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Averagesupernova

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Brief answer: decibel ishow do you know when to use dB= 20log(ratio of input/output quantities) or dB=10log(ratio of output and

inout).

Here is why (aka very long answer): decibel is one of those unfortunate units that they tell you about at the university but spend maybe

only a few minutes explaning what it really means.

The unit is actually called 'Bel' (named after Alexander Graham Bell, the man who is generally recognized as the one who invented a

telephone). It's not technically a unit (as say 1 meter) but rather a ratio of 2 numbers (actually "the order of ratio of 2 numbers"). It

shows the

The "log" calculates (from definition) the order of ratio of those 2 numbers.

Now, say you want to use "bels" in electronics. You have a weak input signal somewhere around 1uV (or 1uW) or any other "micro" level you

can think of, and after processing you have - say - levels somewhere around 10V or 10W or 10 whatever_units. That's a difference of 7

orders or 7 Bel. That gives you a range of values between 0 and 7 (if you pronounced the input level as 0 bel) or generally n .. n+7 bel.

It's a very short range so if you compared 2 values within the circuit you would need to use many decimal points to provide a result that

is exact enough. Say you'd need to say, that the ratio is 3.58bel. It's a number that's supposedly more difficult to remember than for

instance 42 or 42.3. That's where the 'deci' comes in.

"Deci" is a standard (metric) prefix meaning "(one) tenth" much the same way as kilo is a

prefix meaning thousand (i.e. 1000) and micro meaning "one milionth" (10^-6). So 1 decibel is "one tenth of a bel". To get decibels from

bels you just multiply bels by 10 (again, exactly the same way as you get hertz (Hz) from kilohertz (kHz) by multiplying by 1000 and volts

from microvolts by multiplying by 10^-6).

Now, back to our measurement of 3.58B. If we multiply it by 10 we get 35.8 decibels (35.8dB) which is supposedly easier to remember that

3.58B.

Using "tenths of a unit" instead of the full unit expands our range of 7 units or orders (or 7 bel) for a typical electronics circuit to 70

new units (or tenths of the original unit or 70dB). That gives us wider range of many different numbers (easier to remember) and less

numbers behind a decimal point for a given precision.

The question is, if dB is always 10x B (bels) where does the "20"come from? Well, it appears when you calculate power transfer (let's callvoltage seems to be 20, power seems to be 10...

it "T") and derive the values of input and output powers from measurements of voltage at both ends of the circuit. P=U x I=U x (U/R) = U^2

/ R

T=10 log (P_out/P_in)=10 log ( (U_out^2/R) / (U_in^2/R) ) = 10 log (U_out^2 / U_in^2) = 10 log ( U_out/U_in )^2 = 10 x 2 x log (U_out/U_in)

= 20 log (U_out/U_in)

It's always 10x (because of the "deci-" prefix), even when it appears as (20x) (actually 2x10x) as demonstrated above (the "2" comes fromhow about when doing fourier transforms and you want to find out 3dB of maximum, for example?

the squared argument "behind" the logarithm, so moving it in front of the logarithm saves us computing the squares, although nothing

prevents you from calculating the squares or the square of the ratio and then just multiplying the resulting logarithm by 10).

Hopefully, I explained it to you well enough. I wish somebody explained it to me like this years ago when I had the same question(s).

Eventually someone did (actually touched on the explanation and I was able to derive the rest), I passed many exams, thought I had a good

understanding of why using logarithms and bels/decibels makes sense but even now, years later I still find from time to time some usage,

benefits and properties that I wasn't aware of or had a poor understanding of.

Wikipedia has another explanation of decibel.

Good luck.

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I bet there are values where one must calculate 5th root of a cubic of the value but people generally don't call it "

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sophiecentaur

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This is an old chestnut and the rule is easy to remember if you bear in mind that a Bel is fundamentally Log(base 10) of the ratio of POWERS. Clearly, this implies that a DeciBel is one tenth of a Bel so 10dB represents a Power ratio of 10. There is no need to specify anything about the impedance involved in ratios of Power, of course.

As a totally side issue, you can ask what the voltage ratio would be for two powers into arbitrary resistances.*There is NO simple answer to that question.* The question you can answer is "what is the ratio of voltages for a given ratio of powers into the same resistance?"

The answer to that is that the ratio of the Powers is the square of the ratio of the Volts into the same resistance.

Doing the sums will tell you that the 'factor of two' comes from this square factor. So a ratio of powers of 10 will be 10dB and a ratio of volts of 10 on the same resistance will produce a ratio of powers of 100 - which is 20dB. 'Voltage dBs' are not a really valid concept. If they were, then a step-up transformer could be regarded as having Power Gain (!!!???).

People will say I am being too fussy but 'Volts dBs' are a snare and delusion and people who insist on 'using them' are heading for a fall. I am lucky in having spent most of my 'technical life' dealing with RF and matched systems so it's been mostly 50Ω all the time and 20dB has usually**implied** a voltage ratio of 10. But that's all. An OP Amp, [Edit: with feedback to give it ] a voltage gain of 1 probably has a gain (in yer proper actual dB) of 80dB or more. No wonder students get confused when they're exposed to that sort of thing.

As a totally side issue, you can ask what the voltage ratio would be for two powers into arbitrary resistances.

The answer to that is that the ratio of the Powers is the square of the ratio of the Volts into the same resistance.

Doing the sums will tell you that the 'factor of two' comes from this square factor. So a ratio of powers of 10 will be 10dB and a ratio of volts of 10 on the same resistance will produce a ratio of powers of 100 - which is 20dB. 'Voltage dBs' are not a really valid concept. If they were, then a step-up transformer could be regarded as having Power Gain (!!!???).

People will say I am being too fussy but 'Volts dBs' are a snare and delusion and people who insist on 'using them' are heading for a fall. I am lucky in having spent most of my 'technical life' dealing with RF and matched systems so it's been mostly 50Ω all the time and 20dB has usually

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This is typically the case in a filter where you measure an output voltage or an attenuation at different frequencies.

There, -3dB means voltage*0.71 without any sort of doubt, and it is useful since voltages are often measured.

Other example: if you have several stages of op amps, each with a vanishingly small and inconsistent output impedance, a power ratio would be unusable, but a gain in dB is useful.

So the 20*log convention is well-defined (if not easy), useful, and must be known.

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sophiecentaur

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I don't know of anyone who has never encountered a problem due to the confusion in a long life with dBs, even if it was just because someone else got it wrong.

It's along the lines of that awful thing with pounds mass and pounds weight and they both should be stamped out and never allowed to be used by civilised Engineers.

My prejudices are showing here, I'm afraid.

Voltage gain in standard form is just as useful and just as calculable.if you have several stages of op amps, each with a vanishingly small and inconsistent output impedance, a power ratio would be unusable, but a gain in dB is useful.

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