Voltage ratio in decibels: Given Vout = 500 mV and Vin = 1.3 V for a signal chain, compute the gain in dB for the ratio Vout/Vin.

Introduction / Context:
Decibels provide a logarithmic way to express voltage gains or losses. For voltage ratios in linear systems with the same impedance reference, we use 20 * log10(Vout/Vin). This is common in audio, RF, and control systems.

Given Data / Assumptions:

• Vout = 0.500 V.
• Vin = 1.3 V.
• Same impedance conditions; use the 20 * log10 rule.

Concept / Approach:
Gain_dB = 20 * log10(Vout / Vin). A ratio less than 1 will yield a negative dB value, indicating attenuation rather than amplification.

Step-by-Step Solution:
Ratio r = Vout/Vin = 0.5 / 1.3 ≈ 0.384615Gain_dB = 20 * log10(0.384615)log10(0.384615) ≈ -0.415Gain_dB ≈ 20 * (-0.415) ≈ -8.30 dB

Verification / Alternative check:
A -6 dB change corresponds to a factor of ~0.5; since 0.3846 is less than 0.5, the attenuation must be more than -6 dB, consistent with -8.3 dB.

Why Other Options Are Wrong:

• 0 dB: Would imply Vout = Vin.
• +8.30 dB or 0.8 dB: Indicate gain or minimal change, not consistent with Vout < Vin.

Common Pitfalls:
Using 10 * log10 for voltage (that is for power), or mixing units (mV with V) without converting. Always convert to the same unit first and apply 20 * log10 for voltage ratios.

Final Answer:
–8.30 dB