Transconductance (for transfer conductance), also infrequently called mutual conductance, is the electrical characteristic relating the 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 AC equivalent of transconductance.
Definition
Transconductance is very often denoted as a conductance, gm, with a subscript, m, for mutual. It is defined as follows:
g m = Δ I out Δ V in {\displaystyle g_{\text{m}}={\frac {\Delta I_{\text{out}}}{\Delta V_{\text{in}}}}}
For small signal alternating current, the definition is simpler:
g m = i out v in {\displaystyle g_{\text{m}}={\frac {i_{\text{out}}}{v_{\text{in}}}}}
The SI unit for transconductance is the siemens, with the symbol S, as in conductance.
Transresistance Transresistance (for transfer resistance), also infrequently referred to as mutual resistance, is the 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 rm:
r m = Δ V out Δ I in {\displaystyle r_{\text{m}}={\frac {\Delta V_{\text{out}}}{\Delta I_{\text{in}}}}}
The SI unit for transresistance is simply the ohm, as in resistance. Transimpedance (or, transfer impedance) is the AC equivalent of transresistance, and is the dual of transadmittance.
Devices
Vacuum tubes For vacuum tubes, 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 gm for a small-signal vacuum tube are 1 to 10 mS. It is one of the three characteristic constants of a vacuum tube, the other two being its gain μ (mu) and plate resistance rp or ra. The Van der Bijl equation defines their relation as follows:
g m = μ r p {\displaystyle g_{\mathrm {m} }={\frac {\mu }{r_{\mathrm {p} }}}}
Field-effect transistors Similarly, in field-effect transistors, and MOSFETs 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 gm for a small-signal field-effect transistor are 1 to 30 mS. Using the Shichman–Hodges model, the transconductance for the MOSFET can be expressed as (see MOSFET § Modes of operation)
g m = 2 I D V OV , {\displaystyle g_{\text{m}}={\frac {2I_{\text{D}}}{V_{\text{OV}}}},}
where ID is the DC drain current at the bias point, and VOV is the overdrive voltage, which is the difference between the bias point gate–source voltage and the threshold voltage (i.e., VOV ≡ VGS – Vth). The overdrive voltage (sometimes known as the effective voltage) is customarily chosen at about 70–200 mV for the 65 nm process node (ID ≈ 1.13 mA/μm × width) for a gm of 11–32 mS/μm. Additionally, the transconductance for the junction FET is given by
g m = 2 I DSS | V P | ( 1 − V GS V P ) , {\displaystyle g_{\text{m}}={\frac {2I_{\text{DSS}}}{|V_{\text{P}}|}}\left(1-{\frac {V_{\text{GS}}}{V_{\text{P}}}}\right),}
where VP is the pinchoff voltage, and IDSS is the maximum drain current.
Bipolar transistors The gm of bipolar small-signal transistors varies widely, being proportional to the collector current. It has a typical range of 1 to 400 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
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