Figure 4

Our analysis goal is to end up with a gain equation in terms of v_{out}/q_{in}. In the process, we will gain some insight into how a charge amplifier deals with the problem of changing cable capacitance.

First, let's sum all the charge flows:

q_{in} = q_{p} + q_{c} + q_{f}

Because of the relationship q = CV, we can rewrite the above as:

q_{in} = v_{in}C_{p} + v_{in}C_{c} + v_{f}C_{f}

or

q_{in} = v_{in}(C_{p} + C_{c}) + v_{f}C_{f}

We know, however, that v_{in} = 0, because of the *virtual* short across the input terminals of the op amp (assume an ideal op amp). So the equation above simplifies to:

q_{in} = v_{f}C_{f}

Rearranging, we have:

v_{f} = q_{in}/C_{f}

Again, because of the virtual short across the terminals of the op amp, we can say:

v_{out} = v_{f} = q_{in}/C_{f}

Rearranging again, we have:

v_{out}/q_{in} = 1/C_{f} where units are mV/pC.