# Transconductance

**Transconductance** is an electrical parameter that relates a change in a device's input voltage to the resulting change in its output current; formally, it is the partial derivative of output current with respect to input voltage, conventionally denoted \( g_m \) and expressed in siemens (S), equivalent to amperes per volt.<sup>[1](https://technav.ieee.org/topic/transconductance/)</sup> The name combines "transfer" and "conductance," because it is a conductance that relates quantities across the input-output boundary of a device rather than within a single port.<sup>[1](https://technav.ieee.org/topic/transconductance/)</sup> The term derives from "transfer conductance," and the quantity was formerly measured in "mho" (ohm spelled backwards).<sup>[2](https://www.analog.com/en/resources/glossary/transconductance.html)</sup> Transconductance is the basic gain measure of vacuum tubes and field-effect transistors, and it is a fundamental small-signal parameter for transistors generally, central to the analysis and design of amplifiers, oscillators, and analog filters.<sup>[2](https://www.analog.com/en/resources/glossary/transconductance.html)</sup><sup> • </sup><sup>[1](https://technav.ieee.org/topic/transconductance/)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( g_m = \partial I_{\text{out}} / \partial V_{\text{in}} \), evaluated at a bias point; units siemens (A/V)<sup>[1](https://technav.ieee.org/topic/transconductance/)</sup> |
| BJT value | \( g_m = I_C / V_T \), with thermal voltage \( V_T = kT/q \approx 26 \) mV at room temperature<sup>[3](https://library.knu.edu.af/opac/temp/11304.pdf)</sup> |
| MOSFET value (saturation) | \( g_m = \sqrt{2 \mu C_{ox} I_D (W/L)} \), proportional to the square root of drain current<sup>[4](https://web.stanford.edu/~boyd/papers/pdf/opamp_accsc.pdf)</sup> |
| Weak-inversion efficiency | \( g_m/I_D = 1/(n V_T) \), the largest transconductance efficiency a MOSFET reaches; \( V_T = 25.8 \) mV at 300 K<sup>[5](https://iic-jku.github.io/analog-circuit-design/aicd.html)</sup><sup> • </sup><sup>[6](https://analog-electronics.tudelft.nl/webbook/SED/_build/html/ModelingActiveDevices/ActiveDevices_MOS_transistors.html)</sup> |
| BJT vs MOS | For a given DC output current, BJT transconductance tends to be much larger than MOS transconductance<sup>[7](https://blogs.ncl.ac.uk/stephanelegoff/files/2014/12/EEE1002-EEE1010-Lecture-Notes.pdf)</sup> |
| Circuit role | Voltage gain of a loaded stage is \( A_v = g_m R \); higher \( g_m \) means a small input swing produces a larger current change and greater gain<sup>[1](https://technav.ieee.org/topic/transconductance/)</sup> |
| Example device | The OPA860 operational transconductance amplifier provides \( g_m = 95 \) mA/V, adjustable with an external resistor<sup>[8](https://www.ti.com/product/OPA860)</sup> |

## Definition and role in small-signal analysis

Transconductance is defined at a quiescent (DC) operating point. For a BJT, it is the slope of the transfer characteristics curve, a plot of collector current against base-to-emitter voltage at constant collector-to-emitter voltage, evaluated at that operating point.<sup>[9](https://leachlegacy.ece.gatech.edu/ece3050/notes/bjt/BJTBasicsAdv.pdf)</sup> For a field-effect transistor, \( g_m \) is the slope of the drain current versus gate voltage curve at the given bias point.<sup>[1](https://technav.ieee.org/topic/transconductance/)</sup> In the small-signal model of a MOSFET in saturation, the device is represented by a dependent current source of value \( g_m v_{gs} \) between drain and source, alongside an output conductance \( g_o \) and parasitic capacitances.<sup>[4](https://web.stanford.edu/~boyd/papers/pdf/opamp_accsc.pdf)</sup>

The practical meaning is direct: a higher transconductance means a small input voltage swing produces a proportionally larger output current variation, which translates into greater voltage gain when the output is loaded with a resistance. The parameter depends on the operating point, the device geometry, and the semiconductor technology chosen.<sup>[1](https://technav.ieee.org/topic/transconductance/)</sup> A common-emitter or common-source stage loaded by a resistor \( R \) therefore has a small-signal gain of approximately \( A_v = g_m R \).<sup>[1](https://technav.ieee.org/topic/transconductance/)</sup>

## Transconductance in bipolar junction transistors

The collector current of a BJT in the forward-active mode follows an exponential law derived from the [Boltzmann distribution](https://www.edgepedia.org/boltzmann-distribution) of carriers:

\[ I_c = I_s \exp(V_{be}/V_t) \]

where \( I_s \) is the saturation current and \( V_t = kT/q \) is Boltzmann's thermal voltage, 26 mV at room temperature.<sup>[3](https://library.knu.edu.af/opac/temp/11304.pdf)</sup> Differentiating this expression with respect to \( V_{be} \) gives a transconductance directly proportional to the DC collector current:

\[ g_m = \frac{I_C}{V_T} \]

This makes BJT \( g_m \) a simple and predictable function of bias current alone, independent of device geometry.<sup>[1](https://technav.ieee.org/topic/transconductance/)</sup> As a worked scale: at \( I_C = 4 \) mA with \( V_T = 25 \) mV, the transconductance is 160 mA/V.<sup>[10](https://www.wonderslate.com/u/visvesvaraya-technological-university/electronics-and-communication/electronic-principles-and-circuits/notes?siteName=books)</sup>

## Transconductance in MOSFETs

For a MOSFET in saturation, the square-law drain current is

\[ I_D = \tfrac{1}{2} \mu C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2 \]

and differentiating with respect to \( V_{GS} \) (ignoring channel-length modulation, a small error) gives<sup>[4](https://web.stanford.edu/~boyd/papers/pdf/opamp_accsc.pdf)</sup>

\[ g_m = \frac{\partial I_D}{\partial V_{GS}} = \sqrt{2 \mu C_{ox} I_D \frac{W}{L}} \]

The same quantity can be written as \( g_m = \mu C_{ox} (W/L)(V_{GS} - V_{th}) \).<sup>[11](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_V%253A_Amplifiers_and_Oscillators_%28Steer%29/01%253A_Introduction_to_Active_RF_and_Microwave_Circuits/1.03%253A_Transistor_Technology)</sup> Unlike the BJT case, MOS transconductance depends on geometry: increasing channel width or reducing gate length raises \( g_m \), which is why CMOS technology scaling historically improved analog performance alongside digital speed.<sup>[1](https://technav.ieee.org/topic/transconductance/)</sup>

At very short gate lengths the square-law picture breaks down. As gate length decreases or channel mobility increases, carriers travel at the saturation velocity \( v_{sat} \) for most of the channel, and the transconductance no longer increases linearly with gate bias; it turns over due to velocity saturation in the channel.<sup>[12](https://web.ece.ucsb.edu/Faculty/rodwell/publications_and_presentations/theses/adc_thesis.pdf)</sup>

## Transconductance efficiency

Analog designers often compare devices by the ratio \( g_m / I_D \), the transconductance produced per unit of bias current; it also serves as a measure of the level of inversion in a MOS channel.<sup>[6](https://analog-electronics.tudelft.nl/webbook/SED/_build/html/ModelingActiveDevices/ActiveDevices_MOS_transistors.html)</sup> In weak inversion, meaning small gate-source voltage, a MOSFET reaches its largest efficiency:<sup>[5](https://iic-jku.github.io/analog-circuit-design/aicd.html)</sup>

\[ \frac{g_m}{I_D} = \frac{1}{n V_T} \]

where \( n \) is the subthreshold slope factor and \( V_T = kT/q \) is 25.8 mV at 300 K.<sup>[5](https://iic-jku.github.io/analog-circuit-design/aicd.html)</sup> In strong inversion, without short-channel effects, the efficiency drops to \( 2/V_{eff} \), where \( V_{eff} = V_{GS} - V_{T0} \).<sup>[6](https://analog-electronics.tudelft.nl/webbook/SED/_build/html/ModelingActiveDevices/ActiveDevices_MOS_transistors.html)</sup> A design point combining high transconductance per bias current with the fastest operation does not exist; the two requirements trade off.<sup>[5](https://iic-jku.github.io/analog-circuit-design/aicd.html)</sup>

The comparison with the BJT follows from these expressions. Because the gate overdrive \( V_{GS} - V_{TN} \) in a saturated MOSFET is typically much larger than the thermal voltage, the BJT transconductance, set by \( V_T \approx 26 \) mV, tends to be much larger than the MOS transconductance for the same DC output current.<sup>[7](https://blogs.ncl.ac.uk/stephanelegoff/files/2014/12/EEE1002-EEE1010-Lecture-Notes.pdf)</sup>

## Devices built around transconductance

**Operational transconductance amplifiers.** An [operational transconductance amplifier](https://www.edgepedia.org/operational-transconductance-amplifier) (OTA) is a voltage-to-current converter amplifier whose output is a current proportional to a differential input voltage. The OPA860 from [Texas Instruments](https://www.edgepedia.org/texas-instruments), for example, is a wideband bipolar OTA with a transconductance of 95 mA/V, an 80 MHz open-loop bandwidth at a gain of +5, and a 900 V/µs slew rate; its transconductance can be adjusted with an external resistor, allowing bandwidth, quiescent current, and gain trade-offs to be optimized.<sup>[8](https://www.ti.com/product/OPA860)</sup> The OTA behaves like a self-biased ideal transistor, with a high-impedance input, a low-impedance input/output, and a current output that is zero for zero input voltage.<sup>[8](https://www.ti.com/product/OPA860)</sup>

**Vacuum tubes.** For an amplifying vacuum tube, transconductance, historically called mutual conductance, measures the change in plate current resulting from a given change in control grid voltage.<sup>[13](https://www.allaboutcircuits.com/textbook/semiconductors/chpt-13/tube-parameters/)</sup><sup> • </sup><sup>[14](https://manuals.plus/m/109e08cd701e86a60c0faecfd2d439c3a868e7fc5084ffb1816ef65438d6ea2d)</sup> It is one of the two most basic measures of a tube's characteristics, alongside the amplification factor \( \mu \), which is the ratio of a change in plate voltage to a change in grid voltage with plate current held constant.<sup>[13](https://www.allaboutcircuits.com/textbook/semiconductors/chpt-13/tube-parameters/)</sup> The two quantities are often confused, but they describe different relationships: transconductance links grid voltage to plate current, while \( \mu \) compares grid voltage to plate voltage at constant current.<sup>[15](https://www.electronics-notes.com/articles/electronic_components/valves-tubes/transconductance-mutual-conductance-gm-vs-amplification-factor-mu-differences-definitions.php)</sup>

## References

1. [Transconductance, IEEE Technology Navigator](https://technav.ieee.org/topic/transconductance/)
2. [Transconductance, Analog Devices Glossary](https://www.analog.com/en/resources/glossary/transconductance.html)
3. [Bipolar Junction Transistors (BJTs), chapter 16](https://library.knu.edu.af/opac/temp/11304.pdf)
4. [Hershenson, Boyd, and Lee, Optimal CMOS Op-amp Design (Stanford University)](https://web.stanford.edu/~boyd/papers/pdf/opamp_accsc.pdf)
5. [Analog (Integrated) Circuit Design, Institute of Ion Physics, Johannes Kepler University Linz](https://iic-jku.github.io/analog-circuit-design/aicd.html)
6. [MOS transistors, Structured Electronics Design, TU Delft](https://analog-electronics.tudelft.nl/webbook/SED/_build/html/ModelingActiveDevices/ActiveDevices_MOS_transistors.html)
7. [EEE1002/EEE1010 Lecture Notes, Newcastle University](https://blogs.ncl.ac.uk/stephanelegoff/files/2014/12/EEE1002-EEE1010-Lecture-Notes.pdf)
8. [OPA860 Wide Bandwidth Operational Transconductance Amplifier and Buffer, Texas Instruments](https://www.ti.com/product/OPA860)
9. [W. Marshall Leach, Jr., BJT Basics: Advanced Material, Georgia Institute of Technology](https://leachlegacy.ece.gatech.edu/ece3050/notes/bjt/BJTBasicsAdv.pdf)
10. [Electronic Principles and Circuits Study Notes, Visvesvaraya Technological University](https://www.wonderslate.com/u/visvesvaraya-technological-university/electronics-and-communication/electronic-principles-and-circuits/notes?siteName=books)
11. [Transistor Technology, Microwave and RF Design, Engineering LibreTexts](https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_V%253A_Amplifiers_and_Oscillators_%28Steer%29/01%253A_Introduction_to_Active_RF_and_Microwave_Circuits/1.03%253A_Transistor_Technology)
12. [Chapter 2: MOSFET Theory, UCSB thesis](https://web.ece.ucsb.edu/Faculty/rodwell/publications_and_presentations/theses/adc_thesis.pdf)
13. [Tube Parameters, Electronics Textbook, All About Circuits](https://www.allaboutcircuits.com/textbook/semiconductors/chpt-13/tube-parameters/)
14. [Sylvania Tube Tester Models 219 & 220 Operating Manual](https://manuals.plus/m/109e08cd701e86a60c0faecfd2d439c3a868e7fc5084ffb1816ef65438d6ea2d)
15. [Vacuum Tube Transconductance gm vs Amplification Factor µ, Electronics Notes](https://www.electronics-notes.com/articles/electronic_components/valves-tubes/transconductance-mutual-conductance-gm-vs-amplification-factor-mu-differences-definitions.php)

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