Hi. I am confused when calculating Vsd in this figure as the resistor Rgd makes it an unconventional circuit to calculate when I apply DC analysis on it. How should I go around doing this question? Is there anyway that I can thevenin's Rg2, Rg2 and Rgd together or something? Thanks!
If this is a DC analysis, then you can ignore all of the nets with capacitors, and you end up with essentially a voltage divider made up of R\$_{G2}\$, R\$_{G1}\$, R\$_{GD}\$, and R\$_{D}\$.
R\$_{DS}\$ is the resistance between the drain and source; when the MOSFET is off, it is very high, when the MOSFET is on, or saturated, it is very low, the maximum value is generally given in a datasheet as R\$_{DS}\$(on).
The voltage at the drain is then:
$$V_{d} = V_{dd} \space\times\space \frac{R_{D}}{(\frac{(R_{G2}+R_{G1}+R_{GD})\space\times\space R_{DS}}{R_{G2}+R_{G1}+R_{GD}+R_{DS}})}$$
Since the voltage at the source is V\$_{dd}\$, then the voltage V\$_{sd}\$ is:
$$V_{sd} = V_{dd} - (V_{dd} \space\times\space \frac{R_{D}}{(\frac{(R_{G2}+R_{G1}+R_{GD})\space\times\space R_{DS}}{R_{G2}+R_{G1}+R_{GD}+R_{DS}})})$$
$$ \space = V_{dd} \space\times\space (1 - \frac{R_{D}}{(\frac{(R_{G2}+R_{G1}+R_{GD})\space\times\space R_{DS}}{R_{G2}+R_{G1}+R_{GD}+R_{DS}})})$$
The circuit you have has negative feedback to try and ensure the drain is set to about 50% of the power rail - so assume the drain has a quiescent voltage of half Vdd. Next, decide what drain current you wish to be flowing and this gives you the value of Rd.
You then have to calculate what dc voltage is required on the gate to achieve that drain current. Once you have that you can calculate the values of RG1, RG2 and RGD.
If you are looking for a mathematical relationship without constraining yourself to picking a drain current then I'll leave you to it because that's not how anyone would design this to operate in a practical circuit.
In other words Vsd is intentended to be half Vdd in this type of circuit.
