All I Know About Decibel (dB) - Part 2

As Part 1 introduces, we get decibel using the formula

$$l = 10 \times log \frac{p_1}{p_2} (dB)$$

I'd blame my poor English ability that I didn't realise what the name decibel really meant for quite a few years. It's called deci-bel so apparently there's a Bel first.

The bel (B) and the smaller decibel (dB) are units of measurement of sound pressure level (SPL) invented by Bell Labs and named after him

from: Wikipedia

That's him! As stated, using deci-, people can make bel smaller, and meanwhile we have to add the $10 \times$ in the formula, to make bel become decibel.

You might notice that when talking about voltage and current, the formula becomes

$$20 \times log \frac{p_1}{p_2} (dB)$$

So why 20 here? I was very confused. I asked around and got this answer: For measuring something like a magnitude, we use $20 \times$ , otherwise for the power stuffs, use $10 \times$ ... Okay then it's time to test my English again: what kind of thing is a magnitude? And what is not? I reckon it's a correct but not so good answer.

Finally I met a guy, he took out a piece of paper and wrote some high school maths on it:

$$P=U \times I=U \times (U/R) = U^2 / R$$

Yeah that's Ohm's Law, I get it. Then if $P_r$ and $U_r$ are the references and $U$ is the voltage we are measuring, from the original $10 \times$ formula we have:

$$10 \times log (P / P_r)$$

$$= 10 \times log ((U^2/R)/(U_r^2/R))$$

$$= 10 \times log (U/U_r)^2$$

$$= 2 \times 10 \times log (U/U_r) = 20 \times log (U/U_r)$$

That's how $20 \times$ comes up. Same process for the current (Use $P=I^2 \times R$ ). Maths can be scary, but useful.

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from OZLab