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MOS小信号模型

Notations

notations

Understanding the concept of small signal model

Transient response

transient

\[ v_{in} = v_a \sin \omega t \]
\[ v_{out} = -i_dR_d = -\mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{TH})v_a \sin \omega t \]

Amplification factor: \(\mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{TH})\)

Transconductance of MOS

\[ g_m = \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{TH}) = \sqrt{2 \mu_n C_{ox} \frac{W}{L} I_D} = \frac{2I_D}{V_{GS}- V_{TH}} \]

Three elements for calculating \(g_m\)

  • There are only two independent elements in \(W/L\), \(I_D\) and \(V_{GS} - V_{TH}\).
  • Any one can be derived from the other two.

three

Ideal small signal model of MOS

basic

Second order effect

Body effect

当\(V_{S}\)大于\(V_{B}\)时，会发生什么？\(V_{TH}\)变大。

\[ V_{T H}=V_{T H 0}+\gamma\left(\sqrt{\left|2 \Phi_F+V_{S B}\right|}-\sqrt{\left|2 \Phi_F\right|}\right) \]

\(\gamma\) is called the body effect coefficient, 跟工艺有关。

Influence of body effect on small signal model: \(g_{mb}\)

\(g_{mb}\): body transconductance

\[ g_{mb} = \frac{\partial I_D}{\partial V_{SB}} \]

Channel length modulation effect

  • A NMOS operating in the saturation region with large enough length
  • \(I_D\) is independent of \(V_{DS}\).
  • MOS成为理想电流源。
  • Infinite output resistance(因为两端电压变化后，电流却不变)

The influence of \(V_{DS}\) on \(I_D\) is called channel length modulation effect.

\[ I_D \approx \frac{1}{2} \mu_n C_{o x} \frac{W}{L}\left(V_{G S}-V_{T H}\right)^2\left(1+\lambda V_{D S}\right) \]

沟道长度调制效应的结果是，即使在饱和区，漏极电流（\(I_D\)）依然轻微依赖于漏极电压（\(V_{DS}\)）

定义finite output resistance \(r_o\), \(g_{ds} = \frac{1}{r_o}\)

\[ r_{o}=\frac{1}{\lambda I_{D}} \]

Important note:

MOS small-signal output resistance can be changed by adjusting bias current \(I_D\), or transistor length \(L\).