Active filters make use of Op-Amps. This can make it possible to avoid using inductors and get the same effect by "reversing" a capacitor. There are multiple topologies in existance for these active filters:
I can see that they have different role off and amount of ripples in pass band and stop band. What I do not understand is how these were discovered or invented in the first place. I fully understand passive filters made up of RLC. It is evident how they work and why they are the way they are. However, for the active filters it is not clear. Did the designers get a hypethetical transfer function first and then realize it in electronics?