Editing Transconductance (section)

== 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}}.