Common Source MOSFET Amplifier Biasing

Question

While reviewing simple transistor amplifier biasing techniques I came across this paragraph in Microelectronic Circuits by Sedra & Smith.

Here too we show the \$i_D–v_{GS}\$ characteristics for two devices that represent the extremes of a batch of MOSFETs. Superimposed on the device characteristics is a straight line that represents the constraint imposed by the bias circuit [...] The intersection of this straight line with the \$i_D–v_{GS}\$ characteristic curve provides the coordinates (\$I_D \$ and \$V_{GS}\$) of the bias point.

- S&S 7th Ed., Page #457

I understand how the \$R_S\$ resistor provides negative feedback when the MOSFET's gate voltage is held constant. But how does it impose a constraint and provide the coordinates of the bias point? Or do the authors mean something else?

Answer 1

By KVL $$V_G = V_{GS} + I_{D}R_S$$

Rearranging to solve for \$I_{D}\$ as a function of \$V_{GS}\$ we have $$I_D = \frac{V_G - V_{GS}}{R_S}$$

This is a line that has a slope of \$-1/R_S\$, intersects the \$I_D\$ axis at \$V_G/R_S\$, and the \$V_{GS}\$ axis at \$V_G\$, as shown in the graph. This line is a constraint, dependent on \$R_S\$, which the \$(V_{GS}, I_D)\$ bias point must intersect in order to obey KVL. Variations in the MOSFETs' characteristics can result in slightly different bias points (e.g. one might have a higher \$V_{GS}\$ with a lower \$I_D\$, another might have a lower \$V_{GS}\$ with a higher \$I_D\$), but all MOSFETs' bias points must fall on that line regardless of their characteristics since they must obey KVL.