Hi,
Well in our case i think we are looking for the frequency domain solution, which would be like:
H(jw)=1/(j*w*R*C)
or without considering phase:
H(w)=1/(w*R*C)
and in alternate form:
H(w)=w0/w
and what i had found with the simulations i did was that the switched capacitor integrator had a function:
H(w)=K*(C1/C2)*w0/w, {with K a fixed gain}
and that tells me that the 'new' formula is either wrong or involves some other assumptions or something else we dont know about yet.
My suspicion was that the 'new' formula might be due to using an op amp with finite gain and bandwidth, so i proceeded to come up with a formula for a continuous integrator with that kind of op amp and this is what i found:
Vout=-(Aol*Vin*GBW)/(2*pi*Aol*f*j*C2*GBW*R1+2*
pi*f*j*C2*GBW*R1-2*pi*Aol*f^2*C2*R1+GBW+Aol*f*j)
where
Aol is the finite open loop gain of the op amp,
GBW is the finite gain bandwidth of the op amp,
f is frequency,
j is the imaginary operator,
C2 is the feedback capacitor found in integrators,
R1 is the input resistor of the integrator,
pi=3.14159...
Note that i did not verify this formula yet, but one thing that stands out already and seems intuitive is that any formula that contains a non ideal op amp would have to contain information about those non ideals. This one, as expected, contains the finite open loop gain Aol as well as the finite gain-bandwidth GBW. As far as i know, we cant have a formula without something like these because we would never be able to get the right result without knowing what the non ideals are. This is especially true if we allow the non ideals to change drastically, like 1 to 100000 for example. In this case we'd get different outputs just because that alone, so we'd have to include those i believe.
Now if the good professor is imposing some assumptions that we dont know about, then we'll never be able to figure that out. The only way then is to query the professor for more information. So Wizard, since you got this from the professor you'll have to query him to find out more about this. Perhaps the professor talked about things like this in the past and you forgot.
I had not gone thought the z transform solutions yet mostly because the simulations already had shown the gain is what it is, but we can do that if necessary.
For example, the function then is something like:
(C1/C2)*1/(z-1)