The circuit is needed because you need to convert a small signal -which is generated by the microphone- to an higher-level signal, which is used by the *duino.
An operational-amplifier (OPAMP) is a Lego-brick-like amplifier: no easier way to combine a series of amplifiers together, so that you could perform "operations" on analog signals. As for "Operation", I mean addition, subtraction, etc.
However, you are not to deal with design, but you must consider some basic considerations for making the circuit working.
First of all, the circuit takes the "sound" captured by the microphone -which is a sine wave, for simplicity- then amplifies several times. I mean that if the wave's amplitude is 1 mV, the first OPAMP in the circuit outputs the same wave shape, but scaled of a factor 101. Why 101? Because the factor is given by the R4 / R3 ration, plus one.
BTW, that only scales the wave: it does not change the shape, which is still a sine. So, if you're whistling near the microphone, the sine wave could be -let's say- around 1kHz.
Secondly, there's another OPAMP (IC1/2), which amplifies *a lot* the pre-amplifies signal, so that it's output (going to the *duino) is almost a square wave. That's because the resulting sine-wave would be so huge that is cut: the supply of the OPAMP is limited. Try to draw a very-tall sine wave, and cut its edges horizontally: the result is much like a square wave.
Some personal consideration to the circuit.
- I don't understand why they put a capacitor right to the second OPAMP output: it has no sense, and stresses a lot the OPAMP, due the current flowing. I mean the desired goal, but that's not the right way.
- The circuit takes a sine wave and outputs a square wave: it's still a wave, and the Netduino isn't fast enough to process an analog signal. I'd change the circuit so that it reveals the "peak" of the sine wave.
You say that your circuit is not working. Well, some questions:
- have you used the same OPAMP model? Not all are the same.
- the microphone has to be an "electret", which is an active component.
- I would supply the circuit with +5V, not +3.3V. That's because the OPAMP works better, with a wider margin. Also the Netduino won't have problems with a +5V signal.
- There's "C2", which aim is to "decouple" the DC-offset of the microphone output. That's fine: try to detach the microphone, and the rest of the circuit should behave as a total silence. In such a context, try to test the voltage respect to the ground (using a voltmeter) in various points of the circuit. That's really useful. Without microphone (i.e. total silence) the last OPAMP output should be +5V.
About the last OPAMP output...
I'd cut the C3 from the output, and I'd try to read the output voltage by whistling near the microphone. You should have a floating value ranging from 0 to +5V. As soon there's no sound, the output should rise to +5V back.
If that's going right, I'd use a diode (e.g. 1N4148) from the OPAMP output to the Netduino input. The diode strip has to be toward the OPAMP.
The "old" C3 is still useful, and I'd place it across the Netduino input and ground, that is *after* the diode.
A resistor could be also useful, but its value is rather hard to calculate in advance. The better and easier way to select it, is to try to make some sounds and tune the resistor for the best performance respect the Netduino detector program. The resistor has to be placed from the Netduino input and +5V. Due the embedded pull-up of the Netduino, that resistor might be missing. Just try it...
What after this small modification? This modification is a "peak detector": as soon the signal goes low, the diode takes the voltage low also on the Netduino/C3 side. As soon the OPAMP returns high, the diode won't have effect, and the voltage on the Netduino/C3 side will raise "very-slowly" (*). So that the Netduino can read it without pain.
(*) "very-slowly" has to be meant "compared to the input signal. When you whistle, the 1kHz wave changes its value every 500us. With this peak-detector, the signal should keep its low-state for several hundreds of millisecs, or even more.
Hope it helps.
Biggest fault of Netduino? It runs by electricity.