Consider the same series circuit as in Ex. 21.1, but now take the output voltage to be the voltage across the capacitor: DV_out = DV_C.  Find an expression for the gain of the circuit.

We follow through the same five steps to find the gain as we did in the previous example.  Since the circuit is the same as in Ex. 21.1, the total impedance and the current in the circuit are the same:

Step 1:

Step 2:

(Make sure that you can show these on your own!) Again, since the circuit is a series circuit, it follows that I = I_s for Step 3. 

Now, from Step 4 we get that

or

Solving for the ratio DV_out/DV_in gives

(Note that we have again used the relation that 1/j = –j in the previous step, so that 1/(jwC) = –j/(wC). )Finally, from Step 5, the gain is given by the ratio of the magnitudes of the numerator  and denominator, which are given by

and

as found previously in Ex. 21.1. The gain is therefore given by

This is the desired answer.  (Look at the bright side—just think of what an algebra expert you’re going to be!)