Gain (electronics)

<h2 id="logarithmic-units-and-decibels">Logarithmic units and decibels</h2>
<h3>Power gain</h3>
<p><a href="/facts/Power_gain/jRITzwqo">Power gain</a>, in <a href="/facts/Decibel/0LyxAPXh">decibels</a> (dB), is defined as follows:
</p>

gain-db
        
        =
        10
        
          log
          
            10
          
        
        ⁡
        
          (
          
            
              
                P
                
                  out
                
              
              
                P
                
                  in
                
              
            
          
          )
        
         
        
          dB
        
        ,
      
    
    {\displaystyle {\text{gain-db}}=10\log _{10}\left({\frac {P_{\text{out}}}{P_{\text{in}}}}\right)~{\text{dB}},}

<p>where 
  
    
      
        
          P
          
            in
          
        
      
    
    {\displaystyle P_{\text{in}}}
  
 is the power applied to the input, 
  
    
      
        
          P
          
            out
          
        
      
    
    {\displaystyle P_{\text{out}}}
  
 is the power from the output.
</p><p>A similar calculation can be done using a <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a> instead of a decimal logarithm, resulting in <a href="/facts/Neper/nzHnA6ev">nepers</a> instead of decibels:
</p>

gain-np
        
        =
        
          
            1
            2
          
        
        ln
        ⁡
        
          (
          
            
              
                P
                
                  out
                
              
              
                P
                
                  in
                
              
            
          
          )
        
         
        
          Np
        
        .
      
    
    {\displaystyle {\text{gain-np}}={\frac {1}{2}}\ln \left({\frac {P_{\text{out}}}{P_{\text{in}}}}\right)~{\text{Np}}.}

<h3>Voltage gain</h3>
<p>The power gain can be calculated using voltage instead of power using <a href="/facts/Joule%27s_first_law/eANEHVzP">Joule's first law</a> 
  
    
      
        P
        =
        
          V
          
            2
          
        
        
          /
        
        R
      
    
    {\displaystyle P=V^{2}/R}
  
; the formula is:
</p>

gain-db
        
        =
        10
        log
        ⁡
        
          
            
              
                V
                
                  out
                
                
                  2
                
              
              
                R
                
                  out
                
              
            
            
              
                V
                
                  in
                
                
                  2
                
              
              
                R
                
                  in
                
              
            
          
        
         
        
          d
          B
        
        .
      
    
    {\displaystyle {\text{gain-db}}=10\log {\frac {\frac {V_{\text{out}}^{2}}{R_{\text{out}}}}{\frac {V_{\text{in}}^{2}}{R_{\text{in}}}}}~\mathrm {dB} .}

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

gain-db
        
        =
        10
        log
        ⁡
        
          
            (
            
              
                
                  V
                  
                    out
                  
                
                
                  V
                  
                    in
                  
                
              
            
            )
          
          
            2
          
        
         
        
          dB
        
        ,
      
    
    {\displaystyle {\text{gain-db}}=10\log \left({\frac {V_{\text{out}}}{V_{\text{in}}}}\right)^{2}~{\text{dB}},}

gain-db
        
        =
        20
        log
        ⁡
        
          (
          
            
              
                V
                
                  out
                
              
              
                V
                
                  in
                
              
            
          
          )
        
         
        
          dB
        
        .
      
    
    {\displaystyle {\text{gain-db}}=20\log \left({\frac {V_{\text{out}}}{V_{\text{in}}}}\right)~{\text{dB}}.}

<p>This simplified formula, the <a href="/facts/20_log_rule/0LyxAPXh">20 log rule</a>, is used to calculate a voltage gain in decibels and is equivalent to a power gain if and only if the <a href="/facts/Electrical_impedance/I06NSXtN">impedances</a> at input and output are equal.
</p>
<h3>Current gain</h3>
<p>In the same way, when power gain is calculated using current instead of power, making the substitution 
  
    
      
        P
        =
        
          I
          
            2
          
        
        R
      
    
    {\displaystyle P=I^{2}R}
  
, the formula is:
</p>

gain-db
        
        =
        10
        log
        ⁡
        
          
            (
            
              
                
                  
                    I
                    
                      out
                    
                    
                      2
                    
                  
                  
                    R
                    
                      out
                    
                  
                
                
                  
                    I
                    
                      in
                    
                    
                      2
                    
                  
                  
                    R
                    
                      in
                    
                  
                
              
            
            )
          
        
         
        
          dB
        
        .
      
    
    {\displaystyle {\text{gain-db}}=10\log {\left({\frac {I_{\text{out}}^{2}R_{\text{out}}}{I_{\text{in}}^{2}R_{\text{in}}}}\right)}~{\text{dB}}.}

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

gain-db
        
        =
        10
        log
        ⁡
        
          
            (
            
              
                
                  I
                  
                    out
                  
                
                
                  I
                  
                    in
                  
                
              
            
            )
          
          
            2
          
        
         
        
          dB
        
        ,
      
    
    {\displaystyle {\text{gain-db}}=10\log \left({\frac {I_{\text{out}}}{I_{\text{in}}}}\right)^{2}~{\text{dB}},}

gain-db
        
        =
        20
        log
        ⁡
        
          (
          
            
              
                I
                
                  out
                
              
              
                I
                
                  in
                
              
            
          
          )
        
         
        
          dB
        
        .
      
    
    {\displaystyle {\text{gain-db}}=20\log \left({\frac {I_{\text{out}}}{I_{\text{in}}}}\right)~{\text{dB}}.}

<p>This simplified formula is used to calculate a current gain in decibels and is equivalent to the power gain if and only if the <a href="/facts/Electrical_impedance/I06NSXtN">impedances</a> at input and output are equal.
</p><p>The "current gain" of a <a href="/facts/Bipolar_transistor/9BMcGv5Q">bipolar transistor</a>,  
  
    
      
        
          h
          
            FE
          
        
      
    
    {\displaystyle h_{\text{FE}}}
  
 or 
  
    
      
        
          h
          
            fe
          
        
      
    
    {\displaystyle h_{\text{fe}}}
  
, is normally given as a dimensionless number, the ratio of 
  
    
      
        
          I
          
            c
          
        
      
    
    {\displaystyle I_{\text{c}}}
  
 to 
  
    
      
        
          I
          
            b
          
        
      
    
    {\displaystyle I_{\text{b}}}
  
 (or slope of the 
  
    
      
        
          I
          
            c
          
        
      
    
    {\displaystyle I_{\text{c}}}
  
-versus-
  
    
      
        
          I
          
            b
          
        
      
    
    {\displaystyle I_{\text{b}}}
  
 graph, for 
  
    
      
        
          h
          
            fe
          
        
      
    
    {\displaystyle h_{\text{fe}}}
  
).
</p><p>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 <a href="/facts/Ampere/MScqkCgz">amperes</a>. In the case of other devices, the gain will have a value in <a href="/facts/SI/NShmfzlD">SI</a> units. Such is the case with the <a href="/facts/Operational_transconductance_amplifier/kUVmLFtQ">operational transconductance amplifier</a>, which has an open-loop gain (<a href="/facts/Transconductance/lwh8krQa">transconductance</a>) in <a href="/facts/Siemens_(unit)/kLUHhcCj">siemens</a> (<a href="/facts/Mho/kLUHhcCj">mhos</a>), because the gain is a ratio of the output current to the input voltage.
</p>
<h3>Example</h3>
<p>Q. An amplifier has an input impedance of 50 ohms and drives a load of 50 ohms. When its input (
  
    
      
        
          V
          
            in
          
        
      
    
    {\displaystyle V_{\text{in}}}
  
) is 1 volt, its output (
  
    
      
        
          V
          
            out
          
        
      
    
    {\displaystyle V_{\text{out}}}
  
) is 10 volts. What is its voltage and power gain?
</p><p>A. Voltage gain is simply:
</p>

gain
        
        =
        
          
            
              V
              
                out
              
            
            
              V
              
                in
              
            
          
        
        =
        
          
            10
            1
          
        
        =
        10
         
        
          V/V
        
        .
      
    
    {\displaystyle {\text{gain}}={\frac {V_{\text{out}}}{V_{\text{in}}}}={\frac {10}{1}}=10~{\text{V/V}}.}

<p>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, <i>P</i> = <i>V</i>2/<i>R</i>, the power gain is:
</p>

gain
        
        =
        
          
            
              
                V
                
                  out
                
                
                  2
                
              
              
                /
              
              50
            
            
              
                V
                
                  in
                
                
                  2
                
              
              
                /
              
              50
            
          
        
        =
        
          
            
              V
              
                out
              
              
                2
              
            
            
              V
              
                in
              
              
                2
              
            
          
        
        =
        
          
            
              10
              
                2
              
            
            
              1
              
                2
              
            
          
        
        =
        100
         
        
          W/W
        
        .
      
    
    {\displaystyle {\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}}.}

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

gain-db
        
        =
        
          G
          
            dB
          
        
        =
        10
        log
        ⁡
        
          G
          
            W/W
          
        
        =
        10
        log
        ⁡
        100
        =
        10
        ×
        2
        =
        20
         
        
          dB
        
        .
      
    
    {\displaystyle {\text{gain-db}}=G_{\text{dB}}=10\log G_{\text{W/W}}=10\log 100=10\times 2=20~{\text{dB}}.}

<h3>Unity gain</h3>
<p>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 <i><a href="/facts/1_(number)/m3wpSfMf">unity</a> gain</i>.
</p>
<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Active_laser_medium/Gmcharyh">Active laser medium</a></li>
<li><a href="/facts/Antenna_gain/ke1y0LcM">Antenna gain</a></li>
<li><a href="/facts/Aperture-to-medium_coupling_loss/ou1sBehQ">Aperture-to-medium coupling loss</a></li>
<li><a href="/facts/Automatic_gain_control/qY7qWtC7">Automatic gain control</a></li>
<li><a href="/facts/Attenuation/XP4vX360">Attenuation</a></li>
<li><a href="/facts/Complex_gain/KYhyNgba">Complex gain</a></li>
<li><a href="/facts/DC_offset/crU5EHzl">DC offset</a></li>
<li><a href="/facts/Effective_radiated_power/tjnmEd03">Effective radiated power</a></li>
<li><a href="/facts/Gain_before_feedback/j5q92GJg">Gain before feedback</a></li>
<li><a href="/facts/Insertion_gain/dGEMSHGj">Insertion gain</a></li>
<li><a href="/facts/Loop_gain/h2E89W7B">Loop gain</a></li>
<li><a href="/facts/Open-loop_gain/sDngzGDz">Open-loop gain</a></li>
<li><a href="/facts/Net_gain_(telecommunications)/b9tcfS9p">Net gain</a></li>
<li><a href="/facts/Power_gain/jRITzwqo">Power gain</a></li>
<li><a href="/facts/Process_gain/ManRCGj8">Process gain</a></li>
<li><a href="/facts/Transmitter_power_output/8WvTscAX">Transmitter power output</a></li></ul>

<ul><li> This article incorporates <a href="/facts/Copyright_status_of_works_by_the_federal_government_of_the_United_States/Q1jvr96g">public domain material</a> from <a href="https://web.archive.org/web/20220122224547/https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm"><i>Federal Standard 1037C</i></a>. <a href="/facts/General_Services_Administration/Np3opv9x">General Services Administration</a>. Archived from <a href="https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm">the original</a> on 2022-01-22.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Graf, Rudolf F. (1999). Modern Dictionary of Electronics (7 ed.). Newnes. p. 314. ISBN 0080511988. <a href="0080511988" target="_blank">0080511988</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Basu, Dipak (2000). Dictionary of Pure and Applied Physics. CRC Press. p. 157. ISBN 1420050222. <a href="1420050222" target="_blank">1420050222</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Bahl, Inder (2009). Fundamentals of RF and Microwave Transistor Amplifiers. John Wiley and Sons. p. 34. ISBN 978-0470462317. <a href="978-0470462317" target="_blank">978-0470462317</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>White, Glenn; Louie, Gary J (2005). The Audio Dictionary (3 ed.). University of Washington Press. p. 18. ISBN 0295984988. <a href="0295984988" target="_blank">0295984988</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Graf, Rudolf F. (1999). Modern Dictionary of Electronics (7 ed.). Newnes. p. 314. ISBN 0080511988. <a href="0080511988" target="_blank">0080511988</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>White, Glenn; Louie, Gary J (2005). The Audio Dictionary (3 ed.). University of Washington Press. p. 18. ISBN 0295984988. <a href="0295984988" target="_blank">0295984988</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>White, Glenn; Louie, Gary J (2005). The Audio Dictionary (3 ed.). University of Washington Press. p. 18. ISBN 0295984988. <a href="0295984988" target="_blank">0295984988</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>White, Glenn; Louie, Gary J (2005). The Audio Dictionary (3 ed.). University of Washington Press. p. 18. ISBN 0295984988. <a href="0295984988" target="_blank">0295984988</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Basu, Dipak (2000). Dictionary of Pure and Applied Physics. CRC Press. p. 157. ISBN 1420050222. <a href="1420050222" target="_blank">1420050222</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
</ol>

Gain (electronics) open-in-new

Gain (electronics)