Understanding Decibels
April 1974 Popular Electronics

April 1974 Popular Electronics Table of Contents Wax nostalgic about and learn from the history of early electronics. See articles from Popular Electronics, published October 1954 - April 1985. All copyrights are hereby acknowledged.

Since new people are constantly entering the field of electronics, there is a constant need to post articles covering some of the basics of the craft. Just as the seasoned practitioner looks to currently published magazines and books for guidance, so too do contemporary technician and engineer fledglings. Decibels have long been a cause of confusion for many - even some who have been in the field for many years. I have seen on many occasions engineers who are way smarter than me routinely mix units of dB (dimensionless) with units of dBm and dBV (power and volts, respectively) when writing. In this 1974 Popular Electronics magazine piece entitled "Understanding Decibels," author George Board does as good of a job as anyone I've read at explaining the basics.

Understanding Decibels

Here Are Some Rules of Thumb that Make it Easy to Use and Understand dB's

By George Board

Learning to understand the decibel - really understanding it - can be very confusing. Actually, a decibel is nothing more than the logarithm of the ratio of two power levels; but the problem gets complex when the term is used in such diverse areas as antenna gain or microphone voltage. As a result, the only people who toss off decibel ratings with apparent confidence are salesmen and others who have memorized the values without knowing what they really mean.

Actually, understanding decibels requires just a slight amount of general knowledge and two simple rules of thumb. There is no "higher" math involved. The basic expression is

G_dB = 10 log₁₀ (P₂/P₁)

The power ratio has no units since the two powers cancel each other.

The output power, P₂, is some multiple of the input power, P₁. For example, with a power ratio of 2 and an input power of 5 watts, the output power is twice the input, or 10 watts. This power ratio of 2 expressed in decibels is 3 dB. (The output is 3 dB greater than the input.) The input power, then, is usually the starting point or reference level to which the output power is compared. Knowing the reference level defines what the dB measures.

The concept of reference level is not found only in the use of decibels. Voltages are usually referenced to one volt. Multiples of one volt can then be used to define all voltage levels. Similarly, the power ratio in decibels defines the power level in multiples of the reference level. The difference is that the reference for voltage is one volt, while the reference for dB could be any defined power or just an arbitrary test input.

These references are usually understood for each type of measurement. Typically, in measuring acoustical energy, 10^-6 watt/cm² is used as a reference. In antenna measurements, everything is compared to a simple dipole. Hopefully, if the reference is not standard, it is abbreviated as a letter following the dB value. Audio power, dBm, is referenced to 1 milliwatt. Other common references are dBW (1 watt reference) and dBn (thermal noise). There is no set rule for the derivation of these abbreviations; they are originated for special cases.

Why Are They Used?

The beauty of using the decibel system is that decibel figures can be added (as can any logarithm) rather than multiplied, as must be done with regular gain figures. The circuit gains in dB from one stage to the next can simply be added or subtracted from the input reference level. For example, assume that a microphone has an output of 0.00001 mW (-50 dBm) which is attenuated by a factor of 1/10 ( -10 dB) in the cable to the amplifier. The amplifier has a power gain of 10,000,000,000 (100 dB). The output of the amplifier can be found in one of two ways:

(1) 0.00001 mW X 1/10 X 10,000,000,000 = 10,000 mW = 10 W

(2) -50 dBm - 10 dB + 100 dB = 40 dBm = 10 W

Using the dB figures requires an additional step to convert the answer to watts, but the rest of the calculation is much easier. The numbers are of a more manageable size and addition is easier than multiplication.

Any signal level or known characteristic can be used as a reference as long as it is related to power. The power measurements, however, are seldom taken directly. That would require a calorimeter or a direct-reading wattmeter. Usually the voltage or current through a known resistance is measured and the power is calculated from P = V²/R = I²R = VI.

Input and Output Resistances

The power ratio is simplified when both the input and output resistances are equal. By cancelling the like resistance terms, it is only necessary to measure voltage or resistance and square it to obtain the power ratio. The logarithm of the voltage ratio squared is twice the logarithm of the voltage ratio unsquared. Thus,

G_dB = 20 log₁₀ V₂/V₁

The special case of having equal input and output resistances is not unusual. Equipment requiring many signal interconnections usually has input and output resistances of 50 ohms. In such a system, the formula

20 log₁₀ V₂/V₁ is always usable in finding the dB gain.

Instant chaos develops if the input and output resistances are not equal. The voltage ratio squared would then not be equivalent to the power ratio. In this case the power ratio must be calculated using the proper resistance values.

Rules of Thumb

Knowing what decibel ratings are, the next thing to learn is how they are used. When a salesman says, "Instead of the beam, why not get this new kW rig," he knows that the beam antenna gain in dB cannot be directly compared to the power ratio between your old transmitter and a kilowatt. To figure out what he means, using algebra and logarithm tables, would take too long and making a guess is difficult with decibels. Being logarithmic, they don't increase in a normal fashion. A power ratio of 5, for instance, represents 7 dB, but increasing the ratio to 10 only increases the decibels by 3. So some rules of thumb are useful.

The first rule of thumb pertaining to the decibel scale is that adding 3 dB doubles the power ratio. (The factor is actually 1.9953 ... , but doubling is close enough.) Thus the gain represented by 6 dB is twice that of 3 dB just as a gain of 58 dB is twice that of 55 dB. Similarly, a gain of 55 dB is half that of 58 dB and -3 dB is a power ratio of one half.

If powers of two were so easy to remember, things would be easy. But it wouldn't be worth the effort to figure out that 30 dB is 10 times 3 dB for a power ratio of 2¹⁰.

The second rule of thumb is that each 10 dB represents a power-of-ten change in the power ratio. Or, add 10 dB for each zero in the power ratio. If the power ratio is less than one, subtract 10 dB for each zero, including the zero to the left of the decimal. A ratio of 1000 is 30 dB (3 zeros), and a ratio of 0.001 is -30 dB (3 zeros).

Combining the counting of zeros and the multiples of two, the simplified system expands to provide a handy reference for all decibel levels. A power ratio of 20,000 is 40 dB (4 zeros) plus 3 dB for the factor of two, totaling 43 dB. Half (-3 dB) of 100 (20 dB) is 50; so 50 is 20 dB minus 3 dB or 17 dB. Interpolating, 0.0001875 is between 0.0001 (-40 dB) and 0.0002 (-37 dB). But the number is much closer to 0.0002, so it must be -38 dB. A closer figure could only be obtained by using logarithm tables and many calculations.

The two rules of thumb for power ratios can also be used for voltage ratios. Remember that the voltage ratio in dB is 20 log₁₀ V₂/V₁, and the power ratio is the square of the voltage ratio. Thus a power ratio of 16 (12 dB) is the same as a voltage ratio of 4. Now, 4 would be 6 dB as a power ratio - one half of the 12 dB for 16. So, the voltage ratio in dB is twice what the rules for the power ratio would make it. In other words, a voltage ratio of 2 is 6 dB, 4 is 12 dB, and 0.01 is -40 dB.

Summary

(1) Adding decibels is like multiplying power ratios. (2) The reference level must be known. (3) The voltage ratio in dB (resistances equal) is twice what the power ratio would be. (4) Doubling the power ratio adds 3 dB (6 dB for the voltage ratio). (5) Multiplying the power ratio by 10 adds 10 dB (20 dB For the voltage ratio) .

Posted February 6, 2024
(updated from original post on 5/3/2017)