Capacitive load with bridge rectifier

To demonstrate that the improved model can simulate the PZT beam with any electrical load, a bridge rectifier and a capacitive load are connected to the PZT terminals. Figure 6.10 illustrates the circuit with a ceramic capacitor load whose internal series resistance is modelled explicitly by R_esr. This circuit is the most

V2

Figure 6.10: Current source circuit representation of the PZT beam model connects to bridge rectifier followed by a capacitive load.

likely to be used in practice. In energy harvesting systems where power efficiency and simplicity are the top priorities, passive bridge rectifiers are generally preferred over Pulse Width Modulation (PWM) rectifiers. A low loss capacitor is then used as an intermediate energy storage. Only when the capacitor’s voltage is above a threshold the energy is extracted by enabling a DC-DC converter (not shown in circuit) in a very brief time, to transfer the energy to a bulk storage.

This configuration is recommended by the manufacturers of the energy harvesting power conditioning ICs MAX17710 and LTC3588-1. The circuit in Figure 6.10 is the linkage where the mechanical and electrical domain interact. Therefore, understanding the dynamic behaviour of this circuit is important for the design of the overall electrical circuit.

To model this system, modifications have to be made to the equations of motion developed in the earlier section. The forward voltage drop per diode is denoted by 94

VF. Starting from initial conditions as the switch is closed, when the diodes are not conducting, i.e. |vp(t)| − 2VF 6v_dc(t), the second coupled equation of motion (6.10b) needs to be rewritten as

Cpe˙vp(t) − which is just an open circuit condition with the polarisation losses included. The voltage of the capacitor remains the same. When the diodes are conducting, i.e.

|v_p(t)| − 2V_F > v_dc(t), the second coupled equation of motion becomes

Cpe˙vp(t) −

Including the first coupled ODE (6.10a) and (6.14) - (6.18), a system of ODEs can be established and solved by numerical methods. A relatively small step size should be used, or it renders the solution noisy or even unstable.

The numerical values for the experimental set-up can be found in Table B.6 of Appendix B. The diode full bridge rectifier used in the circuit is DF005S, and the ceramic load capacitor measurement is presented in Table 6.1. An experimental

Table 6.1: Parameters & material properties of the PZT beam.

Description Symbol Value Capacitor ESR Resr 0.54 Ω Load capacitance CL 4.554 µF Vol. drop per diode VF 0.6 V

validation of dynamic response in time waveform can be challenging due to the transient response. The transient response of the simulation is ideal but that of the experiment is compounded with signals from the transience of the shaker and the driving electronics in the experiment. Hence a switch is added to the test circuit to synchronise the transient response. The recorded data is processed in a way that at t = 1 s, the switch is turned on. The validation results for the input normalised tip mass velocity and output voltage are presented in Figures

CHAPTER 6. RESONANCE TUNING BY IMPEDANCE MATCHING

Time (s) Abs.velocity˙wabs(t)(m/s)

Figure 6.11: Validation results for the velocity time waveform of the tip mass. Both curves are the envelopes of the sinusoidal waveforms. Base acceleration is 0.2613 g at 171.3 Hz.

6.11 and 6.12, respectively. Unlike any of the previous presented measurement, the recordings in these two figures are not scaled by the base acceleration. This is because the measurement involved a bridge rectifier that has a fixed voltage drop, the output DC voltage cannot be simply scaled by the base acceleration.

Therefore, the actual measured values for 0.2613 g of base acceleration are used.

As can be seen, quantitatively the model agree well with the experiments. Inter-esting observations on electrical and mechanical damping can be made in Figure 6.11. Looking at the energy flow of the system (in Figure 6.13) can help under-stand what has happened to the harvester before and after the switch has closed.

During period A of Figure 6.11, the PZT is an open circuit, hence the damping consists of both mechanical and electrical damping. This can be reflected by the relatively low velocity amplitude. At the moment B when the switch is closed, the PZT is short circuited forcing the electrical damping to zero. However, the veloc-ity of the mass cannot increase instantaneously because it requires time for the kinetic energy to build up and reach steady state. During period C when the load capacitor is charging, the electrical damping starts recovering. Firstly the proof mass starts to build up its momentum due to the reduced electrical damping. The interaction of velocity, voltage and electrical damping causes the velocity to reach maximum and then decrease. Over the whole period C, the electrical damping is changing due to the change of the output voltage. This is continued until the capacitor is fully charged up as can be seen in period D, where the PZT is back to open circuit condition.

Figure 6.12 shows the rectified DC voltage output. A zoom-in of the plot confirms that the pattern of the curves from the model and experiment are very consistent.

From the design point of view, it is important to identify the capacitor voltage at which maximum power transfer occurs. This voltage corresponds to the point where ^dv(t)_dt v(t) is maximum [78] as the capacitor being charged up. Keeping the 96

Time (s) DCvoltagevdc(t)(V)

Figure 6.12: Validation results for the DC output voltage time waveform. Before 1s the plot shows the envelopes of the sinusoidal waveforms. Base acceleration is 0.2613 g at 171.3 Hz.

E = ¹₂mv²

Figure 6.13: Energy flow of the energy harvesting system. Here the harvested energy is the energy stored in the load capacitor.

capacitor around this voltage by a hysteresis loop can significantly improve power extraction.

For the validation results, there are obvious discrepancies between the model and experiment. One of them is the small phase shift found in the zoom-in views, which is caused by the small mismatch of resonant frequencies (less than 0.4 Hz) between the model and the experiment. The other obvious discrepancy is during the period C, when the capacitor is charging. It is deduced that the total damping predicted by the model does not completely match that of the actual system. The validation from the modulus matching circuit demonstrated the model’s ability to predict the electrical damping with the change of the output voltage. This makes the mechanical damping a potential source of mismatch. Figure 6.14 con-firms that the mechanical damping in the model is close to the actual mechanical damping of the PZT beam for the base acceleration of 0.2613 g. This is because mechanical damping ratio can be estimated as the ratio of the FWHM to the nat-ural frequency. However, the bending frequency response of the measured velocity suggests that the non-linearity of the PZT start to take effect. As elaborated by

CHAPTER 6. RESONANCE TUNING BY IMPEDANCE MATCHING

1400 150 160 170 180 190 200 210 220

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

Model Experiment

Frequency (Hz)

Abs.velocity(m/s)

Figure 6.14: Relative velocity of the PZT beam in short circuit condition. Base accelera-tion is 0.2613 g at 171.3 Hz.

Wolf and Gottlieb [79], the amplitude bending to the left is a non-linear softening behaviours. While the mechanical damping used by the model is constant, it is believed that the actual mechanical damping is changing due to the change of velocity. As the velocity decreases, the system becomes less non-linear and hence the mechanical damping decreases. That explains why the measured velocity in period C in Figure 6.11 is higher than expected. Subsequently, this hypothesis is confirmed by experiments, from which the results are shown in Figure 6.15.

Although the experiments were carried out on a PZT beam with a different length

(A) (B) (C)

Frequency (Hz) Frequency (Hz) Frequency (Hz)

Baseacceleration(g) Velocity(m/s) VelocityFRF(m/s/g)

Figure 6.15: Measured velocities for a PZT beam in short circuit. As the base acceleration increases, velocity increases but velocity per g decreases. (A) indicates different level of base acceleration. (B) shows the velocity response for the corresponding base acceleration in (A). (C) shows the FRF velocity response for the corresponding base acceleration in (A). [color only]

(L_b = 23 mm), it is evident that when the base acceleration increases, the velocity of the proof mass increases, but the velocity FRF actually decreases. This is an indication that the mechanical damping indeed increases as the PZT beam be-come more non-linear. If the PZT beam is excited in its linear region, the model’s accuracy should be improved. Overall speaking, the validation results presented 98

in Figures 6.11 and 6.12 are satisfactory. Lastly, by comparing Figures 6.15 and 6.11, the electrical damping has caused the velocity to reduce from 0.8 m/s to 0.3 m/s. This implies the electrical damping has a huge influence to the dynamic of the system. It reconfirms that electrical tuning, which is a way to manipulate the electrical damping, is an effective means to shift the resonant frequency of the system.