Question 7.14: PIEZOELECTRIC SPARK GENERATOR The piezoelectric spark genera...

We need to express the induced voltage in terms of the applied force. If the applied stress is T, then the induced polarization P is

P=d T=d \frac{F}{A}

Induced polarization P leads to induced surface polarization charges given by Q = AP.

If C is the capacitance, then the induced voltage is

V=\frac{Q}{C}=\frac{A P}{\left(\frac{\varepsilon_{o} \varepsilon_{r} A}{L}\right)}=\frac{L P}{\varepsilon_{o} \varepsilon_{r}}=\frac{L\left(d \frac{F}{A}\right)}{\varepsilon_{o} \varepsilon_{r}}=\frac{d L F}{\varepsilon_{o} \varepsilon_{r} A}

Therefore, the required force is

F=\frac{\varepsilon_{o} \varepsilon_{r} A V}{d L}=\frac{\left(8.85 \times 10^{-12} \times 1000\right) \pi\left(1.5 \times 10^{-3}\right)^{2}(3500)}{\left(250 \times 10^{-12}\right)\left(10 \times 10^{-3}\right)}=87.6  N

This force can be applied by squeezing by hand an appropriate lever arrangement; it is the weight of 9 kg. The force must be applied quickly because the piezoelectric charge generated will leak away (or become neutralized) if the charge is generated too slowly; many spark igniters use mechanical impact. The energy in the spark depends on the amount of charge generated. This can increase by using two piezoelectric crystals back to back as in Figure 7.44b, which is a more practical arrangement for a spark generator. The induced voltage per unit force V ∕ F is proportional to d /\left(\varepsilon_{o} \varepsilon_{r}\right) which is called the piezoelectric voltage coefficient. In general, if an applied stress T = F ∕ A induces a field E = V ∕ L in a piezoelectric crystal, then the effect is related to the cause by the piezoelectric voltage coefficient g,

E = g  T               [7.58]                  Piezoelectric voltage coefficient

It is left as an exercise to show that g = d /\left(\varepsilon_{o} \varepsilon_{r}\right) .