Derive an expression for the gain, g, of a CMOS inverter at its switching threshold, Vm.

The gain of a CMOS inverter at its switching threshold, Vm, can be derived using the following steps:

1. The CMOS inverter consists of a PMOS and NMOS transistor. At the switching threshold, the input voltage equals the output voltage. Therefore, we can write: Vm = Vin = Vout.

2. At this point, both transistors are in saturation. Therefore, the current through both transistors is equal. We can write: IDn = IDp.

3. The current through a transistor in saturation can be described by the equation: ID = 1/2 * kn * (VGS - VTn)^2 for NMOS and ID = 1/2 * kp * (VSG - VTp)^2 for PMOS.

4. Substituting VGS = Vin for NMOS and VSG = VDD - Vin for PMOS in the above equations, and equating IDn and IDp, we get: 1/2 * kn * (Vin - VTn)^2 = 1/2 * kp * (VDD - Vin - VTp)^2.

5. Solving the above equation for Vin gives us the expression for the switching threshold Vm.

6. The gain of the inverter is the derivative of Vout with respect to Vin. At Vm, the gain is maximum and can be found by differentiating the above equation and evaluating it at Vin = Vm.

7. The gain g is therefore given by: g = dVout/dVin = -2 * (kn * (Vin - VTn) / kp * (VDD - Vin - VTp)) evaluated at Vin = Vm.

This is the expression for the gain of a CMOS inverter at its switching threshold.