# Laplace transform for a piezoelectric acceletometer

## Homework Statement

FIGURE 4(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 cable2.

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, , of one face of the crystal with respect to the other, as illustrated in FIGURE 4(b). Here , where K is a constant of proportionality. In Laplace form, . ddxt p ddx i K tps s i Ksx  

In FIGURE 4(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 4(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)).

a) Draw the Laplace form of the input portion of the circuit, as represented in FIGURE 4(c).

b) Derive an expression for the Laplace transfer function, , of the circuit of FIGURE 4(c).      
T(s)= ((delta)VL)(s)/((delta)ip)(s)

c) Express delta(vL) as a function of time (i.e. the transient response of the voltage ) 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

RL 5 M

## The Attempt at a Solution

I appreciate I may get a warning for no effort here, but I don't understand what is required for a). The fact they ask me to "DRAW" the input function has confused me. The transducer is an accelerometer, so acceleration is the input?

I started looking at an equation to link F=ma to Ip(s)=K(s)delta X(s) but struggled to see a way to link the two with the data I have.

I have emailed my tutor but I got his out of office, not back in until 2nd august, which is too late for me[/B]

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rude man
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## Homework Statement

a) Draw the Laplace form of the input portion of the circuit, as represented in FIGURE 4(c).
I think they meant fig. (b).?
b) Derive an expression for the Laplace transfer function, , of the circuit of FIGURE 4(c).      
T(s)= ((delta)VL)(s)/((delta)ip)(s)
c) Express delta(vL) as a function of time (i.e. the transient response of the voltage ) 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
RL 5 M
I appreciate I may get a warning for no effort here, but I don't understand what is required for a). The fact they ask me to "DRAW" the input function has confused me. The transducer is an accelerometer, so acceleration is the input?
Fortunately, you are not to be concerned with the output voltage as a function of acceleration. Which is a good thing since this system would never work as described. In particular, the pzt doesn't work at all as described.
But all they're asking for is the electrical part. You need to read the problem carefully.
In (a) you generate a step input of current ip and you're asked to derive the voltage across RL.
Similarly, in (b) thru (d) your input is ip, not acceleration.
Just ignore acceleration and F=ma.
You'll have to assume an amplifier gain Ka.

Hi Rude Man, thanks for the reply.

In a) they ask for the laplace form of the input portion of the circuit, but they give Ip(s)=Ks(delta)x(s) in the text

Have they not given me what they are asking for? or am i looking at it wrong?

In terms of the output I am looking at the formula V=1/c (int) i dt transforming it to laplace form as i/cs^2

Then i was going to find a form of i in terms of ip

sorry if this seams obvious but i am struggling with it

Last edited:
rude man
Homework Helper
Gold Member
Hi Rude Man, thanks for the reply.

In a) they ask for the laplace form of the input portion of the circuit, but they give Ip(s)=Ks(delta)x(s) in the text
Have they not given me what they are asking for? or am i looking at it wrong?
As I said the problem asks for electrical transfer functions only. You are to ignore all mechanical parameters, and that includes Ks(delta)x(s). Your input is ip and your output is either the voltage across RL (part (a) or at the amplifier output VL (part (c). Note that they never refer to fig. 4 (b) in the problem statements (a) thru (d).
I
In terms of the output I am looking at the formula V=1/c (int) i dt transforming it to laplace form as i/cs^2
If you were not introduced to network analysis using frequency impedances (1/sC for a capacitor) then you have to write an equation for the circuit of fig. (c) in the time domain. That would be i = C dV/dt for a capacitor and i = V/R for a resistor, or V = (1/C) ∫ i dt and V = iR, using the Kirchhoff laws. Then you can formally translate the time domain equation to the frequency (s) domain.
Then i was going to find a form of i in terms of ip
Yes.

So for a) what does it mean when it says draw the laplace form of the input portion of the circuit as represented in figure 4C. ive had no look in finding anything to do with this in my lesson notes.