View source for Transconductance

{{Short description|Electrical characteristic}}
'''Transconductance''' (for '''transfer conductance'''), also infrequently called '''mutual [[electrical conductance|conductance]]''', is the electrical characteristic relating the [[Electric current|current]] through the output of a device to the [[voltage]] across the input of a device. Conductance is the reciprocal of resistance.

'''Transadmittance''' (or '''transfer [[admittance]]''') is the [[Alternating current|AC]] equivalent of transconductance.

== Definition ==
[[Image:Image for Transconductance.svg|right|thumb|150px|Model transconductance device]]
Transconductance is very often denoted as a conductance, {{math|''g''<sub>m</sub>}}, with a subscript, {{math|m}}, for ''mutual''. It is defined as follows:
: <math>g_\text{m} = \frac{\Delta I_\text{out}}{\Delta V_\text{in}}</math>

For [[small signal]] [[alternating current]], the definition is simpler:
: <math>g_\text{m} = \frac{i_\text{out}}{v_\text{in}}</math>

The [[SI]] unit for transconductance is the '''[[siemens (unit)|siemens]]''', with the symbol '''S''', as in conductance.

== Transresistance ==

'''Transresistance''' (for '''transfer resistance'''), also infrequently referred to as '''mutual resistance''', is the [[Duality (electrical circuits)|dual]] of transconductance. It refers to the ratio between a change of the voltage at two output points and a related change of current through two input points, and is denotated as {{math|''r''<sub>m</sub>}}:
: <math>r_\text{m} = \frac{\Delta V_\text{out}}{\Delta I_\text{in}}</math>

The SI unit for transresistance is simply the [[ohm (unit)|ohm]], as in resistance.

'''Transimpedance''' (or, '''transfer [[Electrical impedance|impedance]]''') is the AC equivalent of transresistance, and is the [[Duality (electrical circuits)|dual]] of transadmittance.

== Devices ==

=== Vacuum tubes ===
For [[vacuum tube]]s, transconductance is defined as the change in the plate (anode) current divided by the corresponding change in the grid/cathode voltage, with a constant plate (anode) to cathode voltage. Typical values of {{math|''g''<sub>m</sub>}} for a small-signal vacuum tube are 1 to {{val|10|u=mS}}. It is one of the three characteristic constants of a vacuum tube, the other two being its [[Gain (electronics)|gain]] {{mvar|μ}} (mu) and plate resistance {{math|''r''<sub>p</sub>}} or {{math|''r''<sub>a</sub>}}. The [[Hendrik van der Bijl|Van der Bijl]] equation defines their relation as follows:
: <math>g_\mathrm{m} = \frac{\mu}{r_\mathrm{p}}</math><ref>Blencowe, Merlin (2009). "Designing Tube Amplifiers for Guitar and Bass".</ref>

=== Field-effect transistors ===
Similarly, in [[field-effect transistor]]s, and [[MOSFET]]s in particular, transconductance is the change in the drain current divided by the small change in the gate–source voltage with a constant drain–source voltage. Typical values of {{math|''g''<sub>m</sub>}} for a small-signal field-effect transistor are {{val|1|to|30|u=mS}}.

Using the [[Channel length modulation#Shichman–Hodges model|Shichman–Hodges model]], the transconductance for the MOSFET can be expressed as (see ''{{slink|MOSFET#Modes of operation}}'')
: <math>g_\text{m} = \frac{2I_\text{D}}{V_\text{OV}},</math>
where {{math|''I''<sub>D</sub>}} is the DC drain current at the [[bias point]], and {{math|''V''<sub>OV</sub>}} is the [[overdrive voltage]], which is the difference between the bias point gate–source voltage and the [[threshold voltage]] (i.e., {{math|1=''V''<sub>OV</sub> ≡ ''V''<sub>GS</sub> – ''V''<sub>th</sub>}}).<ref name=Sedra>
{{citation
 |last1=Sedra
 |first1=A. S.
 |last2=Smith
 |first2=K. C.
 |title=Microelectronic Circuits
 |year=1998
 |edition=Fourth
 |publisher=Oxford University Press
 |location=New York
 |isbn=0-19-511663-1
 |url=http://worldcat.org/isbn/0-19-514251-9
}}</ref>{{rp|p.&nbsp;395, Eq.&nbsp;(5.45)}} The overdrive voltage (sometimes known as the effective voltage) is customarily chosen at about 70–200&nbsp;mV for the [[65&nbsp;nm process]] node ({{nowrap|{{math|''I''<sub>D</sub>}} ≈ 1.13&nbsp;mA/μm × width}}) for a {{math|''g''<sub>m</sub>}} of 11–32&nbsp;mS/μm.<ref name=Baker>
{{citation
 |last=Baker
 |first=R.&nbsp;Jacob
 |title=CMOS Circuit Design, Layout, and Simulation, Third Edition
 |year=2010
 |publisher=Wiley-IEEE
 |location=New York
 |isbn=978-0-470-88132-3
 |url=http://worldcat.org/isbn/978-0-470-88132-3
}}</ref>{{rp|p.&nbsp;300, Table&nbsp;9.2}}<ref name=Sansen>
{{citation
 |last=Sansen
 |first=W. M. C.
 |title=Analog Design Essentials
 |year=2006
 |publisher=Springer
 |location=Dordrecht
 |isbn=0-387-25746-2
 |url=http://worldcat.org/isbn/0387257462
}}</ref>{{rp|p.&nbsp;15, §0127}}

Additionally, the transconductance for the junction FET is given by 
: <math>g_\text{m} = \frac{2I_\text{DSS}}{|V_\text{P}|} \left(1 - \frac{V_\text{GS}}{V_\text{P}}\right),</math>
where {{math|''V''<sub>P</sub>}} is the pinchoff voltage, and {{math|''I''<sub>DSS</sub>}} is the maximum drain current.

=== Bipolar transistors ===
The {{math|''g''<sub>m</sub>}} of [[Bipolar junction transistor|bipolar]] small-signal transistors varies widely, being proportional to the collector current. It has a typical range of {{val|1|to|400|u=mS}}. The input voltage change is applied between the base/emitter and the output is the change in collector current flowing between the collector/emitter with a constant collector/emitter voltage.

The transconductance for the bipolar transistor can be expressed as
: <math>g_\mathrm{m} = \frac{I_\mathrm{C}}{V_\mathrm{T}}</math>
where {{math|''I''<sub>C</sub>}} is the DC collector current at the [[Q-point]], and {{math|''V''<sub>T</sub>}} is the [[Boltzmann constant#Thermal voltage|thermal voltage]], typically about {{val|26|u=mV}} at room temperature. For a typical current of {{val|10|u=mA}}, {{math|''g''<sub>m</sub> ≈}} {{val|385|u=mS}}. The input impedance is the [[Bipolar_junction_transistor#Transistor_parameters:_alpha_(%CE%B1)_and_beta_(%CE%B2)|current gain ({{mvar|β}})]] divided by the transconductance.

The output (collector) conductance is determined by the [[Early voltage]] and is proportional to the collector current. For most transistors in linear operation it is well below {{val|100|u=μS}}.

== Amplifiers ==

=== Transconductance amplifiers ===
A '''transconductance amplifier''' (''g''{{sub|m}} amplifier) puts out a current proportional to its input voltage. In '''[[Network analysis (electrical circuits)|network analysis]]''', the transconductance amplifier is defined as a ''{{dfn|voltage controlled current source}}'' ({{abbr|VCCS}}). These amplifiers are commonly seen installed in a [[cascode]] configuration, which improves the frequency response.

An ideal transconductance amplifier in a voltage follower configuration behaves at the output like a resistor of value {{math|1/''g''{{sub|m}}}}, between a buffered copy of the input voltage and the output. If the follower is loaded by a single capacitor {{math|''C''}}, the voltage follower transfer function has a single pole with time constant {{math|''C''/''g''{{sub|m}}}},<ref>{{cite web | url=https://hasler.ece.gatech.edu/Courses/ECE6414/Unit3/gmCFilter01.pdf | title=Basics of Transconductance - Capacitance Filters | website=hasler.ece.gatech.edu | first=Paul | last=Hasler}}</ref> or equivalently it behaves as a 1st-order low-pass filter with a [[half-power bandwidth|{{val|-3|u=dB}} bandwidth]] of {{math|''g''{{sub|m}}/2''πC''}}.

==== Operational transconductance amplifiers ====
{{Main article|Operational transconductance amplifier}}
An [[operational transconductance amplifier]] (OTA) is an integrated circuit which can function as a transconductance amplifier. These normally have an input to allow the transconductance to be controlled.<ref name="MAX3724">{{cite web |title=3.2&nbsp;Gbps SFP Transimpedance Amplifiers with RSSI |url=https://datasheets.maximintegrated.com/en/ds/MAX3724-MAX3725.pdf |website=datasheets.maximintegrated.com |publisher=Maxim |access-date=15 November 2018}}</ref>

=== Transresistance amplifiers ===
{{Main article|transimpedance amplifier}}
A '''transresistance amplifier''' outputs a voltage proportional to its input current. The transresistance amplifier is often referred to as a '''transimpedance amplifier''', especially by semiconductor manufacturers.

The term for a transresistance amplifier in network analysis is ''current controlled voltage source'' (''CCVS'').

A basic inverting transresistance amplifier can be built from an [[operational amplifier]] and a single resistor. Simply connect the resistor between the output and the inverting input of the operational amplifier and connect the non-inverting input to ground. The output voltage will then be proportional to the input current at the inverting input, decreasing with increasing input current and vice versa. 

Specialist chip transresistance (transimpedance) amplifiers are widely used for amplifying the signal current from photo diodes at the receiving end of ultra high speed fibre optic links.

== See also ==
* [[Transistor]]
* [[Vacuum tube]]
* [[Electronic amplifier]]
* [[Transimpedance amplifier]]
* [[Fontana bridge]]
* [[Operational transconductance amplifier]]
* [[MOSFET]]

== References ==
{{reflist}}

* {{citation|author1=[[Paul Horowitz|Horowitz, Paul]]  | author2=[[Winfield Hill|Hill, Winfield]]|title=The Art of Electronics|title-link=The Art of Electronics (book)|publisher=Cambridge University Press|year=1989|isbn=0-521-37095-7}}

== External links ==
{{Wiktionary|transconductance}}
* [http://searchsmb.techtarget.com/sDefinition/0,290660,sid44_gci214200,00.html Transconductance] — SearchSMB.com Definitions
* Transconductance in audio amplifiers: article by David Wright of Pure Music  [https://web.archive.org/web/20070206073621/http://www.beauhorn.com/articles/TC_amps_%26_SD_horns.html]

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[[Category:Transfer functions]]
[[Category:Electrical resistance and conductance]]