Transconductance

This note is based on CMOS technology. All the equations are based on n-channel MOS.

I-V Characteristic of MOSFET

$~~$A Metal-Oxide-Semiconductor field effect transistor has 3 regions that it can works in.

1. Cut off region (Sub-threshold region): In Sub-threshold region, there are still some small currents between Drain and Source, whose saturated value has nothing to do with the Gate-Source voltage.

   $$I_{DS}=\frac{W}{L}I_t\exp(\frac{V_{GS}-V_{TH}}{nV_{TH}})[1-\exp(-\frac{V_{DS}}{V_{TH}})]$$

   The $I_t$ relates to diffusion constant, concentration of electrons in P doped region, and anbient temperature.

2. Triode Region: In this region, $V_{DS} < V_{GS}-V_{TH}$, which means that there is no pinch off in the channel of the device. The I-V equation comes as follow:

   $$I_{ds}=\frac{1}{2}\mu_{n}C_{ox}(\frac{W}{L})[2(V_{GS}-V_{TH})V_{DS}-V_{DS}^2]$$

3. Saturation region: Due to the pinch off in the channel, the current can not increase infinitely as the Gate voltage increases. Once $V_{DS} = V_{GS}-V_{TH}$, the channel will pinch off. In another word, there will appear a small depletion region nearby the Drain.

   $$I_{ds}=\frac{1}{2}\mu_{n}C_{ox}(\frac{W}{L})(V_{GS}-V_{TH})^2$$

   $~~$Nevertheless, if we take some small size factors into account, the saturation current will change to:

   $$I_{ds}=\frac{1}{2}\mu_{n}C_{ox}(\frac{W}{L_{eff}})(V_{GS}-V_{TH})^2(1+\lambda V_{DS})$$

Transconductance of different types MOS amplifier circuits

$~~$In DC (Directional Current) analysis, transconductance has been defined as $g_m=\frac{\partial I_{DS}}{\partial V_{GS}}$, and in most of time, we analysis a FET in its saturation region. Thus for n-MOS, the transconductance can be derived as:

$$g_m=\mu_{n}C_{ox}(\frac{W}{L})(V_{GS}-V_{TH})(1+\lambda V_{DS})$$

or in form of (ignore the channel length modulation of the device):

$$g_m=\mu_{n}C_{ox}(\frac{W}{L})(V_{GS}-V_{TH})=\sqrt{2\mu_{n}C_{ox}(\frac{W}{L})I_{DS}}$$

$~~$When we analysis small signal model of a MOS transconductance at a certain bias $V_{GS0}$, we can finally derive the relation: $i_d=g_mv_i$.

$~~$This is where the difference between large signal gain and small signal gain comes from.

MOS small signal models

$~~$Different non-ideal factors such as body effect and channel length modulation make small signal models look different. In some circumstances, we do not need to consider all the non-ideal factors, and we just change the small signal to a simpler one. Here comes the models consider different factors:

2021.3