Abstract:
An innovative piezoelectric transducer apparatus having a piezoelectric workpiece with a modal-shaped actuator electrode for converting an input energy of one form into an output energy of another form. The piezoelectric workpiece comprises an actuator section and a sensor section. The actuator section is formed in the piezoelectric workpiece and has a modal-shaped electrode for exciting the piezoelectric workpiece into mechanical vibration upon driven by the input energy. The sensor section is also formed in the piezoelectric workpiece and has a sensor electrode for delivering the output energy to an external load of the apparatus by picking up the energy generated by the excitation. The modal-shaped actuator electrode has a shape function defined by the mathematical solution function of the governing equation of the piezoelectric workpiece under the condition that the piezoelectric workpiece being vibrating in a selected resonant mode.

Description:
FIELD OF THE INVENTION  
         [0001]    This invention relates in general to a piezoelectric energy conversion apparatus and, in particular, to one with high energy conversion efficiency. More particularly, this invention relates to a piezoelectric transformer having modal-shaped electrode for efficiently converting electric power.  
         BACKGROUND OF THE INVENTION  
         [0002]    Piezoelectricity is widely utilized in various applications. As a transducer for converting energy in one form into another, a piezoelectric material block is useful as a workpiece for converting mechanical energy to electrical or vice versa. A piezoelectric workpiece can be used to convert an electrical power at one AC frequency and voltage into another at different frequency and voltage. This specific application of AC electrical power conversion involving converted (frequently raised) voltage makes a piezoelectric transducer device a useful piezoelectric transformer suitable for various industrial and consumer applications.  
           [0003]    Charles A. Rosen proposed his piezoelectric transformer in the 1950&#39;s. It is not until the 1990&#39;s when piezoelectric transformer has become more and more popular in electronic devices. Piezoelectric transformers have been used such as in the power supply section of various electronic devices, portable ones in particular. Due to benefits such as their size and weight, it is obvious that piezoelectric transformers will become even more popular. Improved safety is another desirable feature of a piezoelectric transformer. A failed piezoelectric transformer normally breaks itself and ceases to function. Fire hazards are virtually of no concern at the part of the piezoelectric workpiece itself.  
           [0004]    Electrodes plated to the surface of a piezoelectric material block are necessary for conveying electric power into and/or out of the workpiece in order for the piezoelectric system to function. Traditionally, dimensioning factors including both shape and size of the electrodes have not been carefully considered parameters when designing a piezoelectric system. Typically, rectangular-shaped electrodes are used for rectangular-shaped workpieces. In disk- or ring-shaped workpieces, electrodes are typically also disk-shaped or having the shape of a section of the disk or ring. These electrode shape configurations are considered uniform electrodes in the description of the present invention.  
           [0005]    Traditionally, piezoelectric systems have been analyzed based on circuit theories. Mechanical considerations of a piezoelectric system are “translated” into parameters that are fit into circuit models. Piezoelectric workpieces thus developed are not optimized in issues such as energy conversion efficiency and parasitic noises—signals at undesirable frequencies with sufficiently significant amplitudes mixed in the output of a piezoelectric workpiece.  
           [0006]    [0006]FIG. 1 is a perspective view of a piezoelectric workpiece outlining the definition of the piezoelectric parameters, both electrical and mechanical, and material orientation and dimensions for the analysis of a piezoelectric system. As is illustrated, the piezoelectric workpiece  100  is, in general, an elongated thin rectangular-shaped plate having a length L, a width W and a thickness, or height, H. Note that in the following theoretical development for the description of the piezoelectric transducer apparatus of the present invention, the IEEE Compact Matrix Notation system is employed.  
           [0007]    By convention, the two largest surfaces of the elongated thin-plate workpiece are the side surfaces, and the two smallest are the end surfaces. The direction along the longitudinal axis of the workpiece is the first orientation, that perpendicular to the first orientation and parallel to the side surfaces is the second, and that perpendicular both to the side surfaces and to the longitudinal axis is the third, as is shown in the drawing by an orientation axis system  110  labeled with orientations 1, 2 and 3 respectively.  
           [0008]    Also, in the theoretical development of the invention in the following detailed description, a variable x is set up along the direction of the first orientation. This coordinate axis, one measuring the longitudinal dimension, or length, of the elongated workpiece  100 , serves as a variable in the analysis of the piezoelectric system according to the teaching of the present invention. One end of the workpiece  100  is set conveniently as the origin of the x coordinate axis, as is seen in FIG. 1.  
           [0009]    The workpiece  100  has a side-plated electrode  102  and another not shown in the perspective view. That electrode which opposes electrode  102  and provides for a complete electrical circuit path is plated on the other side of the workpiece opposite to the side for the electrode  102 .  
           [0010]    Each of the electrodes is connected to a corresponding terminal of an electric circuit  120 , represented in the drawing by a voltage sign V that signifies a voltage across the electrodes of the piezoelectric workpiece. The current corresponding to the voltage V in the circuit  120  is identified by I. P represents the electrical power arising from voltage V and current L External mechanical forces applied to the workpiece  100  is identified in the drawing by the symbol F, the corresponding mechanical velocity induced as a result of an applied force F is identified by U.  
           [0011]    The piezoelectric system of FIG. 1, which has a workpiece  100  with side-plated electrodes including  102 , is in the 31 mode, i.e., the polarization is in orientation 3, and the mechanical vibration is in orientation 1. By contrast, a system such as that illustrated in FIG. 3 which has a workpiece  300  with end-plated electrodes  302  and  304  is in 33 mode. Both its electrical polarization and mechanical vibration are along orientation 3 of its orientation axis system  310 .  
           [0012]    Based on the traditional method of analysis that relies on circuit theories, a piezoelectric system such as that depicted in FIG. 1 which has side-plated electrodes and operates in the 31 mode has the following constitutive equations:  
             S   1   =s   11   E   T   1   +d   31   E   3   (1)  
             D   3   =d   31   T   1 +ε 33   T   E   3 .  (2)  
           [0013]    In Equations (1), (2) and those to be discussed below, the parameters and constants are in accordance with their respective definitions in the IEEE notation system mentioned above.  
           [0014]    Based on the relationship between the piezoelectric constants and between the longitudinal strain u and the longitudinal stress S 1 , considering the longitudinal motion equation for the system, the following governing equation may be derived for the 31 mode piezoelectric workpiece:  
                   -   ρ              ∂   2        u       ∂     t   2           +       c   11   E              ∂   2        u       ∂     x   2             =       E   3              ∂     e   31         ∂   x       .               (   3   )                               
 
           [0015]    Since a conventional piezoelectric workpiece employs uniform electrodes characterized by its rectangular-shaped electrode patterns, Equation (3) may thus be reduced to:  
                   -   ρ              ∂   2        u       ∂     t   2           +       c   11   E              ∂   2        u       ∂     x   2             =   0.           (   4   )                               
 
           [0016]    Equation (4) is a homogeneous one-dimensional wave equation, which has a general solution for the longitudinal strain u that can be expressed as  
             u =( B   1  sin β x+B   2  cos β x ) e   jwt   (5)  
           [0017]    wherein  
           β=ω/ c   (6)  
           [0018]    and  
             c={square root}{square root over (c 11   E /ρ)},   (7)  
           [0019]    in which j={square root}{square root over (−1)}, ω is the angular frequency, β is the wave number, c is the velocity of wave propagation, and B 1  and B 2  are, respectively, coefficients to be determined by the boundary condition of the examined system.  
           [0020]    In addition to the mechanical piezoelectric parameters as incorporated into the analysis above, electrical ones also need to be considered in the analysis as well. In a 31 mode piezoelectric system of FIG. 1, the electromechanical coupling coefficient can be defined to be  
               k   31   2     =           d   31   2          c   11   E         ɛ   33   T       .             (   8   )                               
 
           [0021]    Assuming that the 31 mode piezoelectric workpiece in the system is clamped at its both ends, the mechanical characteristic impedance of the workpiece transmitting mechanical waves can be described as  
             Z   0   =HWρc=HW{square root}{square root over (ρc 11   E )}.   (9)  
           [0022]    In a traditional analysis approach based on circuit theories, mechanical parameters of force and velocity in a piezoelectric system are considered in an analogy and treated as the equivalents of another parameter pair of voltage and current respectively. Based on such an analogy, a 31 mode piezoelectric system can be shown to have an equivalent transformer step-up ratio of  
             N=Wd   31   c   11   E .  (10)  
           [0023]    [0023]FIG. 2 shows the equivalent circuit of the 31 mode piezoelectric system of FIG. 1. The circuit configuration in this equivalent circuit is obtained utilizing the conventional technique of circuit analysis as described above. Characteristics of the equivalent circuit, namely, those of a 31 mode piezoelectric transformer as defined by the system of FIG. 1 are outlined in the drawing. These include the mechanical characteristic impedance Z 0 , the wave number β, the velocity of wave propagation c, the electromechanical coupling coefficient k 31   2 , and the equivalent step-up ratio N.  
           [0024]    Based on the same circuit theories, another piezoelectric system, one in the 33 mode as outlined in FIG. 3, may be shown to have an equivalent circuit illustrated in FIG. 4, which is derived from the following governing equation  
                   -   ρ              ∂   2        w       ∂     t   2           +       c   33   D              ∂   2        w       ∂     z   2             =   0.           (   11   )                               
 
           [0025]    To construct a piezoelectric transformer, a known method is to combine the 31 and 33 mode piezoelectric systems of FIGS. 1 and 3. In such a piezoelectric transformer, a Rosen-type piezoelectric transformer for example, the side electrodes can be used as the input terminals and the end electrodes as the output terminals. Note the same terminology of the side and end electrodes as that used in the above description of the systems of FIGS. 1 and 3. The side and end electrodes are those plated on the largest and smallest surfaces of the elongated thin-plate workpiece respectively.  
           [0026]    Traditional circuit analysis for a Rosen-type piezoelectric transformer may be developed based on the equivalent circuits of FIGS. 2 and 4 as both are combined to construct the transformer. For example, for a Rosen-type piezoelectric transformer put to operate on its second resonant frequency, an equivalent circuit can be derived as shown in FIG. 5. However, the equivalent circuit of FIG. 5 is derived by employing a considerable level of simplifications in the process of its theoretical development. These mathematical simplifications based on physically reasonable assumptions are necessary in order to avoid a mathematical system that may become excessively complex to analyze and to seek solution.  
           [0027]    Though, this traditional approach has at least a few drawbacks. As the mechanical parameters of a piezoelectric workpiece are incorporated into the equivalent circuit for analysis based purely on circuit theories, some critical mechanical parameters diminish or even disappear in the process. These lost characteristics of a piezoelectric system that should otherwise have been preserved in the mathematical model of the system imply less flexible or even lost design parameters. Those characteristics, if preserved, may lead to the construction of better piezoelectric systems.  
           [0028]    For example, the system performance characteristics such as energy conversion efficiency can not be optimized. This is due to the fact that circuits interfacing with the piezoelectric workpiece including both the driving circuit and the load have to be designed by considering the circuit configuration based on the equivalent circuit of the piezoelectric workpiece. However, the equivalent circuit is not the best model of the piezoelectric body as its mechanical characteristics are not fully encompassed in circuit theory-based models.  
           [0029]    The result obtained from the conventional circuit theory in the analysis of a piezoelectric transducer system thus is not the best possible result. This is a fact reflected by the harmonics existing in the output of a piezoelectric system constructed in accordance with the developed result. The presence of these harmonics reduces overall system efficiency, and also incurs other problems such as load matching due to the existence of the undesirable harmonics.  
         SUMMARY OF THE INVENTION  
         [0030]    Accordingly, it is an object of the present invention to provide a piezoelectric transducer apparatus having a modal-shaped electrode that operates in a resonant mode without significant undesirable harmonics of non-resonant frequencies.  
           [0031]    It is another object of the present invention to provide a piezoelectric transducer apparatus having a modal-shaped electrode that achieves high operating efficiency in converting energy.  
           [0032]    It is a further object of the present invention to provide a piezoelectric transformer having a modal-shaped electrode that outputs an AC electric power without significant undesirable harmonics.  
           [0033]    These and other objects are achieved in an innovative piezoelectric transducer apparatus having a piezoelectric workpiece with a modal-shaped actuator electrode for converting an input energy of one form into an output energy of another form. The piezoelectric workpiece comprises an actuator section and a sensor section. The actuator section is formed in the piezoelectric workpiece and has a modal-shaped electrode for exciting the piezoelectric workpiece into mechanical vibration upon driven by the input energy. The sensor section is also formed in the piezoelectric workpiece and has a sensor electrode for delivering the output energy to an external load of the apparatus by picking up the energy generated by the excitation. The modal-shaped actuator electrode has a shape function defined by the mathematical solution function of the governing equation of the piezoelectric workpiece under the condition that the piezoelectric workpiece being vibrating in a selected resonant mode. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0034]    [0034]FIG. 1 is a perspective view of a piezoelectric workpiece with side-plated electrodes outlining the definition of the piezoelectric parameters and material dimensioning for the analysis of a piezoelectric system.  
         [0035]    [0035]FIG. 2 is the equivalent circuit of the piezoelectric system of FIG. 1 obtained utilizing the prior-art technique of circuit analysis.  
         [0036]    [0036]FIG. 3 is a perspective view of a piezoelectric workpiece with end-plated electrodes outlining the definition of the piezoelectric parameters and material dimensioning for the analysis of a piezoelectric system.  
         [0037]    [0037]FIG. 4 is the equivalent circuit of the piezoelectric system of FIG. 3 obtained utilizing the prior-art technique of circuit analysis.  
         [0038]    [0038]FIG. 5 is an equivalent circuit of a Rosen-type piezoelectric transformer operating at the second resonant frequency obtained utilizing the prior-art technique of circuit analysis.  
         [0039]    [0039]FIGS. 6A, 6B and  6 C illustrate patterns of the actuator electrode for a first mode edge-driving quasi-modal actuator section of a piezoelectric transformer having an electrode length ratio of 2/9, 1/2 and 2/3 respectively.  
         [0040]    [0040]FIG. 7 illustrates the side view of a piezoelectric transformer constructed utilizing the first mode edge-driving quasi-modal actuator with an electrode pattern of FIG. 6B.  
         [0041]    [0041]FIGS. 8A, 8B and  8 C illustrate patterns of the actuator electrode for a second mode edge-driving quasi-modal actuator section of a piezoelectric transformer having an electrode length ratio of 4/9, 1/2 and 2/3 respectively.  
         [0042]    [0042]FIG. 9 illustrates the side view of a piezoelectric transformer constructed utilizing the second mode edge-driving quasi-modal actuator with an electrode pattern of FIG. 8C.  
         [0043]    [0043]FIGS. 10A, 10B and  10 C illustrate patterns of the actuator electrode for a first mode center-driving quasi-modal actuator section of a piezoelectric transformer having an electrode length ratio of 4/9, 5/9 and 2/3 respectively.  
         [0044]    [0044]FIG. 11 illustrates the side view of a piezoelectric transformer constructed utilizing the first mode center-driving quasi-modal actuator with an electrode pattern of FIG. 10A.  
         [0045]    [0045]FIGS. 12A, 12B and  12 C illustrate patterns of the actuator electrode for a second mode center-driving quasi-modal actuator section of a piezoelectric transformer having an electrode length ratio of 2/9, 5/18 and 1/3 respectively.  
         [0046]    [0046]FIG. 13 illustrates the side view of a piezoelectric transformer constructed utilizing the second mode center-driving quasi-modal actuator with an electrode pattern of FIG. 12C. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0047]    The underlying concept leading to the development of the present invention is, in summary, the added dimension of consideration in the mechanical aspect in the mathematical modeling of the piezoelectricity in energy conversion. Traditional modeling for a piezoelectric system has been concentrated on electrical circuit theories. With the mechanical characteristics of a piezoelectric workpiece incorporated into the system modeling considerations in accordance with the teaching of the present invention, the mathematical solutions to the system governing equations lead to and open up an entire new category of piezoelectric transducer devices.  
         [0048]    The inventive piezoelectric transducer apparatuses according to the present invention are capable of substantially more efficient operation as they are used in various applications for energy conversion, including the transformation of voltage and current of AC electrical power. These innovative piezoelectric transducer devices can be characterized by having distinguished patterns for their electrodes defined by the solution of the governing equation, which has included in its establishment, in addition to the obvious electric circuit factors, those necessary mechanical considerations.  
         [0049]    Electrodes for traditional piezoelectric workpieces have been shaped with simple geometry, such as those rectangular-shaped ones. In contrast, the electrode for the innovative piezoelectric transducer apparatuses of the present invention has the shape of mathematical functions including trigonometric functions. Electrode patterning thus becomes an additional parameter that can be conveniently manipulated for designing piezoelectric transducer apparatuses such as a piezoelectric transformer.  
         [0050]    In accordance with the teaching of the present invention, a piezoelectric workpiece in a piezoelectric system is considered as a complete and total structure and is modeled mathematically in a set of state equations. In the state equations, characteristics of the piezoelectric system as observed from the mechanical aspect are included for detailed consideration and mathematical analysis. According to the invention, to implement the mathematical analysis of a piezoelectric system, concepts employed for the analysis of pseudo-modal piezoelectric actuators and the theory of wave propagation are incorporated into the entire modeling system.  
         [0051]    As an embodiment of a piezoelectric transducer apparatus that can be analyzed in accordance with the teaching of the invention, consider, for example, a Rosen-type piezoelectric transformer. The transformer can be modeled by combining a 31 and a 33 mode piezoelectric workpiece similar to that of FIGS. 1 and 3 respectively. The governing equation for such a transformer can be shown to be  
                   -   ρ              ∂   2        u       ∂     t   2           +       c   11   E              ∂   2        u       ∂     x   2             =       E   3   in              ∂     e   31         ∂   x       .               (   12   )                               
 
         [0052]    Note that in the governing equation, variable t is time and x is the longitudinal dimension for the piezoelectric workpiece, as shown in the model of FIG. 1.  
         [0053]    Derived from the constitutive equation set of the generalized piezoelectric system, the governing Equation (12) describes a relationship between the concerned functioning conditions in that system that is established for a sought piezoelectric workpiece. Equation (12) specifically addresses the condition for a 31 mode piezoelectric workpiece having a structural configuration of elongated thin plate. Equation (12) describes that the partial derivatives of the longitudinal strain u in time and spatial dimension are related to the partial derivative of the piezoelectric stress/charge constant e kp  in the spatial dimension.  
         [0054]    Equation (12) is a standard non-homogeneous one-dimensional wave equation. The term to the right of the equal sign represents the spatial external force in the model system. In the case of the Rosen-type piezoelectric transformer, boundary conditions for the wave equation are  
                       F   1               j                 ω                 t         =         -     HW        (         c   11   E            ∂   u       ∂   x         -         e   31          (   x   )            E   3   in         )              |     x   =   0         =   0                     →       ∂   u       ∂   x              |     x   =   0         =                      e   31          (   0   )              E   3   in          (   t   )           c   11   E                     (   13   )                               
 
         [0055]    and  
                       F   2               j                 ω                 t         =         -     HW        (         c   33   D            ∂   u       ∂   x         +       h   33          D   3   out         )              |     x   =   l         =   0                     →       ∂   u       ∂   x              |     x   =   l         =     -                    h   33            D   3   out          (     l   ,   t     )           c   33   D                       (   14   )                               
 
         [0056]    wherein the effective length of the Rosen-type piezoelectric transformer is  
           l=L+L   e   ′=L ( 1+{square root}{square root over (1− k   33   2 )}).   (15)  
         [0057]    According to the teaching of the present invention, the piezoelectric stress/charge constant e kp , which presents itself as e 31  in the above governing Equation (12) (to the right of the equal sign in the equation) for the Rosen-type piezoelectric transformer system, is treated as a variable that represents an adjustable factor when designing the piezoelectric system. It becomes a design factor for piezoelectric systems (such as a Rosen-type piezoelectric transformer) that conveys the spatial weight over system parameters including the geometrical shape of the electrode and the polarization level achieved by the electrode.  
         [0058]    Due to this additional system design parameter, or, in another perspective, an additional dimension of system consideration, a piezoelectric transducer apparatus developed via the manipulation of this additional design parameter is able to enjoy more optimized system characteristics. By contrast, in the traditional approach employed for the analysis of piezoelectric system, mathematical integration in the circuit theory-oriented considerations practically eliminates this system design factor. This will be shown in the following theoretical development of the underlying concept of the present invention. Experimental results supporting the achieved improvements are also available.  
         [0059]    Theoretically, the analysis of a piezoelectric transducer apparatus in accordance with the teaching of the present invention calls for solving the governing equation via the approach of eigenfunction expansion. To obtain a solution to the governing Equation (12) by eigenfunction expansion, the concerned governing equation for the eigenfunction φ i (x) can be outlined as  
                       d   2          ϕ   i         d                   x   2         +       λ   i          ϕ   i         =   0     ,           (   16   )                               
 
         [0060]    and the boundary conditions are  
                              ϕ   i            x            (   0   )       =   0                          ϕ   i            x            (   l   )       =   0.                 (   17   )                               
 
         [0061]    Based on Equation (16) and the boundary conditions specified in (17), the characteristic function (eigenfunction) φ i (x) is found to be  
               ϕ   i     =         2   l          cos          λ   i          x             (   18   )                               
 
         [0062]    wherein the characteristic value (eigenvalue) λ i  is  
                 λ   i     =           (       i                 π     l     )     2                   i     =   1       ,   2   ,     3                 …             (   19   )                               
 
         [0063]    When looking into a piezoelectric transducer apparatus from the mechanical perspective, the concept of modal filtering can be incorporated into the mathematical analysis of the system. In accordance with the present invention, it is possible to arrive at a piezoelectric apparatus that is more optimized for many desired applications when modal filtering is included in consideration. Here, modal filtering refers to the phenomenon that non-resonant frequencies in a piezoelectric system functioning in a resonant mode are substantially filtered out, which is a virtual function of frequency filtering.  
         [0064]    When the thinking of modal filtering is included in the analysis, the first derivative of the function representing the mathematical shape of the electrodes of a piezoelectric workpiece (∂e 31 /∂x) has to comply with the condition that the internal mode of the piezoelectric system matches that of a modal actuator. In other words, the first derivative of the electrode shape has to be one of the modal shapes at the desired mode n. This can be expressed by the following condition:  
                 ∂     e   31         ∂   x       =     k                     ϕ   n          (   x   )                 (   20   )                               
 
         [0065]    wherein k is a scale factor.  
         [0066]    Here, modal shape is referring to the shape of the electrode for the piezoelectric workpiece that brings the workpiece into the resonant mode of operation. In a similar sense, a modal actuator is referring to the actuator section in a piezoelectric transformer that drives the actuator section of the transformer into the resonant mode of operation. Thus, Equation (20) specifies the condition that the piezoelectric workpiece satisfies the operating condition of being functioning in a resonant mode.  
         [0067]    As an example, consider an edge-driving modal actuator that can be used for the construction of a Rosen-type piezoelectric transformer system. Here, note that for a complete piezoelectric transformer, there is another “sensor” section to correspond to the “actuator” section in the system. In other words, it can be envisioned that the actuator is the section that receives energy input to the piezoelectric transformer system so that the sensor can deliver the “transformed” energy to an external load.  
         [0068]    In the edge-driving modal actuator to be examined below, the actuator can be considered to be a 31 mode piezoelectric body that has its actuating electrode formed at the edge, or end, of the elongated thin-plate workpiece. Note that in a 31 mode workpiece, the actuator electrode is formed on the side surfaces, the largest surfaces of the workpiece.  
         [0069]    In accordance with the present invention, the shape function of the actuator electrode for an edge-driving modal actuator can be obtained by considering the above Equations (18) and (20) and outlined as follows:  
                   e   31          (   x   )         e   31   0       =     sin                       n                 π                 x     l     .               (   21   )                               
 
         [0070]    This resolved shape function is a trigonometric function of the variable x. As will be described in the following paragraphs, the dependent variable as determined by the trigonometric function of the lengthwise variable x of the workpiece represents the width of the electrode along the direction normal to the longitudinal axis of the workpiece.  
         [0071]    In a modal actuator having an edge-driving electrode with the shape specified by Equation (21), the actuation of the workpiece is only possible in the designated operating mode. If the system is not driven at the selected mode, the workpiece will not be actuated at all.  
         [0072]    It should be noted that when a modal actuator is utilized for the construction of a piezoelectric transducer system such as a piezoelectric transformer, it is not considered a true modal system. This is because of the fact that the system requires the integration of the sensor section onto the same workpiece. Due to the presence of the sensor section, there has to be reserved space on the workpiece for the sensor electrode. As a result, it is mathematically impossible to implement the full-domain integration to achieve in a true modal actuator. Less the co-existing sensor section, the actuator becomes what is referred to as a quasi-modal actuator.  
         [0073]    An electrode length ratio a is defined as the ratio of the length of the actuator electrode to that of the entire length of the piezoelectric workpiece in the longitudinal direction. This ratio becomes another parameter for implementing the design of a piezoelectric transducer apparatus in accordance with the present invention.  
         [0074]    Consider a piezoelectric transformer having a quasi-modal actuator operating in its first mode. Based on the characteristic Equation (18) of the system and the necessary condition of the modal actuator as outlined in Equation (20), a quasi-modal actuator in first mode should have the following electrode shape:  
                   e   31          (   x   )         e   31   0       =         sin                     π                 x     l       k          [       H        (   x   )       -     H        (     x   -     a                 l       )         ]               (   22   )                               
 
         [0075]    wherein a is the electrode length ratio described above and k is an adjustable scale factor defined as follows:  
                   k   =     sin                 a                 π             if            a   ≤     1   2                 k   =   1           if            a   ≥     1   2                   (   23   )                               
 
         [0076]    The term H(x)−H(x−al) in the electrode shape Equation (22) is a Heaviside Step function reflecting the fact that the workpiece can not have a full-domain integration as required by a true modal actuator.  
         [0077]    [0077]FIGS. 6A, 6B and  6 C illustrate patterns of the actuator electrode for a first mode edge-driving quasi-modal actuator section of a piezoelectric transformer having an electrode length ratio of 2/9, 1/2 and 2/3 respectively. The patterns are produced according to the corresponding electrode shape function resolved. An electrode can be considered to be formed on the surface of the piezoelectric workpiece using the width of the electrode along orientation 2 (FIG. 1) as a function of the x coordinate along orientation 1.  
         [0078]    For example, the edge-driving actuator electrode  614  of FIG. 6A with an electrode length ratio a=2/9 can be envisioned as formed on the side surface  612  of the workpiece  610 , which originates from the end  616  of the workpiece  610 , and last for 2/9 the total length of the workpiece. Along orientation 2 (FIG. 1), the width of the electrode  614  is determined by the value obtained from Equation (22). It should be noted that since the analytical system is one-dimension along the x axis of orientation 1 (FIG. 1), therefore the width on orientation 2 can be scale-less so long as it conforms to the solution equation. This means that the shape of an electrode needs not be symmetrical along longitudinal axis of the workpiece, although the patterns exemplified in FIGS. 6A, 6B and  6 C are symmetrical.  
         [0079]    [0079]FIG. 7 illustrates the side view of a piezoelectric transformer constructed utilizing the first mode edge-driving quasi-modal actuator with an electrode pattern of FIG. 6B. The side view is taken along orientation 2 so that the workpiece  620  reveals its quasi-modal actuator electrode  624  only as a thin layer. The electrode  627  on the right end of the workpiece  620  is the sensor, or output electrode of the transformer system. The electrode  625  formed on the side surface opposite to that on which the actuator electrode  624  adheres to is utilized in the example as the common ground, or return path for both the actuator and the sensor electrodes.  
         [0080]    The drawing also shows the directions of polarization for both sections of the transformer system. Specifically, the polarization direction identified by reference numeral  628  is the actuation polarization and that by  629  the sensing polarization.  
         [0081]    The system of FIG. 6C is similar to those of FIGS. 6A and 6B except for the difference of the electrode length ratio a. As mentioned, this ratio can be used as another parameter for adjustment when designing a piezoelectric transducer system.  
         [0082]    When a piezoelectric transformer employing a quasi-modal actuator section featuring the actuator electrode with a shape determined by Equation (22) is compared to one having a conventional uniform actuator electrode, available experimental results demonstrated improved transformer performance characteristics. Improvements were shown to include higher transformation efficiency and much less undesirable noise signals.  
         [0083]    As another example, consider the piezoelectric transformers employing quasi-modal actuator sections operating in their second resonant mode. FIGS. 8A, 8B and  8 C illustrate patterns of the actuator electrode for a second mode edge-driving quasi-modal actuator section of a piezoelectric transformer having an electrode length ratio of 4/9, 1/2 and 2/3 respectively. The patterns are produced according to the corresponding electrode shape function resolved.  
         [0084]    For example, the edge-driving actuator electrode  824  of FIG. 8B with an electrode length ratio a 1/2 can be envisioned as formed on the side surface  822  of the workpiece  820 , which originates from the end  826  of the workpiece  820 , and last for one-half the total length of the workpiece. Along orientation 2 (FIG. 1), the width of the electrode  824  is determined by the value obtained from Equation (22). Again, the width of the resolved electrode on orientation 2 (FIG. 1) is scale-less. Shape of the electrode is not necessarily symmetrical along longitudinal axis of the workpiece, although the patterns exemplified in FIGS. 8A, 8B and  8 C are symmetrical.  
         [0085]    [0085]FIG. 9 illustrates the side view of the piezoelectric transformer  830  constructed utilizing the second mode edge-driving quasi-modal actuator with an electrode pattern of FIG. 8C. The side view is taken along orientation 2 (FIG. 1) in which the workpiece  830  reveals its quasi-modal actuator electrode  834  only as a thin layer. Note that the electrode  834  can be seen as comprising two sub-sections  834 A and  834 B. The electrode  837  on the right end of the workpiece  830  is the sensor, or output electrode of the transformer system. The electrode  835  formed on the side surface opposite to that for the actuator electrode  834  is utilized in the example as the common ground, the return path for both the actuator and the sensor electrodes.  
         [0086]    The drawing also illustrates the directions of polarization for both the actuator and the sensor section of the transformer system. Specifically, the polarization direction identified by reference numerals  838 A and  838 B are the actuation polarization and that by  839  the sensing polarization. This polarization pattern in the actuator section is different from that for the first mode operation as described above. For second mode implementations in which the electrode length ratio a is greater than 1/2, two mutually reversed actuation polarizations  838 A and  838 B exist in the piezoelectric body. Each polarization is corresponding to the sub-section,  834 A and  834 B, of the actuator electrode  834 . Care should be taken when implementing such designs as the phenomenon of mechanical stress concentration arises inside the piezoelectric body near the vicinity where the polarization reverses, as is in the area generally indicated by reference numeral  833  in FIG. 9.  
         [0087]    The following resolves the electrode shape function for an edge-driving quasi-modal actuator operating in its second mode:  
                         e   31          (   x   )         e   31   0       =         sin                     2      π                 x     l       k          [       H        (   x   )       -     H        (     x   -     a                 l       )         ]                         k   =     sin                 2      a                 π             if            a   ≤     1   4                 k   =   1           if            a   ≥     1   4                         (   24   )                               
 
         [0088]    Aside from the edge-driving actuator implementations described above, a different approach for placing the actuator electrode is worth examining. In center-driving designs, the resolved actuator electrode is placed at the center of the surface of the workpiece in a piezoelectric transducer system. Center-driving designs are not different from edge-driving when the governing equation is concerned. The only difference being the boundary conditions set up for the resolution of the electrode shape function.  
         [0089]    Based on the modal shape equation as outlined in Equation (18) and the design rule equation for the electrode shape of modal actuators expressed in Equation (20), the mathematical expression representing a center-driving quasi-modal actuator operating in its first mode can be expressed as  
                   e   31          (   x   )         e   31   0       =     sin                     π                 x     l            {       H        [     x   -       (     1   -   a     )          l   2         ]       -     H        [     x   -       (     1   +   a     )          l   2         ]         }     .               (   25   )                               
 
         [0090]    [0090]FIGS. 10A, 10B and  10 C illustrate patterns of the actuator electrode for a first mode center-driving quasi-modal actuator section of a piezoelectric transformer having an electrode length ratio of 4/9, 5/9 and 2/3 respectively. Note that in a center-driving quasi-modal actuator, the shape function puts the electrode at the center of the side surface of the elongated thin-plate workpiece along the longitudinal axis, namely, x coordinate along orientation 1.  
         [0091]    [0091]FIG. 11 illustrates the side view of a piezoelectric transformer constructed utilizing the first mode center-driving quasi-modal actuator with an electrode pattern of FIG. 10A. In the case of a piezoelectric transformer system constructed utilizing a first mode center-driving quasi-modal actuator section such as that illustrated in FIG. 10A, the corresponding sensor section of the system may utilize a pair of end-plated electrodes, such as  1016  and  1017  in the side view. Note the respective polarization directions for the actuator,  1018 , and sensor sections,  1019 A and  1019 B, respectively. Also note that the entire system as viewed in FIG. 11 is symmetrical with respect to orientation 2 (FIG. 1) of the workpiece  1010 .  
         [0092]    The following shape function resolves the electrode shape for a center-driving quasi-modal actuator operating in its second mode:  
                   e   31          (   x   )         e   31   0       =         sin                     2      π                 x     l       k          {       H        [     x   -       (     1   -   a     )          l   2         ]       -     H        [     x   -       (     1   +   a     )          l   2         ]         }               (   26   )                               
 
         [0093]    wherein  
                   k   =     sin                   (     1   -   a     )        π             if            a   ≤     1   2                 k   =   1           if            a   ≥     1   2                   (   27   )                               
 
         [0094]    [0094]FIGS. 12A, 12B and  12 C illustrate patterns of the actuator electrode for a second mode center-driving quasi-modal actuator section of a piezoelectric transformer having an electrode length ratio of 2/9, 5/18 and 1/3 respectively. In these center-driving quasi-modal actuator operating at second mode, as are in the case of FIGS. 10A, 10B and  10 C, the shape function positions the electrode at the center of the side surface of the elongated thin-plate workpiece along the longitudinal axis, i.e., x coordinate along orientation 1.  
         [0095]    [0095]FIG. 13 illustrates the side view of a piezoelectric transformer constructed utilizing the second mode center-driving quasi-modal actuator with an electrode pattern of FIG. 12C. If a piezoelectric transformer system is constructed utilizing a second mode center-driving quasi-modal actuator section such as that illustrated in FIG. 12C, the corresponding sensor section of the system may utilize a pair of end-plated electrodes,  1236  and  1237  in the side view. Note the respective polarization directions for the actuator,  1238 A and  1238 B, and sensor sections,  1239 A and  1239 B, respectively. Meanwhile, the entire system as viewed in FIG. 13 is symmetrical with respect to orientation 2 (FIG. 1) of the workpiece  1230 .  
         [0096]    The above examples of the inventive piezoelectric transformer systems of the present invention are superior in performance than the conventional transformers with uniform electrodes. Notably, significantly less undesirable harmonics are present in the output of the piezoelectric systems. For example, in a center-driving piezoelectric transformer of the invention, the system does not function in even modes if the assigned mode of operation is odd. In contrast, if a center-driving transformed is operated in an even mode, the workpiece will not be driven into any odd mode. This substantially reduces the noise harmonics. In turn, the overall operating efficiency is improved as well.  
         [0097]    The solutions resolved for the shape of the actuator electrodes for the piezoelectric transducer apparatuses of the present invention, as outlined, are trigonometric functions. This is because the governing equation for the piezoelectric workpiece based on an elongated thin-plate configuration is a second order partial differential equation. If the basic structural configuration of the piezoelectric workpiece is changed into other forms, the resolved shape function will also be different, not necessarily be trigonometric function.  
         [0098]    On the other hand, typical sensor electrode for a piezoelectric transducer apparatus in accordance with the present invention may be a conventional uniform sensor electrode, for example, one with a rectangular shape. This is convenient as, in many occasions such as in a Rosen-type piezoelectric transformer, the sensor electrode in accordance with the present invention is plated on the small end surface(s) of an elongated thin-plate piezoelectric workpiece.  
         [0099]    While the above is a full description of the specific embodiments, various modifications, alternative constructions and equivalents may be used. For example, although piezoelectric workpieces having the structural configuration of an elongated thin plate are used as example for the theoretical development of the underlying principle of the present invention, use of workpieces of other physical dimensions are possible. For example, a piezoelectric workpiece in the shape of a thin and circular disk can be considered mathematically as a one-dimensional device in the polar coordinate system and analyzed in accordance with the teaching of the present invention. Therefore, the above description and illustrations should not be taken as limiting the scope of the present invention which is defined by the appended claims.