Small Signal Model

Building up the model

Fig. 15 Small-signal model where the signal is added to the gate.

Since these models are so nonlinear, we linearize around specific bias points. The \(\Delta I_D\) and \(\Delta V_{\text{GS}}\) is the signal riding on top of the large signal, \(V_{\text{GS}_0}\) and \(I_{D_0}\).

Fig. 16 Linearization of the IV curve.

In the above figure, \(\Delta V_{\text{GS}}\) and the operating point are added to the signal \(V_{\text{GS}_0}\).

Fig. 17 Small-signal equivalent circuit.

In the above figure, \(\Delta I_D = g_m \Delta V_{\text{GS}}\).

Fig. 18 Small-signal model where the signal is added to the drain.

Fig. 19 Linearization of the IV curve.

Fig. 20 Small-signal model adds the resistor to include the channel length modulation effect.

Note the addition of the resistor \(R_0\) in the above circuit. This includes the effects of channel length modulation into the small-signal mode and models the change in \(I_D\) in response to the change in \(V_{\text{DS}}\).

Fig. 21 We normally connect bulk to the lowest voltage to avoid any current flow.

Fig. 22 We can bias the MOSFET to saturation.

\[ \frac{\partial I_D}{\partial V_{\text{BS}}} = g_{\text{mb}} = \frac{\partial I_D}{\partial V_{\text{th }}} \frac{\partial V_{\text{th}}}{\partial V_{\text{BS}}} \]

\[ \frac{\partial I_D}{\partial V_{\text{th}}} = - \mu c_{\text{ox}} \frac{W}{L} (V_{\text{GS}} - V_{\text{th}}) = -g_m \]

\[ \frac{\partial V_{\text{th}}}{\partial V_{\text{BS}}} = - \frac{\partial V_{\text{th}}}{\partial V_{\text{SB}}} = \frac{\gamma}{2} (2 \phi _F + V_{\text{SB}})^{- \frac{1}{2}} \]

(7)\[\begin{split} g_{\text{mb}} &= g_m \frac{\gamma}{2} (2 \phi _F + V_{\text{SB}})^{- \frac{1}{2}} \\ &= g_m \eta \end{split}\]

where \(\eta \approxeq 0.2\), typically. We often set \(g_{\text{mb}}\) to 0, since its effects are 5x smaller.

NMOS model

Fig. 23 Small-signal model of an NMOS.

Fig. 24 Complete NMOS model with capacitances.

PMOS model

Fig. 25 Small-signal model of a PMOS.

Summary

Region of Operation	Condition
Cutoff	\(V_{\text{SG}} \lt V_{\text{th}}\)
Saturation	\(V_{\text{SD}} \gt V_{\text{SG}}, V_{\text{SG}} \geq V_{\text{th}}\)
Triode	\(V_{\text{SD}} \leq V_{\text{SG}} - V_{\text{th}}, V_{\text{SG}} \geq V_{\text{th}}\)