I am having trouble understanding how this relaxation oscillator works.
Can someone explain, why it produces a square wave form, how do the currents flow in this circuit and what is the op amp output each time?
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How it works:
What do you mean by switches high and switches low?
Figure 1. Internals of the ancient 741 opamp. Source: Wikipedia.
Most opamps will have an output arrangement similar to the push-pull arrangement of the old 741. Others will have FET transistors rather than BJTs. In either case if the top transistor (red oval) is turned on the output will be pulled to positive rail (switches high). If the bottom transistor (green oval) is turned on the output will be pulled to negative rail (switches low). How close they get depends on the exact output configuration and the driving circuitry.
So the op amp chooses if it will be for example +15V(switches high) or -15V (switches low) depending on the input.
Yes, if a +/-15 V supply is used. I suspect that your circuit uses only a +15 V supply because R4 is connected to 0 V (ground).
However what do you mean by saying " V+ is biased at towards half-supply by R3 and R4"? Why is it half supply?
I will explain more below but for now think that R3 is connected to V+ and R4 to 0 V so the connection is half-way or mid-supply, 7.5 V for a 15 V supply.
... and where is the supply in this circuit?
Don't ask me - you posted the schematic! It's not there. Neither are the component values. It is a poor schematic.
The oscillator relies on Schmitt trigger operation. A little "hysteresis" is added by the positive feedback to change the switching point.
Figure 2. (a) With a dual +/- supply the hysteresis is provided by R2 and R4 and is connected to ground. (b) With a single-ended supply (+ only) a "mid-supply" must be generated.
Maybe you don't know how the op-amp works- here it is intended to be used as a comparator and should be easy to analyze in the two possible output states, however the 'designer' has made a classic error and used an inappropriate component.
The OP-07 is a bipolar design with active bias current cancellation and has 2 series pairs of diodes back-to-back internally across the inputs. So once the differential input voltage exceeds two diode drops the inputs will present about a 1K load between them and the oscillator will work poorly and at a much higher frequency than would be expected.