Analysis and Design of a Piezoelectric Transformer AC/DC Converter in a Low Voltage Application Gregory Ivensky, Svetlana Bronstein, and Sam Ben-Yaakov* Power Electronics Laboratory Department of Electrical and Computer Engineering Ben-Gurion University of the Negev P.O. Box 653, Beer-Sheva 84105, ISRAEL Phone: +972-8-646-1561; Fax: +972-8-647-2949; Email: [email protected] bgu.ac.il; Website: www.ee.bgu.ac.il/~pel v Co i and a current source r (n is the transformer turn n n ratio) [7]. A halfway rectifier Rect and an inductor Lo are connected at the output of the PT. The rectifier includes two diodes ( D1 , D 2 ) and L f , C f filter. RL is the load resistance. The function of Lo in this circuit is to provide a path for the DC current and also to cancel the circulating energy caused by Co [2]. The resonant frequency of the parallel circuit Lo, Co has to be equal to the series resonant frequency of the PT. The simulated current and voltage waveforms of the rectifier operating in overlapping and non-overlapping modes are given at Fig. 2 a, b ( ϑ = ωt is the normalized time, ω is the operating angle frequency). The PT current i r is assumed to be sinusoidal in both modes due to the

Abstract - steady-state processes in a power PT converter with a halfway two diodes output rectifier are studied. It is shown that such a power converter can operate in two modes: with or without overlap of the conduction intervals of the diodes. The analytical equations and characteristics derived in this study were found to be in good agreement with simulation and experimental results. The study highlights the effect of the load and parasitic resistances on the output voltage and the losses for both modes. Expressions obtained in the paper can be used in engineering designs.

I. INTRODUCTION Piezoelectric Transformers (PT) convert electric power into electric power via a mechanical resonant link. PT’s have important advantages over magnetic transformers in a few applications that require a low DC output voltage [1-6]. In this case a halfway output rectifier with two diodes is the preferred choice. However, in earlier investigations only one operational mode of this rectifier was studied: when the conducting intervals of both diodes of the rectifier do not overlap. The presented paper shows that this operation mode is not unique. In a wide range of load resistances the rectifier operates in another mode in which the conduction intervals of the diodes overlap. The paper presents a generalized detailed analysis of both operating modes. The results of this analysis reveal new important characteristics of this rectification scheme.

high Q of practical PTs. The capacitor C o voltage is practically sinusoidal in the non-overlapping mode but has a discontinuous nature (intervals of zero voltage) in the overlapping mode. We study here in detail the overlapping mode. During the interval ϑ1 ϑ 2 = λ the voltage vCo across the capacitor Co is positive, therefore the diode D1 is ON and the diode D2 – OFF. At ϑ 2 , the polarity of the voltage v Co changes and therefore D2 turns on. If just before ϑ 2 , the instant current of D1 ( i D1 ) is higher than the absolute value of the capacitor Co current ( i Co ) the rectifier will operate under overlapping conditions. In this case the diodes D1 and D2 are conducting simultaneously during the interval ϑ 2 ϑ 3 , the current i D 2 is

II. PRINCIPLE OF OPERATION OF THE HALF-WAY TWO DIODES RECTIFIER The conventional equivalent circuit of the PT was used to analyze the low voltage AC/DC converter topology (Fig. 1). It includes L r , C r , R m , C in , C O , a voltage source

increasing while the current i D1 is decreasing (due to changing i r ).

Rect

PT ir

Vin

Cin

D1

Lr Cr Rm vCo n

+ -

ir n

iCo

Lf

i Lo

Co Lo vCo

D2

Fig. 1. The PT converter loaded by a halfway two diodes rectifier.

*Corresponding author

Io Cf

RL

vout

A. Lossless PT (Rm=0)

ir n v CO

VCO p

γ

λ

iLO

ϑ

π v Co (ϑ) = VCO p cos ϑ λ

ϑ

γ1 γ 2

iCO iD1 iD2

ϑ

We assume that the impulses of v Co have a sinusoidal shape and that the duration of each impulse is approximately λ (Fig. 2). According to this assumption:

ϑ

ϑ1

where VCO p is the peak of the voltage across the capacitor Co.

iD2

iD1

0 ϑ2 ϑ3

ϑ

ϑ4

(a)

ir n

ϑ

The peak of the first harmonic of this voltage: λ cos 4 2 = VCO p 2 λ π −1 λ

VCO (1) p

ϑ

λ =π

iLO

λ π2 Combining (3) in (2) we obtain:

ϑ VCO (1) p = 4Vout

iCO

ϑ iD2

iD1

ϑ1

0 ϑ2 = ϑ3

ϑ4

ϑ

(b) Fig. 2. The rectifier’s current and voltage waveforms in the overlapping (a) and non-overlapping (b) modes.

During this overlapping interval, the current iCo=0, because diodes short the capacitor. At ϑ 3 , i D1 = 0 and D1 turns off. During ϑ 3 ϑ 4 , only D2 is ON and v Co < 0 . Inductance Lo charges by the input supply and it’s current increases. At ϑ 4 , v Co changes polarity and an overlapping interval begins again, but now i D1 is increasing while i D 2 is decreasing, etc. III. THE UNIFIED ANALYSIS OF THE OVERLAPPING AND NON-OVERLAPPING MODES Analysis is carried out under the assumptions: a) Diodes and reactive elements are ideal; b)

The output filter is ideal ( L f = ∞ , C f = ∞ );

c)

The current ir has a sinusoidal waveform.

d)

following

The operation frequency ω is equal to the resonant 1 1 frequency: ω = . = LrCr LoCo

(2)

The average output voltage Vout is equal to the DC component of the positive value of v Co : Vout = VCO p

v CO

iD1 iD2

(1)

λ cos 2 2 λ 1− π

(3)

(4)

Since the operating frequency is equal to the resonant frequency of the series circuit C r , L r : VCO (1) p

= Vin .p (5) n The transfer ratio of the output voltage of the rectifier Vout to the peak of the input voltage Vin .p for the

ideal PT is: 2

ko ideal =

Vout nVin .p

λ 1− π = λ 4 cos 2

In the non-overlapping mode: 1 NOM ko ideal = lim(ko ideal ) = λ →π π

(6)

(7)

At the instant ϑ 2 the diode D 2 carries a current that was flowing via capacitor C o . This current that starts as a step can be found from (1): I ch = ωC o

dv Co dϑ

= ϑ2

π ωC o VC O p λ

(8)

or applying (3): I ch =

π3 Vout ωC o λ2

(9)

λ

The output current I O at ϑ 2 may be written as I O = I ch

(10) 2

I i where: m sin γ 1 is the current r (Fig.1) at the instant ϑ 2 , n n I m is the peak of the input current, γ 1 - see Fig. 2 (a), and i L O is the inductor L O current. The inductor L O is shorted

by the diodes D1 and D 2 during the whole interval ϑ 2 ϑ 3 . That is why its voltage is practically zero and its current i L O has a constant value up to the instant ϑ 3 when D1 turns off. Taking into account this condition, we use ϑ 3 to derive i L O . At ϑ 3 i CO = i D1 = 0 and therefore: ir Im = sin (γ 2 ) n n where γ 2 - see Fig. 2 (a). i LO =

(11)

Consequently, the current balance at ϑ 2 (10) may be written as: I π3 γ γ − γ2 I O = 2 Vout ωC o + 2 m sin cos 1 n λ 2 2 where γ = γ 1 + γ 2 is the overlapping angle: γ = π−λ

(12)

(13)

Taking into account that

γ λ sin = cos 2 2

and

γ − γ2 assuming that cos 1 ≈ 1 equation (12) can be 2 simplified to:

IO =

Vout π 3 I λ = 2 Vout ωC o + 2 m cos RL n λ 2

(14)

From the energy balance it follows that: Vin .p I m 2

V = out RL

π

3

I + m sin (γ 1 ) + i LO n

NOM 1

OM 0

-2

ωC O R L

10 -1

1

π

10 0

Fig. 3. Duration λ as a function of the normalized load factor ωCoRL. OM - overlapping mode, NOM- non-overlapping mode.

input voltage of the rectifier are in phase with the input voltage of the PT. Under these assumptions: VC (1) p Vin .p − I m R m = O (17) n or, according to (4): Vin .p − I m R m =

4Vout n

λ cos 2 2 λ 1− π

(18)

It can be shown that equation (14) derived above for the ideal PT, is also correct for the lossy PT. Taking λ cos from (14) and inserting it to (18) we obtain: 2 2

Vin .p I m 2

2

−

Im R m = 2

Vout π3 2 − Vout ωC O 2 RL λ λ 1− π

2

(19)

The left side of this equation is the output power, hence (19) may be rewritten as: 2

2

(15)

Solving (6), (14) and (15) together, we obtain the equation for duration of the impulses of the voltage v Co in the overlapping mode: λ = 4 π 5 ωC o R L

10

(16)

In the non-overlapping mode λ = π . Duration of the capacitance C o voltage impulses λ as a function of the load factor ωC o R L based on (16) is depicted in Fig. 3. The rectifier operates in overlapping 1 mode when ωC o R L < . π B. Real PT (Rm>0). We assume the followings:

2

Vout = RL

Vout π3 2 − Vout ωC o 2 RL λ λ 1− π

2

(20)

Equation (20) can now be used to obtain the angle λ . The solution is identical to (16). We find here that even in the lossy case, λ is independent on the equivalent losses resistance of the PT - Rm . C. The Diodes Losses In low output voltage rectifiers the voltage drops of the diodes are significant and therefore should be taken into account in the converter’s design. The power losses PD in each diode are calculated from equation given in [2] PD = VF I D ( ave) + R F I 2D ( rms)

(21)

a) The voltage across the resonant circuit L r , C r is zero.

where: I D ( ave) and I D ( rms) are the average and the rms diode

b)

currents, and R F and VF are the diode parameters: R F is

The voltage across R m and the first harmonic of the

the equivalent forward resistance and VF is the forward voltage:

η rect ≈

According to Kirchoff’s law, the diodes’ currents i D1 and i D 2 during the overlapping intervals can be described by the following equations: i D1 = i L O +

ir n

(22)

i i D2 = I O − i LO − r (23) n During the non-overlapping intervals, the current of one of the diodes is I O and the current of the other diode is zero. The average diode current ID(ave) is approximately equal to the half output DC current I D ( ave) = 0.5I O . The rms diode

current I D ( rms) was determined using the assumption that during the overlapping intervals the diode currents are changing linearly. Applying (9), (13) and (20), following expression of ID(rms) was obtained: I D ( rms ) =

IO 2

ϕ(λ)

(24)

where: 1 λ2 2 λ2 + 1 − π π 2 3 π 2

ϕ(λ ) =

2

(π − λ ) + λ

(25)

Hence, the general equation of the power losses in each diode PD (21) can be rewritten in the following form: 1 1 2 I O VF + I O2 R F (ϕ(λ )) (26) 2 2 In non-overlapping mode this equation is simplified:

PD =

2 O

I RF 1 I O VF + (27) 2 2 Assuming that the diode’s power dissipations is the major power loss, the rectifier efficiency is expressed as: PDNOM =

η rect

Pout = Pout + 2PD

(28)

where: Pout = Vout I O

(29)

is the output power. Applying (26) we transform (28) into: η rect =

1

V R (ϕ(λ )) 1+ F + F Vout RL

2

(30)

The value of RF in modern diodes is usually small and in most practical cases it is much lower than the load resistance of the rectifier RL. This and the fact that the function ϕ(λ)≤1 for every λ suggest that the term RF(ϕ(λ))2/RL in the last equation can be neglected. Hence,

1 V 1+ F Vout

(31)

As this equation shows, the rectifier efficiency ηrect decreases as the output voltage Vout gets lower. It is especially noticeable when Vout is getting closer to the diodes forward voltage VF. D. Voltage Range and Power Characteristics of a Real Converter Applying the power balance condition we can replace the loaded output rectifier by an equivalent resistance R eq connected to the output of the PT (Fig. 4). The power balance condition including the rectifier losses can be expressed as: V C2

O (1) p

=

2R eq

2 2 Vout Vout + VF I O = RL R L η rect

(32)

from which: R eq

VC (1) p = O Vout

2

R L η rect 2

(33)

It was found that the voltage ratio of the last equation could be calculated by an expression, which is similar to (4), but takes into account the rectifier’s efficiency: * cos λ 4 2 = η rect λ* 2 1 − π

VC O (1) p Vout

(34)

The angle λ* is defined by an expression similar to (16): λ* = 4

π 5 ωC o R L η rect

(35)

In non-overlapping mode, expressions (33) and (34) converges to: NOM R eq =

π2 RL 2η rect

(36)

Applying (31)-(34) we obtain the output to input voltage ratio of a converter with losses in the rectifier and the PT: λ* 2 0.25η rect 1 − π ko = n 2 R m λ* 1 + cos R eq 2

(37)

The dependences of ko ideal (6) and ko (37) on the normalized load factor ( ωC o R L ) are depicted in Fig. 5 a, b. Fig. 5 shows that a decrease in the load factor ωC o R L causes a decrease of the voltage ratio ko in the overlapping mode even in ideal case of a lossless PT ( R m = 0 ), but this

decrease is much more pronounced in the real case ( R m > 0 , PD > 0 ). Fig. 6 depicts the normalized output power Lr 2 2 C V r PO* = out as a function of the normalized load R L V in2 .p

P*

150

100 50

PT Lr

ir

0

Cr Rm vCO

Cin

n

iCO iL

ir n Co Lo Req O

+ -

Fig. 4. A simplified equivalent circuit of the converter.

1 π

0.3 0.25 Simulation

0.15

Calculation

0.1 0.05

OM

10-2

NOM

10-1 ωC O R L

100

1 π

(a)

Simulation Calculation

0.25

10-1

100

Efficiency of the PT η PT and efficiency of the whole converter η can be also easily calculated if we know Req:

η=

R eq n R m + R eq 2

Pout = η PT η rect Pin

(38) (39)

Philips PT (RT 35x8x2 PXE43-S) with thickness polarization mode was used in the experiment. Its main parameters (Fig. 1) were measured to be Lr=165mH, Cr=15.1pF, Cin=Co=510pF, Rm=105 Ω , n=1. Operation frequency was equal to the resonant frequency. The parallel resonant inductance was L o = 4.88mH and the diodes were a Schottky diode (MBR160P). The experiments were carried out for a load resistance RL range of 30Ω to 1kΩ. The range of the peak of the input voltage was Vin .p = 5 − 30V . The experimental and the theoretical

V. DISCUSSION AND CONCLUSIONS

0.2 0.15 0.1 NOM

OM

0.05 10-4

10-2

ωC O RL

voltage ratios as a function of the load resistance are shown in Fig. 7. It is evident that the analytical equations and characteristics, derived in this study, are in excellent agreement with simulation and experimental results

ko

0

10-3

IV. EXPERIMENTAL RESULTS

0.2

0.3

10-4

Fig. 6. The normalized output power P* as a function of the normalized load factor ωCORL. Real PT and real diodes (the same parameters as in Fig. 5 b).

η PT =

koideal

0

Model

200

factor ωC o R L .

Vin

Simulation

10-3

10-2 ωC O R L

10-1 1

π

100

(b) Fig. 5. The voltage ratio as a function of the normalized load factor ωCORL: (a) Lossless PT, ideal diodes, (b) Real PT (ωCoRm=0.034), real diodes (MBR160P). The peak input voltage Vin.p=30V. OM – overlapping mode, NOM – non-overlapping mode.

It was shown that the PT power converter with a halfway two diodes rectifier can operate in two modes: NOM - the operational mode described earlier in which there is no overlap between the conduction intervals of the diodes, and OM – the newly uncovered mode with overlap between the conduction intervals of the diodes. It was found that the operation mode is depending on the normalized load factor ωC o R L . The load factor is a function of the PT parameters and the load resistance. The study highlights the effect of the load and parasitic resistances on the output voltage and the losses for both modes. The analytical investigations were supported by simulation and experiment results, and found to be in excellent agreement.

Vout Vin.p

[5] C.Y. Lin and F.C. Lee, “Design of a Piezoelectric Transformer Converter and Its Matching Networks,” IEEE PESC Record, pp. 607-612, 1994. [6] T. Zaitsu, T. Shigehisa, M. Shoyama, and T. Ninomiya, “Piezoelectric Transformer Converter with PWM Control,” IEEE Intelec Proc., pp. 279-283, 1996. [7] G. Ivensky, M. Shvartsas, and S. Ben-Yaakov, “Analysis and modeling of a piezoelectric transformer in high output voltage applications,” IEEE APEC record, pp. 1081-1087, 2000.

Experiment Calculation

0.3

0.2

0.1

0

1

10

100

RL [Ω]

1000

Fig. .7 Experimental and theoretical voltage ratio Vout/Vin.p as a function of the load resistance RL. Vin.p=30V, ωCoRm=0.034.

It is important to note that a design that is based on the known NOM may be in a considerable error if the rectifier operates in OM. The analytical equations derived in this study can be used in engineering design. An example of a design procedure is given in the Appendix. In this design example we assume that the PT is given and calculate the required input voltage to meet the out specifications (Pout, Vout). If the calculated input voltage is not equal to the available input source, a trial and error procedure need to be following until the correct PT is found. Alternatively, the PT parameters (e.g. n) can be adjusted to meet the requirement and by this define the set of parameters of the PT that will meet the given specifications. [1] [2] [3] [4]

REFERENCES T. Zaitsu, T. Inoue, O. Ohnishi, and A. Iwamoto, “2 MHz Power Converter with Piezoelectric Ceramic Transformer,” IEEE Intelec Proc., pp.430-437, 1992. M. Kazimerczuk and D. Czarkowski, Resonant power converters, John Wiley & sons, Inc., 1995. C. Y. Lin and F. C. Lee, ”Development of a Piezoelectric Transformer Converter,” VPEC Seminar Proc., pp. 79-85, 1993. T. Zaitsu, O. Ohnishi, T. Inoue, M. Shoyama, T. Ninomiya, F.C. Lee, and G.C.Hua, “Piezoelectric Transformer Operating in Thickness Extensional Vibration and Its Application to Switching Converter,” IEEE PESC Record, pp. 585-589, 1994.

APPENDIX Design Guidelines Given: the AC/DC PT converter output requirements ( Vout , Pout ), the PT parameters ( L r , C r , C in , C o , R m , n ), and the diodes to be used in converter. Variables to be evaluated: the input voltage Vin that ensures the required output attributes of the power converter and the input power Pin . Design steps: 1) Define the resonant frequency ω r , the load resistance R L , the resonant inductance L O =

LrCr and the load Co

factor ωC o R L from the output specifications and the PT parameters. 2) Find the forward diode voltage VF from data sheet and evaluate the rectifier efficiency ηrect (31). 3) Calculate the parameter λ* (35). The rectifier is operating in non-overlapping mode when λ* = π and in overlapping mode when λ* < π . 4) Define the equivalent load resistance of the converter R eq (33) and (34). 5) Obtain the voltage ratio ko of the real converter (37). V 6) Calculate the peak input voltage as Vin ( p ) = out . ko 7) Find the PT and the whole converter efficiencies η PT and η (38), (39). P 8) Determine the input power Pin = out . η

Abstract - steady-state processes in a power PT converter with a halfway two diodes output rectifier are studied. It is shown that such a power converter can operate in two modes: with or without overlap of the conduction intervals of the diodes. The analytical equations and characteristics derived in this study were found to be in good agreement with simulation and experimental results. The study highlights the effect of the load and parasitic resistances on the output voltage and the losses for both modes. Expressions obtained in the paper can be used in engineering designs.

I. INTRODUCTION Piezoelectric Transformers (PT) convert electric power into electric power via a mechanical resonant link. PT’s have important advantages over magnetic transformers in a few applications that require a low DC output voltage [1-6]. In this case a halfway output rectifier with two diodes is the preferred choice. However, in earlier investigations only one operational mode of this rectifier was studied: when the conducting intervals of both diodes of the rectifier do not overlap. The presented paper shows that this operation mode is not unique. In a wide range of load resistances the rectifier operates in another mode in which the conduction intervals of the diodes overlap. The paper presents a generalized detailed analysis of both operating modes. The results of this analysis reveal new important characteristics of this rectification scheme.

high Q of practical PTs. The capacitor C o voltage is practically sinusoidal in the non-overlapping mode but has a discontinuous nature (intervals of zero voltage) in the overlapping mode. We study here in detail the overlapping mode. During the interval ϑ1 ϑ 2 = λ the voltage vCo across the capacitor Co is positive, therefore the diode D1 is ON and the diode D2 – OFF. At ϑ 2 , the polarity of the voltage v Co changes and therefore D2 turns on. If just before ϑ 2 , the instant current of D1 ( i D1 ) is higher than the absolute value of the capacitor Co current ( i Co ) the rectifier will operate under overlapping conditions. In this case the diodes D1 and D2 are conducting simultaneously during the interval ϑ 2 ϑ 3 , the current i D 2 is

II. PRINCIPLE OF OPERATION OF THE HALF-WAY TWO DIODES RECTIFIER The conventional equivalent circuit of the PT was used to analyze the low voltage AC/DC converter topology (Fig. 1). It includes L r , C r , R m , C in , C O , a voltage source

increasing while the current i D1 is decreasing (due to changing i r ).

Rect

PT ir

Vin

Cin

D1

Lr Cr Rm vCo n

+ -

ir n

iCo

Lf

i Lo

Co Lo vCo

D2

Fig. 1. The PT converter loaded by a halfway two diodes rectifier.

*Corresponding author

Io Cf

RL

vout

A. Lossless PT (Rm=0)

ir n v CO

VCO p

γ

λ

iLO

ϑ

π v Co (ϑ) = VCO p cos ϑ λ

ϑ

γ1 γ 2

iCO iD1 iD2

ϑ

We assume that the impulses of v Co have a sinusoidal shape and that the duration of each impulse is approximately λ (Fig. 2). According to this assumption:

ϑ

ϑ1

where VCO p is the peak of the voltage across the capacitor Co.

iD2

iD1

0 ϑ2 ϑ3

ϑ

ϑ4

(a)

ir n

ϑ

The peak of the first harmonic of this voltage: λ cos 4 2 = VCO p 2 λ π −1 λ

VCO (1) p

ϑ

λ =π

iLO

λ π2 Combining (3) in (2) we obtain:

ϑ VCO (1) p = 4Vout

iCO

ϑ iD2

iD1

ϑ1

0 ϑ2 = ϑ3

ϑ4

ϑ

(b) Fig. 2. The rectifier’s current and voltage waveforms in the overlapping (a) and non-overlapping (b) modes.

During this overlapping interval, the current iCo=0, because diodes short the capacitor. At ϑ 3 , i D1 = 0 and D1 turns off. During ϑ 3 ϑ 4 , only D2 is ON and v Co < 0 . Inductance Lo charges by the input supply and it’s current increases. At ϑ 4 , v Co changes polarity and an overlapping interval begins again, but now i D1 is increasing while i D 2 is decreasing, etc. III. THE UNIFIED ANALYSIS OF THE OVERLAPPING AND NON-OVERLAPPING MODES Analysis is carried out under the assumptions: a) Diodes and reactive elements are ideal; b)

The output filter is ideal ( L f = ∞ , C f = ∞ );

c)

The current ir has a sinusoidal waveform.

d)

following

The operation frequency ω is equal to the resonant 1 1 frequency: ω = . = LrCr LoCo

(2)

The average output voltage Vout is equal to the DC component of the positive value of v Co : Vout = VCO p

v CO

iD1 iD2

(1)

λ cos 2 2 λ 1− π

(3)

(4)

Since the operating frequency is equal to the resonant frequency of the series circuit C r , L r : VCO (1) p

= Vin .p (5) n The transfer ratio of the output voltage of the rectifier Vout to the peak of the input voltage Vin .p for the

ideal PT is: 2

ko ideal =

Vout nVin .p

λ 1− π = λ 4 cos 2

In the non-overlapping mode: 1 NOM ko ideal = lim(ko ideal ) = λ →π π

(6)

(7)

At the instant ϑ 2 the diode D 2 carries a current that was flowing via capacitor C o . This current that starts as a step can be found from (1): I ch = ωC o

dv Co dϑ

= ϑ2

π ωC o VC O p λ

(8)

or applying (3): I ch =

π3 Vout ωC o λ2

(9)

λ

The output current I O at ϑ 2 may be written as I O = I ch

(10) 2

I i where: m sin γ 1 is the current r (Fig.1) at the instant ϑ 2 , n n I m is the peak of the input current, γ 1 - see Fig. 2 (a), and i L O is the inductor L O current. The inductor L O is shorted

by the diodes D1 and D 2 during the whole interval ϑ 2 ϑ 3 . That is why its voltage is practically zero and its current i L O has a constant value up to the instant ϑ 3 when D1 turns off. Taking into account this condition, we use ϑ 3 to derive i L O . At ϑ 3 i CO = i D1 = 0 and therefore: ir Im = sin (γ 2 ) n n where γ 2 - see Fig. 2 (a). i LO =

(11)

Consequently, the current balance at ϑ 2 (10) may be written as: I π3 γ γ − γ2 I O = 2 Vout ωC o + 2 m sin cos 1 n λ 2 2 where γ = γ 1 + γ 2 is the overlapping angle: γ = π−λ

(12)

(13)

Taking into account that

γ λ sin = cos 2 2

and

γ − γ2 assuming that cos 1 ≈ 1 equation (12) can be 2 simplified to:

IO =

Vout π 3 I λ = 2 Vout ωC o + 2 m cos RL n λ 2

(14)

From the energy balance it follows that: Vin .p I m 2

V = out RL

π

3

I + m sin (γ 1 ) + i LO n

NOM 1

OM 0

-2

ωC O R L

10 -1

1

π

10 0

Fig. 3. Duration λ as a function of the normalized load factor ωCoRL. OM - overlapping mode, NOM- non-overlapping mode.

input voltage of the rectifier are in phase with the input voltage of the PT. Under these assumptions: VC (1) p Vin .p − I m R m = O (17) n or, according to (4): Vin .p − I m R m =

4Vout n

λ cos 2 2 λ 1− π

(18)

It can be shown that equation (14) derived above for the ideal PT, is also correct for the lossy PT. Taking λ cos from (14) and inserting it to (18) we obtain: 2 2

Vin .p I m 2

2

−

Im R m = 2

Vout π3 2 − Vout ωC O 2 RL λ λ 1− π

2

(19)

The left side of this equation is the output power, hence (19) may be rewritten as: 2

2

(15)

Solving (6), (14) and (15) together, we obtain the equation for duration of the impulses of the voltage v Co in the overlapping mode: λ = 4 π 5 ωC o R L

10

(16)

In the non-overlapping mode λ = π . Duration of the capacitance C o voltage impulses λ as a function of the load factor ωC o R L based on (16) is depicted in Fig. 3. The rectifier operates in overlapping 1 mode when ωC o R L < . π B. Real PT (Rm>0). We assume the followings:

2

Vout = RL

Vout π3 2 − Vout ωC o 2 RL λ λ 1− π

2

(20)

Equation (20) can now be used to obtain the angle λ . The solution is identical to (16). We find here that even in the lossy case, λ is independent on the equivalent losses resistance of the PT - Rm . C. The Diodes Losses In low output voltage rectifiers the voltage drops of the diodes are significant and therefore should be taken into account in the converter’s design. The power losses PD in each diode are calculated from equation given in [2] PD = VF I D ( ave) + R F I 2D ( rms)

(21)

a) The voltage across the resonant circuit L r , C r is zero.

where: I D ( ave) and I D ( rms) are the average and the rms diode

b)

currents, and R F and VF are the diode parameters: R F is

The voltage across R m and the first harmonic of the

the equivalent forward resistance and VF is the forward voltage:

η rect ≈

According to Kirchoff’s law, the diodes’ currents i D1 and i D 2 during the overlapping intervals can be described by the following equations: i D1 = i L O +

ir n

(22)

i i D2 = I O − i LO − r (23) n During the non-overlapping intervals, the current of one of the diodes is I O and the current of the other diode is zero. The average diode current ID(ave) is approximately equal to the half output DC current I D ( ave) = 0.5I O . The rms diode

current I D ( rms) was determined using the assumption that during the overlapping intervals the diode currents are changing linearly. Applying (9), (13) and (20), following expression of ID(rms) was obtained: I D ( rms ) =

IO 2

ϕ(λ)

(24)

where: 1 λ2 2 λ2 + 1 − π π 2 3 π 2

ϕ(λ ) =

2

(π − λ ) + λ

(25)

Hence, the general equation of the power losses in each diode PD (21) can be rewritten in the following form: 1 1 2 I O VF + I O2 R F (ϕ(λ )) (26) 2 2 In non-overlapping mode this equation is simplified:

PD =

2 O

I RF 1 I O VF + (27) 2 2 Assuming that the diode’s power dissipations is the major power loss, the rectifier efficiency is expressed as: PDNOM =

η rect

Pout = Pout + 2PD

(28)

where: Pout = Vout I O

(29)

is the output power. Applying (26) we transform (28) into: η rect =

1

V R (ϕ(λ )) 1+ F + F Vout RL

2

(30)

The value of RF in modern diodes is usually small and in most practical cases it is much lower than the load resistance of the rectifier RL. This and the fact that the function ϕ(λ)≤1 for every λ suggest that the term RF(ϕ(λ))2/RL in the last equation can be neglected. Hence,

1 V 1+ F Vout

(31)

As this equation shows, the rectifier efficiency ηrect decreases as the output voltage Vout gets lower. It is especially noticeable when Vout is getting closer to the diodes forward voltage VF. D. Voltage Range and Power Characteristics of a Real Converter Applying the power balance condition we can replace the loaded output rectifier by an equivalent resistance R eq connected to the output of the PT (Fig. 4). The power balance condition including the rectifier losses can be expressed as: V C2

O (1) p

=

2R eq

2 2 Vout Vout + VF I O = RL R L η rect

(32)

from which: R eq

VC (1) p = O Vout

2

R L η rect 2

(33)

It was found that the voltage ratio of the last equation could be calculated by an expression, which is similar to (4), but takes into account the rectifier’s efficiency: * cos λ 4 2 = η rect λ* 2 1 − π

VC O (1) p Vout

(34)

The angle λ* is defined by an expression similar to (16): λ* = 4

π 5 ωC o R L η rect

(35)

In non-overlapping mode, expressions (33) and (34) converges to: NOM R eq =

π2 RL 2η rect

(36)

Applying (31)-(34) we obtain the output to input voltage ratio of a converter with losses in the rectifier and the PT: λ* 2 0.25η rect 1 − π ko = n 2 R m λ* 1 + cos R eq 2

(37)

The dependences of ko ideal (6) and ko (37) on the normalized load factor ( ωC o R L ) are depicted in Fig. 5 a, b. Fig. 5 shows that a decrease in the load factor ωC o R L causes a decrease of the voltage ratio ko in the overlapping mode even in ideal case of a lossless PT ( R m = 0 ), but this

decrease is much more pronounced in the real case ( R m > 0 , PD > 0 ). Fig. 6 depicts the normalized output power Lr 2 2 C V r PO* = out as a function of the normalized load R L V in2 .p

P*

150

100 50

PT Lr

ir

0

Cr Rm vCO

Cin

n

iCO iL

ir n Co Lo Req O

+ -

Fig. 4. A simplified equivalent circuit of the converter.

1 π

0.3 0.25 Simulation

0.15

Calculation

0.1 0.05

OM

10-2

NOM

10-1 ωC O R L

100

1 π

(a)

Simulation Calculation

0.25

10-1

100

Efficiency of the PT η PT and efficiency of the whole converter η can be also easily calculated if we know Req:

η=

R eq n R m + R eq 2

Pout = η PT η rect Pin

(38) (39)

Philips PT (RT 35x8x2 PXE43-S) with thickness polarization mode was used in the experiment. Its main parameters (Fig. 1) were measured to be Lr=165mH, Cr=15.1pF, Cin=Co=510pF, Rm=105 Ω , n=1. Operation frequency was equal to the resonant frequency. The parallel resonant inductance was L o = 4.88mH and the diodes were a Schottky diode (MBR160P). The experiments were carried out for a load resistance RL range of 30Ω to 1kΩ. The range of the peak of the input voltage was Vin .p = 5 − 30V . The experimental and the theoretical

V. DISCUSSION AND CONCLUSIONS

0.2 0.15 0.1 NOM

OM

0.05 10-4

10-2

ωC O RL

voltage ratios as a function of the load resistance are shown in Fig. 7. It is evident that the analytical equations and characteristics, derived in this study, are in excellent agreement with simulation and experimental results

ko

0

10-3

IV. EXPERIMENTAL RESULTS

0.2

0.3

10-4

Fig. 6. The normalized output power P* as a function of the normalized load factor ωCORL. Real PT and real diodes (the same parameters as in Fig. 5 b).

η PT =

koideal

0

Model

200

factor ωC o R L .

Vin

Simulation

10-3

10-2 ωC O R L

10-1 1

π

100

(b) Fig. 5. The voltage ratio as a function of the normalized load factor ωCORL: (a) Lossless PT, ideal diodes, (b) Real PT (ωCoRm=0.034), real diodes (MBR160P). The peak input voltage Vin.p=30V. OM – overlapping mode, NOM – non-overlapping mode.

It was shown that the PT power converter with a halfway two diodes rectifier can operate in two modes: NOM - the operational mode described earlier in which there is no overlap between the conduction intervals of the diodes, and OM – the newly uncovered mode with overlap between the conduction intervals of the diodes. It was found that the operation mode is depending on the normalized load factor ωC o R L . The load factor is a function of the PT parameters and the load resistance. The study highlights the effect of the load and parasitic resistances on the output voltage and the losses for both modes. The analytical investigations were supported by simulation and experiment results, and found to be in excellent agreement.

Vout Vin.p

[5] C.Y. Lin and F.C. Lee, “Design of a Piezoelectric Transformer Converter and Its Matching Networks,” IEEE PESC Record, pp. 607-612, 1994. [6] T. Zaitsu, T. Shigehisa, M. Shoyama, and T. Ninomiya, “Piezoelectric Transformer Converter with PWM Control,” IEEE Intelec Proc., pp. 279-283, 1996. [7] G. Ivensky, M. Shvartsas, and S. Ben-Yaakov, “Analysis and modeling of a piezoelectric transformer in high output voltage applications,” IEEE APEC record, pp. 1081-1087, 2000.

Experiment Calculation

0.3

0.2

0.1

0

1

10

100

RL [Ω]

1000

Fig. .7 Experimental and theoretical voltage ratio Vout/Vin.p as a function of the load resistance RL. Vin.p=30V, ωCoRm=0.034.

It is important to note that a design that is based on the known NOM may be in a considerable error if the rectifier operates in OM. The analytical equations derived in this study can be used in engineering design. An example of a design procedure is given in the Appendix. In this design example we assume that the PT is given and calculate the required input voltage to meet the out specifications (Pout, Vout). If the calculated input voltage is not equal to the available input source, a trial and error procedure need to be following until the correct PT is found. Alternatively, the PT parameters (e.g. n) can be adjusted to meet the requirement and by this define the set of parameters of the PT that will meet the given specifications. [1] [2] [3] [4]

REFERENCES T. Zaitsu, T. Inoue, O. Ohnishi, and A. Iwamoto, “2 MHz Power Converter with Piezoelectric Ceramic Transformer,” IEEE Intelec Proc., pp.430-437, 1992. M. Kazimerczuk and D. Czarkowski, Resonant power converters, John Wiley & sons, Inc., 1995. C. Y. Lin and F. C. Lee, ”Development of a Piezoelectric Transformer Converter,” VPEC Seminar Proc., pp. 79-85, 1993. T. Zaitsu, O. Ohnishi, T. Inoue, M. Shoyama, T. Ninomiya, F.C. Lee, and G.C.Hua, “Piezoelectric Transformer Operating in Thickness Extensional Vibration and Its Application to Switching Converter,” IEEE PESC Record, pp. 585-589, 1994.

APPENDIX Design Guidelines Given: the AC/DC PT converter output requirements ( Vout , Pout ), the PT parameters ( L r , C r , C in , C o , R m , n ), and the diodes to be used in converter. Variables to be evaluated: the input voltage Vin that ensures the required output attributes of the power converter and the input power Pin . Design steps: 1) Define the resonant frequency ω r , the load resistance R L , the resonant inductance L O =

LrCr and the load Co

factor ωC o R L from the output specifications and the PT parameters. 2) Find the forward diode voltage VF from data sheet and evaluate the rectifier efficiency ηrect (31). 3) Calculate the parameter λ* (35). The rectifier is operating in non-overlapping mode when λ* = π and in overlapping mode when λ* < π . 4) Define the equivalent load resistance of the converter R eq (33) and (34). 5) Obtain the voltage ratio ko of the real converter (37). V 6) Calculate the peak input voltage as Vin ( p ) = out . ko 7) Find the PT and the whole converter efficiencies η PT and η (38), (39). P 8) Determine the input power Pin = out . η