*2.1. Small-Signal Transconductances*

Small-signal transconductances are essential for both the design of integrated circuits and the extraction of the four transistor parameters. Figure 2 presents the low-frequency small-signal model for MOSFET in which the variation of the drain current is expressed by using (8), where *gmg*, *gms*, *gmd* and *gmb* are, respectively, the gate, source, drain and bulk transconductances given by using (9); *vg*, *vs*, *vd* and *vb* represent small variations in the gate, source, drain and bulk voltages, respectively.

$$\dot{\mathbf{u}}\_d = \mathbf{g}\_{m\mathbf{\mathcal{S}}} \mathbf{v}\_{\mathbf{\mathcal{S}}} - \mathbf{g}\_{ms} \mathbf{v}\_{\mathbf{s}} + \mathbf{g}\_{md} \mathbf{v}\_d + \mathbf{g}\_{mb} \mathbf{v}\_b \tag{8}$$

$$\mathbf{g}\_{\rm mg} = \frac{\partial I\_D}{\partial V\_G}; \mathbf{g}\_{\rm ms} = -\frac{\partial I\_D}{\partial V\_S}; \mathbf{g}\_{\rm md} = \frac{\partial I\_D}{\partial V\_D}; \mathbf{g}\_{\rm mb} = \frac{\partial I\_D}{\partial V\_B} \tag{9}$$

**Figure 2.** Low-frequency small-signal model of the MOSFET.

The relationships between the transconductances and the inversion levels are obtained by applying the partial derivatives of (9) to the UICM along with (1).

The transconductance-to-current ratios, in terms of inversion level, are given by using expressions (10)–(12) in which *ID*,*sat* stands for the approximation of the drain current in the saturation region, where *ir* << *if* [13].

$$\phi\_t \frac{g\_{ms}}{I\_{D,sat}} = \left(1 - \frac{\sigma}{n}\right) \frac{2}{\sqrt{1 + i\_f} + 1} \tag{10}$$

$$\phi\_t \frac{\mathcal{G}\_{md}}{I\_{D, \text{sat}}} = \left(\frac{\sigma}{n}\right) \frac{2}{\sqrt{1 + i\_f} + 1} \tag{11}$$

$$\phi\_l \frac{g\_m}{I\_{D, \text{sat}}} = \left(\frac{1}{n}\right) \frac{2}{\sqrt{1 + i\_f} + 1} \tag{12}$$
