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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 pi ft))

Current

I(t) = UoC pi f sin (2 pi ft)

UO max. supply voltage
f frequency
C actuator capacitance

Peak Current

Ip = pi Umax Cf

Average Current

Ia = Umax Cf

Peak Current exceeds average current by factor pi. 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 excitation

Operating 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 voltage

Power balance
Energy content E of a charged capacitance

E = CUo/2

Average power consumption PA during cycling with repetition rate w

PA = CUow/2

Dissipated power (selfheating problem)
During the charging/discharging cycles, the transferred power is partially dissipated into heat according

Pdis = CUow tan

tantan dissipation factor
5-10% of total power with common PZT actuator ceramics

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