Frequency-domain conversion: Which formula correctly converts an amplitude ratio (AR) into decibels (dB)?

Introduction / Context:
Engineers often express frequency-response magnitudes on a logarithmic scale to simplify multiplication into addition and to align with human perception. The decibel (dB) scale is standard in Bode plots and control/communications. Choosing the correct conversion from amplitude ratio (AR) to decibels is essential for proper gain and margin calculations.

Given Data / Assumptions:

• Amplitude ratio AR is a non-negative real number representing |G(jω)|.
• Base-10 logarithms are used for dB in engineering (unless explicitly stated otherwise).
• We are converting an amplitude (not power) ratio.

Concept / Approach:
By definition for amplitudes, dB = 20 * log10(AR). For power ratios, the coefficient is 10 because power ∝ amplitude^2, so 10 log10(PR) = 20 log10(AR). Using natural logs is possible but then a conversion factor must be included (20/ln(10)). Without that factor, expressions using loge are incorrect for straight substitution.

Step-by-Step Solution:

Verification / Alternative check:
Check a known case: AR = 10 → dB = 20 log10(10) = 20 dB; AR = 0.1 → dB = −20 dB. These match standard results.

Why Other Options Are Wrong:

Common Pitfalls:
Mixing up amplitude vs. power ratio formulas; always use 20 for amplitude and 10 for power, both with log base 10 unless stated otherwise.

Final Answer:
Decibel = 20 log10(AR)