solid state switches which cut and chop the power supplies sinusoidal waveform to control motor power, or to convert the sinusoidal AC supply to DC. Theses switching circuits tend to draw current only at the peak values of the AC supply and since the switching current waveform is non-sinusoidal the resulting load current is said to contain Harmonics.

Non-sinusoidal complex waveforms are constructed by “adding” together a series of sine wave frequencies known as “Harmonics”. Harmonics is the generalised term used to describe the distortion of a sinusoidal waveform by waveforms of different frequencies.

Then whatever its shape, a complex waveform can be split up mathematically into its individual components called the fundamental frequency and a number of “harmonic frequencies”. But what do we mean by a “fundamental frequency”.

Fundamental Frequency

A Fundamental Waveform (or first harmonic) is the sinusoidal waveform that has the supply frequency. The fundamental is the lowest or base frequency, ƒ on which the complex waveform is built and as such the periodic time, Τ of the resulting complex waveform will be equal to the periodic time of the fundamental frequency.

Let’s consider the basic fundamental or 1st harmonic AC waveform as shown.

fundamental waveform

Where: Vmax is the peak value in volts and ƒ is the waveforms frequency in Hertz (Hz).

We can see that a sinusoidal waveform is an alternating voltage (or current), which varies as a sine function of angle, 2πƒ. The waveforms frequency, ƒ is determined by the number of cycles per second. In the United Kingdom this fundamental frequency is set at 50Hz while in the United States it is 60Hz.

Harmonics are voltages or currents that operate at a frequency that is an integer (whole-number) multiple of the fundamental frequency. So given a 50Hz fundamental waveform, this means a 2nd harmonic frequency would be 100Hz (2 x 50Hz), a 3rd harmonic would be 150Hz (3 x 50Hz), a 5th at 250Hz, a 7th at 350Hz and so on. Likewise, given a 60Hz fundamental waveform, the 2nd, 3rd, 4th and 5th harmonic frequencies would be at 120Hz, 180Hz, 240Hz and 300Hz respectively.

So in other words, we can say that “harmonics” are multiples of the fundamental frequency and can therefore be expressed as: , , , etc. as shown.

Complex Waveforms Due To Harmonics

harmonics and harmonic waveforms

Note that the red waveforms above, are the actual shapes of the waveforms as seen by a load due to the harmonic content being added to the fundamental frequency.

The fundamental waveform can also be called a 1st harmonics waveform. Therefore, a second harmonic has a frequency twice that of the fundamental, the third harmonic has a frequency three times the fundamental and a fourth harmonic has one four times the fundamental as shown in the left hand side column.

The right hand side column shows the complex wave shape generated as a result of the effect between the addition of the fundamental waveform and the harmonic waveforms at different harmonic frequencies. Note that the shape of the resulting complex waveform will depend not only on the number and amplitude of the harmonic frequencies present, but also on the phase relationship between the fundamental or base frequency and the individual harmonic frequencies.

We can see that a complex wave is made up of a fundamental waveform plus harmonics, each with its own peak value and phase angle. For example, if the fundamental frequency is given as; E = Vmax(2πƒt), the values of the harmonics will be given as:

For a second harmonic:

E2 = V2(max)(2*2πƒt) = V2(max)(4πƒt), = V2(max)(2ωt)

For a third harmonic:

E3 = V3(max)(3*2πƒt) = V3(max)(6πƒt), = V3(max)(3ωt)

For a fourth harmonic:

E4 = V4(max)(4*2πƒt) = V4(max)(8πƒt), = V4(max)(4ωt)

and so on.

Then the equation given for the value of a complex waveform will be:

harmonic frequency harmonics equation

Harmonics are generally classified by their name and frequency, for example, a 2nd harmonic of the fundamental frequency at 100 Hz, and also by their sequence. Harmonic sequence refers to the phasor rotation of the harmonic voltages and currents with respect to the fundamental waveform in a balanced, 3-phase 4-wire system.

A positive sequence harmonic ( 4th, 7th, 10th, …) would rotate in the same direction (forward) as the fundamental frequency. Where as a negative sequence harmonic ( 2nd, 5th, 8th, …) rotates in the opposite direction (reverse) of the fundamental frequency.

Generally, positive sequence harmonics are undesirable because they are responsible for overheating of conductors, power lines and transformers due to the addition of the waveforms.

Negative sequence harmonics on the other hand circulate between the phases creating additional problems with motors as the opposite phasor rotation weakens the rotating magnetic field require by motors, and especially induction motors, causing them to produce less mechanical torque.

Another set of special harmonics called “triplens” (multiple of three) have a zero rotational sequence. Triplens are multiples of the third harmonic ( 3rd, 6th, 9th, …), etc, hence their name, and are therefore displaced by zero degrees. Zero sequence harmonics circulate between the phase and neutral or ground.

Unlike the positive and negative sequence harmonic currents that cancel each other out, third order or triplen harmonics do not cancel out. Instead add up arithmetically in the common neutral wire which is subjected to currents from all three phases.

The result is that current amplitude in the neutral wire due to these triplen harmonics could be up to 3 times the amplitude of the phase current at the fundamental frequency causing it to become less efficient and overheat.

Then we can summarise the sequence effects as multiples of the fundamental frequency of 50Hz as:

Harmonic Sequencing

Name Fund. 2nd 3rd 4th 5th 6th 7th 8th 9th
Frequency, Hz 50 100 150 200 250 300 350 400 450
Sequence + 0 + 0 + 0

Note that the same harmonic sequence also applies to 60Hz fundamental waveforms.

Sequence Rotation Harmonic Effect
+ Forward Excessive Heating Effect
Reverse Motor Torque Problems
0 None Adds Voltages and/or Currents in Neutral Wire causing Heating

Harmonics Summary

Harmonics are higher frequency waveforms superimposed onto the fundamental frequency, that is the frequency of the circuit, and which are sufficient to distort its wave shape. The amount of distortion applied to the fundamental wave will depend entirely on the type, quantity and shape of the harmonics present.

Harmonics have only been around in sufficient quantities over the last few decades since the introduction of electronic drives for motors, fans and pumps, power supply switching circuits such as rectifiers, power converters and thyristor power controllers as well as most non-linear electronic phase controlled loads and high frequency (energy saving) fluorescent lights. This is due mainly to the fact that the controlled current drawn by the load does not faithfully follow the sinusoidal supply waveforms as in the case of rectifiers or power semiconductor switching circuits.

Harmonics in the electrical power distribution system combine with the fundamental frequency (50Hz or 60Hz) supply to create distortion of the voltage and/or current waveforms. This distortion creates a complex waveform made up from a number of harmonic frequencies which can have an adverse effect on electrical equipment and power lines.

The amount of waveform distortion present giving a complex waveform its distinctive shape is directly related to the frequencies and magnitudes of the most dominant harmonic components whose harmonic frequency is multiples (whole integers) of the fundamental frequency. The most dominant harmonic components are the low order harmonics from 2nd to the 19th with the triplens being the worst.

119 Comments

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  • Dan Bullard

    Hilarious! Free energy huh? “Generally, positive sequence harmonics are undesirable because they are responsible for overheating of conductors, power lines and transformers due to the addition of the waveforms.”
    So harmonics are free energy that didn’t come from the wall socket or battery, and thus they cause overheating? If this is true, why haven’t you patented the creation of harmonics and made a fortune by extracting free energy from the harmonics rather than writing articles that are hilariously wrong! You could get filthy rich no?

  • Gary Harstvedt

    I wanted an antenna which could accept multiple harmonic sequences by a master / slave cable ready tuner receptor and find the variable for changing reception along with the remote control unit.

  • Philip

    Just purchased a EcoFlow Delta 2 a weeks ago on Amazon. Has from 1800 watts to 2200 watt output. User manual describes consistent ac 119.7 volts 50/60HZ output from 6 receptacles. Notice when plugged into 1 – 3 prong receptacle and running my 1550 watt microwave. I noticed a sound that my microwave seemed to be bogging down and straining to operate to warm a cup of coffee. I became concerned so I meter tested the receptical it was plugged into on the solar generator it showed a 119.3V at 50Hz consistant output. I checked the other 5 remaining receptical on the solar generator. Yet my house recepticals any one I tested put out 120.3 volts at 60Hz consistently. Im concerned that the 50Hz 119.3 this solar generator puts out may be detrimental to my major appliances Fridg, microwave, large Toaster and even my Furnace Blower motor and other appliances and electronics and may cause some of them to fail I I keep this EcoFlow Delta3 solar generator. I just don’t know. Should I bee concerned or is this much to do about nothing. Your expert opinion would be greatly appreciated. Thanks. Philip of 73 yrs. Thanks

    • William P Albright

      Philip, if your power source is providing 50hz and your appliances are North American 60hz, they won’t run as efficiently as they were designed. Transformers are wound a particular way to make the best inductive conversion at a given frequency. Same with motor windings. It will cause excessive heat and a low power effect, such as with the microwave. The high voltage transformer is whats taking the brunt of the effect from lower frequency. In turn its output voltage which gets rectified to close to 3000 vdc in a doubler circuit using a high voltage capacitor and rectifier off the secondary of that transformer. So your supply to the magnetron will be less than what it needs. This will be the result of taking longer to cook. Also running low frequency AC can make a noise like its “laboring”, the normal inductive hum will be more prominent can even cause physical vibration of inductive components. A good example of problems with AC not being on frequency or sinusoidal in wave shape, is the output of inexpensive power inverters. Basically they are alternating a DC voltage 60 times a second by swapping polarity with the use of power transistors. That type of power doesn’t have the linear gradual ramp as its cycle runs. True AC climbs up to its peak in a linear fashion and comes back down in that same linear fashion. Switching DC goes peak to peak instantaneously and there’s no gradual ramp up or down. For transformer action to take effect most efficiently it needs that sine wave and at the frequency it was designed for.
      You said the user manual states 50/60hz. Is it selectable? It should be if they’re referring to an inverter output. That kind of nomenclature usually means a product is capable of operating at 50 or 60 hz and generally will also state 120/240 vac most computer power supplies for laptops and cellphone chargers will automatically detect the input voltage and set itself to run on either without user selection. If you are running an inverter for your power, the base frequency or the “fundamental” frequency should be 60hz. As a matter of fact the frequency of AC coming from the utility company to residential homes is ‘always’ dead on 60hz all the time. The voltage may swing up/down especially during peak use periods, but that 60hz will always be 60hz. Kinda gives an idea of its importance in AC power. Its probably directly dependent on the RPM of the megawatt generators/alternators the power companies use to make the electric current for our homes. Some old school electric clocks like you find on a stove, use that 60 hz for the time base in the design of those clocks that had a basic induction motor in them. Being 60 min in hour and 60 seconds in a minute, this simplified designing them. If your inverter cannot be set to put out 60hz you may want to see if you can return that one and get a 60hz inverter. You’ll have less possible problems, some equipment it will ruin it. Some plug in wall transformers like ones that take 120vac and have a 9vdc output for a radio, or a Wi-Fi router, etc. They will run hot, and that just takes life off the unit and possibly complete failure. Dremel tool motors and the speed control circuit doesn’t like non-standard AC at all. Low speed, gets hot. With those problems of inefficiency going on, this means your using up more of your battery runtime because you’re having to run a microwave for longer times, probably is pulling a bit more current than normal also, so thats dragging your battery source down. Best thing get 60hz AC !!! Hope that explains whats going on when the frequency isn’t correct. I wouldn’t put my appliances or electronics on it. Lamps incandescent no problem, electric heater without digital thermostat no problem, although I wouldn’t use electric heat with battery/inverter. There’s way better ways for heat. Gas! Fireplace! B4 using electric.

  • Pathmanaban Deenadayalan

    Very informative.

  • Sunil Bhatt

    I interest in power quality improvement .

  • Ashwini Pandey

    Very informative content

  • Vidhusharani R

    Nill

  • Abdulkadir A. Ohiare

    The reading is very educative and informative.

  • Marlene

    My neighbors AC vibration is making me crazy and they don’t care. The vibration stretches the entire length of my house on the side where their AC is located. I bought MLV soundproofing for outside use and would like to know how best to apply it to the fence (overlap, height, Lee, thickness, etc) .

    I need a lot of MLV material to cover my fence and it is exspensive. Can you please recommend what thickness, height and length, overlap, etc I should use and the most important thing to consider when soundproofing. Thank you for your help!

    • William P Albright

      Marlene, I know you said your neighbors don’t care, but your dilemma sounds like they need their AC unit serviced. It could be a bearing in the condenser fan motor or simply the metal grill is loose and vibrating. Both can cause an annoying sound. Neighbors probably don’t hear it from where they’re located inside or it’s on the side of a garage. What I am getting at is, try to solve the problem from the source. Do you think the neighbor would object to you having an AC tech look at it? Maybe go halves on the cost. I think it would be cheaper going this route. The insulating the problem doesn’t have any guarantee that it will block it satisfactorily. While fixing the problem would eliminate it. I’m not familiar with MLV material, it may be unpleasant to look at lining your fence? I don’t know but something to take into consideration.

  • Afonso Pereira

    you just save my college essay, thanks a lot

  • Mohamed Salim Abdalla

    Thank you very much.
    The way you have written is very clear and understandable.

  • Stanley

    Someone should kindly help out with

    A complex voltage wave is given by
    200Sin(((5)/(9))fPi^(2))+80Sin(((5)/(3))fPi^(2))+40Sin(((25)/(9))fPi^(2))

    Determine
    A. Which harmonics are present
    B. The RMS value of the fundamental
    C. The frequency of the fundamental
    D. The periodic time of the fundamental
    E. The frequency of the harmonics

  • Ryan

    Can you make a detail about harmonic sequence?
    Thanks

  • Abdirisak Abdukadir

    thanks.
    could you make a lesson of calculations?

  • Shahid Lateef

    I am really confused that for the real power to flow in harmonics frequences must be same but is the same true for reactive power also ( i.e for the reactive power to flow frequences must be same)

  • Dada oluwagbenga

    Very knowledgeable lectures, but please do guys have any ideas how to model a nonlinear load with Matlab using an equivalent resistor and set of current sources.

  • Heramb Mayadeo

    First of all, thank you for posting this fundamental article. I wish to know if we take the ratio of the magnitude of nth order harmonic to the magnitude of the fundamental harmonic, what that ratio is called? Is there any such quantity defined? For example V2max/V1max and V3max/V1max.

  • Ajoy Ghosh

    Nice

  • Manish Shukla

    Nice and informative article.

  • Harish Chandra Barai

    Good