Voltage gain to decibels — quick check: An amplifier has a voltage gain Av = 210 (ratio). Evaluate the statement: “The voltage gain expressed in decibels is 46.4 dB.”

Introduction / Context:
Engineers often express gain in decibels (dB) for convenience, especially when cascading stages. Knowing the correct conversion from a simple voltage ratio to dB is essential for quick design checks and documentation.

Given Data / Assumptions:

• Voltage gain Av = Vout/Vin = 210 (dimensionless).
• dB conversion for voltage ratios uses 20 * log10(Av) when impedances are matched or when just expressing ratios.

Concept / Approach:
The general formulas are: Gv_dB = 20 * log10(Av) for voltage, and Gp_dB = 10 * log10(Pout/Pin) for power. Here we use the voltage formula directly because Av is given as a voltage ratio.

Step-by-Step Solution:

Verification / Alternative check:
Sanity check: 40 dB corresponds to a ratio of 100; each additional 6 dB approximately doubles voltage. 100 → 200 is roughly +6 dB; 40 dB + 6 dB ≈ 46 dB, matching 210 ≈ 46.4 dB.

Why Other Options Are Wrong:

• Incorrect: Contradicts the precise calculation.
• Only true for power gain: dB conversion for voltage explicitly uses the 20 multiplier.
• Insufficient information: The ratio is enough for dB conversion.
• Approximately 20.0 dB: Would imply a ratio near 10, not 210.

Common Pitfalls:
Using 10 * log10 for voltage ratios (that is for power), or mis-rounding logarithms leading to large errors.

Final Answer:
Correct