Question

a micromechanical resonator is vibrating at the frequency fo = 20 kHz with vibration aplitude x0 = 10 nm. The vibrations are measured using the rate-of-change measuremnent with a parallel plate capacitor that has the area A - 1000 um^2 and the gap d = 1um. Calculate the motional current if the bias voltage over the capacitor is Vdc = 5 V.

Answer 1

Answer:

Motional current is -8.85 x 10 ^-6

Explanation:

As the data is given in the question :

     V = 5 V

     A = 1000  μm2

     f = 20 kHz

Amplitude = 10nm

As we know, the formula of the capacitance is :

C =q/v

C = ε0A /d

dc/dt = 1/V dq/dt

ε0A d/dt  (1/d) = 1/V i

-ε0A/d2 d(d)/dt = 1/V i

As we know,

            Amplitude = 10nm

            frequency = 20kHZ

            d(d) /dt = 10 x$10^{9}$ x 20 x $10^{-3}$

                         =0.2 x $10^{-3}$

          i = -ε0AV / $d^{2}$ x 0.2x $10^{-3}$

          i = $\frac{8.85*10^{-12} *1000*10^{-6} *5*0.2*10^{-3} }{(10^{-6} )^{2} }$

          i = -8.85 x $10^{-6}$