Parameter extraction from BVD electrical model of PZT actuator of micropumps using time-domain measurement technique

Abstract

To minimize the power consumption of the piezoelectric micropumps utilized in the microfluidics field, suitable models are required to enable the optimization of the lead zirconate titanate (PZT) actuator driving circuits. The recent research shows that the electromechanical parameters of piezoelectric materials can be obtained from Butterworth Van-Dyke (BVD) model. The current study presents a novel time-domain measurement technique for extracting the parameters of a BVD electrical model describing a PZT actuator of the micropump excited by a square pulse with a relatively high voltage and low frequency. The validity of the BVD model is evaluated by using MATLAB software to solve the ordinary differential equations of the electrical model and then comparing the numerical results obtained for the current response of the PZT actuator with the experimentally-observed results. The BVD model has been found to be valid for this specific application of piezo actuators. A good agreement is observed between the two sets of result, and thus the validity of the BVD model and the proposed time-domain measurement technique is confirmed.

Appendix: Procedure of solving the RLC branch

Applying the KVL to this simplified circuit, the RLC series circuit, it can be shown that

$$ i_{s} R^{\prime} + v_{{C_{1} }} + L_{1} \frac{{{\text{d}}i_{s} }}{{{\text{d}}t}} = A + N(1 - {\text{e}}^{{ - {t \mathord{\left/ {\vphantom {t {\tau_{1} }}} \right. \kern-\nulldelimiterspace} {\tau_{1} }}}} ) $$

(34)

where R ^′ = R ₁ + R _L ,  $v_{{C_{1} }}$ is the voltage difference across C ₁, and A, N and τ ₁ represent the minimum voltage intensity, the peak-to-peak voltage and the rising time constant, respectively, of the amplified square pulse voltage (v ₁). Since $i_{s} = C_{1} ({{{\text{d}}v_{{C_{1} }} }}/{{{\text{d}}t}}),$ and given the assumption of an initial condition$v_{{C_{1} }} (0) = A,$ Eq. 34 can be rewritten in the form of the following second-order differential equation:

$$ \frac{{{\text{d}}^{2} v_{{C_{1} }} }}{{{\text{d}}t^{2} }} + \frac{{R^{\prime}}}{{L_{1} }}\frac{{{\text{d}}v_{{C_{1} }} }}{{{\text{d}}t}} + \frac{{v_{{C_{1} }} }}{{L_{1} C_{1} }} = \frac{{A + N(1 - {\text{e}}^{{ - {t \mathord{\left/ {\vphantom {t {\tau_{1} }}} \right. \kern-\nulldelimiterspace} {\tau_{1} }}}} )}}{{L_{1} C_{1} }} $$

(35)

in which the general solution of $v_{{C_{1} }}$ has the form

$$ v_{{C_{1} }} (t) = v_{\text{h}} (t) + v_{\text{p}} (t) $$

(36)

The homogeneous term in Eq. 34 is given by

$$ v_{\text{h}} (t) = (B_{1} \cos \omega_{\text{d}} t + B_{2} \sin \omega_{\text{d}} t){\text{e}}^{ - \alpha t} $$

(37)

with

$$ \alpha = \frac{{R^{\prime}}}{{2L_{1} }} $$

(38)

and

$$ \omega_{\text{d}} = \sqrt {\frac{1}{{L_{1} C_{1} }} - \alpha^{2} } = \frac{{2{{\uppi}}}}{T} $$

(39)

where α is the damping coefficient, ω _d is the damped angular frequency and T is the damped period. Note that the value of ω _d can be easily determined from the experimental data and is discussed in Sect. 3.2. However, α, B ₁ and B ₂ are all unknowns with constant values.

The particular term in Eq. 34 has the form

$$ v_{\text{p}} (t) = a + b{\text{e}}^{{ - {t \mathord{\left/ {\vphantom {t {\tau_{1} }}} \right. \kern-\nulldelimiterspace} {\tau_{1} }}}} $$

(40)

where a and b are constants and unknown.