Understanding Bels and Decibels: From Power Gain to Voltage Conversion

The Bel is Used to Represent Gain

In its most straightforward definition, an amplifier’s gain is the ratio of output power to input power. Because it is a ratio, it is dimensionless. Nevertheless, engineers use a dedicated unit—called the bel—to express gain in a more readable format.

The bel was originally devised to quantify signal loss in telephone cabling rather than gain in amplifiers. Named after Alexander Graham Bell, the unit captured the power attenuation over a standard cable length. Today it is defined as the base‑10 logarithm of a power ratio:

Understanding Bels and Decibels: From Power Gain to Voltage Conversion

The Bel is Nonlinear

Because the bel is logarithmic, it behaves nonlinearly. The following table illustrates how power losses and gains translate between simple ratios and bel values:

Moving from the Bel to the Decibel

The bel is a relatively large unit, so a metric prefix—deci—was applied to create the decibel (dB). The decibel is now the ubiquitous unit for expressing logarithmic gains and losses. Below is a comparison between power ratios and decibel values:

Logarithmic scales are especially useful because human hearing is logarithmic: doubling perceived loudness requires a ten‑fold increase in physical sound power. A 1‑bel loss corresponds to a 50 % perceived loss, and a 1‑bel gain doubles perceived intensity.

Other Logarithmic Scale Examples: Richter Scale and Chemical pH

Richter Scale

The Richter scale for earthquakes is a direct analogue. A 6.0 reading is ten times more powerful than a 5.0; a 7.0 is one hundred times more powerful than a 5.0; and a 4.0 is one‑tenth as powerful as a 5.0.

Chemical pH

Similarly, the pH scale is logarithmic: each unit represents a ten‑fold difference in hydrogen‑ion concentration. The power of logarithmic scales lies in representing vast ranges with a compact set of numbers.

Using the Bel to Express System Gains and Losses

When multiple amplifier stages are cascaded, the overall power gain is the product of the individual ratios:

Overall gain = (3)(5) = 15

If each stage’s gain is expressed in bels, the total system gain is simply the sum of the individual bel values:

In this example, 0.477 B + 0.699 B = 1.176 B, demonstrating the additive property of logarithmic units.

Gains using Decibels

Applying the same logic to decibels yields:

4.77 dB + 6.99 dB = 11.76 dB.

Mathematically, the sum of logarithms corresponds to the product of the original numbers—a principle exploited by slide rules and still fundamental when working with logarithmic scales.

Conversion of Decibels and Unitless Ratio

To revert from a decibel or bel value back to a plain power ratio, apply the antilogarithm (10 raised to the exponent). For decibels, the exponent is the decibel value divided by 10:

Example: If a 1 W input yields 10 W output, the power gain in dB is:

AP(dB) = 10*log10(PO / PI) = 10*log10(10 /1) = 10*1 = 10 dB

Example: A 20 dB power gain corresponds to a ratio of 10^(20/10) = 100.

AP(dB) = 20 = 10*log10(AP_ratio)\n20/10 = log10(AP_ratio)\n10^(20/10) = 10^2 = 100 = AP_ratio

Converting Power Gain to Voltage/Current Gain

Because a bel measures power, translating a voltage or current gain into bels or decibels requires converting that gain to an equivalent power gain. With a constant load, a voltage or current gain of 2 yields a power gain of 4 (2²); a gain of 3 yields 9 (3²). Consequently, when converting, an exponent of 2 is applied to the ratio inside the logarithm:

Because multiplying by 2 inside the log is equivalent to taking the log of the squared ratio, the formula simplifies to:

To recover a unitless voltage or current ratio from a decibel value, use:

While the bel is tailored for power, the neper (symbol Np) uses the natural logarithm and directly expresses voltage or current gains. In practice, the neper is rarely used in U.S. engineering; decibels remain the standard.

Example: A 600 Ω audio line amplifier receives 10 mV and delivers 1 V. The power gain in dB is:

A(dB) = 20*log10(VO / VI) = 20*log10(1 /0.01) = 20*2 = 40 dB

Example: For a 20 dB voltage gain with equal 50 Ω input and output impedances, the voltage ratio is 10^(20/20) = 10.

A Review of the Decibel

Gains and losses can be expressed as a unitless ratio or in bels (B) or decibels (dB). A decibel is one‑tenth of a bel.
To convert a power ratio to bels or decibels, use the equations shown above.
When applying the bel or decibel to voltage or current ratios, include a factor of 2 in the logarithm to account for the squared relationship to power.
A positive value indicates amplification; a negative value indicates attenuation. Zero denotes unity gain.
For multi‑stage amplifier systems, multiply individual power ratios to obtain the overall ratio, but add the bel or decibel figures to find the total logarithmic gain.

RELATED WORKSHEETS:

Decibel Measurements Worksheet
Elementary Amplifier Theory Worksheet

Understanding Amplifier Gain: Voltage, Current, and Power Understanding Absolute Decibel Scales in Sound, Power, and Telecommunications

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