Preamble
FIGURE (a) represents a system to measure acceleration (i.e. an accelerometer). It shows a piezoelectric crystal that is connected to an amplifier and display via a length of coaxial cable.
A piezoelectric current is produced when the crystal is distorted by an applied force. In this application the force is produced by the reactive force of a mass when the accelerometer undergoes a change in velocity. The current ip is directly proportional to the rate of displacement, dx/dt, of one face of the crystal with respect to the other, as illustrated in FIGURE (b). Here ip = Kdx/dt, where K is a constant of proportionality.
In Laplace form, ip (s) = KsΔx(s).
In FIGURE(a) the piezoelectric crystal is modelled by a Norton current generator as a current ip in parallel with a capacitance Cp. The capacitance is due to the parallel-plate capacitor formed by the metallised contact plates placed on opposite faces of the crystal and the crystal itself forming the dielectric (see FIGURE (b)).
When a force F is applied across the face of the crystal, the current ip is generated. The interconnecting cable can be represented by a lumped capacitance Cc. The input resistance of the amplifier, RL, acts as a load to the crystal. The output of the amplifier drives a display (a moving coil voltmeter that is calibrated in units of acceleration (ms-2)).
Question
a) Draw the Laplace form of the input portion of the circuit, as represented in FIGURE (c).
b) Derive an expression for the Laplace transfer function, T (s) = ΔVL(s)/Δip(s), of the circuit of FIGURE (c).
c) Express ΔVL, as a function of time (i.e. the transient response of the voltage ΔVL) when ip is subject to a step change.
d) Using the values given in TABLE A, estimate the time taken for the voltage vL, to reach its steady state value if the current ip is subject to a step change of 2 nA.
| Cp |
1400 pF
|
| Cc |
250 pF
|
| RI |
5 MO
|
Table A
Please provide a detailed, plagiarism free solution for question.