the difference between K coefficient and gm in the mosfet

Question

i know that K in mosfet is as beta in Bjt but since it represents the change in Id in relative to the change in Vgs isn't it supposed to be the transconductance gm ? but why is it different and the mosfet has K and gm so what is the difference between the two IN THE MOSFET ? another question : does tempreture affect K as it affects beta ?

Answer 1

The letter \$K\$ is typically used to isolate the process-dependent parameters from the design parameters. It is not the same as the transconductance gain \$g_m\$.

For example, the most simple model equations of a mosfet can be stated in the triode and saturation region as

$$\begin{align} i_{DS} &= K\cdot\frac{W}{L}\left((v_{GS}-v_{TH})v_{DS} - \frac{v_{DS}^2}{2}\right) &\text{triode}\\ i_{DS} &= K\cdot \frac{W}{2\cdot L}(v_{GS}-v_{TH})^2 & \text{saturation} \end{align}$$

In these equations, the letter \$K\$ means the following

$$K = \mu\cdot C_{OX}' = \mu\cdot \frac{\epsilon_{ox}}{t_{ox}}$$

where \$\mu\$ is the electron or hole mobility, \$\epsilon_{ox}\$ is the permittivity of the gate material (usually Silicon-dioxide) and \$t_{ox}\$ is the oxide thickness. None of these parameters can be changed by the IC designer.

The transconductance gain \$g_m\$ is defined as the derivative of the current to the gate-source voltage, or

$$g_m = \frac{\partial i_{DS}}{\partial v_{GS}}$$

If you used the model equations from before, you can calculate them in the linear and saturation region

$$\begin{align} g_m &= K\cdot \frac{W}{L}\cdot v_{DS} & \text{triode}\\ g_m &= K\cdot \frac{W}{L}\cdot (v_{GS}-v_{TH}) & \text{saturation} \end{align}$$

The transconductance gain \$g_m\$ will depend on its biasing and size and is similar to a BJT's \$\beta\$.

Answer 2

In the long-channel FET math model, the Drain current is

Idd = K/2 * W/L * Ve^2

where Ve = that voltage above threshold, or (Vgs - Vt); Ve means "effective voltage".

The Transconductance of the (long channel) FET is the derivative of Idd, or

2 * (K/2 * W/L * Ve^1)

or

Transconductance = K * W/L * Ve

Regarding temperature effects: K = MU * Cox, where MU is mobility of carriers; that is temperature dependent. Any impact of temperature upon Vt is not what I've studied. In my silicon work, I matched FETs with careful interdigitated layouts, and ran sims from -55 to +125, always examining the highspeed waveforms to ensure signal fidelity was preserved even at extreme corner cases. Can one speak of a pulse having "signal fidelity"? Of course.