An introductory column for measurement beginners: Calling all those who don't understand! Part 7: "What is dB (Decibel)? Part 2"

In the previous lesson, we explained that dB (decibel) and B (bel) are comparative quantities of power expressed logarithmically (common logarithm). This time, we will continue our explanation of the necessity of the SI prefix d (deci), and the power gain and voltage gain of dB.

The necessity of the SI prefix d (deci).

As explained in the previous column, "What is dB? Part 1," B (Bel) is the common logarithm of the ratio of a measured physical quantity to a reference physical quantity. For example, if the measured value is 10 times the reference value, it is ^10¹⁻¹, so it is 1B (1 Bel); if it is 100 times, it is ^10²⁻¹, so it is 2B (2 Bels); and if it is 1/10 times, it is ^10⁻¹, so it is -1B (-1 Bel). However, actual measured values are not such neat numbers, but are numbers such as twice or five times the reference value, so as shown in Table 1, when expressed logarithmically, fractional parts occur.
Table 1. Relationship between natural numbers and logarithms (common logarithms).

Multiplying by 1 (0B) and 10 (1B), which are the same as the reference values, results in round numbers without fractions. However, multiplying by 2 to 9 results in numbers with fractional parts after the decimal point, as shown above. When expressed in B (Bell), the representative values in the table above are approximately 0.3B for 2 times, 0.6B for 4 times, 0.7B for 5 times, and 0.9B for 8 times.
​^{Note 1)} The notation uses a numerical value of 1 or more, but looking at the corresponding B (Bell) value, one might be misled by the numerical image and mistakenly think it's less than 1. Therefore, it was thought that the B (Bell) notation should be multiplied by 10 by adding a 'd' (deci) so that the values up to 10 times, which are relatively frequently used, are greater than or equal to 1. If we express the above representative values in dB notation, then 2 times is approximately 3 dB, 4 times is approximately 6 dB, 5 times is approximately 7 dB, and 8 times is approximately 9 dB. This gives the impression of being greater than 1 intuitively. However, strictly speaking, even in dB notation, values less than 1 dB (0.1 B), that is, 10 to the power of 0.1 (= approximately 1.26), will be expressed as a decimal less than 1. However, since there will likely be few cases where values less than 1 dB are used, in practice, values of 2 dB (0.2 B, approximately 1.58 in numerical value) or more should be easy to understand.
Due to these circumstances, dB (decibel), which includes the SI prefix d (deci), came to be used.
Note 1) The argument of a logarithm, log _a x, refers to the value of x.
Table 1 shows the case where the argument is a natural number.

Power gain (power ratio), voltage gain (voltage ratio)

Gain refers to the ratio of input to output in an electrical circuit. It describes how the output changes when an input of 1 is applied to a system. The quantity used can be power, voltage, or current, and these are called power gain, voltage gain, and current gain, respectively. While gain can simply be expressed as a multiple, dB (decibels) is more commonly used, so this section will focus on explaining this unit.
To put it simply, while we have explained that B (Bell) in relation to power gain is related to the comparative expression of power, the multiple notation changes when it comes to voltage (current) gain.
The table below shows the conversion values for power gain (power ratio) and voltage gain (voltage ratio).
Table 2. Relationship between dB (decibels) values and their corresponding true values (power ratio and voltage ratio).

At a 0dB value, both the power ratio and voltage ratio are 1, but the other values are different. This is due to the difference between power and voltage. While this column has tried to avoid using mathematical formulas as much as possible, we will now use a few. Even readers who are not good at math should try their best to follow along.
Do you, dear readers, remember learning about Ohm's Law in middle school science class?
Voltage: E (V) = Current: I (A) × Resistance: R (Ω)
This is the gist of it. The formula means that if you know two of the three values—voltage, current, and resistance—you can find the remaining one. In other words, the formula "E (voltage) = I (current) × R (resistance)" can also be written as "R (resistance) = E (voltage) ÷ I (current)" or "I (current) = E (voltage) ÷ R (resistance)".
The formula for calculating electrical power, which represents the amount of electrical energy, is as follows:
Power: Watts (W) = Voltage: E (V) × Current: I (A)
Substituting "I (current) = E (voltage) ÷ R (resistance)," which is rewritten using Ohm's Law, into the above formula for calculating power, we obtain the following formula:
Power: Watts (W) = Voltage: E (V) × {Voltage: E (V) ÷ Resistance: R (Ω)}
= {Voltage: E(V)} ² ÷ Resistance: R(Ω)
In other words, if the resistance (Ω) is constant, the power (W) is proportional to the square of the voltage (V).
Let's verify this. Suppose we have this circuit with a voltage of 1V and a current of 1A. According to Ohm's law, the resistance of this circuit is 1Ω (1V ÷ 1A). The power is 1W (1V × 1A). Now, let's apply a voltage of 2V to the same circuit. Since we previously found the resistance to be 1Ω, the current will be 2A (2V ÷ 1Ω) according to Ohm's law. The power will be 4W (2V × 2A), which is four times the power at 1V. Similarly, if we apply 3V, the current will be 3A and the power will be 9W, which is nine times the power at 1V.
Although this explanation has been lengthy, what I want to say here is that power gain (power ratio) is proportional to the square of the voltage gain (voltage ratio). Also, whether it's power gain or voltage gain, the corresponding dB value must be the same. The conversion formulas for each gain are shown in (Equation 1) and (Equation 2). Since the power gain is the square of the voltage gain, while the power gain is calculated by taking the common logarithm and multiplying by 10, the voltage gain is calculated by multiplying by 20.
The conversion formula for power gain is given by: where L is the dB value of the power gain and is the power gain (real value),

Furthermore, since the voltage gain is the square of the power gain, if we let be the voltage gain (true value), then

The power ratio and voltage ratio values in Table 2 are calculated from the second equation of equation (1) and equation (2), respectively.
Generally, signals to a standard analyzer are often input as voltage values via a detector, so voltage gain (voltage ratio) is frequently used. However, it is necessary to carefully examine and determine which case is being used.
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Some readers may have found the latter half of this article difficult. Ohm's law and logarithmic calculations are topics covered in middle school science and high school mathematics. Readers who are interested should definitely review these topics on their own.

Next time, in the final installment of our introductory series on dB, we will explain why we use dB and what its advantages are.

Below are our technical reports on dB and a selection of past measurement columns. Please read them if you are interested in dB after reading this document.
【reference】
Ono Ono Sokki Technical Report: "What is dB (Decibel)?"
Ono Sokki Measurement Column: "About dB (Decibel)"

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