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5

Assuming ideal diodes, for $V_{in} \lt 0$ this is an inverting op amp with gain $\frac{R_3}{R_1}$, and $R_2$ keeps the non inverting input from floating. For $V_{in} \gt 0$, this is a buffer. The circuit probably makes the most "sense" if $R_3=R_1$, but if $R_3 \ll R_1$, than you are getting something akin to half wave rectification instead of ...

3

The opamp circuit is a full wave rectifier with some level of noise rejection when the input peak amplitude starts to fall below about 0.6 volts. Positive voltages are amplified by D2 (D1 is blocking) and negative voltages are amplified and inverted via D1 (D2 is now blocking). R2 is needed to bias the non-inverting opamp input when the device is inverting i....

2

In engineering or science speak unity gain is 0.0000000'db. Get your head out of your theoretical cloud. You need to decide in terms of how you are applying this where close enough to 0db is good enough.

2

C2's purpose is providing negative feedback for higher-frequency signals, thereby attenuating them. In other words, U1's configuration works as an active low-pass filter. Not by any means a steep one. I have several problems with the shown schematic, though. They range from aesthetic (GND should always point downwards, full stop) to the functional (the ...

2

The diodes on the input are intended to compensate for the diode on the output. They face different directions so that each compensates for the positive drop on the output diode.

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