How to calculate ratio from dB without a calculator.. made EASY.


Everything you have to remember is:


10 * log(10) = 10               - if something is 10 times the size, it's +10dB larger

                                            - If something is 10 times smaller, it is -10db smaller

10 * log(2) = 3                   - if something is twice the size, it is 3dB larger

                                            - if someting is half the size, it is -3dB smaller

 

That's it. Everything else depends on your skills with a multiplication table.


How to use simple steps to get ratio out of a dB

  1. We write the number with a sum of 10s and 3s.
  2. We play with the equation a bit.
    • 3 becomes 2
    • 10 stays 10
    • + becomes *
    •  – becomes /

Examples:

32dB               32 = 10 + 10 + 12 = 10 + 10 + 3 +3 + 3 +3

                              Now we change the sign + in to a *, number 10 into a 10 and number 3 in to a 2.

                              X = 10 * 10 * 2 * 2 * 2 * 2 = 100 * 16 = 1600. 32dB means an increase by 1600 times.

                             Test: 10*log(1600) is 32.041 dB

48dB               48 = 10 + 10 + 10 + 3 + 3 + 3 + 3 + 3 + 3 

                              X = 10 * 10 * 10 * 2 * 2 * 2 * 2 * 2 * 2  = 1000 * 64 = 64000

                              Test:  10*log(64000) =  48.06 dB

There are many ways to do the same number.

                         48 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 16 * 3

                              X = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2  = 2 ^ 16 = 64000

                         48 = 60 – 12 = 10 + 10 + 10 + 10 + 10 + 10 - 3 - 3 - 3 - 3

                            X = 10 * 10 * 10 * 10 * 10 * 10 / 2 / 2 / 2 / 2 = 1000000 / 2 / 2 / 2 / 2 = 500000 / 2 / 2 / 2 = 64000

 

 

The hardest numbers to write like this are the ones you'd think are the easiest – numbers from 1 to 10.

1dB                  1 = 10 – 9 = 10 – 3 – 3 – 3             Now we change – into a /,    10 into a 10 and 3 into a 2.

                              X = 10 / 2 / 2 / 2 = 10 / 8 = 5 / 4 = 1.24

                              Test: 10*log(1.24) = 0.93dB,  which is close to 1

2dB                  2 = 12 – 10 = 3 + 3 + 3 + 3 – 10

                              X = 2 * 2 * 2 * 2 / 10 = 16 / 10 = 1.6

                              Test: 10*log(1.6) = 2.04dB

4dB                  4 = 24 – 10 – 10 = 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 - 10 - 10

                              X = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 / 10 / 10 = 256 / 10 / 10 = 2.56

                              Test: 10*log(2.56) = 4.08dB

5dB                  5 = 15 – 10 = 3 + 3 + 3 + 3 + 3 - 10

                              X = 2 * 2 * 2 * 2 * 2 / 10 = 32 / 10 = 3.2

6dB                  6 = 3 + 3

                              X= 2 * 2 = 4

7dB                  7 = 10 – 3

                              X = 10 / 2 = 5  Test: 10*log(5) = 6.98dB

8dB                  8 = 18 – 10 = 3 + 3 + 3 + 3 + 3 + 3 - 10

                              X = 2 * 2 * 2 * 2 * 2 * 2 / 10 = 64 / 10 = 6.4


Below (on the very bottom of the page) i've written examples for every number up to 120dB. Try thinking of a random number and transform it in to a ratio. If you believe it can not be done, check the file below and you'll se it can be done.

Why only till 120dB? Engineering wise, dB's go maximum up to 180dB, you won't need more 99% of cases.

 

How about it I want to calculate 30,5dB?

·        The answer is: 

                        Calculate for 30dB and sum 10% to it.

o   30dB       x = 1000        

o   30,5dB    x = 1000 + 10% out of 1000 = 1100       (10*log(1100) = 30.4), close enough

Or Calculate for 31dB and remove 10%

o   21dB     à x=1250

o   30,5dB à x= 1250 – 10% out of 1250 = 1125       (10*log(1125) = 30.5, even closer

 

Confusion with the number 6:

Sometimes the professors will say that +6dB means double the power.

All of this confusion comes from a simple equation: P=U^2/R. Your professor forgot to say that a gain of +6dB in voltage results in doubling (+3dB) of the power.

Try to follow and you confusion should be gone:

·        3dB ALLWAYS means double the ratio, no exceptions or special cases.

·           However, you are using 2 variables. One is voltage U and the other one is the power P. 

       P is proportional to U^2; thus if you double the power (+3dB), you quadruple the voltage (+6dB).

10Log(P/P0) == 10Log((U/U0)^2) == 20Log(U/U0)   

                    Don't forget this simple rule. If you have a squared entity inside the logarithm, you can out this (number 2) in front of the logarithm. That's why logartihms were invented in the first place.. but who uses them now when we have calucaltors...

 

All the possibilites with the number 3, 6, Ohm's law, power and logratihms

  • o   If you add 3dB to the power, the power will double, voltage qill quadruple
  • o   If you add 6dB to the power, the power will quadruple, voltage will *8
  • o   If you add 6dB to the voltage, the power will double, the voltage will quadruple
  • o   If you add 3dB to the voltage, the power will be for factor Sqrt(2) bigger, the voltage will double
  • o   If you substract 3dB from the power, the  power will half in size, voltage will /4
  • o   If you substract 6dB from the power, the power will /4, votage will half.
  • o   If you substract 6dB from the voltage, the power will half, voltage will /8
  • o   If you substract 3dB from the voltage, the power wil Sqrt(3), voltage will half.

It is fairly simple once you know how to use these 2 equations hand in hand.

 

The crib note:

dB set apart                                                     the switchover        ratio
11 10+10-3-3-3                                                 10*10/2/2/2         12,5
12 3+3+3+3                                                      2*2*2*2                  16
13 10+3                                                           10*2                         20
14 10+10-3-3                                                    10*10/2/2                 25
15 3+3+3+3+3                                                  2*2*2*2*2                 32
16 10+3+3                                                        etc...
17 10+10-3                                  
18 3+3+3+3+3+3
19 10+3+3+3
20 10+10
21 10+10+10-3-3-3
22 10+3+3+3+3
23 10+10+3
24 10+10+10-3-3
25 10+3+3+3+3+3
26 10+10+3+3
27 10+10+10-3
28 10+3+3+3+3+3+3
29 10+10+3+3+3
30 10+10+10
31 10+10+10+10-3-3-3
32 10+10+3+3+3+3
33 10+10+10+3
34 10+10+10+10-3-3
35 10+10+3+3+3+3+3
36 10+10+10+3+3
37 10+10+10+10-3
38 10+10+3+3+3+3+3+3
39 10+10+10+3+3+3
40 10+10+10+10
41 10+10+10+10+10-3-3-3
42 10+10+10+3+3+3+3
43 10+10+10+10+3
44 10+10+10+10+10-3-3
45 10+10+10+3+3+3+3+3
46 10+10+10+10+3+3
47 10+10+10+10+10-3
48 10+10+10+3+3+3+3+3+3
49 10+10+10+10+3+3+3
50 10+10+10+10+10
51 10+10+10+10+10+10-3-3-3
52 10+10+10+10+3+3+3+3
53 10+10+10+10+10+3
54 10+10+10+10+10+10-3-3
55 10+10+10+10+3+3+3+3+3
56 10+10+10+10+10+3+3
57 10+10+10+10+10+10-3
58 10+10+10+10+3+3+3+3+3+3
59 10+10+10+10+10+3+3+3
60 10+10+10+10+10+10
61 10+10+10+10+3+3+3+3+3+3+3
62 10+10+10+10+10+3+3+3+3
63 10+10+10+10+10+10+3
64 10+10+10+10+10+10+10-3-3
65 10+10+10+10+10+3+3+3+3+3
66 10+10+10+10+10+10+3+3
67 10+10+10+10+10+10+10-3
68 10+10+10+10+10+3+3+3+3+3+3
69 10+10+10+10+10+10+3+3+3
70 10+10+10+10+10+10
71 10+10+10+10+10+10+10+10-3-3-3
72 10+10+10+10+10+10+3+3+3+3
73 10+10+10+10+10+10+10+3
74 10+10+10+10+10+10+10+10-3-3
75 10+10+10+10+10+10+3+3+3+3+3
76 10+10+10+10+10+10+10+3+3
77 10+10+10+10+10+10+10+10-3
78 10+10+10+10+10+10+3+3+3+3+3+3
79 10+10+10+10+10+10+10+3+3+3
80 10+10+10+10+10+10+10+10
81 10+10+10+10+10+10+10+10+10-3-3-3
82 10+10+10+10+10+10+10+3+3+3+3
83 10+10+10+10+10+10+10+10+3
84 10+10+10+10+10+10+10+10+10-3-3
85 10+10+10+10+10+10+10+3+3+3+3+3
86 10+10+10+10+10+10+10+10+3+3
87 10+10+10+10+10+10+10+10+10-3
88 10+10+10+10+10+10+10+3+3+3+3+3+3
89 10+10+10+10+10+10+10+10+3+3+3
90 10+10+10+10+10+10+10+10+10
91 10+10+10+10+10+10+10+10+10+10-3-3-3
92 10+10+10+10+10+10+10+10+3+3+3+3
93 10+10+10+10+10+10+10+10+10+3
94 10+10+10+10+10+10+10+10+10-3-3
95 10+10+10+10+10+10+10+10+3+3+3+3+3
96 10+10+10+10+10+10+10+10+10+3+3
97 100-3
98 80+18
99 90+9
100 100                                                         10^10                        10.000.000.000
101 80+10+10+10-3-3-3                                 10^11/2^3 = 10^11/8 = 12,500,000,000    (still without a calculator!)
102 90+3+3+3+3
103 100+3
104 110-3-3
105 90+3+3+3+3+3
106 100+3+3
107 110-3
108 90+3+3+3+3+3+3
109 100+3+3
110 100+10
111 120-3-3-3
112 100+3+3+3+3
113 110+3
114 120-3-3
115 100+3+3+3+3+3
116 110+3+3
117 120-3
118 100+3+3+3+3+3+3
119 110+3+3
120 100+10+10                                         10^12                        1.000.000.000.000