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  Logarithmic units and decibels

Power gain

Power gain, in decibels (dB), is defined by the 10 log rule as follows:

Where P in and P out are the input and output powers respectively.

A similar calculation can be done using a natural logarithm instead of a decimal logarithm,

and without the factor of 10, resulting in nepers instead of decibels:

Voltage gain

When power gain is calculated using voltage instead of power, making the

substitution ( P =V  2/ R), the formula is: 

In many cases, the input and output impedances are equal, so the above equation can be

simplified to:

And then the 20 log rule: 

This simplified formula is used to calculate a voltage gain in decibels, and is equivalent to a

 power gain only if the impedances at input and output are equal.

Current gain

In the same way, when power gain is calculated using current instead of power, making the

substitution ( P  = I  2 R), the formula is:

In many cases, the input and output impedances are equal, so the above equation can be

simplified to:

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And then:

This simplified formula is used to calculate a current gain in decibels, and is equivalent to

the power gain only if the impedances at input and output are equal.

The "current gain" of a  bipolar transistor , hFE or hfe, is normally given as a dimensionless

number, the ratio of I c to I  b (or slope of the I c-versus- I b graph, for hfe).

In the cases above, gain will be a dimensionless quantity, as it is the ratio of like units

(Decibels are not used as units, but rather as a method of indicating a logarithmic

relationship). In the bipolar transistor example it is the ratio of the output current to the input

current, both measured in Amperes. In the case of other devices, the gain will have a value

in SI units. Such is the case with the  operational transconductance amplifier ,  which has an

open-loop gain (transconductance) in Siemens (mhos), because the gain is a ratio of the

output current to the input voltage.

Example 

Q. An amplifier has an input impedance of 50 ohms and drives a load of 50 ohms. When its

input ( ) is 1 volt, its output ( ) is 10 volts. What is its voltage and power gain?

A. Voltage gain is simply:

The units V /V  are optional, but make it clear that this figure is a voltage gain and not a power

gain. Using the expression for power, P  = V 2/ R, the power gain is:

Again, the units W/W are optional. Power gain is more usually expressed in decibels, thus:

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A gain of factor 1 (equivalent to 0 dB) where both input and output are at the same voltage

level and impedance is also known as unity gain.