Converters Measurements And Characterization

Let us now start using converters modeling (based on previously established equivalent models) for the purpose of converters characterization, optimization and defining converters quality parameters (using only electrical measurements data).

For the purpose of illustrating measured and calculated converters’ quality parameters, in the table T 1.1 are given (low-signal) measurement data of BRANSON’s converter, model 502/932R, max. 3000 Watts, 20 kHz (see also Fig. 12 and Fig. 13 with impedance-phase curves of the same, non-loaded and fully-loaded 502/932R converter), measured using HP 4194A Network Impedance Analyzer. The converter example chosen here is neither the best nor the worst case of 502/932 BRANSON converters, and it is only taken for the purpose of having numerical illustration/s following converters modeling presented in this paper.

Converter loading (regarding data in T 1.1) is realized by placing its front emitting surface on a thick rubber-foam full with water (with wooden-plate backing), and by pressing the converter towards its load (applying a force of about 20 kg, on the converter housing). Practically, this kind of load simulation is measured by HP 4194A under low current and voltage signals (sweeping frequency signal of 1 v, rms, produced internally in HP 4194A). In reality loading measurements should be made with operating converter driven high power, but from long experience in this field, we know that it is sufficiently representative to make low-signal measurements (similar to above explained situation) in order to get useful results that would be in the error-limits of max. 10%, compared to a situation if measurements were made operating converter full power. Here we shall follow the strategy to explain qualitative changes during converters (mechanical or acoustical) loading, and until standardized loading situations are not specified, loading would stay application-dependant and descriptively explained process.

T 1.1 (BRANSON 502/932R, 3000 Watts, 20 kHz converter; measurement data)

Assembled Converter	In Series Resonance	In Parallel Resonance
Non-Loaded (modest quality) Converter Fig. 12
	Coupling factor of the fully assembled, modest quality converter (PZT8, k33=0.64) (the ideal converter design would be when ( ))

Fully-Loaded (modest quality) Converter (still well operational and able to deliver maximal output power), Fig. 13
	(the ideal converter design would be when ( ))

Fig. 12 Impedance curve of non-loaded BRANSON 502, 932R converter

Fig. 13 Impedance curve of fully-loaded BRANSON 502, 932R converter

Converters Quality Parameters

The most important quality quantifications of a high power ultrasonic converter (operating in certain selected resonant mode) are its mechanical quality factors (related and found only for the mechanical oscillatory circuit-part/s of the equivalent converter models, Fig. 6), equal to,

(1.1)

and can be expressed (similar as in Electric Circuit Theory) for series resonance and for non-loaded piezoelectric converter (hanging in air) as,

(1.2)

and for parallel resonance (for non-loaded piezoelectric converter, hanging in air) as,

(1.3)

When we are talking about converters’ mechanical quality factors, this is usually related to non-loaded (fully free) converters, hanging in air, and in all above given expressions (as well as in other parts of this paper) every indexing with "o" indicates this situation ("o" = non-loaded). Index "m" indicates that we are talking about mechanical-circuit related parameter/s, and indexing with "1" and "2" is related to series and parallel resonances. Since the same parameters can be found for loaded converters, indexing "L" is reserved for loading and/or load-influenced parameters.

A well operating and highly efficient (high power) converter, operating in a continuous regime should be designed to have as higher as possible mechanical quality factors (either being non-loaded, or fully loaded), and in this paper we shall mostly address only such kind of converters (what will significantly simplify different mathematical expressions and formulas related to such converters).

In (1.2) and (1.3) we can see two of unusual mechanical parameters (not found in literature), which are here addressed as "characteristic mechanical impedances of series and parallel resonance, of non-loaded converters", defined as,

(1.4)

Another couple of unusual mechanical parameters (invented in this paper for the purpose of converters characterization and comparison), are "characteristic average, axial wave-velocities of non-loaded half-wavelength converters, operating in series and/or parallel resonance", defined as,

(1.5)

In all previously given expressions, it is obvious that we are talking about series and parallel resonant impedance characteristics, where series and parallel resonant frequencies can be approximated (in the case of high quality converters) as,

(1.6)

Also, the absolute values of converter’s impedance in its series and parallel resonance (for non-loaded converter) are,

(1.7)

High quality converters usually have very low series impedance (10 Ω range, and lower values), and very high parallel impedance ( 100 KΩ range, and higher values).

Now, after introducing definitions and symbolic related to most important converter "static" parameters, we can address the other very-important, dynamic (loading) parameters of high power piezoelectric converters. Converters loading process is changing its electrical impedance on the way that (under loading) minimal series impedance is increasing, and that maximal parallel impedance is decreasing (coincidently). Both mechanical quality factors ((1.2) & (1.3)) of series and parallel resonance mechanical-circuits are also dropping down coincidently following mechanical load increase.

It will be shown that the mechanical quality-factors ratio, between the state of non-loaded and fully-loaded converter, is one of the best measure of converter’s dynamic (loading) performances (see T 1.2). Let us establish the following convention and symbolic regarding converters mechanical quality factors (in non-loaded and loaded conditions):

(1.8)

Under fully-loaded converter’s quality factors, (1.8), we shall understand that converter is measured when heavily (maximally) loaded, but being still well operational and able to deliver (safely) 100% of its maximal operating power. This time we shall not describe what means heavy-loaded converter, since this is an application-dependent situation, but in general certain loading situations could be standardized, such as process of immersing the front emitting face of the converter in water, or contacting and pressing converter’s front emitting face to some other material (until it properly operates full-power). In reality, we should first test maximal load situation, by operating converter full-power (to see where is maximal loading level, when converter delivers its maximal output power), and then we should keep the same loading (the same kind of contact between converter and its load), and make low-signal impedance measurements using Impedance Analyzer (this time converter is only connected to Impedance Analyzer). Of course, the best would be to have an impedance analyzer able to operate in high power conditions, measuring high voltage and high current values, and to avoid load simulation, until all loading standards and correlations between Low-signal and High-signal measurements become well known (but such instruments are still not available).

Using symbolic from (1.8) we can formulate the following, most important dynamic loading parameters of ultrasonic high power converters, related separately to converters’ efficiency ( η in %) in series and parallel resonance operation/s:

(1.9)

Practically, the higher the ratio between non-loaded and fully-loaded mechanical quality factors (1.9) is, the better efficiency η , and better dynamic (loading) converter performances are (respecting that fully-loaded converter is still well operational and able to deliver its maximal power). The ratio/s (1.9) also present the measure/s of converters capacity to accumulate and exchange potential elastomechanical energy with input electric energy (when converter operates in resonance and when electrical and elastomechanical energy are mutually transforming, producing converter oscillations).

In the close relation with converters dynamic performances are relations presenting resonant frequency shift (or frequency deviation) from non-loaded until fully-loaded situation, expressed as,

(1.10)

High quality converters should have frequency intervals and central frequencies ratios (1.10) close to 1 (or saying differently, as this ratio is closer to 1, the converter is more stable and performing better).

Regarding frequency stability between loaded and non-loaded converter states we can also calculate the central frequency-shift ratios on the following way,

(1.11)

High quality converters should have all central frequency shifts (1.11) as lower as possible. Relations given in (1.10) and (1.11) are also useful for designing ultrasonic power supplies and circuits for automatic, PLL resonant frequency tracking.

Since in certain operating regimes converters can operate between series and parallel resonant frequency, and in any case series and parallel resonance (of the same converter) are mutually coupled and dependant (both, minimal and maximal converter’s impedances are coincidently changing under loading), we can also define "mixed-mode, static and low signal quality parameters", formulating a kind of effective parameters, practically creating average value/s between every quality parameter defined for series and for parallel resonance (see the last column in T 1.2). The table T 1.2 is presenting an overview of converters quality parameters (as already introduced above), and all numerical values in T 1.2 are calculated for Branson 502/932R converter (see measured impedance-phase curves and model parameters presented on Fig. 12 & Fig. 13 and in T 1.1).

There is another important quality parameter of a piezoelectric converter, dominantly related to its mechanical design, and to elastomechanical material properties of all components and parts of that converter, known as Converter’s Electromechanical Coupling Factor, k_c. If we take converter/s from T 1.1 (either modest or high quality, non-loaded and/or loaded, one), we shall see that Coupling Factor in all mentioned cases remains almost unchanged,

(1.12)

If we now compare the Converter Coupling factor k_c with the Coupling Factor, k₃₃ of the single and load-free piezoceramics (that is the part of the same converter), we shall see how good (or optimal) is the mechanical design of that converter,

(1.13)

since the ideal mechanical converter design would be when converter is not changing the initial Coupling Factor of its piezoceramics, , meaning that elastomechanical properties of converter metal parts would perfectly match such properties of piezoceramics.

T 1.2 (All numerical values in this table are related to BRANSON 502/932R converter: see T 1.1 & Figs. 12,13)

Parameter ß	Series Resonance ß	Parallel Resonance ß	Mixed-mode ß
Mechanical Quality Factors Þ (non-loaded converter)
Mechanical Quality Factors Þ (fully-loaded converter)
Ratio of Mechanical Quality Factors Þ & Efficiency (non-loaded): (fully-loaded converter)
Characteristic Mechanical Impedance Þ (non-loaded converter)
Characteristic Mechanical Impedance Þ (fully-loaded converter)
Characteristic Axial Wave Velocity Þ (non-loaded converter)
Characteristic Axial Wave Velocity Þ (fully-loaded converter)
Resonant Frequency Þ (non-loaded converter)
Resonant Frequency Þ (fully-loaded converter)
Converter’s Resonant Frequency Stability Parameters
Coupling factor of the assembled converter from T 1.1 (PZT8, k33=0.64)	(the ideal converter design would be when ( ))
The other important quality parameters of power ultrasonic converters Static & Low-signal parameters (converter is operating in air = no-load)
		Total input capacitance
		Piezoceramics Young modulus
		Relations between Mechanical Quality Factors of assembled, non-loaded converter and Quality factor of piezoceramics (that is the part of the same converter)
		Effective mechanical quality factor (here invented parameter)
		Effective electromechanical quality factor (here invented parameter)
		Dielectric loss factor Qe = Dielectric quality factor
		Piezoelectric charge constant
		Maximum to minimum impedance ratio
		Curie temperature
		Frequency gap between series and parallel resonant frequency

You have been reading a sample chapter of this comprehensive book on the technology of ultrasonic transducer / converter design and mathematical modelling - available to download or mailed to you

Piezoelectric Converters Modelling and Characterization

Miodrag Prokic, Author

240 pages, Copyright ©2004 MPI. All international distribution rights reserved.