Explain the white noise and flicker noise in MOSFET. Derive equation for output and input referred noise voltage of CS stage
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i) For MOS devices operating in saturation region the channel noise can be modeled by a current source connected between the drain and source terminal and expressed as $\bar{T_n^2}=4KT\gamma g_m$
where, K= boltzmann constant=$1.38*10^{-23}$ J/K
$\gamma$=2/3 for long channel device in saturation
$\hspace{0.3cm}$=2 for sub-micron devices.

ii) The dependance of thermal noise upon Temperature(T) suggests that low temperature operation decrease the noise in analog circuit.

iii) Flicker Noise:
As charge carrier moves at the interface the random charge trapping by the energy states introduces a noise in the drain current called flicker noise. Flicker noise is modelled by a current source across the drain and source and expressed as

$\bar{V_n^2}=\frac{K}{C_{ox}WL}\, \frac{1}{f}$

enter image description here
where, K= process dependant constant.
W,L= width and length of MOS device.

iv) Flicker noise reduces with increasing frequency and at a point it starts falling much below the thermal noise. The frequency at which flicker noise is equal to thermal noise is called corner frequency($f_c$) of flicker noise.

enter image description here

$4KT(2/3 gm)= \frac{K}{C_{OX}WL}\, \frac{1}{f_c}g^2_m$
i.e $f_c=\frac{K}{C_{OX}WL}gm \frac{3}{8KT}$

This result implies that $f_c$ generally depends on devices dimensions and bias current.

Equation for output noise voltage of CS stage

enter image description here

We model the thermal and flicker noise of $M_1$ by 2 current ->

We also represent thermal noise of $R_D$ by current source.

The output noise voltage per unit BW=
$\bar{V_{n,out}^2}=\Big( 4KT\frac{2}{3}g_m+\frac{Kg_m^2}{C_{OX}WLf}+\frac{4KT}{R_D} \Big)R_D^2 \hspace{2cm}$.......(1)

We have,
$=\Big( 4KT\frac{2}{3}g_m+\frac{Kg_m^2}{C_{OX}WLf}+\frac{4KT}{R_D} \Big)R_D^2 \,\, \frac{1}{g_m^2\,R_D^2}\,\hspace{2cm}$

Input reffered noise:
$\therefore\,\,\,V_{n,in}^2=\Big( 4KT\frac{2}{3g_m}+\frac{K}{C_{OX}WLf}+\frac{4KT}{g_m^2\,R_D} \Big)\hspace{2cm}$.......(2)

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