I tried to draw this schematic into recognizable blocks so lets go trough those blocks see what they do in circuit build them and test each block and its the order we build each block.
this block couldn't get any simpler 220VAC comes into primary of transformer and on the secondary we get 24VAC this is rectified by bridge rectifier consisting of 4 diodes then its fed to the high value capacitors for smoothing once we build this part time to turn it on and measure DC voltage across capacitors without load it should read ~32.44v �6v .
Now you might ask how come we get 33v on the output when we had 24VAC this is because on the AC scale of your multimeter multimeter takes only effective value of the AC voltage or RMS value real peak of the AC is increased by √2 or 1.41 so you take effective value in our case 24VAC and multiply it with 1.41 and we get 33.84 now we need to reduce this further for the amount of 2 diodes voltage drop, now in school i was taught that this voltage drop for silicon diode is about 0.7v but later you will find out that this value varies with temperature but lets just call it 0.7 for now, so 33.84-2x0.7=33.44v but in reality it should be close to that you will never get exact value calculated �6v for our application is fine. If it works proceed to next block.
Again simple block this block is just stable supply for the op-amps and voltage inverter it takes any higher voltage and spits out voltage set by the resistors. it consists of 3-Terminal Positive Adjustable Regulator chip LM317T 2 resistors which will set our output voltage and smoothing capacitors.
On schematic it is labeled that in that node we should get 22v but in reality i measured 23.5v which is totally fine. but here is how to calculate the output voltage, in datasheet we can find that Vout=Vref*(1+R2/R1) Vref is chips internal voltage reference which is 1.25v in case of LM317T now to calculate for our case Vout=1.25*(1+5100/300)=22.5v but again resistors aren't perfect and in this application they don't need to be precise they will also vary with temperature its why they are somewhat higher power there is no other reason, and again couple of capacitors to keep things smooth as possible
This block is voltage inverter it means that it takes voltage that is more positive then the GND and on its output gives voltage that is more negative in respect to the GND.
But why do we need such a contraption? Well its due to a fact that op-amps cant go to supply rails let me explain that if you supply op-amp with 22v on its output you will never get 22v or 0v it will always be close to those 2 extremes but in our application we need 0-20 volts on the output of the op-amp 22v is high enough to get 20v out but if we want to go to 0 volts output we would have to go lower then the GND thats why we need negative voltage anything lower then -2 and within a limit of what op-amp can handle is fine in schmatic is -6.6v which is fine