# Why is total harmonic distortion defined in the following manner?

So the total harmonic distortion is defined as

$$THD = \sqrt{\frac{V_{2}^2 + V_{3}^2 + V_{4}^2...+V_{n}^2}{V_{1}^2}}$$

which means Total harmonic distortion is defined as the square root of the ratio of the average power delivered by harmonic components divide power delivered by non-harmonic component. So why include the square root when it would make more intuitive sense not to include it in the first place?

• when it would make more intuitive sense not to include it That might be so in your intuition but in mine it does not. The sum of squares makes more sense to me as it is the way that uncorrelated signals need to be summed up. – Bimpelrekkie Jan 6 at 14:21
• The root sum of squares term is added because when the terms are negative and others are positive it would not make sense to have these terms cancel out when they both contribute to the signals distortion. – m-walker95 Jan 6 at 14:23
• Apologies I have edited the formula – cojoye Jan 6 at 14:29
• @Bimpelrekkie Without the squareroot the definition of the equation becomes total power delivered by harmonic components/total power delivered by fundamental frequency. Which begs the question, why take the squareroot when this is a valid measure of harmonic distortion? – cojoye Jan 6 at 14:36
• it doesn't happen when each is squared – Tony Stewart Sunnyskyguy EE75 Jan 6 at 15:04

Because the voltages are in rms. Remember that the rms voltage is the equivalent DC voltage source that delivers the same amount of power to a resistive load. The rms is given by

$$v_\mathrm{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} v^2 \ \mathrm{d}t}.$$

In our case $$\v\$$ is a sum of pure sine waves. Hence it is easier to work in the frequency domain, if you do not have a burning desire of evaluating integrals containing products of sine waves.

The rms in the frequency domain is given by

$$V_\mathrm{rms} = \sqrt{\frac{1}{N^2} \sum_{i = 1}^N |V_i |^2},$$

where $$\V_i\$$ are the frequency components and $$\N\$$ is the number of samples. Now remember that a sine wave satisfies

$$|V| = \frac{A}{2} (\delta(f - f_0) + \delta(f + f_0)).$$

Hence a single harmonic has two components with amplitude $$\A / 2\$$. Thus

$$V_\mathrm{rms}^2 = \frac{1}{N^2} \frac{A^2}{2}.$$

Consider what happens when we take the rms of a waveform comprising two distinct harmonics with amplitudes $$\A_1\$$ and $$\A_2\$$. Then you have

$$V^2_\mathrm{rms} = \frac{1}{N^2} \left( \frac{A_1^2}{2} + \frac{A_2^2}{2} \right).$$

Substituting the rms of the first harmonic $$\V_1\$$ and the rms of the second harmonic $$\V_2\$$ yields

$$V_\mathrm{rms} = \sqrt{V_1^2 + V_2^2}.$$