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Abstract— A multi-mesh loop matrix model for a piezoelectric

transformer operating with switched circuit excitation is

presented. The modeling requires the inclusion of harmonics and

the coupling of vibration in different directions of the

piezoelectric transformer. Simulation and experimental results

are used to validate the proposed model using matrix circuits.

The proposed method has also compared with other conventional

modeling method, a good agreement is obtained.

Keywords—Modeling, Nonlinear magnetics, Reluctance motors,

Simulation

I. INTRODUCTION

Piezoelectric transformer is considered as one of the most

important power transfer device in the next decade because of

its low profile, no EMI problem and high efficiency. It can

also achieve voltage conversion ratio using frequency control

easily whereas thee magnetic transformer cannot. Therefore

piezoelectric transformer can be used to construct a regulated

power converter easily with a frequency control as the

parameters. Today, piezoelectric transformer has found

applications in many power conversions in consumer

products, medical electronics and lighting system [1-2],

rectifier [3], miniature power supplier [4] and low power

source [5].

Recently the use of multilayer piezoelectric transformer is

getting more attention because of the requirement of larger

voltage conversion ratio. All the layers can be connected in

either parallel or series internally in order to follow similar

method in magnetic transformers connection. The resultant

multilayer can give very high output voltage. The typical

applications are high voltage power supply, electrostatic

precipitator, and ionizer.

Common modeling of the piezoelectric transformer is based

on a simple LRC circuit [6] which can only describe the

vibrating structure in only one direction, such as the

longitudinal direction. However, the coupling between

different vibration directions will have significant influence

upon the energy transmission and operating efficiency along

the axes in many cases [7].

Finite element model (FEM) is a powerful tool to study the

operation of piezoelectric transformers [8]. However,

modeling of the piezoelectric transformer using FEM requires

a complete set of the electromechanical properties for each

constituent material. Such properties include the dielectric and

coupling constants. The loss mechanics including the

attenuation is also very complex [9]. Moreover, an accurate

calculation of the voltage conversion ratio is of paramount

importance in the circuit design for power conversion

applications. Conventional, the modeling of piezoelectric

transformer is based on circuit model which the resistance,

capacitance and inductance are used. The modeling of multi-

layer type is especially important because the model exhibits

small higher order effect and the coupling between the layers

make the simple model of the RLS not sufficient to represent

the whole piezoelectric transformer. Hence it is necessary to

develop a model that can give simple and accurate results on

the coupling, energy loss and conversion ratio. A multilayer

type is used as an example to demonstrate the model of the

piezoelectric transformer. In this paper, a mutual coupling is

introduced in the circuit model so that the parasitic effect inn

different layers of the piezoelectric transformer can be

modeled accurately.

II. PROPOSED MODEL

The model is based on a matrix resonant circuit having

similar conventional LRC resonant elements together with the

inclusion of different coupling mechanics in different

directions. Fig. 1 shows the proposed model that represents the

three directional couplings in a realistic manner. The mesh

current as shown in Fig. 2 in each facet of the matrix model is

Ii, where 1≤i≤6 is the number of each facet. The element in

Modeling and Analysis of Piezoelectric

Transformer using Multi-mesh Loop Matrix

Circuit under Square-wave Excitation

Conditions

1K. W. E. CHENG, 1Y.L.Ho, 1S. L. Ho, 2K.W.Kwok, 2X.X.Wang, 2H.Chan and 1X.D.Xue

1Department of Electrical Engineering, 2Department of Applied Physics,

The Hong Kong Polytechnic University

The financial support of the Research Committee , The Hong Kong

Polytechnic University under project G-YD65 is appreciated

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each of the edge gives rise to the more physical meaning of

the piezoelectric transformer. Each edge gives the connection

between the longitudinal vibration and the coupling between

the longitudinal vibrations. They are therefore contributes to

the resonant tank in each edge and the coupling between the

components will be added in the later session of the paper. The

impedance of each edge of the matrix can be represented by

Zuvw-xyz, where uvw and xyz are the coordinates of the Zuvw-xyz

to which it is connected. The matrix model of the Fig 1 can

be expressed as:

V =

(1)

k ij ijk

IAC

where

the subscript has the range of

1<k<6; 1<i<6; 1<j<6. (2)

k is the number of the mesh loops. i and j are also equal to the

number of mesh loop. For the present model, Aij is the 6×6

matrix. Vk is the voltage source in the circuit. It includes the

input and output voltages. Cij is a connection matrix which is

1 when there is an element in the loop and 0 when there is no

element in the loop. The element Aij is the impedance matrix.

Ik is the loop current. Also,

Aij ∈ Zuvw-xyz

The impedance Zuvw-xyz is a series connection of R, L and C.

(3)

xyzuvw

xyzuvwxyzuvw

R

xyz uvw

sC

sLZ

−

−−−

++=

1

(4)

The voltage conversion ratio is then:

00001041

00010051

)(

)(

−

−

+

+

=

ZII

ZII

V

V

in

o

(5) (1)

Fig. 1. Matrix model for the piezoelectric transformer

Fig. 2. Mesh current of each facet of the matrix model

II. OLD MODEL

Conventional model [10] for modeling the piezoelectric

transformer can be revisited in Fig 3. The model only gives

the fundamental resonance of the transformer and a 2nd order

resonant frequency. However, for multilayer case, the

coupling between the layers cannot be realized.

Fig 3: Conventional model of piezoelectric transformer

Fig 4 shows a modified model which some dielectric loss

components has been added to facilitate some of the loss and

coupling. The loss and the higher order frequencies are

expressed by use of the capacitance Cd1, Cd2 and Cp1. It now

uses three models presented here to examine which model is

more accurate for the captioned applications.

Fig 4: Modified Piezoelectric transformer model

III. EXPERIMENT RESULT OF THE MULT-LAYER

A multilayer piezoelectric transformer is used for the analysis.

The dimension of the multilayer piezoelectric transformer is

8mm x 3mm x 36mm. The primary side is multilayer but the

secondary side is single layer. The load resistance for this test

is 101kΩ.

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Fig 5: Measured results of the multilayer transformer

Ch1= input voltage (2V/div) & Ch2=input current (0.5A/div),

Ch3=output voltage (200V/div) & Ch4=output current (10mA/div)

The voltage conversion characteristics versus frequency is

measured and shown in Fig 6. It can be seen that there are 4

resonance frequencies which are at: 46kHz, 85kHz, 135.5kHz

and 172.8kHz. It can also be seen that there is no direct

relationship between each resonant frequencies such as the

multiple relation. The resonant frequency is the self-resonant

of its energy storage components and therefore using circuit

model, usually a connection of the R-L-C is used to model the

piezoelectric transformer. To be accurate, a more systemic

approach is needed to express the frequency,

0

20

40

60

80

100

120

0 50 100150200

Frequency (kHz)

Voltage Conversion ratio

Fig 6: Measured voltage conversion ratio under sinusoidal input and load =

101kΩ

IV. MODELLING RESULTS

Three modeling methods are used to model the piezoelectric

transformer. They are the proposed matrix model,

conventional and modified methods. The experiment was

conducted using a high frequency amplifier with an accurate

signal generator and the resonant peak is recorded. The

frequency of interest is from 0Hz to 200kHz. Experimental

measurement of multilayer transformer with the proposed

matrix model was compared and present in Fig 7. It can be

seen that a reasonable agreement can be found in the model.

However, the 3rd and 4th resonant frequencies cannot be

predicted very well with 10-30% error.

0

20

40

60

80

100

120

0 50100150200

Frequency (kHz)

Voltage Conversion ratio

Fig 7: Comparison between the calculated and measured result.

Fig 8 and 9 shows the results of the ratio for the modified

model and the conventional model. It can be seen that the

modified model can describe the voltage conversion ratio very

well and is different from the measurement by 10% for the

resonant peaks. However, the 4th resonant frequency cannot

be predicted very well and there is 80% error. The

conventional model give up to 20% error in the resonant peak

and the 4th resonant frequency at 172kHz cannot be retrieved

from the model. It is quite obvious because the conventional

model does not implemented with the 4th order self resonant

frequency and therefore it is out of score for the capability.

The thee resonance frequencies are generated from the

combination of the frequency resonant tank from the Cd1, Cd2

and CsS1.

Therefore it is necessary that a simple circuit model is unable

to predict the complicate self-resonant frequency of the

piezoelectric transformer and a model with higher order of

resonant element as proposed is needed to order to give higher

order of resonance frequency for the proper modeling.

0

20

40

60

80

100

120

050100150200

Frequency (kHz)

Voltage Conversion ratio

Fig.8. Calculated results using modified model

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0

20

40

60

80

100

120

0 50100150200

Frequency (kHz)

Voltage Conversion ratio

Fig 9: Calculated Piezoelectric transformer model using conventional model.

V. COUPLED MODEL

The above matrix method is suffered from the high frequency

accuracy because of the coupled between the each branch has

not been activated. Equations (1-4) are modified as follows:

∑

+=

ijxyk ijijk

IMCIACV

k

(6)

where 1≤x≤6; and 1≤y≤6. Cxy is the connection matrix to

connect the Mij to the voltage equation. Cxy defines the

switching function is either 0 when no coupling between the

current and the inductance and equal to 1 when there is a

mutual couple between Mij and Ik. As the coupling between

the inductance and current are small when the layer are far

away and large when the layer are close. Therefore the

modified equations give a more realistic representation of the

multilayer piezoelectric transformer. The model is closer to

the realistic situation because the coupling is the same effect

as the mechanical coupling between the layers of the

piezoelectric transformer.

0

20

40

60

80

100

120

050100150200

Frequency (kHz)

Voltage Conversion ratio

Fig 10: Matrix modeling with mutual coupling

VI. ERROR OF THE MODELS AND DISCUSSION

The conventional model can model the 1st and 2nd resonant

frequencies very well. For high values of resonance, the

model cannot meet the requirement because the inter-coupled

and parasitic capacitance cannot be revealed in the model.

The modified one is constructed with additional parasitic

capacitance. The modeling of the higher order resonance is

good, but still the 4th order resonant frequency is not modeled

very well.

By introduction the proposed matrix model, it allows a more

network connection of each of the layers of the piezoelectric

transformer. All the R-L-C networks are now connected to

give coupling between each. However, the matrix model does

not give mutual coupling between the layers that are needed

for the multi-layer piezoelectric transformer. Finally, the

matrix model with mutual coupling is used. It can now be

seen that the new model give a better modeling method for the

transformer.

Under square wave operation, eqn (6) can be used to represent

square wave accordingly. Table 1 shows the error of the

model as compared with other model and the experimental

results. It is clearly seen that the proposed modeling method

give a good agreement in both sinusoidal and switching square

wave applications. The switching square wave is produced by

a full-bridge converter which gives the same rms voltage as

the sine waveform as in Section III and IV. The rms error of

1st 4 resonant peaks is calculated. It is clearly seen that again

the matrix method with mutual coupling give the best solution

to model the multi-layer piezoelectric transformer.

Table 1: RMS Error compared with experiment

Sinusoidal

Conventional 32%

Modified

conventional

Matrix 10%

Matrix with

mutual coupling

The method has also been tested for another piezoelectric

transformer. It can also model the proposed 4 peaks of

resonance frequencies. The error of the measurement is for

the matrix with mutual coupling as compared with the

theoretical calculation is also 2%.

Square wave

35%

25% 20%

14%

3% 2%

VII. CONCLUSION

This paper proposes an improved model of the piezoelectric

transformer. The model is first expressed as impedance of

each R-L-C network and it is then using mesh calculation to

compute the characteristics. Four modeling method have

been presented. They are the conventional, modified version

which have layer capacitances, matrix model and coupled

matrix model. The conventional model cannot predict the high

resonant frequencies. The modified model cannot predict the

peak value of the resonant peak. Therefore both models have

the difficulty to express the coupling effect between layer and

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the non-longituitual energy transfer.

The matrix model is based on the intra-connection of the

reactive components and the agreement has been improved.

The model is further extended to cover the mutual coupling

among the reactive component and the completed model

namely matrix model using mutual coupling is developed. It

can give very good agreement to the experimental results. It is

clearly shown that after the coupling between the impedance

edges are considered, the resonant peaks can be accurately

calculated.

This paper demonstrates an alternative approach to model

the piezoelectric transformer such that only R, L and C are

used. Using the mutual coupling, the model gives improved

results as compared with conventional method. The method

also give a remarkable result as compared to the conventional

modeling method that is only based on simple circuit mode.

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"Generic operational

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