Gain (electronics)

Not to be confused with Antenna gain.

In electronics, gain is a measure of the ability of a two-port circuit (often an amplifier) to increase the power or amplitude of a signal from the input to the output port 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 to the amplitude or power at the input port. It is often expressed using the logarithmic decibel (dB) units ("dB gain"). 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.

The term gain alone is ambiguous, and can refer to the ratio of output to input voltage (voltage gain), current (current gain) or electric power (power gain). In the field of audio and general purpose amplifiers, especially operational amplifiers, the term usually refers to voltage gain, but in radio frequency amplifiers it usually refers to power gain. Furthermore, the term gain is also applied in systems such as sensors 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_FE ("beta", the static ratio of I_c divided by I_b at some operating point), or sometimes h_fe (the small-signal current gain, the slope of the graph of I_c against I_b 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 design; antenna gain is the ratio of radiation intensity from a directional antenna to $P_{\text{in}}/4\pi$ (mean radiation intensity from a lossless antenna).

File:Amplification2.svg

Graph of the input $v_{i}(t)$ (blue) and output voltage $v_{o}(t)$ (red) 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 decibels (dB), is defined as follows:

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

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\text{in}} is the power applied to the input, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\text{out}} is the power from the output.

A similar calculation can be done using a natural logarithm instead of a decimal logarithm, resulting in nepers instead of decibels:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{gain-np} = \frac{1}{2} \ln\left(\frac{P_\text{out}}{P_\text{in}}\right)~\text{Np}.}

Voltage gain

The power gain can be calculated using voltage instead of power using Joule's first law Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = V^2/R} ; the formula is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 impedance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_\text{in}} and output impedance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_\text{out}} are equal, so the above equation can be simplified to:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{gain-db} = 10 \log \left(\frac{V_\text{out}}{V_\text{in}}\right)^2~\text{dB},}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{gain-db} = 20 \log \left(\frac{V_\text{out}}{V_\text{in}}\right)~\text{dB}.}

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 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = I^2 R} , the formula is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{gain-db} = 10 \log \left(\frac{I_\text{out}}{I_\text{in}}\right)^2~\text{dB},}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{gain-db} = 20 \log \left(\frac{I_\text{out}}{I_\text{in}}\right)~\text{dB}.}

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

The "current gain" of a bipolar transistor, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_\text{FE}} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_\text{fe}} , is normally given as a dimensionless number, the ratio of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_\text{c}} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_\text{b}} (or slope of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_\text{c}} -versus-Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_\text{b}} graph, for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_\text{in}} ) is 1 volt, its output (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_\text{out}} ) is 10 volts. What is its voltage and power gain?

A. Voltage gain is simply:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{gain-db} = G_\text{dB} = 10 \log G_\text{W/W} = 10 \log 100 = 10 \times 2 = 20~\text{dB}.}

Unity gain

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.

See also

• 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
• Power gain
• Process gain
• Transmitter power output

References

•  This article incorporates public domain material from the General Services Administration document: "Federal Standard 1037C".