Understanding an active full-wave rectifier

[300b]

Hi Guys,

Since you were all very kind and helpful last time I had some trouble understanding a circuit, I guess it'll be a good idea to ask for your help once more.

I'm having some trouble understanding the following circuit:

Its from 'The Art of Electronics' page 222. Can you please help me understand it (a detailed explanation will be very much appreciated).

I've started by saying there is 0V at both inverting inputs (since the op-amp will do whay ever it can for it to happen), but I'm not sure where to go to next.

[djoffe]

top op-amp is a summer...the gain from Vin is -1.
The gain from the HW node is -2.

bottom op-amp is a half wave rectifier. The gain from the input to its output, the HW node, is -1 for positive input signals, and 0 for negative input signals.

Now, consider Vin>0 at the input. The overall output is
Vin*-1+Vin*-1*-2=+1*Vin.

For Vin <0 at the input, we have:
Vin*-1+0*Vin=-Vin

For positive Vin (Vin>0), the gain is +1, so the output is positive.
So for negative Vin (Vin<0), the gain is -1, so the output is positive.

[300b]

Sorry for the delayed response.

Thank you very much for the explanation, its a very nice way of looking at this circuit, didn't cross my mind