1. Piezoelectric materials are often used as sensor materials because they transform mechanical deformation to electrical charge. Figure 1 shows an electric circuit of a piezoelectric sensor. The piezoelectric material is modeled as a current source in parallel with a capacitor. The resistance models the impedance of an amplifier. The goal is to make the output voltage v(t) proportional to the input charge q(t), so that we can use the voltage v(t) to measure the charge q(t). The ODE governing the input and output is dv RC +v=R- dt Ddq dt (1) where q(t) is the input and v(t) is the output. (a) Determine the frequency response function G(w). You can leave it as a fraction of two complex numbers. (b) Find the magnitude of the frequency response function G(w). (c) Show the asymptotic behavior of G(w) when w <1/RC and when w≫ 1/RC. (d) Plot the magnitude of G(w). (Not necessarily the Bode plot.) (e) For what frequency range can you use this sensor? x(t) R k с ġ(t) v(t) →f(t) Figure 1: A model of piezosensors Figure 2: A load cell to measure force