Piezo Ceramics Tutorial 15 of 15

Useful Electro-Mechanical Relationships:
Dynamic Operation

Formula 17 (Plates)
Formula 18 (Discs)	kp2 = (1 - kp2) (Bessel Function)
Formula 19 (Rods)

In addition to the coupling coefficient, the total efficiency of a transducer depends on the mechanical and dielectric losses. The dielectric losses, tan , are given by the dissipation factor, D.F., as described in Formula 20.

Formula 20	D.F. = tan = 1/QE
Where QE is the electrical damping.

The mechanical losses can be determined from the mechanical quality or damping factor, Q_M, from Formula 21.

Formula 21
Where C is the low frequency capacitance and Zr is the minimum impedance.

Q_M can also be determined approximately from the frequency response curve as follows:

(only where Q>3)

The frequency difference f₂ - f₁ is the frequency bandwidth at about 3dB where the amplitude is 1/SQR(2) of its maximum value.

Of these losses, the dielectric losses are usually the most significant. Therefore it is recommended that materials with a low dissipation factor be used for high power applications, particularly since these losses increase with power. For high intensity transducers, the overall electroacoustical efficiency is given approximately by Formula 22.

Formula 22
Where QA is the mechanical quality factor due to the acoustic load alone.

It should be noted that at high drive levels QE and QM are not constants. They are usually lower than the low drive level values.

The dielectric permittivity of the material. and therefore the dielectric constant and capacitance, decreases as the applied frequency (mechanical or electrical) exceeds each resonant frequency of the particular ceramic part. For static operation, well below the first resonance frequency, the dielectric permittivity is ^T₃₃ (free).

For dynamic operation well above all resonance frequencies of the ceramic part, the material behaves as if it was clamped (strain = 0), and the electric permittivity is ^S₃₃ (clamped). Between each, the permittivity is the product of the static permittivity and a loss term based on the coupling of the resonance mode each resonance point the applied frequency has exceeded, as described in Formula 23 (above first resonance), Formula 24 (above second resonance), and Formula 25 (above third resonance).

Formula 23 (above first resonance)	T33 (1 - k12)
Formula 24 (above second resonance)	T33 (1 - k12) (1 - k22)
Formula 25 (above third resonance)	T33 (1 - k12) (1 - k22) (1 - k32)

where k₁, k_Z and k₃ represent the coupling factors for the particular resonance For a thin plate, k₁ and k₂ are k₃₁ and k'₃₁ (length and width, respectively), and k₃ is k_t (thickness) For a thin disc, k₁ is k_p (radial), k₂ is Kt (thickness), and there is no third resonance. For a rod, k₁ is k₃₃ (length), k₂ is k'_p, and there is no third resonance.

In addition to FA and f_r (series and parallel resonance frequencies), there is a frequency, f_m at which the transducer's electromechanical transduction is maximized This frequency represents the maximum sensitivity for receivers or the maximum output for drivers This frequency, the bandwidth, and the output are all dependent on the external resistive load, R_eX .

When k<<1, f_m may be calculated using Formula 26.

Formula 26
Where
fm = FA for (Q<<1, Rex small, short circuit condition) fm = f for (Q>>1, Rex large, no load condition)

The maximum bandwidth, B, obtainable by electrical tuning, is approximately equal to the product of the coupling coefficient and the series or parallel resonance frequency. Refer to Formula 27.

If the mechanical quality factor is high (QM>Q), the external Formula 37 resistance R_ex for a fairly flat frequency response can be approximated by Formula 28 for parallel inductance, or Formula 29 for series inductance.

Formula 28
Formula 29

Many of the calculated parameters above are interrelated Thus, many useful relationships can be derived A few of the most useful relationships are described in Formulas 30 through 37.

Formula 30 (Rods)
Formula 31 (Rods)
Formula 32 (Rods)
Formula 33 (Plates)
Formula 34 (Plates)
Formula 35 (Plates)
Formula 36 (Hydrostatic charge constant)
Formula 37