How Big Is A dB?

MapYourTech March 26, 2025 No Comments Free Standards Technical

Last Updated: August 16, 2025

4 min read

Introduction To The dB

Describing Power

Signal stages are cascaded, so powers are multiplied by gain or loss. This yields a lot of multiplications. This suggests the need for a logarithmic representation of power.

A logarithmic scale is used to

Condense wide range of numbers
Ease multiplication

Logarithms

Log(x) = power to which base must be raised to give x. The base is chosen to be 10.

Log(x) = y means that x = 10^y

Log(A x B) = Log(A) + Log(B)
Hence: Log(x^N) = N x Log(x)

Some example logarithm values:

Log(100) = 2 because 10²= 100.
Log(1000) = 3 because 10³ = 1000.
Log(1000000) = 6 because 10⁶ = 1000000.
Log(10) = 1 because anything to the power of 1 is itself.
Log(1) = 0 because anything to the power of 0 is 1.
Log(1/10) = -1 because 10^-1 = (1/10)
Log(1/1000) = -3

The deciBel

Represent gains or attenuations logarithmically (base 10) (the Bel)

But to make numbers more convenient, scale by a factor of 10 (the deciBel or dB)

Then, G = 10Log(P_out/ P_in) in dB

Examples:

An amplifier has a power gain of 1000. What is this in dB?G = 10Log(1000) = 10 x 3 = 30 dB
An attenuator has its output power 1/10th of its input. What is its transfer function in dB?G = 10Log(1/10) = 10 x -1 = -10 dB. (Note – dB can be negative)

Since Log(A x B) = Log(A) + Log(B) we can add gains and losses.

P_R = P_T + 20 – 1 + 30 – 2 – 204 + 30 -1 + 60 = P_T – 68 dB

For converting from a power ratio to dB, first work out powers of 10, e.g:

Ratio		dB
1000	= 10³	30 dB
1	= 10⁰	0 dB
1/1000000	= 10⁶	-60 dB

Then note the smaller factors:

Factor of 2 is 3 dB (remember this!)
Factor of 4 = 2 x 2 is 3 + 3 = 6 dB etc.

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Ratio		dB
20	2 x 10 is 3 + 10	13 dB
1/400	4 x 100 is 6 + 20	-26 dB

Examples of converting from dB to a Ratio (or more generally, ratio = 10^dB/10):

dB		Ratio
23	3 + 20 is 2 x 100	200
-3		1/2
-63	-60 – 3 is 1/10⁶ x 1/2	1/2000000
-160		10^-16
-167	-170 + 3 is 10^-17 x 2	2 x 10^-17
7	10 – 3 is 10/2	5
9	3 + 3 + 3 is 2 x 2 x2	8
1	10-9 is 10/8	1.25

Applying dB to Other Units

By default, dB is a power ratio. But it can be other things, for example, dB banana = dB relative to 1 banana.

dBW = dB relative to 1 watt, so:

3 dBW = 2 W
-30 dBW = 1/1000 W = 1 mW (1 milli-watt) = 0 dBm (m here – milliwatt)
-60 dBW = 1 µW (1 micro-watt) = -30 dBm

Bandwidth in Hz can be expressed in dB-Hz

1 MHz = 60 dB-Hz

Similarly, Noise Temperature:

200 K = 23 dB-K

By default, with dBs we are dealing with power.

P = V²/ R where V is the root mean square voltage, V_RMS

Thus a change in power (e.g. due to amplification) can be represented by:

10Log(P₂ / P₁) = 10Log(V₂² / V₁²) = 20Log(V₂ / V₁) since Log(x^N) = NLog(x)

TIP: Take care with “Voltage gain in dB” which is usually a power gain, i.e 20Log(V₂ / V₁)