
Power and Voltage ratios in DECIBELS
The decibel notation is basically a method of expressing power ratio, the number of decibels (N dB)
being given by
where P _{1} and P_{2} are the two power levels under consideration.
On the assumption of constant impedance the notation is also used for voltage ratio, giving
where V_{1} and V_{2} are the two voltage levels.
In the above expressions the denominator (P_{1} or V_{1}) is the reference level, so that a power or voltage
may be expressed as "NdB with respect to X watts or volts"
This is often contracted to dB μV (to mean decibels with respect to 1 μV).
dBm (with respect to 1 mW),
dB W (with respect to 1 watt), etc.
Table I
Voltage 
Power 
Decibels 
Voltage 
Power 
1.0 
1.0 
0 
1.0 
1.0 
.9886 
.9772 
.1 
1.012 
1.023

.9661 
.9333 
.3 
1.035 
1.072 
.9550 
.9120 
.4 
1.047 
1.096
 .9441 
.8913 
.5 
1.059 
1.122 
.9333 
.8710 
.6 
1.072 
1.148
 .9226 
.8511 
.7 
1.084 
1.175 
.9120 
.8318 
.8 
1.096 
1.202
 .9016 
.8128 
.9 
1.109 
1.230 
.8913 
.7943 
1.0 
1.122 
1.259
 .8710 
.7586 
1.2 
1.148 
1.318 
.8511 
.7244 
1.4 
1.175 
1.380
 .8318 
.6918 
1.6 
1.202 
1.445 
.8128 
.6607 
1.8 
1.230 
1.514
 .7943 
.6310 
2.0 
1.259 
1.585 
.7762 
.6026 
2.2 
1.288 
1.660 
.7586 
.5754 
2.4 
1.318 
1.738 
.7413 
.5495 
2.6 
1.349 
1.820 
.7244 
.5248 
2.8 
1.380 
1.905 
.7079 
.5012 
3.0 
1.413 
1.995 
.6683 
.4467 
3.5 
1.496 
2.239 
.6310 
.3981 
4.0 
1.585 
2.512 
.5957 
.3548 
4.5 
1.679 
2.818 
.5623 
.3162 
5.0 
1.778 
3.162 
.5309 
.2818 
5.5 
1.884 
3.548 
.5012 
.2512 
6 
1.995 
3.981 
.4467 
.1995 
7 
2.239 
5.012 
.3981 
.1585 
8 
2.515 
6.310 
.3548 
.1259 
9 
2.818 
7.943 
.3162 
.1 
10 
3.162 
10.00 
.2818 
.07943 
11 
3.548 
12.59 
.2512 
.06310 
12 
3.981 
15.85 
.2239 
.05012 
13 
4.467 
19.95 
.1995 
.03981 
14 
5.012 
25.12 
.1778 
.03162 
15 
5.623 
31.62 
.1585 
.02512 
16 
6.310 
39.81 
.1413 
.01995 
17 
7.079 
50.12 
.1259 
.01585 
18 
7.943 
63.10 
.1122 
.01259 
19 
8.913 
79.43 
.1000 
.01000 
20 
10.000 
100.00 

Where the numerator of the ratio is less than the denominator N becomes negative; this may be expressed
as "N dB" or "N dB down",
positive values on N being regarded as "N dB up" by analogy.
Tables I and II facilitate rapid conversion between decibels and voltage or power ratios.
To convert decibels to voltage or power ratios use Table I;
to convert voltage or power ratio to decibels use Table II.
Table I covers dB ratios up to 20dB only. For values of N greater than 20 proceed as follows.
Let X be the whole number of times that 20 can be divided into N, to leave a remainder that falls
within the range of the table.
Look up the voltage or power ratio corrseponding to this remainder on the appropriate side of the table.
For voltage ratio "up" multiply by 10 ^{x }
For power ratio "down" divide by 10 ^{x }
For power ratio multiply or divide, as appropriate, by 10 ^{2x }

Table II
Voltage or Power ratio 
dB Voltage 
dB Power 
1.0 
0.0 
0.0 
1.1 
0.83 
0.41 
1.2 
1.58 
0.79 
1.3 
2.28 
1.14 
1.4 
2.92 
1.46 
1.5 
3.52 
1.76 
1.75 
4.86 
2.43 
2.0 
6.02 
3.01 
2.5 
7.96 
3.98 
3.0 
9.54 
4.77 
3.5 
10.88 
5.44 
4.0 
12.04 
6.02 
4.5 
13.06 
6.53 
5.0 
13.98 
6.94 
5.5 
14.81 
7.40 
6.0 
15.56 
7.78 
6.5 
16.26 
8.13 
7.0 
16.90 
8.45 
7.5 
17.5 
8.75 
8.0 
18.06 
9.03 
8.5 
18.58 
9.29 
9.0 
19.08 
9.54 
9.25 
19.32 
9.16 
9.5 
19.55 
9.27 
9.6 
19.65 
9.32 
9.7 
19.74 
9.87 
9.8 
19.83 
9.91 
9.9 
19.91 
9.95 
10.0 
20.0 
10.0 

Table II covers voltage and power ratios from 1 to 10.
For power ratios outside this range move the decimal point to the left or right to bring the figure
within the range, then add 10 dB for each position that it is moved to the left, or subtract 10 dB for each position that it is moved to the right.
For voltage ratios use the same procedure, but add or subtract 20 dB instead of 10 dB.
Ratios close to 1 and to 10 are given in smaller increments than the rest of the table for convenience
of conversion when errors are to be expressed in decibels.
Click to enlarge

Reference: Marconi Electronics Instrumentation Catalogue 79/80
