Distortion factor, DF and total harmonic distortion THD are related by:

Explanation:

Distortion Factor (DF) and Total Harmonic Distortion (THD) Relationship

Definition: Distortion factor (DF) and total harmonic distortion (THD) are metrics used to quantify the distortion in a waveform, typically an electrical signal, caused by the presence of harmonics. These metrics are vital in analyzing the quality of an electrical signal in power systems, audio applications, and communication systems.

The distortion factor (DF) is a measure of the waveform's deviation from a pure sine wave due to harmonics, while total harmonic distortion (THD) quantifies the extent of harmonic distortion as a ratio of the harmonic content to the fundamental frequency.

Correct Formula:

The relationship between DF and THD is given by the formula:

\(\mathrm{DF} = \sqrt{\frac{1}{1+\mathrm{THD}^{2}}}\)

This formula (Option 2) correctly relates DF and THD. It signifies that the distortion factor decreases as the total harmonic distortion increases, reflecting the increasing deviation of the waveform from a pure sine wave.

Derivation:

To derive the relationship between DF and THD:

• Let the fundamental frequency component of the waveform be \(V_1\), and the RMS value of the harmonic components be \(V_{\text{harmonics}}\).
• The total RMS value of the waveform, \(V_{\text{total}}\), is given by: \[ V_{\text{total}} = \sqrt{V_1^2 + V_{\text{harmonics}}^2} \]
• Total harmonic distortion (THD) is defined as: \[ \mathrm{THD} = \frac{V_{\text{harmonics}}}{V_1} \] Substituting \(V_{\text{harmonics}} = \mathrm{THD} \cdot V_1\) into the total RMS equation: \[ V_{\text{total}} = \sqrt{V_1^2 + (\mathrm{THD} \cdot V_1)^2} = V_1 \sqrt{1 + \mathrm{THD}^2} \]
• The distortion factor (DF) is defined as the ratio of \(V_1\) to \(V_{\text{total}}\): \[ \mathrm{DF} = \frac{V_1}{V_{\text{total}}} = \frac{V_1}{V_1 \sqrt{1 + \mathrm{THD}^2}} = \sqrt{\frac{1}{1 + \mathrm{THD}^2}} \]

Thus, the formula \(\mathrm{DF} = \sqrt{\frac{1}{1+\mathrm{THD}^{2}}}\) is derived, confirming Option 2 as the correct answer.

Applications:

• Analyzing power system quality in electrical grids.
• Evaluating signal quality in audio systems and communication networks.
• Designing filters and circuits to minimize distortion.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\mathrm{THD} = \sqrt{\frac{1}{1+\mathrm{DF}^{2}}}\)

This formula is incorrect. The relationship between DF and THD does not support this formulation. Substituting \(V_{\text{harmonics}} = \mathrm{THD} \cdot V_1\) and \(V_{\text{total}} = V_1 \sqrt{1 + \mathrm{THD}^2}\) into the definition of DF demonstrates that Option 1 does not align with the derived formula.

Option 3: \(\mathrm{DF} = \sqrt{\frac{1}{1-\mathrm{THD}^{2}}}\)

This formula is invalid because it implies a subtraction of \(\mathrm{THD}^2\), which can yield nonsensical or negative values when \(\mathrm{THD} > 1\). The correct formula involves addition (\(1 + \mathrm{THD}^2\)) in the denominator.

Option 4: \(\mathrm{THD} = \sqrt{\frac{1}{1-\mathrm{DF}^{2}}}\)

Similar to Option 3, this formula is incorrect due to the subtraction term (\(1 - \mathrm{DF}^2\)) in the denominator. The derived relationship between DF and THD involves addition, not subtraction.

Conclusion:

The correct formula, \(\mathrm{DF} = \sqrt{\frac{1}{1+\mathrm{THD}^{2}}}\), accurately describes the relationship between distortion factor and total harmonic distortion. This formula is essential in applications where understanding and minimizing waveform distortion is critical, such as in power systems, audio engineering, and signal processing. The incorrect options fail to align with the derived mathematical relationship and can lead to erroneous interpretations if applied.