Editing Gain (electronics) (section)

==Logarithmic units and decibels==

===Power gain===
[[Power gain]], in [[decibel]]s (dB), is defined as follows:

:<math>\text{gain-db}=10 \log_{10} \left(\frac{P_\text{out}}{P_\text{in}}\right)~\text{dB},</math>

where <math>P_\text{in}</math> is the power applied to the input, <math>P_\text{out}</math> is the power from the output.

A similar calculation can be done using a [[natural logarithm]] instead of a decimal logarithm, resulting in [[neper]]s instead of decibels:

:<math>\text{gain-np} = \frac{1}{2} \ln\left(\frac{P_\text{out}}{P_\text{in}}\right)~\text{Np}.</math>

===Voltage gain===
The power gain can be calculated using voltage instead of power using [[Joule's first law]] <math>P = V^2/R</math>; the formula is:

:<math>\text{gain-db} = 10 \log{\frac{\frac{V_\text{out}^2}{R_\text{out}}}{\frac{V_\text{in}^2}{R_\text{in}}}}~\mathrm{dB}.</math>

In many cases, the input impedance <math>R_\text{in}</math> and output impedance <math>R_\text{out}</math> are equal, so the above equation can be simplified to:

:<math>\text{gain-db} = 10 \log \left(\frac{V_\text{out}}{V_\text{in}}\right)^2~\text{dB},</math>

:<math>\text{gain-db} = 20 \log \left(\frac{V_\text{out}}{V_\text{in}}\right)~\text{dB}.</math>

This simplified formula, the [[20 log rule]], is used to calculate a '''voltage gain''' in decibels and is equivalent to a power gain if and only if the [[Electrical impedance|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 <math>P = I^2 R</math>, the formula is:

:<math>\text{gain-db} = 10 \log{\left(\frac{I_\text{out}^2 R_\text{out}}{I_\text{in}^2 R_\text{in}}\right)}~\text{dB}.</math>

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

:<math>\text{gain-db} = 10 \log \left(\frac{I_\text{out}}{I_\text{in}}\right)^2~\text{dB},</math>

:<math>\text{gain-db} = 20 \log \left(\frac{I_\text{out}}{I_\text{in}}\right)~\text{dB}.</math>

This simplified formula is used to calculate a '''current gain''' in decibels and is equivalent to the power gain if and only if the [[Electrical impedance|impedances]] at input and output are equal.

The "current gain" of a [[bipolar transistor]],  <math>h_\text{FE}</math> or <math>h_\text{fe}</math>, is normally given as a dimensionless number, the ratio of <math>I_\text{c}</math> to <math>I_\text{b}</math> (or slope of the <math>I_\text{c}</math>-versus-<math>I_\text{b}</math> graph, for <math>h_\text{fe}</math>).

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 [[ampere]]s. 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 (unit)|siemens]] ([[mho]]s), 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 (<math>V_\text{in}</math>) is 1 volt, its output (<math>V_\text{out}</math>) is 10 volts. What is its voltage and power gain?

A. Voltage gain is simply:

:<math>\text{gain} = \frac{V_\text{out}}{V_\text{in}} = \frac{10}{1} = 10~\text{V/V}.</math>

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''<sup>2</sup>/''R'', the power gain is:

:<math>\text{gain} = \frac{V_\text{out}^2/50}{V_\text{in}^2/50} = \frac{V_\text{out}^2}{V_\text{in}^2} = \frac{10^2}{1^2} = 100~\text{W/W}.</math>

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

:<math>\text{gain-db} = G_\text{dB} = 10 \log G_\text{W/W} = 10 \log 100 = 10 \times 2 = 20~\text{dB}.</math>

=== Unity gain ===
A gain of factor 1 (equivalent to 0&nbsp;dB) where both input and output are at the same voltage level and impedance is also known as ''[[1 (number)|unity]] gain''.