Estimated Output Voltage Theoretical Analysis of Multilayer Structures

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3.5 Analysis and Discussion on Calculation Results

A cantilever as shown in Figure 3-6 with standard dimensions as summarised in Table 3-2 is used to verify the model as derived in section 3.4. The standard parameters for the calculation are also incorporated into the same table. Figure 3-6: Diagram of a multimorph cantilever structure. The theoretical model was based on a unimorph sandwiched structure of PZT, lower and upper electrodes. Table 3-2: Standard dimensions of a cantilever used to verify theoretical model. The calc ulat ion take s into acc oun t of the effe ct of cant Dimension Unit Value Length l b mm 18 Width w b mm 9 PZT thickness h p µm 80 Lower electrode thickness h e1 µm 15 Upper electrode thickness h e2 µm 20 PZT density kgm 3 7400 Electrode density kgm 3 10900 Base excitation A in ms 2 10 PZT elasticity e e GPa 116 Electrode elasticity e p GPa 60 Piezoelectric charge constant magnitude d 31 pCN 1 50 Dielectric permittivity nFm 1 4 Potential Free-Standing Structure Length, l b Width, w b Solder Pad Sacrificial Layer 57 ilever length on the mechanical damping, coupling factor and matching resistive load. These parameters were measured experimentally in Chapter 6 and were used to fit in the model. Mechanical damping involves complex damping loss factors which will be discussed in Chapter 6, but for a good approximation, the mechanical damping ratio is proportional to the length of the cantilever as shown in Figure 3-7 a. Typically a proof mass is attached at the end of a cantilever in order to reduce the resonant frequency and induce greater stress on the structure, which increases the mechanical damping of the structure. Figure 3-7 b shows the experimental results of the mechanical damping ratio for a standard cantilever with the dimensions as shown in Table 3-2. It seems that changing the proof mass has a greater effect on damping ratio than changing the beam length. For example, doubling the proof mass increases the damping ratio by nearly an order of magnitude more than the effect that doubling the beam length has on damping ratio. Figure 3-7: Experimental data of mechanical damping ratio as a function of cantilever length a and proof mass b. The dotted lines are a fitting line to illustrate that the mechanical damping ratio is proportional to the cantilever and proof mass. As the mechanical damping ratio increases, the electrical damping will increase to match the mechanical damping. Figure 3-8 a shows the experimental results of optimum resistive load, which is proportional to the length of the cantilever and shows a similar effect on the mechanical damping ratio to that of changing the length. The presence of proof mass, however, does not produce a linear effect on optimum resistive load. Figure 3-8 b shows that the optimum resistive load l evel off at about 240 kΩ. a b