# TIL: Decibels, Power, and Root Power

I’ve been an EE to one degree or another for 30-ish years now. And one thing I never understood was when a decibel was 10*log_10(x) and when it was 20*log_10(x). My EE books describe it as 20*, but common use has always been 10*.

## A brief refresher

Skip this part if you already know how to use decibels.

A decibel is a unitless quantity, often used to describe one measurement relative to another: “the output is +17dB up from the input” or “this is a 3dB attenuator.” Or as a unit multiplier attached to a unit: “the output is 10dBm, or 10dB relative to 1mW.” (In this last use, it’s very similar to SI unit multipliers, like kilo, mega, centi, etc.)

Remembering just two simple dB values will let you figure out almost every other dB quantity:

• 3dB is 2x (or, very close to it)
• 10dB is 10x (exactly)

Adding/subtracting decibels is the same as multiplying/dividing the quantity. So, 6dB is 3dB+3dB, which is the same as 2x * 2x, or 4x. Similarly, 1dB is 10dB-3dB-3dB-3dB, or 10/2/2/2, or 1.25x.

With those two quantities (3dB and 10dB), you can figure out all values from 1 to 10, except 2dB, 5dB, and 8dB (which is just 10dB-2dB). The way to remember those three is only slightly more complex.

2dB is 1dB+1dB, so 1.25*1.25. I think of that as 1.25 + a quarter of 1.25. A quarter of 1.0 is .25, so add that and you get 1.5. A quarter of .25 is … well, .25*.25 is going to have the same numbers as 25*25, and I know that’s 625. So slide the decimal point over by 4 positions, you get .0625. Add that to the 1.5, you get 1.5625. So, 2dB = 1.56. I usually remember it as “a bit more than 1.5x.”

You can then use that to figure out 8dB, which is 10dB-2dB, or “10x divided by a bit more than 1.5x”. That’s the same as “a little less than 2/3 of 10x.” Two thirds of 10x is 6.67x, a little less is 6.5x. So 8dB = 6.5x.

5dB is the only stand-alone. Let 5dB be N. 5dB+5dB=10dB, so N*N=10x, so N squared is 10, so N is the square root of 10. The square root of 10x is about 3.16x. So 5dB =3.16x

The complete table of dB values between 1dB and 10dB is:

• 1dB = 1.25x: 10dB-3dB-3dB-3dB, 10/2/2/2, 10->5->2.5->1.25x.
• 2dB = 1.56x: 1dB+1dB, 1.25*1.25, 1.56x
• 3dB = 2x (by very close approximation)
• 4dB = 2.5x: 10dB-3dB-3dB, 10/2/2, 10->5->2.5x
• 5dB = 3.16x: sqrt(10)
• 6dB = 4x: 3dB+3dB, 2*2, 2->4x
• 7dB = 5x: 10dB-3dB, 10/2, 10->5x
• 8dB = 6.5x: 10dB-2dB, “a bit less than 2/3 of 10”, 6.5x
• 9dB = 8x: 3dB+3dB+3dB, 2*2*2, 2->4->8x
• 10dB = 10x (by definition)

The tens place of the dB value just describes how many zeros to append, or how many places to move the decimal. So 20dB is 100x, -30dB is 1/1000x, etc.

Put that together with the table, and you can figure out any dB value. For example, 37dBm is: 30dB = 1000x, +7dB = 5x so 5000x. The unit “m” in this context means “milli Watt.” So 37dBm is 5000mW, or 5W.

It’s worth noting that 0dB = 1x. A bit of a special case.

## The complication

I’d always seen the formal definition of a decibel to be: 20*log_10(x). Meaning 10x would be 20dB, not 10dB… What gives?

This always confused me, but I just ignored the confusion because I always saw it used as 10*, as described above…

Until this week when formalizing some bits of the Open Headset Interconnect Standard. Audio signal levels in general, and microphone sensitivity ratings in specific, tend to use one of two different units:

• dBV: decibel Volts RMS. Simple.
• dBu: decibel relative to the voltage needed to drive 1mW into a 600 ohm load. That is, decibels relative to 0.775Vrms. dBu = dBV – 2.21dB.

Bla bla bla, that’s not important to todays discussion. It’s just the context to my epiphany moment.

The electret microphones I was looking at were listing their sensitivity between -40dBV and -47dBV. As we know from the previous section, that’s about (-40dB = 1/10,000x or .1milli, -7dB = 1/5x) .1mV to .02mV. But I know from experience that electret microphones put out closer to 10mV!

Wait a second, 10mV, I think of that as -20dBV… which is half of -40dBV. IS THIS WHERE THAT 20* COMES IN?!

## The Epiphany

Turns out, yes. This is exactly where that 20* comes in.

Turns out, the Bel is a 10x increase IN POWER. Not just a 10x increase in whatever number it’s applied to. So, when using Bels to describe power, 10 deciBels is a 10x increase in power. Whew! I haven’t been using it incorrectly this whole time.

But voltage is different! For a given system (read: for a constant load impedance), Power = V^2/R. It’s that V^2, V squared, that’s the key. If I increase the voltage by 10x the current will also increase by 10x (ohms law: V=IR), so the total power delivered will increase by 100x (power law: P=IV)

That is, a 10x increase in voltage results in a 100x increase in power. So a 10x increase in voltage is a 20dB increase in power. So a 10x increase in voltage is 20dBV.

THERE’S THE 20*log_10(V)

In all cases when calculating dB, do it IN POWER:

• If the number you’re changing relates to power directly (like… power), then it’s 10*log_10(x).
• If the number you’re changing relates to power via a square root (like voltage, which changes as the square root of power), then it’s 20*log_10(x).

## No Duh…

Many engineers are probably laughing at me right now, thinking something along the lines of “Didn’t you already know this?”

Nope! I sure didn’t! I knew it was something along these lines, but I was always hazy on the details.

And from the responses I’ve gotten by asking around, I think a lot of other really smart people don’t fully understand this either. (Hence this post.) So, person in my head saying “no DUH”, stop being such a jerk about it, ok?

It’s a good reminder than even smart people who know a lot, don’t know everything. Sometimes, we don’t even know the simple things, fundamental things, like what a decibel REALLY is. This is why I enjoy reading “Today I Learned (TIL)” or “I was today years old when…” posts. Not only do I often learn something new myself, they also remind me that it’s not just me. 🙂

Cheers, 73 de N6MTS.

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