# Two Stage Amplifier

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This problem studies the two-stage MOSFET amplifier shown below. The first stage is built with an N-channel MOSFET. The second stage is built with a P-channel MOSFET. The P-channel MOSFET is identical to the N-channel MOSFET except for a reversal of some signs, which results in its source and drain being turned upside down. For the N-channel MOSFET, a sufficiently positive gate-source voltage (v_{GSN}) permits a positive drain current (i_{DN} > 0); for a P-channel MOSFET a sufficiently positive source-gate voltage (v_{SGP}) permits a negative drain current (−i_{DP} \gt 0). Thus the current flows from the source to the drain through a P-channel MOSFET which results in a negative i_{DP} (defined positive into the drain as always). A comparison of the N-channel and P-channel MOSFET behavioral equations are shown below. Note that K_N, K_P and usually V_{TN} are all positive, while V_{TP} is usually negative. Note too that the voltage subscript order indicates the definition of sign. For simplicity, let V_{TN} = V_{TP} = 0 in this problem.

i_{DN} = \begin{cases} 0, & \text{for } 0\gt v_{GSN}- V_{TN} \\ 0.5K_N\left(v_{GSN}-V_{TN}\right)^2, & \text{for } v_{DSN}\geq v_{GSN}-V_{TN}\geq 0\\ K_N\left(v_{GSN}-V_{TN}-v_{DSN}/2\right)v_{DSN}, &\text{for } v_{GSN}-V_{TN}\gt v_{DSN} \\ \end{cases}

-i_{DP} = \begin{cases} 0, & \text{for } 0\gt v_{SGP}+ V_{TP} \\ 0.5K_P\left(v_{SGP}+V_{TP}\right)^2, & \text{for } v_{SDP}\geq v_{SGP}+V_{TN}\geq 0\\ K_P\left(v_{SDP}+V_{TP}-v_{SDP}/2\right)v_{SDP}, &\text{for } v_{SGP}+V_{TP}\gt v_{SDP} \\ \end{cases}

###Part 1

Assuming that both MOSFETs operate in their saturation region, determine v_{OUT} as a function of v_{IN}. Use RP, KP, RN, KN, and vIN for R_P, K_P, R_N, K_N, and v_{IN}, respectively.

v_{OUT}=

###Part 2

Using your result from above determine the small-signal voltage gain of the amplifier as a function of the input bias voltage vIN. That is, determine v_{OUT}/v_{IN} = dv_{OUT}/dv_{IN} evaluated at V_{IN}. Use RP, KP, RN, KN, and VIN for R_P, K_P, R_N, K_N, and V_{IN}, respectively.

\frac{dv_{OUT}}{dv_{IN}}=

###Part 3

Again assume that the MOSFET is biased so that it operates in its saturation region. Develop a small-signal circuit model for the amplifier that can be used to determine vout as a function of v_{in}. In doing so, assume that V_{IN} defines the operating point around which the small- signal model is constructed, and evaluate all small-signal model parameters in terms of VIN as necessary.

 g_{mN} = g_{mP} =

###Part 4 Use the small-signal model to determine the small-signal voltage gain v_{out}/v_{in}. Compare this small-signal gain to that found in Part (1) and explain any differences.

v_{out}/v_{in}=

###Part 5 Determine the small-signal Thevenin equivalent of the amplifier when it is viewed through its output port.

 v_{th} = r_{th} =