why are the coefficient to converting power to value in dB is different with the coefficient for converting RL

Question

why we used 10 log P to convert power into dB and why we used 20 log P to covert return loss in dB? why does the coefficient differ?

Answer 1

The bel unit was originally intended for sound power levels. A decibel is 1/10 of a bel. However, sound is often measured in pressure levels (sound pressure level = SPL). Since the power in a sound wave is proportional to the square of the pressure level, the values are equivalent if the pressure number is multiplied by 2. This is because the bel is a logarithmic unit:

bel = log(base 10) of Power Ratio

decibel = 10 * log(base 10) of Power Ratio

If Power Ratio ~ Pressure Ratio ^2, and you take the log of both sides you get

power decibels ~ 10 * 2* log(pressure ratio) = 20* log (pressure ratio)

The same concept has been adapted to non-acoustic measurements, such as voltage. Voltage decibels are 20*log(voltage ratio); since power is proportional to the square of voltage. Power decibels are 10*log(power ratio).

Answer 2

20 log is only used when the measurement is voltage, not power. It comes from the fact that the definition is based on power, and arbitrarily assigned a factor of 10 by Mr Bell, I believe. One consideration for that factor of 10 is that 10 mW is required to power an earpiece, which comes out to 0 dBm, a nice reference.

Anyway, if you measure two voltages and compare them, you need to take 20 log (V2/V1) because Power is proportional to V^2.

The only other thing I can think of is that if you have a cable with 15 loss, its return loss is 30 dB becasue it travels both ways.

Answer 3

The coefficients are arbitrarily chosen. We choose 10 as the multiplier in front of the logarithm for decibels because we like an increase of 10 decibels to correspond to 10 times greater sound intensity, 20 decibels to correspond to 100 times greater sound intensity, etc.

I have no idea why anyone would use a scale that's off by a factor of 2 for other sound computations. Very mysterious.