(Solved): A capacitive level sensor is shown in Fig. 1. It consists of two concentric cylinders with diameter ...

A capacitive level sensor is shown in Fig. 1. It consists of two concentric cylinders with diameters \( d_{1} \) and \( d_{2} \). The \( A C \) bridge shown in Fig. 2 is used to detect the fluid level inside a tank \( \left(h_{2}\right) \). The variable capacitance \( C_{x} \) is function of the fluid level \( h_{2} \) and is given by Ag. 2 \[ C_{x}=C_{0}\left(1+a h_{2}\right) \] where \( C_{0}=c \) a pacitance value when the tank is empty (air capa citor) = \( \frac{2 \pi \varepsilon_{0} H}{\ln \left(d_{2} / d_{1}\right)}=40 \mathrm{pF} \) \( \alpha= \) positive constant \( =\frac{c_{r t}-1}{N} \) \( H= \) height of the tank \( =h_{1}+h_{2} \). \( \varepsilon_{1}=\varepsilon_{0}= \) permittivity of air (fre e space) \( =8.85 \times 10^{-12} \mathrm{Fm}^{-1} \) \( \varepsilon_{2}=\varepsilon_{o} \varepsilon_{r 2}= \) permittivity of the fluid. \( \varepsilon_{r 2}= \) relative permittivity of the fluid. The fluid used in this experiment is the acetone which has a relative \( p \) ermittivity of \( \varepsilon_{r 2}=20.7 \) at \( 25^{*} C \), the tank height \( H=2 \mathrm{~m} \) and the acetone level \( h_{2} v a r i e s \) between \( 0.0 \mathrm{~m} \) and \( 1.8 \mathrm{~m} \). 1)Prove the expression of \( C_{x} \) given in equation (1) 2) Find the value of the capacitor \( C_{1} \) to balance the bridge (i.e. \( V_{o}=0 \) when the tank is e mpty) if \( R_{2}=R_{3}=5 \mathrm{k} \Omega \) 3) Find the expression of the bridge voltage \( V_{o} \) in terms of the bridge parameters \( R_{2}, R_{3}, C_{1}, C_{x} \) and the supply voltage \( V_{e} \) 4) If \( V_{e}=10 \sin \omega t V \), find the minimum and the maxim um values of \( V_{o} \).