An introductory column for measurement beginners: Calling all those who don't understand! Part 8: "What is dB (Decibel)? Part 3"

In the previous two installments, we explained the term dB (decibels) and how the conversion calculations to true values differ depending on whether you're using power gain (power ratio) or voltage gain (voltage ratio). This time, as the final installment of our dB series for beginners, we'll introduce some of the advantages of using dB and explain why dB is used to report measurement results. For this explanation, we will use the relatively common method of calculating voltage ratios.

Benefits of using dB

Reduce the risk of misreading measurements.

Using dB allows you to represent large values with fewer digits. It's generally said that humans can reliably handle numbers with three or four digits. For example, the PINs for cash cards and credit cards are four digits long, and telephone numbers are divided into areas, exchanges, and individual numbers using hyphens, resulting in a limited number of digits.
In actual measurements, it's not uncommon to get results with extremely large numbers, such as 10,000 times or 1,000,000 times. By expressing such large numbers in dB notation, the results can be represented with fewer digits. For example, 1 to 10,000,000,000 (10 billion) times in dB notation can be expressed in the range of 0 to 200 dB.
As explained last time, if we let be the voltage gain (true value) and its dB value be L, then the following relationship is obtained.

If L = 200dB,
The true value of the voltage gain is 10 ^(200/20) = 10 ¹⁰ = 10,000,000,000 (10 billion).
If L = -200dB,
The true value of the voltage gain is 10 ^(-200/20) = 10^{- 10} = 1/10,000,000,000 (1/10 billion).
This allows us to handle extremely small numbers, such as 1/10 billion, and extremely large numbers, such as 10 billion, using three-digit numbers ranging from -200 to 200 (dB), which helps prevent misreading of values during measurement.

It makes it easier to compare values with vastly different scales on the same graph.

If the difference in measured values is only a few to tens of times, you can compare their relative magnitudes by displaying them on the same graph, such as a bar graph. However, if the data being compared is more than 100 times greater than the original data, it becomes difficult to accurately compare their relative magnitudes on the same scale using a numerical graph. In such cases, converting to dB values and graphing them has the advantage of making it easier to compare their relative magnitudes.
Let's illustrate this using the size of the stars in our solar system as an example.

Table 1. Size (diameter) of stars in the solar system and the diameter ratio of each star relative to Earth.

Table 1 shows the size (diameter) of each star, its diameter ratio relative to Earth, and its dB value. Graph 1 shows these values as a bar graph.
Because the sun is extraordinarily large in the solar system, in a real-time graph (left), objects like the Earth, which are less than 1/100th the size of the sun, appear almost at zero. On the other hand, in a dB graph (right), the moon, which is about 1/300th the size of the sun, can be represented on the same scale.
In actual measurements, it's not uncommon to compare data with values exceeding 100 times. In such cases, converting the raw values to dB values before comparing the data makes it easier to compare their relative magnitudes.

In FFT analyzers used for frequency analysis of sound, vibration, and other phenomena, power spectrum graphs are frequently used, and the relationship between linear and logarithmic displays in this context is the same.

Making rough calculations of addition and division easier

In dB calculations, multiplication becomes addition, and division becomes subtraction. Many readers may be wondering what this means. Simply put, it's as follows:
To find the product of the arguments, calculate the sum of their dB values.
To find the quotient of the numbers, calculate the difference in their dB values.
In the previous column (What is dB? Part 2), we explained the relationship between dB values and power ratios and voltage ratios in the explanation of power gain and voltage gain (Table 2). This time, we will use this table again to verify the relationship.

Table 2. Relationship between dB (decibels) values and their corresponding true values (power ratio and voltage ratio).

First, let's look at multiplication of the numbers.
The product of the voltage ratios 3.16 and 10 is 3.16 × 10 = 31.6
If we perform the same calculation using dB values, we will be calculating the sum: 10(dB) + 20(dB) = 30(dB).
As you can see in Table 2, a voltage ratio of 31.6 corresponds to 30 dB.
Next is division of the argument.
The quotient of a voltage ratio of 100 times and 10 times is 100 ÷ 10 = 10
If we perform the same calculation using dB values, we will be calculating the difference: 40(dB) - 20(dB) = 20(dB).
As you can see in Table 2, a voltage ratio of 10 corresponds to 20 dB.

Furthermore, if you remember that a voltage ratio of 20dB is 10 times, 10dB is approximately 3 times, and 6dB is 2 times, you can easily perform simple calculations when converting from dB values to numerical values.
For example, 26 dB is 26(dB) = 20(dB) + 6(dB), so in numerical calculations, it can be converted to 20 times the reference value by multiplying by 10 and then by 2. Similarly, for 58 dB, after decomposing it as 58(dB) = 20(dB) + 20(dB) + 6(dB) + 6(dB) + 6(dB), we can easily calculate the numerical value by multiplying by 10, 10, 2, 2, and then by 2, resulting in 800 times the reference value.
Recently, this calculation is frequently used in the fields of transmission systems and servo control systems.
For example, if transmission system 1 has a gain of 20 dB (10 times) and transmission system 2 has a gain of 14 dB (5 times), the gain when the two transmission systems are connected in series can be easily calculated as 20 (dB) + 14 (dB) = 34 (dB).
Unfortunately, there are no convenient, simplified calculation methods for adding or subtracting dB values as described above; calculations must be performed after converting to the true values. However, 100 years ago, when dB was invented, the use of scientific calculators and PCs was unimaginable. The fact that dB could be easily used for multiplication and division alone must have been a source of pride for researchers at the time.
This concludes our introductory series on dB, where we've discussed the effectiveness of dB.
The information I've provided in this column is only a small part of the whole picture.
We hope this column will spark your interest in dB and deepen your knowledge of the subject. Below are our technical reports on dB and links to previous measurement columns. Please refer to them as reference materials.
【reference】
Ono Ono Sokki Technical Report: "What is dB (Decibel)?"
Ono Sokki Measurement Column: "About dB (Decibel)"

(Excerpt from the email newsletter issued on March 16, 2022)