Frequency Response of the Voltage Output

Frequency Response of the Voltage Output Подпись: (3.85)

In the simulations given here, the base of the cantilever is assumed to be not rotating (i. e., h(t) = 0 in Figure 3.1) and the series connection case is considered first. The multi­mode voltage FRFs (per base acceleration) shown in Figure 3.5 are obtained from

where 6r and Cef are as given in the first column of Table 3.1 for the series connection of the piezoceramic layers. Note that, here and hereafter, the electromechanical FRFs are given in the modulus form and the base acceleration in the frequency response graphs is normalized with respect to the gravitational acceleration for a convenient representation (i. e., the voltage FRF given by the foregoing equation is multiplied by the gravitational acceleration, g = 9.81 m/s2). The set of electrical load resistance considered here ranges from 100 to 10 M^. As far as the fundamental vibration mode of this particular bimorph is concerned, the lowest resistance (Ri = 100 Й) used here is very close to short-circuit conditions, whereas the largest load (Ri = 10M^) is very close to open-circuit conditions.

As the load resistance is increased from the short-circuit to open-circuit conditions, the volt­age output at every frequency increases monotonically. To be precise, the voltage output for the exact short-circuit condition with zero external resistance (Ri = 0) will be zero, which would not allow a voltage FRF to be defined. Consequently, throughout this text, the short-circuit condition is defined as Ri ^ 0. At the other extreme, the open-circuit condition (Ri ^ to), the voltage output at every frequency converges to its maximum value. Another important aspect of the voltage FRFs plotted in Figure 3.5 is that, with increasing load resistance, the

Frequency Response of the Voltage Output

Frequency [Hz]

Figure 3.5 Voltage FRFs of the bimorph for a broad range of load resistance (series connection of the piezoceramic layers)

resonance frequency of each vibration mode moves from the short-circuit resonance frequency (rnsrc for R ^ 0) to the open-circuit resonance frequency (rn°rc for Rt ^ to). The short – and open-circuit resonance frequencies of the first three modes read from Figure 3.5 are listed in Table 3.4 (where fS = rnsrc/2л and froc = rn°rc/2л). The direct conclusion based on this ob­servation is that the resonance frequency of a given piezoelectric energy harvester depends on the external load resistance. Moreover, depending on the external load resistance, the resonance frequency of each mode can take a value only between the short – and open-circuit resonance frequencies fS and fOc. Closed-form identification of the frequency shift (A fr = ffc – frsc) based on the single-mode approximation is given in Chapter 5. Here, the data in Table 3.4 is read from the resulting frequency response graph given by Figure 3.5.

Two enlarged views of the voltage FRFs with a focus on the first two vibration modes are shown in Figure 3.6 in order to display clearly the resonance frequency shift from the short – circuit to the open-circuit conditions. Note that the voltage FRFs of the largest two values of load resistance are almost indifferent, especially for the second vibration mode, implying a convergence of the curves to the open-circuit voltage FRF. That is, if the voltage FRF for the 100 M^ case was also plotted, it would not be any different from that of the 10 M^ case. Again, these numbers are for this particular cantilever. For a different configuration, it might be the case that even a load of 100 Ш could be sufficient to represent the open-circuit conditions.

Table 3.4 First three short-circuit and open-circuit resonance frequencies read from the voltage FRF of the bimorph piezoelectric energy harvester

Mode (r)

fSc (Hz)

fOc (Hz)










Frequency Response of the Voltage Output

(a) io2


100 о 1 ко 10 ко 100 ко 1 мо 10 мо


R| increases









Frequency Response of the Voltage Output





160 180 200 220


1100 1150 1200 1250


Frequency [Hz]


Frequency Response of the Voltage OutputFrequency Response of the Voltage Output

Frequency [Hz]

Figure 3.6 Voltage FRFs of the bimorph with a focus on the first two vibration modes: (a) mode 1; and (b) mode 2 (series connection)

Подпись: 102 104 106 Load resistance [Q]

As far as the fundamental vibration mode is concerned, the short – and open-circuit resonance frequencies are 185.1 Hz and 191.1 Hz, respectively. Excitation at these two frequencies will be of particular interest in this section as well as in the experimental validations to be discussed in the next chapter. Variation of the voltage output for excitations at the fundamental short-circuit resonance frequency and at the fundamental open-circuit resonance frequency are plotted in Figure 3.7. As can be seen from the figure, for low values of load resistance, the voltage output at the short-circuit resonance frequency is larger since the system is close to short-circuit conditions. With increasing load resistance, the curves intersect at a certain point (around 120 Ш) and for the values of load resistance larger than the value at the intersection point, the voltage output at the open-circuit resonance frequency is larger. It is important at this stage to notice the linear asymptotic trends at the extrema of Rl ^ 0 and Rl (which will be shown mathematically in Chapter 5). The graph given here in log-log scale shows a linear increase in the voltage output with increasing load resistance for low values of load resistance

(both at the short – and open-circuit resonance frequencies). The voltage output becomes less sensitive to variations in the load resistance for its large values due to the horizontal asymptotes of the Rl extremum.

Updated: September 27, 2015 — 5:22 am