No.8 Various measurement instrument: What is dB? (Part 3)

“What is a decibel?” in the Part 1 and Part 2, I have explained the terminology and the difference in conversion calculation to an antilogarithm depending on the power gain (power ratio) and voltage gain (voltage ratio). In the final, Part 3, I will introduce some of the advantages of using dB and explain why dB is used so often. For this explanation, I will use the voltage ratio calculation method, which is used more frequently.

< Advantages of using dB >

1) Reducing the risk of misreading measured values

Using dB enables to express large values with a small number of digits. It is said that the number of digits that human less misunderstand is usually 3 to 4 digits. For example, the PIN code of cash cards and credit cards is set to 4 digits, and the telephone number is also divided into a small number of digits by- (hyphen) between the area code, city code, and individual number. In actual measurement, it is not uncommon for the result to be a large number of digits, such as 10,000 times or 1 million times.　 By using dB for numbers with such a large number of digits, the result can be expressed with a small number of digits. For example, 1 to 10,000,000,000 (10 billion) times can be expressed in 0 to 200 dB.

As explained earlier, if  is the voltage gain (antilogarithm) and L is its dB value, the conversion formula is shown in Equation 1.

If L = 200dB,  is 10 ^(200/20) = 10¹⁰ = 10,000,000,000

If L = -200dB,   is 10 ^(-200/20) = 10^-10 = 1/10,000,000,000

Even numbers with many digits can be represented by only 3-digit numbers of -200 to 200 (dB), which helps prevent misreading of values during measurement.

2) Easier to compare the numbers whose scales are significantly different on the same graph

If the difference between the values of the measurement results is about a few times to a few tens of times, you can compare by displaying them on the same graph. However, if the data to be compared is more than 100 times the data, it will be difficult to accurately compare them on the antilogarithm graph of the same scale. Even in such a case, there is an advantage that it is easy to compare the differences by converting them to dB value. Let's take an example of comparing the sizes of stars in the solar system.

Table 1 Size (diameter) of the stars in the solar system and Diameter ratio of each star with respect to the earth

Graph 1 Diameter ratio of each star with respect to the earth

The sun is extraordinarily large, thus in the antilogarithm graph (left), the earth, which is 1/100 or less size of the sun, can be only seen near zero. While, in the dB graph (right), the moon, which is about 1/300 size of the sun, can be seen well on the graph of the same scale.

Even in actual measurements, it is often to compare data that exceeds 100 times the difference. In such a case, it is easier to compare the differences by converting the antilogarithm to a dB value.

In the FFT analyzer used for frequency analysis of sound and vibration, the graph of power spectrum is often used. The relationship between displaying values in linear and in log is the same as above.

3) Easy calculation of integration and division

In dB calculations, multiplication is addition and division is subtraction. You may wonder what I’m saying…. Briefly describe as follows:
 - To obtain the product of antilogarithm, calculate the sum of the dB values.
 - To obtain the quotient of antilogarithm, calculate the difference between the dB values.
In the previous column (What is dB? Part 2), I explained the relationship between the dB value and the power ratio/ the voltage ratio (Table 2). I will explain using this table this time as well.

Table 2. Relationship between ｄB value and power ratio/voltage ratio

dB	-20	-6.02	0	3.01	6.02	10	20	30	40
Power ratio	0.01	0.25	1	2	4	10	100	1,000	10,000
Voltage ratio	0.1	0.5	1	1.41	2	3.16	10	31.6	100

First, take a look at the multiplication of antilogarithm.
The product of the voltage ratio 3.16 and 10 is 3.16 × 10 = 31.6.
If the same calculation is performed with the dB value, 10 (dB) + 20 (dB) = 30 (dB).
As you can see in Table 2, the voltage ratio of 31.6 is equivalent to 30 dB.
Second, the division of antilogarithm.
The quotient of the voltage ratio 100 and 10 is 100 ÷ 10 = 10.
If the same calculation is performed with the dB value, 40 (dB) – 20 (dB) = 20 (dB).
As you can see in Table 2, the voltage ratio of 10 is equivalent to 20 dB.

If you remember that 20dB is 10 times, 10dB is about 3 times, and 6dB is 2 times in terms of voltage ratio, you can easily calculate when converting a dB value to an antilogarithm.
For example, 26dB is 26 (dB) = 20 (dB) + 6 (dB), thus the antilogarithm can be converted to 10 x 2 = 20 times. Similarly, in the case of 58 dB, 58 (dB) = 20 (dB) + 20 (dB) + 6 (dB) + 6 (dB) + 6 (dB), thus the antilogarithm can be converted to 10 x 10 x 2 x 2 x 2=800 times.

These days, this calculation is often used in the fields of transmission systems and servo control systems.
For example, if transmission system 1 has a gain of 20 dB and transmission system 2 has 14 dB, the gain when two transmission systems are connected in series is 20 (dB) + 14 (dB) = 34 (dB).

In this column as the final part of basic introduction of dB, I have explained how dB is effective. I hope that this column will make you more interested in dB.

 

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(KH)

[ Reference ]
For further detail information, please refer to the page: What is dB?