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 i_p 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 i_p = Kdx/dt, where K is a constant of proportionality.

In Laplace form, i_p (s) = KsΔx(s).

In FIGURE(a) the piezoelectric crystal is modelled by a Norton current generator as a current i_p in parallel with a capacitance C_p. 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 i_p is generated. The interconnecting cable can be represented by a lumped capacitance C_c. The input resistance of the amplifier, R_L, 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) = ΔV_L(s)/Δi_p(s), of the circuit of FIGURE (c).

c) Express ΔV_L, as a function of time (i.e. the transient response of the voltage ΔV_L) when i_p is subject to a step change.

d) Using the values given in TABLE A, estimate the time taken for the voltage v_L, to reach its steady state value if the current i_p is subject to a step change of 2 nA.

Table A

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