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7. Useful formulas

Notice: the following relations are dealing with ideal capacitors, where the capacitance is invariable under the driving conditions. But Piezoelectric actuators show to some extent deviations from this ideal behavior due to their ferroelectric nature. Their capacitances depend on electrical fieldstrength (voltage level), temperature and other parameters and may exceed the nominal values by 50%, which are stated in the data sheet.

 General: capacitor relation C=Q/U charging/discharging current I(t) = C dU/dt Average current I, t repetition rate, Uo maximum voltage Ia = UoC/t Sinuoidal excitation Unipolar signal U(t) = Uo/2(1-cos(2 ft)) Current I(t) = UoC f sin (2 ft) UO max. supply voltage f frequency C actuator capacitancePeak Current Ip = Umax Cf Average Current Ia = Umax Cf Peak Current exceeds average current by factor . Current booster needed for optimum power efficiency. Symmetric triangular signal Peak current Ip = Umax Cf Average current Ia = Umax Cf No current booster necessary. Pulse excitationOperating voltage Ua(t) of actuator: Ua(t) = Uo (1-e-t/RC) Charging current Ic(t) Ic(t) = (Uo-Ua(t))/R R load resistor of pulse generator (see sec. 2.8)Peak current at pulse onset: Icmax = Uo/R Average current Ia = UoCw w repetition rate, Uo supply voltagePower balance Energy content E of a charged capacitance E = CUo²/2 Average power consumption PA during cycling with repetition rate w PA = CUo²w/2 Dissipated power (selfheating problem) During the charging/discharging cycles, the transferred power is partially dissipated into heat according Pdis = CUo²w tan tan dissipation factor 5-10% of total power with common PZT actuator ceramics

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