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.

Where: V_{max} 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: 2ƒ, 3ƒ, 4ƒ, etc. as shown.

### Complex Waveforms Due To Harmonics

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 1^{st} 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 = V_{max}(2πƒt), the values of the harmonics will be given as:

For a second harmonic:

E_{2} = V_{2(max)}(2*2πƒt) = V_{2(max)}(4πƒt), = V_{2(max)}(2ωt)

For a third harmonic:

E_{3} = V_{3(max)}(3*2πƒt) = V_{3(max)}(6πƒt), = V_{3(max)}(3ωt)

For a fourth harmonic:

E_{4} = V_{4(max)}(4*2πƒt) = V_{4(max)}(8πƒt), = V_{4(max)}(4ωt)

and so on.

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

Harmonics are generally classified by their name and frequency, for example, a 2^{nd} 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 2^{nd} to the 19^{th} with the triplens being the worst.