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In Analog CMOS integrated circuit book، it is stated that:

In order to restore the dynamic range, the transconductance of the transistors must be increased by a factor of square of x because thermal noise and currents scaled with root of gm(transconductance).

Why does thermal noise in short-channel transistors depend on root of transconductance? Can you please introduce some article about it?

UPDATE

A more detailed quote:

The greatest impact of scaling on analog circuits is the reduction of the supply voltage. With ideal scaling, the maximum allowable voltage swings decrease by a factor of α, lowering the dynamic range1 of the circuit. For example, if the lower end of the dynamic range is limited by thermal noise, then scaling VDD by α decreases the dynamic range by the same factor because gm and hence thermal noise remain constant. Of course, since for analog circuits (VDD/α)(IDD/α) = VDD.IDD/(α)^2, the power dissipation drops by α^2. In order to restore the dynamic range, the transconductance of the transistors must be increased by a factor of α^2 because thermal noise voltages and currents scale with √gm. Thus, since voltage scaling requires that VGS − VT H decrease by a factor of α, we note from gm = 2×ID/(VGS − VTH ) that ID must increase by the same factor, leading to a power dissipation of (VDD/α)(α×ID) = VDD×ID. Also, from gm = μCox×(W/L)×(VGS−VTH ), we conclude that if Cox is scaled up by α and L and VGS−VTH are scaled down by α, then W must increase by α (whereas in ideal scaling, it would decrease by this factor). That is, for a constant (thermal-noise limited) dynamic range, ideal scaling of linear analog circuits requires a constant power dissipation and a higher device capacitance, e.g., (α×W)(L/α)(α×Cox ) = αWLCox . Interestingly, if the lower end of the dynamic range is determined by kT/C noise, then to maintain a constant slew rate in switched-capacitor circuits, the bias current must scale up by a factor of α^2, resulting in an increase in the power dissipation. (Problem 17.17.3)

mohammad rezza
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