The position x(t) of the step front in the sample thickness direction can be easily calculated from the sound velocity v_{c} of the sample:

$\mathrm{x(t)}={v}_{c}\times t$

The phenomenological piezoelectric coefficient e_{33} is:

${e}_{33}\left(x={v}_{c}\times t\right)=\frac{I\times d}{A}\times {v}_{c}={e}_{i}-P$

I - current

d - sample thickness

A - sample area

e
_{i} - intrinsic piezoelectric coefficient

P - polarisation

If the piezoelectric constant of the quartz, the voltage amplitude, and the acoustic impedance of the quartz and of the sample are known, an absolute calibration of the PPS method is possible:

${e}_{33}=\frac{G\left(x={v}_{c}\times t\right)\times L\times u}{n\times U\times {d}^{2}}$

G - calibration factor (µC/cm^{2})

u - measured voltage signal (mV)

L - sample thickness (µm)

n - gain

U - applied voltage step (V)

d - effective sample diameter (mm)

For an x-cut-quartz we obtain:

$G=\mathrm{282,8}\left({Z}_{q}+{Z}_{p}\right)/{Z}_{q}$

where Z_{q} and Z_{p} are the acoustic impedances of the qartz and the sample

$Z=r\times {v}_{c}$

At 20°C is
$G=362{\mathrm{\mu C/cm}}^{2}$ for PVDF.

For PVDF the intrinsic piezoelectric effect can be neglected, so that

e_{33} = -P.

The measured voltage signal for a PVDF sample is proportional to the polarisation distribution in the thickness direction of the sample.

See the following page for some examples.