THD to Decibels⚠ unverified

Understanding the Parameters

• THD (linear) — $\mathrm{THD} = V_{harm}/V_1$ (or the equivalent current form), a positive number typically $\ll 1$ for clean audio or power waveforms. Enter 0.05 for 5%, not 5. Values $> 1$ are possible for heavily clipped signals and give positive dB.

• $\mathrm{THD_{dB}}$ — almost always negative for good systems (distortion below the fundamental). More negative = lower distortion: e.g. 1% → $\approx -40\,\text{dB}$; 0.1% → $\approx -60\,\text{dB}$.

Derivation (Approaching a Proof)

From Total Harmonic Distortion, the linear THD is an RMS voltage (or current) ratio:

$$\mathrm{THD} = \frac{\sqrt{V_{rms}^2 - V_1^2}}{V_1} = \frac{V_{harm}}{V_1}.$$

The general decibel map for a ratio of like field quantities (voltage, current, pressure) is

$$(X_2/X_1)_{\mathrm{dB}} = 20\log_{10}(X_2/X_1),$$

because power ratios use $10\log_{10}(P_2/P_1)$ and $P \propto X^2$ at fixed impedance, so $10\log_{10}(X_2^2/X_1^2) = 20\log_{10}(X_2/X_1)$. Substituting $X_2/X_1 = \mathrm{THD}$ yields

$$\mathrm{THD_{dB}} = 20\log_{10}(\mathrm{THD}).$$

Contrast with SNR: SNR to Decibels uses $10\log_{10}$ because its input is already a power ratio $P_s/P_n$. Using 10 here for THD, or 20 for power-SNR, is a common unit error.

Inverse: $\mathrm{THD} = 10^{\mathrm{THD_{dB}}/20}$.

History

Expressing distortion in dB became standard in audio and instrumentation as THD+N and related specs matured mid-20th century. Power-quality standards more often quote THD in percent, but the dB form is convenient when comparing levels on a logarithmic scale with SNR and spur floors.

Related Concepts: Total Harmonic Distortion, SNR to Decibels, Signal-to-Noise Ratio, Rectifier Ripple Factor, Power Factor

Notes: Registry calculator thd-db (unverified). Requires $\mathrm{THD} > 0$. Input is the linear ratio, not percent. Factor 20 matches amplitude-ratio convention used with the Power-catalog THD definition.