Exercise 5.6 A common-gate amplifier and a something else

Below, the circuit schematic of the input stage of a dynamic microphone pre-amplifier is shown. Note that it resembled the amplifier in the previous exercise, but now uses MOS transistors. The microphone voltage is indicated by $V_{\mathit{mic}}$. The power supply voltage $V_{\mathit{DD}}=12V$. The transistor has a threshold voltage $V_T=1V$, and $K=40\frac{\mathit{mA}}{V^2}$. For the signal frequencies of interest all capacitors are low impedant.

a)

Draw the small signal equivalent circuit of the MOS pre-amplifier input stage, loaded by a next stage that has input resistance $r_{\mathit{in},\mathit{next}}$.

b)

Derive an expression for the voltage gain $a_v$ of the circuit, including the input resistance $r_{\mathit{in},\mathit{next}}$ of the next stage.

c)

Derive an expression for the input resistance of the amplifier, as seen by the microphone.

d)

Calculate the required drain current for an input resistance of $250\mathrm{\Omega }$.

The next figure shows the schematic of the second amplifier stage that is used to further amplify the voltage from the input stage. Again, $V_{\mathit{DD}}=12V$, $V_T=1V$, $K=40\frac{\mathit{mA}}{V^2}$ and for the signal frequencies of interest all capacitors may be considered as low ohmic.

e)

Draw a small signal equivalent circuit for the second amplifier stage.

f)

Derive an expression for the voltage gain $a_v$ of the second amplifier stage.

g)

Derive an equation for $R_{S2}$ to get a specific $I_D$, assuming square law operation and assuming that the other component vales are known.

h)

The second stage is dimensioned for a drain current of $I_D=5\mathit{mA}$ with $R_D=1.2k\mathrm{\Omega }$, $R_{S1}=250\mathrm{\Omega }$, $R_{G1}=R_{G2}=200k\mathrm{\Omega }$. Calculate (numerically) the required value of $R_{S2}$.

i)

Calculate the numerical value of the voltage gain of the second amplifier stage.