NapkinCalc

Electronics 1 — Filters & Op-Amps

Decibels — the language of gain

ELI5: signal chains multiply gains stage after stage, but multiplying big numbers is annoying — so engineers switch to decibels, where multiplying becomes adding. dB = 20·log₁₀(V_out/V_in): +6 dB ≈ ×2, and −3 dB ≈ ×0.707, the famous "half-power point."

Gratio=10G_{ratio} = 10 a ×10 voltage amplifier
GdB=20log(Gratio)log(10)G_{dB} = \frac{20 \cdot \mathrm{log}\left(G_{ratio}\right)}{\mathrm{log}\left(10\right)} = 2020 = 20 dB
✓ pass abs(20log(0.7071)log(10)3.01)<0.01\mathrm{abs}\left(\frac{20 \cdot \mathrm{log}\left(0.7071\right)}{\mathrm{log}\left(10\right)} - -3.01\right) < 0.01 −3 dB really is ×1/√2

Real-world hook: decibels measure audio volume, Wi-Fi signal strength, and antenna gain — anywhere quantities span a huge range.