Piezoelectric Converters Modeling and Characterization

There are many possibilities to present and analyze equivalent models of piezoelectric converters. For electrical engineering needs (as for instance: when optimizing ultrasonic power supplies, in order to deliver maximal ultrasonic power to a mechanical load) we need sufficiently simple and practical (lumped parameter), equivalent models, expressed only using electrical (and easy measurable or quantifiable) parameters (like resistance, capacitances, inductances, voltages and currents). Of course, in such models we should (at least) qualitatively know which particular components are representing purely electrical nature of the converter, and which components are representing mechanical or acoustical nature of the converter, as well as to know how to represent mechanical load. For here described purpose, the best lumped parameter equivalent circuits that are fitting a typical piezoelectric-converter impedance (the couple of series and parallel resonance of an isolated vibration mode) are shown to be Butterworth-Van Dyke and/or its electrical dual-circuit developed by Redwood (both of them derived by simplifying the Mason equivalent circuit and/or making the best piezoelectric impedance modeling based on experimental results and electromechanical analogies). In this paper the two of mutually equivalent (above mentioned, and slightly modified), dual electrical models will be used to present a piezoelectric converter operating in its series and/or parallel resonance.

Above described objective has been extremely simplified after electronics industry developed Network Impedance (Gain-Phase) Analyzers (such as HP 4194A and similar instruments). Practically, for the purpose of modeling, it is necessary to select one single converter's operating mode (to select a frequency window which captures only the mode of interest, or the single couple of series and parallel resonance belonging to that mode) and let Impedance Analyzer to perform electrical impedance measurements by producing sweeping frequency signal in the selected frequency interval (and by measuring voltage and current passing on the converter connected to the input of Impedance Analyzer). The next step (implemented in Impedance Analyzers) is to compare the measured impedance parameters with theoretically known converter model (lumped parameters model, already programmed as an modeling option inside of Impedance Analyzer), and to calculate model parameters (practically performing the best curve fitting that places measured impedance values into theoretical impedance model). This way, in a few (button pressing) steps we are able to get numerical values of all (R, L, C) electrical components relevant (only) for selected converter mode and selected frequency range (and this is in most cases the most important for different engineering purposes, such as: optimizing ultrasonic power supplies, realizing optimal resonant frequency and output power control, optimizing converters quality…). We are also able to compare (using modern impedance analyzers) how close are measured impedance values (of a real converter), and values resulting from impedance curve fitting process. In cases of well-designed converters (and converters with sufficiently high mechanical quality factor) we are able to get almost 100% correct modeling (meaning that all measured and calculated R, L and C, lumped model parameters, are numerically almost 100% correct).

1. To explain the most important (and simplified) electrical lumped-parameters equivalent circuits suitable to represent piezoelectric converter in its series and/or parallel resonance, for different electrical design purposes, as well as to explain qualitatively converter models regarding higher frequency harmonics, and models when converter transforms mechanical input into electrical output (operating as a receiver or sensor).
2. To establish the very general concept of mechanical loading of piezoelectric converters (where mechanical load is presented in normalized form using the mechanical-load units comparable to internal resistance of the converter-driving electric circuit, or ultrasonic power supply).
3. To analyze the optimal power transfer of piezoelectric converters operating in series and parallel resonance, and to explain mechanical loading process and losses in both situations.

The Fig. 1 presents two of the most widely used lumped-parameters piezoelectric converter impedance models (mutually equivalent), valid for isolated couple of series and parallel resonances (of a non-loaded). In fact, on the fig Fig. 1 are presented the simplest models applicable for relatively high mechanical quality factor piezoelectric converters, where thermal dissipative elements in piezoceramics could be neglected. The more general models (again mutually equivalent), representing real piezoelectric (non-loaded) converters with dissipative dielectric losses and internal resistive electrode-elements (R_op (=) Leakage AC and DC resistance, R_{os1 »} R_os2 (=) Dielectric resistive loss of piezoceramics) in piezoceramics are presented on the Fig. 2. For high quality piezoceramics R_op, is in the range of 10 MW - 50 MW , and R_os1, R_os2 are in the range between 50 W and 100 W , measured at 1 kHz, low signal (and can be calculated from piezoceramics tgd value, or using HP 4194A, Impedance Analyzer). In most of the cases of high quality piezoceramics we can neglect R_op as too high resistance, and R_os1, R_os2, as too low values, but we should also know that dielectric and resistive losses are becoming several times higher when converter is driven high power, in series or parallel resonance, comparing them to low signal measurements.

The other dissipative power losses (R₁ and R₂) are belonging to the mechanical circuit branch and come from converter joint losses, from planar friction losses between piezoceramics and metal parts, from mounting elements and from material hysteresis-related losses (internal mechanical damping in all converter parts). The models from the Fig. 2 can be schematically simplified if we introduce abbreviated electric-elements symbolic presenting dissipative (real) inductances and capacitances together with their belonging resistances, using only one symbol, as for instance: For any electric combination (or connection) between one capacitance and one resistance we shall introduce the symbol C*, and for any electric combination between one inductance and one resistance we shall introduce the symbol L* (since we can always find exact circuit transformations between two elements in serial and parallel connection). Doing this way, models presented on Fig. 2 will be simplified as given on the Fig. 3, and applicable circuit equivalents (used in Fig. 3) are presented on the Fig. 4.

The other elements on the Fig. 2 are: C_{os »} C_op (=) Clamped, static capacitance/s of piezoceramics, C_1/2, L_1/2 (=) motional mass and stiffness elements of converter’s mechanical oscillating circuit/s (see Fig. 7 to find approximate mathematical relations between all model parameters). We could also add in series to any of input converter terminals the cable (and winding) resistance, since every real converter has input electrodes, soldered or bonded (electrical) joints, and a cable (presently neglected parameters).

The influence of an external acoustic load on the converters’ modeling is presented on the Fig. 5, by introducing loading resistances R_L1 and R_L2, as the closest and very much simplified equivalent of the real converter loading (in reality loading resistances R_L1 and R_L2, sometimes should be treated as complex impedances as the most general case).

Based on equivalent electric circuits presented on Fig. 4, we can easily place parallel-loading resistances from Fig. 5 in series with inductances, just by calculating new equivalent frequency-dependant elements-values. In literature regarding the same problematic it is very usual to see that left-side piezoelectric converter-model from Fig. 5 has loading resistance in series with motional inductance and capacitance, and for the model on the right side of the Fig. 5 is usual that loading resistance is found in parallel with motional inductive and capacitive circuit elements (but using Electric Circuit Theory we can easily play with any of parallel or series elements combination, as presented on the Fig. 4). It is also clear that loading nature or load-resistance would change, depending how and where we place it (in situations, like in series connection/s with motional inductance, load resistance would increase with load-increase (starting from very low value), and in case of placing it in parallel with motional inductance (as presented on Fig. 5), load resistance would decrease with load-increase (starting from very high value)).

It is also important to underline which circuit-elements (in all above found circuits, Figs. 1,2,3,5) are representing purely electrical elements of a piezoelectric converter, and which elements are only given as functional (and analog) electrical equivalents of converter’s mechanical parts and its mechanical properties (including loading elements), see Fig.6.

It is very important to know that mechanical converter-loading, presented on Figs. 5 & 6, is equally and coincidently influencing changes, both in series and parallel converter impedance, basically reducing equivalent mechanical quality factor/s of a loaded converter (or coincidently increasing its series resonant-impedance and decreasing parallel resonant-impedance). This is the principal reason why (in this paper) an isolated couple of series and parallel resonances is treated as the same, single and unique oscillating-mode that can be driven in its current or voltage resonance, and produce force or velocity-dominant mechanical output. Since there is certain frequency shift between each couple of series and parallel resonances, in literature regarding converters modeling and converters measurements, many authors are talking about different vibration modes or different harmonics.

Also in different literature regarding piezoelectric converters modeling, we can find very similar equivalent circuits (as previously presented on Figs. 1,2,3,5 & 6), where some of circuit elements found in this paper are not present. Here is accepted the strategy that all mutually equivalent piezoelectric models (Figs. 1,2,3,5 & 6) should present 100% dual (analog) electric circuit-structures, having the same number of elements, and presenting the same (electrical) impedances for the same input DC and/or AC electrical currents and voltages, connected to their input terminals (regardless frequency). Following this strategy, it was necessary to introduce certain electric elements that are not found in other literature sources regarding the same problematic (in order to satisfy circuits symmetry and DC and AC balance).

Practically, all of above presented equivalent electric circuits (Figs.1,2,3,5,6,7) are based on the evolution of mutually analog (or dual) circuits (known in modern filter design), presented on Fig. 7, combined with circuit equivalents from Fig. 4.

On the Fig. 7 we can see the evolution of two (basic) dual electric circuits towards converter models presented here.

In order to give the full picture of the modeling strategy presented in this paper, it would be interesting to explain how model of a (single) piezoelectric converter transforms when we connect a horn or booster to it (of course, added horn or booster should have almost the same resonant frequency as a converter). Practically we should know where in the basic converter model we place one more equivalent circuit-model presenting added horn. The most important background related to this situation is to know that added horn also presents one mechanical resonant structure that can be replaced with an electrical (equivalent) resonant circuit. In most of the literature regarding converters modeling, added horns are treated as converter loads, but here we shall (primarily) treat them as added resonant boxes, resonant circuits, or added filter circuits (all of them operating at the same resonance). Starting from evolution-equivalent circuits presented on Fig. 7, and from basic converter models presented on Fig. 3 we can illustrate what means "converter+horn" modeling with new step-by-step circuit-evolution models given in Fig. 8. We know that regardless how many boosters and horns we add to a converter (all of them having the same resonant frequency); -finally, after making impedance measurements (with HP 4194A, for instance) we get principally the same models as models on Fig. 3 (just particular circuit parameters and mechanical quality factor are changed). Of course this is a kind of over-simplified statement, since added sonotrodes with complex shapes can create new resonant frequencies, but here we accept (in advance) to limit our observations only to a well isolated and separated couple of series and parallel resonances. What is happening (Fig. 8) is that by adding horns to a converter we create higher-orders electrical and mechanical filter-circuits that can be again (going backwards and applying Electric Circuits Theory) transformed into basic converter models presented on Figs. 1,2,3,5,6. We also see that in simplified circuit evolution chart on Fig. 8, we deal with strongly mutually-coupled resonant-circuits (either the same current or the same voltage are coupling such circuits), operating on the same resonant frequency, and that we can place the Load where it would be physically (loading should again be treated on the same way as presented on Figs. 5 & 6). The internal losses of added horns are also not neglected, since we can find dissipative elements in added-horn-related resonant circuits.

We also know (from many years of experience in ultrasonic engineering) that when we add the horn, booster or some other (well designed) sonotrode to a converter, new, resulting mechanical quality factor of the combination "converter+horn" will become much higher than it was in the case of a single converter, meaning that mechanical resonant circuits ("converter+horn") should be mutually connected in a connection that creates higher mechanical quality factors. This is in agreement with filter theory knowledge (regarding connecting multiple filter blocks in series (or sometimes parallel, or more complex) connection, having the same resonance/s), and also strongly supports the evolution chart on Figs. 8 & 9.

We can also create another simplified circuit evolution chart (see Fig. 9), equivalent to one of Fig. 8, in order to explain "converter+horn" modeling (when converter and horn operate on the same resonant frequency), starting from a dual circuit model where motional capacitance and inductance are in parallel connection. It is also important to underline that motional inductances of "converter+horn" combination, both given on Fig. 8 and Fig. 9 are strongly (acoustically, or mechanically) coupled, oscillating on the same resonant frequency, what makes presented circuit transformations easier to understand.

It remains also to explain converter harmonics (higher frequency resonances) in the frame of already established lumped-parameters modeling. On the Fig. 10-a and Fig. 10-b are presented different circuit possibilities using series and parallel connection of basic resonant-circuit configurations. The cases on Fig. 10-a also present (simplified) direct resistive loading on the first resonant mode.

The much more general case of converter (dual) models with harmonics (than the case on Fig 10-a), including different loading situations (which are covering all previously discussed models), is presented on the Fig. 10-b. All possible mechanical loading situations (Fig. 10-b), are presented by (dual) load circuits marked with (1), (2), (3) and (4).

For instance, if we take only the upper converter model (Fig. 10-b), when load (1) is connected to mechanical output terminals (short-circuit connection), converter is operated in idle, or no-load condition, and it is called a piezoelectric resonator. If the load (2) is connected to mechanical output terminals, this is the case of resistive mechanical loading. If the load (3) is connected to mechanical output terminals, this is the case of a simple mechanical resonator similar to a single-resonant-frequency sonotrode or booster, or some other simple mechanical oscillating system. The load (4) presents the most general case of arbitrary and complex mechanical impedance.

For the lower converter model (Fig. 10-b) we again have the same (analogue and dual-models) situation, as already explained for the upper model. The only exception is that when load (1) is connected to mechanical output terminals (in fact nothing, or open-contacts are connected), converter is operated in idle, or no-load condition, and it is called a piezoelectric resonator. The loads (2), (3) and (4) have the same meaning as already explained for the upper model (but presented as dual circuits).

Every piezoelectric converter is in the same time able to detect mechanical vibrations, serving as an accelerometer, microphone, receiver or sensor (producing an electrical output signal, proportional to its mechanical excitation). If we again consider only simplified converter models (without harmonics; Figs. 1, 2, 3, 5 and 6), we can present converter’s mechanical excitation by dual models given on Fig. 10-c. Later on, if we would like to include harmonics into Fig. 10-c, we can simply use the modeling structures presented on Fig. 10-b and keep the mechanical excitation in the same place/s as presented on the Fig. 10-c. It is important to have a feeling how external mechanical excitation influences converter’s operation (and modeling) because in many situation of interest (like ultrasonic cleaning, welding, liquid processing…), when we drive a converter electrically in order to produce mechanical output, in the same time its load can produce (or reflect) mechanical excitation and generate additional charges, currents and voltages inside of a converter’s structure. When piezoelectric converter is used only as the sensor of mechanical excitation, we also need to know what are its most convenient electrical models in order to qualify and quantify sensor parameters and sensor output signals. Models presented on the Fig. 10-c are also important if we would like to operate a piezoelectric converter as an (lock-in) active vibration source able in the same time to detect external mechanical excitation in the well-selected frequency zone (or using piezoelectric converters and sensors in interactive and impedance sensitive operating regimes).

Piezoelectric converters when operating as sensors are also able to detect static and constant mechanical excitation (external force), generating certain constant charge and/or voltage, because of the presence of permanent (frozen) electrical field inside of the crystal structure of piezoelectric materials (created during production of that piezoelectric material by separating internal positive and negative electric charges in the form of oriented electrical dipoles). When external (static) force is applied on a piezoelectric sensor, internally separated electrical charges (belonging to polarized piezoelectric crystal structure) will slightly change their equilibrium position and release certain amount of free electrical charge (proportional to the applied force), which will appear on the external sensor electrodes. In cases when there is only a static, or low frequency mechanical excitation applied to a piezoelectric sensor, we can safely apply two sensor models presented on the left side of Fig. 10-c, and treat charge source Q_a and voltage source u_a as static electric sources.

The same situation becomes more interesting when external mechanical excitation is originated from a dynamical, high frequency vibration source, which is in the same frequency range where sensor has its resonant frequencies, because this type of excitation usually has very low, or close to zero average-level of amplitudes (regarding its average force or average velocity). Sensor will still remain able to react on constant external force, and in the same time be able to detect external dynamic excitation. For such situations we can create models where static and dynamic (or variable) excitation sources are separated, as presented on Fig. 10-d (or we could also separate them in the same positions where they are already placed on Fig. 10-c). Dynamic external excitation (Fig. 10-d) is introduced in the form of inductive coupled voltage sources with motional inductances.

Using Circuit Theory, we could create many of equivalent models, similar to models presented on Fig. 10-c and Fig. 10-d, playing with equivalent and/or dual circuits, with transformations between voltage and current/charge generators, etc.

The meaning of characteristic impedance parameters and frequencies (on Fig. 11-a) is as follows:

f_m – frequency where absolute value of impedance reaches its minimum,

f_s – series motional current-force, mechanical resonant frequency,

f_r – series electrical, input-current resonance,

f_a – parallel, electrical, input-voltage resonance,

f_p – parallel motional voltage-velocity, mechanical resonant frequency,

f_n – frequency where absolute value of impedance reaches its maximum.

A piezoelectric converter when operating at f_s, presents the source of a high motional force and relatively low motional velocity (comparing it with the same converter operating in f_p and producing the same power). If we operate the same converter slightly increasing its operating frequency towards f_r, the converter would start decreasing its motional force and increasing its motional velocity. Since all of the frequencies, f_m, f_s and f_r are mutually very close (for high mechanical quality-factor converters), in most of analyzes here we shall replace all of them with f_{1 »} (f_m< f_s< f_r). Something similar (but in much wider frequency range) would also happen if we start operating converter (high power) from f_s, decreasing its operating frequency towards lower frequencies.

Also, a piezoelectric converter when operating at f_p, presents the source of a high motional velocity and relatively low motional force (comparing it with the same converter operating in f_s and producing the same power). If we operate the same converter slightly decreasing its operating frequency towards f_a, the converter would start decreasing its motional velocity and increasing its motional force. Since all of the frequencies, f_a, f_p and f_n are mutually very close (for high mechanical quality-factor converters), in most of analyzes here we shall replace all of them with f_{2 »} (f_a< f_p< f_n). Operating in the vicinity of f₂ towards higher frequencies is not beneficial regarding producing high power, since the amount of accumulated elastomechanical (potential) energy in that frequency area is on the very low level.

The converter output mechanical power is equal to the product between its motional velocity and motional force, or to the product between its motional voltage and motional current. An optimal power transfer (from electrical to oscillatory, mechanical power) is achieved when motional current and motional voltage are mutually in phase, producing only the active or real output power (and consequently converter should operate either in f_s or in f_p). Some ultrasonic companies are also operating their converters in the frequency area between f₁ and f₂, this way realizing the output power regulation by moving the impedance-frequency operating point (or effectively controlling the phase difference between motional current and motional voltage). The same phase-impedance regulation concept (moving the impedance-frequency operating point) can also be realized for the frequency area in close vicinity to f₁ and left from f₁, towards lower frequencies.

The simplest circuit replacements (equivalent circuits) for complex impedance of a piezoelectric converter, in the function of operating frequency interval/s,are presented on Fig. 11-b. For the most of design needs regarding electronics where piezoelectric converters and sensors are involved, it is very important to know the parameters of the circuits presented on Fig. 11-b in order to select the proper converter or sensor regime.

Obviously the piezoelectric converter impedance is complex and changing its character (depending on frequency), being dominantly capacitive, or dominantly inductive and/or certain combination of capacitive, resistive and inductive elements. On the Fig. 11-c are presented different aspects of converter impedance vs. frequency that can be measured using Network Impedance Analyzer (programmed to detect only particular impedance element, such as input converter capacitance, or input inductance or input resistance).

Based on Mobility system of electromechanical analogies (Current = Force, Voltage = Velocity), electrical equivalent circuits and particular electrical impedance curves presented on Fig. 11-b and Fig. 11-c can be directly transformed into corresponding mechanical equivalents, as given on Fig. 11-d..

In ultrasonic engineering, regarding (isolated) resonant operating regimes of piezoelectric converters (and sensors), the most interesting is to determine the simplest and sufficiently applicable equivalent circuit models in order to realize optimal converter driving, optimal power transfer, the best signal reception and proper electrical and mechanical converter impedance matching. Let us take the most widely used converter (dual) models presented on Fig. 1 (used later on, too). If we now imagine that converter is operating precisely in its series resonance, the corresponding motional reactive-impedance part would be equal to zero.

In case if converter would operate exactly in its parallel resonance, the corresponding motional reactive-impedance part would disappear becoming extremely high impedance.

Now, for resonant operating regimes of sufficiently high mechanical quality factor converters (where we can apply all reasonable approximations and circuit simplifications, as already explained in this chapter), the models presented on Fig. 1 can be even more simplified, as presented on Fig. 11-e.

If we apply equivalent circuit transformations as presented on Fig. 4, it will be possible to additionally transform both models from Fig. 11-e into two more of mutually corresponding series and parallel connection/s between equivalent resistance/s and capacitance/s. It is also important to underline that two circuit models presented on Fig. 11-e are no more mutually equivalent (as their predecessors), since each of them is previously simplified for different operating frequency. In order to maximize real (or active) output power of a piezoelectric converter operating in one of its resonances it would be necessary to neutralize the capacitive components belonging to circuits on Fig. 11-e, by connecting externally one inductance in parallel or series connection (respectively). In reality, when we analyze the optimal power conversion, we should upgrade models on Fig. 11-e with other circuit-elements visible on Figs. 2,3,5,6…that are presently neglected. We should also know that for non-loaded piezoelectric converters, in most cases of practical interests (Fig. 11-e, Figs. 1,2,3,5,6…), R₁ is in the range between 5 Ω and 100 Ω , and R₂ is in the range between 10 kΩ and 100 kΩ (or higher than 100 kΩ). Coincidently with converter load increase we can find that, R₁ would also proportionally increase (maximally until approx. 10 times higher value), and R₂ would proportionally decrease (maximally until approx. 10 times lower value).

In certain circuit configurations, piezoelectric converter can be conveniently compensated (by adding external inductive and capacitive components to its input electric terminals) to become pure resistive impedance in relatively large frequency intervals.

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Piezoelectric Converters Modelling and Characterization

Miodrag Prokic, Author

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