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Gain (electronics)

From Wikipedia, the free encyclopedia
Ability of a circuit to increase the power or amplitude of a signal
Not to be confused withAntenna gain.

Inelectronics,gain is a measure of the ability of atwo-portcircuit (often anamplifier) to increase thepower oramplitude of asignal from the input to the output port[1][2][3][4] by adding energy converted from somepower supply to the signal. It is usually defined as the meanratio of thesignal amplitude or power at the outputport to the amplitude or power at the input port.[1] It is often expressed using thelogarithmicdecibel (dB) units ("dB gain").[4] A gain greater than one (greater than zero dB), that is, amplification, is the defining property of anactive device or circuit, while a passive circuit will have a gain of less than one.[4]

The termgain alone is ambiguous, and can refer to the ratio of output to inputvoltage (voltage gain),current (current gain) or electric power (power gain).[4] In the field of audio and general purpose amplifiers, especiallyoperational amplifiers, the term usually refers to voltage gain,[2] but inradio frequency amplifiers it usually refers to power gain. Furthermore, the term gain is also applied in systems such assensors where the input and output have different units; in such cases the gain units must be specified, as in "5 microvolts per photon" for theresponsivity of aphotosensor. The "gain" of abipolar transistor normally refers to forward current transfer ratio, eitherhFE ("beta", the static ratio ofIc divided byIb at some operating point), or sometimeshfe (the small-signal current gain, the slope of the graph ofIc againstIb at a point).

The gain of an electronic device or circuit generally varies with thefrequency of the applied signal. Unless otherwise stated, the term refers to the gain for frequencies in thepassband, the intended operating frequency range of the equipment. The termgain has a different meaning inantenna design;antenna gain is the ratio ofradiation intensity from a directional antenna toPin/4π{\displaystyle P_{\text{in}}/4\pi } (mean radiation intensity from a lossless antenna).

Graph of the inputvi(t){\displaystyle v_{i}(t)}(blue) and output voltagevo(t){\displaystyle v_{o}(t)}(red) of an ideal linearamplifier 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

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Power gain

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Power gain, indecibels (dB), is defined as follows:

gain-db=10log10⁡(PoutPin) dB,{\displaystyle {\text{gain-db}}=10\log _{10}\left({\frac {P_{\text{out}}}{P_{\text{in}}}}\right)~{\text{dB}},}

wherePin{\displaystyle P_{\text{in}}} is the power applied to the input,Pout{\displaystyle P_{\text{out}}} is the power from the output.

A similar calculation can be done using anatural logarithm instead of a decimal logarithm, resulting innepers instead of decibels:

gain-np=12ln⁡(PoutPin) Np.{\displaystyle {\text{gain-np}}={\frac {1}{2}}\ln \left({\frac {P_{\text{out}}}{P_{\text{in}}}}\right)~{\text{Np}}.}

Voltage gain

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The power gain can be calculated using voltage instead of power usingJoule's first lawP=V2/R{\displaystyle P=V^{2}/R}; the formula is:

gain-db=10log⁡Vout2RoutVin2Rin dB.{\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} .}

In many cases, the input impedanceRin{\displaystyle R_{\text{in}}} and output impedanceRout{\displaystyle R_{\text{out}}} are equal, so the above equation can be simplified to:

gain-db=10log⁡(VoutVin)2 dB,{\displaystyle {\text{gain-db}}=10\log \left({\frac {V_{\text{out}}}{V_{\text{in}}}}\right)^{2}~{\text{dB}},}
gain-db=20log⁡(VoutVin) dB.{\displaystyle {\text{gain-db}}=20\log \left({\frac {V_{\text{out}}}{V_{\text{in}}}}\right)~{\text{dB}}.}

This simplified formula, the20 log rule, is used to calculate avoltage gain in decibels and is equivalent to a power gain if and only if theimpedances at input and output are equal.

Current gain

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In the same way, when power gain is calculated using current instead of power, making the substitutionP=I2R{\displaystyle P=I^{2}R}, the formula is:

gain-db=10log⁡(Iout2RoutIin2Rin) 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}}.}

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

gain-db=10log⁡(IoutIin)2 dB,{\displaystyle {\text{gain-db}}=10\log \left({\frac {I_{\text{out}}}{I_{\text{in}}}}\right)^{2}~{\text{dB}},}
gain-db=20log⁡(IoutIin) dB.{\displaystyle {\text{gain-db}}=20\log \left({\frac {I_{\text{out}}}{I_{\text{in}}}}\right)~{\text{dB}}.}

This simplified formula is used to calculate acurrent gain in decibels and is equivalent to the power gain if and only if theimpedances at input and output are equal.

The "current gain" of abipolar transistor,hFE{\displaystyle h_{\text{FE}}} orhfe{\displaystyle h_{\text{fe}}}, is normally given as a dimensionless number, the ratio ofIc{\displaystyle I_{\text{c}}} toIb{\displaystyle I_{\text{b}}} (or slope of theIc{\displaystyle I_{\text{c}}}-versus-Ib{\displaystyle I_{\text{b}}} graph, forhfe{\displaystyle h_{\text{fe}}}).

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 inamperes. In the case of other devices, the gain will have a value inSI units. Such is the case with theoperational transconductance amplifier, which has an open-loop gain (transconductance) insiemens (mhos), because the gain is a ratio of the output current to the input voltage.

Example

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Q. An amplifier has an input impedance of 50 ohms and drives a load of 50 ohms. When its input (Vin{\displaystyle V_{\text{in}}}) is 1 volt, its output (Vout{\displaystyle V_{\text{out}}}) is 10 volts. What is its voltage and power gain?

A. Voltage gain is simply:

gain=VoutVin=101=10 V/V.{\displaystyle {\text{gain}}={\frac {V_{\text{out}}}{V_{\text{in}}}}={\frac {10}{1}}=10~{\text{V/V}}.}

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 =V2/R, the power gain is:

gain=Vout2/50Vin2/50=Vout2Vin2=10212=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}}.}

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

gain-db=GdB=10log⁡GW/W=10log⁡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}}.}

Unity gain

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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 asunity gain.

See also

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References

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  1. ^abGraf, Rudolf F. (1999).Modern Dictionary of Electronics (7 ed.). Newnes. p. 314.ISBN 0080511988.
  2. ^abBasu, Dipak (2000).Dictionary of Pure and Applied Physics. CRC Press. p. 157.ISBN 1420050222.
  3. ^Bahl, Inder (2009).Fundamentals of RF and Microwave Transistor Amplifiers. John Wiley and Sons. p. 34.ISBN 978-0470462317.
  4. ^abcdWhite, Glenn; Louie, Gary J (2005).The Audio Dictionary (3 ed.). University of Washington Press. p. 18.ISBN 0295984988.
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