An amplifier for a piezoelectric force sensor is shown, where the sensor $\backslash (\backslash$mathrm$\{$C$\}\_\{\backslash$mathrm$\{$s$\}\}\backslash )$ polarizes with applied stress. $\backslash (\backslash$mathrm$\{$C$\}\_\{\backslash$text $\{$ref $\}\}=0\mathrm{.}07\backslash$mathrm$\{$pF$\}\backslash )$

a$)$ Add a large resistor to ensure the op$-$amp has DC bias current.

b$)$ What is the transfer function for this circuit as a function of sensor charge $\backslash (\backslash$mathrm$\{$Q$\}\_\{\backslash$mathrm$\{$s$\}\}\backslash )($assuming R is very large and an ideal amplifier$)?$

c$)$ The crystal is circular with diameter $1$ cm $,$ thickness $1$ mm $,$ Young's modulus $90$ GPa, sensitivity $\backslash (2\backslash$mathrm$\{$pC$\}/\backslash$mathrm$\{$N$\}\backslash ),$ relative permittivity $5,$ and a resistivity of $\backslash (10^\{14\}\backslash$Omega$-\backslash$mathrm$\{$cm$\}\backslash ).$ A force of $0\mathrm{.}01$ N at $1$ kHz is applied uniformly across the electrode surface. What is the open circuit sensor voltage $\backslash (\backslash$mathrm$\{$V$\}\_\{\backslash$mathrm$\{$s$\}\}\backslash )?$ What is the amplifier output voltage?

d$)$ How does the input impedance of this amplifier influence the output voltage?