MOS equations

The well known equations for the MOS transistors are (for the n-type and p-type transistors)[1]:

below saturation

$$I_{DSn,p} = \beta_{n,p}\left[(V_{GS}-V_{T_{n,p}})V_{DS}-\frac{V_{DS}^2}{2}\right]$$ (4.2)

above saturation

$$I_{DS_{n,p}}=\frac{\beta_{n,p}}{2}\left[V_{DSsat_{n,p}}\right]^2$$ (4.3)

where $\beta_{n,p} = \frac{\mu_{n,p}C_{ox}W}{L}$, with $\mu_{n,p}$ modified by the carrier velocity saturation effect:

$$\mu_n = \frac{\mu_{n0}}{1+\frac{V_{DS}}{L {\mathcal E}_{c}}} \quad \mu_p = \frac{\mu_{p0}}{1-\frac{V_{DS}}{L {\mathcal E}_{c} }}$$

The saturation voltage (drain-source), not including the carrier velocity saturation effect, is given by the well known formula:

$$V_{DS_{n,p}} = V_{GS_{n,p}}-V_{T_{n,p}}$$

while considering the effect above-mentioned:

$$V_{DS_{n,p}} = \pm V_c \left[\sqrt{1\pm \frac{2(V_{GS_{n,p}}-V_{T_{n,p}})}{V_c}} - 1\right]$$ (4.4)

where the plus signs are for n-MOSFETs and the minus signs are for the p-MOSFETs, and $V_c = \vert{\mathcal E}_{c} L\vert$