dB, dBW, dBm, Logarithm, ...
What is a Decibel ?
Decibel is the unit used to express relative differences in signal strength.
Decibel is expressed as the base 10 logarithm of the ratio of the power of two signals : dB = 10 x Log10 (P1/P2)
Where Log10 is the base 10 logarithm and P1 and P2 are the powers to compare
(Log10 is different from Ln or LN = Neparian Logarithm, base e logarithm)
Signal amplitude can also be expressed in dB. Since power is proportional to the square of a signal's amplitude, dB is expressed as follows : dB = 20 x Log10 (V1/V2)
Where V1 and V2 are the amplitudes to compare
1 Bell (not really used in current) = Log10(P1/P2)
1 decibel (dB) = 1 Bell / 10 = 10 * Log10(P1/P2)
dBr = dB (relative) = dB = 10 * Log10(P1/P2)
| base 10 Logarithm rules |
|---|
| Log10 (AxB) = Log10 (A) + Log10 (B) |
| Log10 (A/B) = Log10 (A) - Log10 (B) |
| Log10 (1/A) = - Log10 (A) |
| Log10 (0,01) = - Log10 (100) = -2 |
| Log10 (0,1) = - Log10(10) = - 1 |
| Log10 (1) = 0 |
| Log10 (2) = 0,3 |
| Log10 (4) = 0,6 |
| Log10 (10) = 1 |
Log10 (20) = 1,3
Log10(2 x 10) = Log10(2) + Log10(10) = 1 + 0,3 |
| Log10 (100) = 2 |
| Log10 (1 000) = 3 |
| Log10 (10 000) = 4 |
| Logarithm and dB (decibel) |
|---|
| Power Ratio |
dB = 10 x log10 (Power Ratio) |
| AxB | x dB = 10 x Log10(A) + 10 x Log10(B) |
| A/B | x dB = 10 x Log10(A) - 10 x Log10(B) |
| 1/A | x dB = + 10 x Log10(1/A) = - 10 x Log10(A) |
| 0,01 | - 20 dB = - 10 x Log10(100) |
| 0,1 | - 10 dB = 10 x Log10(1) |
| 1 | 0 dB = 10 x Log10(1) |
| 2 | 3 dB = 10 x Log10(2) |
| 4 | 6 dB = 10 x Log10(4) |
| 10 | 10 dB = 10 x Log10(10) |
| 20 | 13 dB = 10 x (Log10(10) + Log10(2)) |
| 100 | 20 dB = 10 x Log10(100) |
| 1 000 | 30 dB = 10 x Log10(1 000) |
| 10 000 | 40 dB = 10 x Log10(10 000) |
| dBm = dB milliwatt = 10 x Log10 (Power in mW / 1 mW) |
|---|
| Power |
Ratio |
dBm = 10 x
Log10 (Power in mW / 1 mW) |
|
1 mW | 1 mW / 1 mW = 1 | 0 dBm = 10 x Log10(1) |
| 2 mW | 2 mW / 1 mW = 2 | 3 dBm = 10 x Log10(2) |
| 4 mW | 4 mW/1mW=4 | 6 dBm = 10 x Log10(4) |
| 10 mW | 10 mW/1mW=10 | 10 dBm = 10 x Log10(10) |
| 0,1 W | 100 mW/1mW=100 | 20 dBm = 10 x Log10(100) |
| 1 W | 1000 mW/1mW=1000 | 30 dBm = 10 x Log10(1 000) |
| 10 W | 10 000mW/1mW=10 000 | 40 dBm = 10 x Log10(10 000) |
| dBW = dB Watt = 10 x Log10 (Power in W / 1 W) |
|---|
| Power |
Ratio |
dBW = 10 x Log10 (Power in W / 1 W) |
| 1 W | 1 W / 1 W = 1 | 0 dBW = 10 x Log10(1) |
| 2 W | 2 W / 1 W = 2 | 3 dBW = 10 x Log10(2) |
| 4 W | 4 W / 1 W = 4 | 6 dBW = 10 x Log10(4) |
| 10 W | 10 W / 1 W = 10 | 10 dBW = 10 x Log10(10) |
| 100 mW | 0,1 W / 1 W = 0,1 | -10 dBW = -10 x Log10(10) |
| 10 mW | 0,01 W / 1 W = 1/100 | -20 dBW = -10 x Log10(100) |
| 1 mW | 0,001W/1W=1/1000 | -30 dBW = -10 x Log10(1000) |
| Power/Voltage Gain |
|---|
| dB |
Power Ratio |
Voltage Ratio |
|
dB |
Power Ratio |
Voltage Ratio |
| 0 | 1,00 | 1,00 |
|
10 | 10,00 | 3,16 |
| 1 | 1,26 | 1,12 |
|
11 | 12,59 | 3,55 |
| 2 | 1,58 | 1,26 |
|
12 | 15,85 | 3,98 |
| 3 | 2,00 | 1,41 |
|
13 | 19,95 | 4,47 |
| 4 | 2,51 | 1,58 |
|
14 | 25,12 | 5,01 |
| 5 | 3,16 | 1,78 |
|
15 | 31,62 | 5,62 |
| 6 | 3,98 | 2,00 |
|
16 | 39,81 | 6,31 |
| 7 | 5,01 | 2,24 |
|
17 | 50,12 | 7,08 |
| 8 | 6,31 | 2,51 |
|
18 | 63,10 | 7,94 |
| 9 | 7,94 | 2,82 |
|
19 | 79,43 | 8,91 |
| 10 | 10,00 | 3,16 |
|
20 | 100,00 | 10,00 |
Attenuation (dB) = 10xLog10(Pin/Pout) = 20xLog10(Vin/Vout)
Gain (dB) = 10xLog10(Pout/Pin) = 20xLog10(Vout/Vin)
Gain (dB) = - Attenuation (dB)
Power (W) = voltage (V) x Voltage (V) x Z (Ohm) = V x V / Z = V x I
So Gain(dB) = 10 x Log10(Pout/Pin) = 10 x Log10((Vout x Vout / Zout) / (Vin * Vin / Zin))
if Zin = Zout, Gain(dB) = 10 x Log10(Vout x Vout / Vin /Vin) =
10 x (Log10(Vout / Vin) + Log10(Vout / Vin)) = 10 x 2 x Log10(Vout / Vin) = 20 x Log10(Vout / Vin)
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