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## Q. 7.7

Find the maximum thermal noise voltage that the gate resistance of a single MOSFET can generate. Neglect the device capacitances.

## Verified Solution

If the total distributed gate resistance is $R_G$, then from Fig. 7.24, the output noise voltage due to $R_G$ is given by

$\overline{V_{n, \text { out }}^2}=4 k T \frac{R_G}{3}\left(g_m r_O\right)^2$                                                   (7.32)

An important observation here is that, for the gate resistance noise to be negligible, we must ensure that (7.32) is much less than (7.30), and thus

$\overline{V_n^2}=\left(4 k T \gamma g_m\right) r_O^2$                              (7.30)

$\frac{R_G}{3} \ll \frac{\gamma}{g_m}$                              (7.33)

The number of gate fingers is chosen large enough to guarantee this condition.