Cascaded amplifier gain

[shredder929]

I'm trying to derive an expression for the magnitude of Vout/Vin for this circuit from Razavi. I know that Vout/Vin = (Vx/Vin) * (Vout/Vx), but does the same apply to the magnitude? Can I derive an expression for the magnitude of gain for each stage and multiply together such that |Vout/Vin| = |Vx/Vin| * |Vout/Vx|? That would simplify things greatly but mathematically I don't think that actually works out.

This isn't homework, I'm just self-studying/brushing up before going back to school.

 

[FvM]

You forgot to mention which circuit parameters are specified respectively ignored for the exercise problem. gm, Cgs, Cgd, Cds, ro?

 

[shredder929]

Sorry, no other parameters are specified. ro is infinite, no parasitics, and gm is to be used as a constant variable. Those don't really matter either way. I know how to derive the expression of the total transfer function given the circuit info, that's not the issue.

I guess this is more of a math question than circuit question. If I have two LTI systems, where the output of one is the input of another, can I multiply the *magnitude* of their transfer functions, or do I need to multiply their transfer functions and then find the magnitude of that? Similarly can I add their phase shifts together to get the total phase shift, or do I need to first derive the expression for the transfer function and then find its phase shift?

 

[The Electrician]

Sorry, no other parameters are specified. ro is infinite, no parasitics, and gm is to be used as a constant variable. Those don't really matter either way. I know how to derive the expression of the total transfer function given the circuit info, that's not the issue.

I guess this is more of a math question than circuit question. If I have two LTI systems, where the output of one is the input of another, can I multiply the *magnitude* of their transfer functions, or do I need to multiply their transfer functions and then find the magnitude of that? Similarly can I add their phase shifts together to get the total phase shift, or do I need to first derive the expression for the transfer function and then find its phase shift?

The magnitude of the product of two complex numbers is equal to the product of the magnitudes of the numbers.

A bigger problem could be that the second system input impedance is not infinite and therefore loads the output of the first system. In that case the product of the transfer functions is not the same as the overall transfer function. The circuit you posted shows the input of the second stage being the gate of a FET so presumably it won't load the output of the first, but in general this won't necessarily be true.