2. This problem is a real application of the impedance method derived from Prof. Shen's research. His research team built a piezoelectric microactuator to be placed in inner ears as a hearing aid. As shown in Fig. 2, the piezoelectric microactuator was submerged in fluid and its impedance was measured. When the microactuator deteriorated, its impedance would change. Therefore, the impedance could be used as an indicator of the wellbeing of the microactuator. We developed a circuit model of the piezoelectric microactuator (see Fig. 3), where Cp was the capacitance from the microactuator, Rp was the resistance of the protection layer of the microactuator, R, and L, were the resistance and inductance of the wiring circuit. A voltage 1 Vs(s)est drove the circuit and a corresponding current I,(s)est flowed into the circuit. Answer the following questions. Examining glass Drilled hole for Ероху fluid injection The PZT fixture Probe Drilled hole for wiring Current Is(s) Artificial Perilymph (should be filled to full) 100 The circuit of diaphragm The circuit of wiring 50 www 11 R Voltage Vs(s) Magnitude mulating curve 3 Frequency (radis in decade) Phase angle -100 3 Frequency (radis in decade) Figure 2: A piezoelectric mi- croactuator in fluid Figure 3: Circuit model of the microactuator Figure 4: Measured and cal- culated impedance (a) Based on Fig. 3, derive the driving point impedance Z(s) as seen from the electric source Vs(s)est. (b) The current Ic,(s)est flowing through the capacitor Cp is related to the source current Is(s)est via Ic,(8)= SRpCp sR,C,+1 -Is(s) (2) i. Find the transfer function Hc(s) from the voltage Vs(s) to the capacitor current Ic,(s), that is, Hc(s) = Ic₂(s)/V₂(s). ii. How are the poles of Hc(s) related to the impedance Z(s) derived in part (a)? (c) One can measure the impedance Z(s) experimentally along the pure imaginary axis by sending a sinusoidal voltage [V(s)est] = sinwt to the microactuator. In this case, sjw, where j = √1. Figure 4 shows the measured impedance, both magnitude Z(jw) and phase LZ(jw) with respect to the driving frequency w in a log-log scale. At the high frequency range log 10 w > 7, the impedance can also be approximated as Z(jw)≈ R₂+j wCp (3) Determine the frequency w where the minimum magnitude occurs. [Hint: the minimum magnitude occurs when the phase angle is zero.]