Editing Gain (electronics)

{{short description|Ability of a circuit to increase the power or amplitude of a signal}}
{{distinguish|Antenna gain}}

In [[electronics]], '''gain''' is a measure of the ability of a [[two-port]] [[electrical network|circuit]] (often an [[amplifier]]) to increase the [[Electric power|power]] or [[amplitude]] of a [[signal]] from the input to the output port<ref name="Graf">{{cite book
 | last1  = Graf
 | first1 = Rudolf F. 
 | title  = Modern Dictionary of Electronics
 | publisher = Newnes
 | edition = 7
 | date   = 1999
 | pages  = 314
 | url    = https://books.google.com/books?id=AYEKAQAAQBAJ&q=amplifier+gain&pg=PA314
 | isbn   = 0080511988
 }}</ref><ref name="Basu">{{cite book
 | last1  = Basu
 | first1 = Dipak 
 | title  = Dictionary of Pure and Applied Physics
 | publisher = CRC Press
 | date   = 2000
 | pages  = 157
 | url    = https://books.google.com/books?id=-QhAkBSk7IUC&q=gain&pg=PA157
 | isbn   = 1420050222
 }}</ref><ref name="Bahl">{{cite book
 | last1  = Bahl
 | first1 = Inder  
 | title  = Fundamentals of RF and Microwave Transistor Amplifiers
 | publisher = John Wiley and Sons 
 | date   = 2009
 | pages  = 34
 | url    = https://books.google.com/books?id=u9kVm8RG4k4C&q=gain+%22two-port&pg=PA34
 | isbn   = 978-0470462317
 }}</ref><ref name="White">{{cite book
 | last1  = White
 | first1 = Glenn  
 | last2  = Louie
 | first2 = Gary J
 | title  = The Audio Dictionary
 | publisher = University of Washington Press
 | edition = 3
 | date   = 2005
 | pages  = 18
 | url    = https://books.google.com/books?id=DulVm8t88QkC&q=amplifier+%22voltage+gain%22+%22current+gain%22++%22power+gain%22&pg=PA18
 | isbn   = 0295984988
 }}</ref> by adding energy converted from some [[power supply]] to the signal. It is usually defined as the mean [[ratio]] of the [[signal]] amplitude or power at the output [[port (circuit theory)|port]] to the amplitude or power at the input port.<ref name="Graf" /> It is often expressed using the [[logarithm]]ic  [[decibel]] (dB) units ("dB gain").<ref name="White" />  A gain greater than one (greater than zero dB), that is, amplification, is the defining property of an [[active device]] or circuit, while a passive circuit will have a gain of less than one.<ref name="White" />

The term ''gain'' alone is ambiguous, and can refer to the ratio of output to input  [[voltage]] (''voltage gain''), [[Electric current|current]] (''current gain'') or electric power (''power gain'').<ref name="White" />  In the field of audio and general purpose amplifiers, especially [[operational amplifier]]s, the term usually refers to voltage gain,<ref name="Basu" />  but in [[radio frequency]] amplifiers it usually refers to power gain.  Furthermore, the term gain is also applied in systems such as [[sensor]]s where the input and output have different units; in such cases the gain units must be specified, as in "5 microvolts per photon" for the [[responsivity]] of a [[photosensor]]. The "gain" of a [[bipolar transistor]] normally refers to forward current transfer ratio, either ''h''<sub>FE</sub> ("beta", the static ratio of ''I''<sub>''c''</sub> divided by ''I''<sub>b</sub> at some operating point), or sometimes ''h''<sub>fe</sub> (the small-signal current gain, the slope of the graph of ''I''<sub>''c''</sub> against ''I''<sub>''b''</sub> at a point).

The gain of an electronic device or circuit generally varies with the [[frequency]] of the applied signal.  Unless otherwise stated, the term refers to the gain for frequencies in the [[passband]], the intended operating frequency range of the equipment. 
The term ''gain'' has a different meaning in [[antenna (radio)|antenna]] design; [[antenna gain]] is the ratio of [[radiation intensity]] from a directional antenna to <math>P_\text{in}/4\pi</math> (mean radiation intensity from a lossless antenna).

[[File:Amplification2.svg|thumb|Graph of the input <math>v_i(t)</math> ''<span style="color:blue;">(blue)</span>'' and output voltage <math>v_o(t)</math> ''<span style="color:red;">(red)</span>'' of an ideal linear [[amplifier]] with a voltage gain of 3 with an arbitrary input signal.   At any instant the output voltage is three times the input voltage.]]

==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''.

==See also==
{{div col|colwidth=20em}}
* [[Active laser medium]]
* [[Antenna gain]]
* [[Aperture-to-medium coupling loss]]
* [[Automatic gain control]]
* [[Attenuation]]
* [[Complex gain]]
* [[DC offset]]
* [[Effective radiated power]]
* [[Gain before feedback]]
* [[Insertion gain]]
* [[Loop gain]]
* [[Open-loop gain]]
* [[Net gain (telecommunications)|Net gain]]
* [[Power gain]]
* [[Process gain]]
* [[Transmitter power output]]
{{div col end}}

==References==
{{Reflist}}
* {{FS1037C}}

[[Category:Antennas (radio)]]
[[Category:Electronics concepts]]
[[Category:Transfer functions]]
[[Category:Electrical parameters]]