inverting configuration

The "inverting configuration" is one of the simple methods of using an operational amplifier to perform actual amplification of an input signal. To create an inverting configuration you configure a circuit like this:

             i
 <-------------------------------
                 
              -Rf+
            +-/\/\--+
            |       |
      -Rs+  | |\    |
   +--/\/\--+-|-\   |  
   |          |  \__+--o Vo
   |          |  /                  
   |+     +---|+/            
Vs(~)     |   |/   
   |-     |                  
   |      |                  
   |      |                  
   +------+
          |
        -----
         ---
          -

We will now analyze the operation of this circuit:

1. Due to the concept of a virtual short, the voltage at the node attached to the (-) terminal of the op amp will be the same as the that at the node attached to the (+) terminal of the operational amplifier:
   V_- = V₊.
2. Since the (+) terminal is wired directly to ground, a potential of 0 exists at both the (+) and (-) terminals, as stated above:
   V_- = V₊ = 0.
3. A current (i) across R_s (The "source resistance") can now be calculated using Ohm's Law:
   i = (V_- - V_s)/R_s = 0 - V_s/R_s = -V_s/R_s
4. This current applied over R_f (The "feedback resistance") yields a voltage drop over R_f. Again, using Ohm's Law:
   V_f = i * R_f.
5. Substitute in i:
   V_f = -V_s*(R_f/R_s).
6. We can now determine the voltage at the output (V_o) of the op amp, relative to ground, by adding V_f to the voltage the (-) terminal:
   V_o = V_- + V_f = 0 + V_f = V_f = -V_s*(R_f/R_s).
7. The gain of an amplifier is a_v = V_out/V_in, so for our circuit:
   V_out = V_o
   V_in = V_s
   a_v = V_o/V_s = -V_s*(R_f/R_s)/V_s = -R_f/R_s

You can now attach various loads to the output of the op amp and be assured that the signal input in V_s, be it a sinusoidal or other signal, will be properly amplified.

This configuration allows one not only negative gains, but also gains smaller than 1, which is something the noninverting configuration does not allow.

Also, note that you can flip the op amp around, so that the (-) and (+) terminals are in the opposite positions, and it will still work, the magic of this configuration is in the fact that V_s has the same current flowing through it as through the resistors R_s and R_f, and in the virtual short to ground between V₊ and V_-.