Definition of harmonics

The presence of harmonics in electrical systems means that current and voltage are distorted and deviate from sinusoidal waveforms.

Harmonic currents are caused by non-linear loads connected to the distribution system. A load is said to be non-linear when the current it draws does not have the same waveform as the supply voltage. The flow of harmonic currents through system impedances in turn creates voltage harmonics, which distort the supply voltage.

On Figure M1 are presented typical current waveforms for single-phase (top) and three-phase non-linear loads (bottom).

Fig. M1 – Examples of distorted current waveforms

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.

The harmonic of order h (commonly referred to as simply the hth harmonic) in a signal is the sinusoidal component with a frequency that is h times the fundamental frequency.

The equation for the harmonic expansion of a periodic function y (t) is presented below:

[math]\displaystyle{ y(t) = Y_0 + \sum_{h=1}^{h=\infty} Y_h \sqrt 2 sin \left( h \omega t - \varphi_{h} \right) }[/math]

where:

• Y₀: value of the DC component, generally zero and considered as such hereinafter,
• Y_h: r.m.s. value of the harmonic of order h,
• ω: angular frequency of the fundamental frequency,
• φ_h: displacement of the harmonic component at t = 0.

Figure M2 shows an example of a current wave affected by harmonic distortion on a 50Hz electrical distribution system. The distorted signal is the sum of a number of superimposed harmonics:

• The value of the fundamental frequency (or first order harmonic) is 50 Hz,
• The 3^rd order harmonic has a frequency of 150 Hz,
• The 5^thorder harmonic has a frequency of 250 Hz,
• Etc…

Fig. M2 – Example of a current containing harmonics and expansion of the overall current into its harmonic orders 1 (fundamental), 3, 5, 7 and 9

Individual harmonic component (or harmonic component of order h)

The individual harmonic component is defined as the percentage of harmonics for order h with respect to the fundamental. Particularly:

[math]\displaystyle{ U_h(\%)= 100 \frac {U_h}{U_1} }[/math] for harmonic voltages

[math]\displaystyle{ i_h(\%)= 100 \frac {I_h}{I_1} }[/math] for harmonic currents

Total Harmonic Distortion (THD)

The Total Harmonic Distortion (THD) is an indicator of the distortion of a signal. It is widely used in Electrical Engineering and Harmonic management in particular.

For a signal y, the THD is defined as:

[math]\displaystyle{ THD = {\sqrt{\sum_{h=2}^{h=H}\left(\frac{Y_h}{Y_1}\right)^2} = \frac {\sqrt {Y_2^2 + Y_3^2 + \dots + Y_H^2} }{Y_1} } }[/math]

THD is the ratio of the r.m.s. value of all the harmonic components of the signal y, to the fundamental Y₁.

H is generally taken equal to 50, but can be limited in most cases to 25.

Note that THD can exceed 1 and is generally expressed as a percentage.

Current or voltage THD

For current harmonics the equation is:

[math]\displaystyle{ THD_i = \sqrt {\sum_{h=2}^{h=H}\left(\frac{I_h}{I_1}\right)^2} }[/math]

By introducing the total r.m.s value of the current: [math]\displaystyle{ I_{rms} = \sqrt {\sum_{h=1}^{h=H}I_h^2 } }[/math]

we obtain the following relation:

[math]\displaystyle{ THD_i = \sqrt {\left(\frac{I_{rms} }{I_1}\right)^2 - 1 } }[/math]

equivalent to:

[math]\displaystyle{ I_{rms} = I_1 \sqrt {1 + THD_i^2} }[/math]

Example: for THDi = 40%, we get:

[math]\displaystyle{ I_{rms} = I_1 \sqrt {1 + \left(0.4 \right)^2} = I_1 \sqrt {1 + 0.16} \approx I_1 \times 1.08 }[/math]

For voltage harmonics, the equation is:

[math]\displaystyle{ THD_u = \sqrt {\sum_{h=2}^{h=H}\left(\frac{U_h}{U_1}\right)^2} }[/math]