Need help with basic low-pass filter design

According to this article about classification and application of filters: https://www.apogeeweb.net/article/35.html. I start up disigning my own basic filter circuit. Firstly there is a low-pass filter(pictured below).


I am supposed to derive an expression for the amplitude of the output voltage as a function of R1R1, R2R2, CC and ωω. I am kinda lost at how to do this, I know you are supposed to add the impedance of the parallel CC and RR, but how do I factor this into the voltage equation? I know the RloadRload isn't in the equation but does it do anything at all or is it just meant to show you there is a load connected?

From the equation of the amplitude of the voltage I'm supposed to then design a filter that meets the following requirements.

For ω=0ω=0 (which corresponds to DC), the amplitude must be A(0)=1A(0)=1. I'm confused because I thought the expression was in terms of AC? When I plug in ω=0ω=0 does it automatically turn it into a DC circuit?
For ω=1,000ω=1,000 rad/s, the amplitude must be A(1,000)=0.7A(1,000)=0.7. For this and the one above am I just supposed to plug them in and then do a guessing game with the values of the capacitor and the resistances?
Your element values must be physically realistic. Any ranges for physically realistic values? I guess ones I can find in our laboratory.
The second filter is a just an alternative circuit for filter design that is shown below.



The first part of this is to choose realistic values for resistors and capacitors. so that A(ω)=10A(ω)=10 for all ωω, I'm really lost on how to do this exactly. I know it's similar to the first one where I calculate the impedance but how would I make it so that A(ω)=10A(ω)=10 for all ωω?

Any help would be much appreciated, I then have to write Matlab code that solves the circuits for different frequencies. Pretty sure I can do this. I'm just looking for a point in the right direction.

I have done a fair amount of work on this already, solved for what I think is A(ω)A(ω) for the first circuit and when running the Matlab code I created it seems to work, however I can't be sure that I'm right and would love someone to post what they get. I can post work if needed however my phone is old school so the image quality is terrible. I'm just looking for a point in the right direction, any help is much appreciated . Thanks.

An ideal op amp can drive any load and still maintain its output voltage. therefore, Rload is not important. if Rload demands 100A out of the op amp, then an ideal op amp will be able to supply it. Therefore Rload does not feature in your equation for gain.

OK, you know the gain of an inverting op amp is -Rf/Ri where Rf is the parallel combination of R2 and C and Ri is equal to R1 in your circuit above. We know that a capacitor has an infinite resistance at dc... so therefore it will not have any impact on R2. It is effectively not there (open circuit). Therefore your gain equation at dc ignores the capacitor. Therefore with, say, R2 = 10k and R1 = 10k, the gain will be -1, regardless of the value of the capacitor

Now, the capacitor decreases in resistance (impedance) as frequency goes up. Therefore as the frequency goes up, the gain of your circuit will go down as the feedback resistor, R2, slowly gets shorted out by the decreasing resistance of the capacitor. When the capacitor's impedance equals the value of the resistance, the gain of the circuit (for equal R1 and R2) will be 0.707. This sounds like an exam (homework) question, so I will not explain why it is 0.707 and not 0.5.

This line of thinking should set you on the right course to work out the rest of the question

-Simon

First, if you just want an answer, go to the Analog's filter wizard. TI's is also good, but not as easy to use IMHO.

If you want to understand it, get or access (i.e., on the Internet), Rolf Schaumann and Mac E. Van Valkenburg's text, Design of Analog Filters, Oxford University Press , ISBN 0-19-511877-4 . It has a orangish cover and is available on the Internet free. I hope I will not be flagellated for giving this link: https://www.academia.edu/15449434/Design_of_Analog_Filters_Rolf_Schaumann_Mac_E_Van_Valkenburg

It really is a wonderful text and I bought it, despite it being free on line. My net cost (after a little negotiation) was $10 for a new, unmarked version.

John

FILTERLAB by Microchip is good to implement real circuits. I used it with good results.