th Annual IEEE Power Electronics Specialists Conference Aachen, Germany, Parallel Connection of Piezoelectric Transformers

Transcription

1 004 35th Annual IEEE ower Electronics Specialists Conference Aachen, Gerany, 004 arallel Connection of iezoelectric Transforers Svetlana Bronstein, Gregory Ivensky and Sa Ben-Yaakov* ower Electronics Laboratory Departent of Electrical and Coputer Engineering Ben-Gurion University of the Negev. O. Box 653,Beer-Sheva 8405, ISRAEL, hone: ,Fax: Eail: Website: Abstract - The behavior of parallel connected iezoelectric Transforers (Ts was analyzed by applying the generic T odel to represent the cobined network in the ost general way. The influence of a deviation in each of the T paraeters was exained with respect to total output and losses in each T. It was found that the requireent for identity in the equivalent circuit paraeters R and C o in the two Ts is not strong. The ost critical paraeter for equal loss in the Ts is ω r the echanical resonant frequency of Ts, which has to be equal in the parallel units, even though the values of the paraeters C r and L r can be slightly different. Since the resonant LCC type inverters are practically identical to the equivalent electrical circuit of Ts, the results of this study are relevant to cobined connected LCC inverters. The analytical results of this study were verified by SICE siulation and experients. The experients were done on parallel-connected Ts (T-, Face Co., VA, USA and LCC inverters built by passive eleents. I. INTRODUCTION As iezoelectric Transforer (T technology is developing, Ts ay becoe a viable alternative to agnetic transforers in various applications. resently, however, the power that can be handled through coercial Ts is still liited. ossible solution to this liitation is joint connection of a nuber of Ts [-]. However, since two different Ts are never absolutely identical, the ain proble of joint connection of Ts is ensuring current sharing when the Ts are feeding the sae load. The objective of the study was to delineate the requireents for joint connection of Ts, and in particular, to identify and quantify the effect on current sharing and overall efficiency due to non-identities in the paraeters of the Ts. This is necessary to avoid the overheating of one or ore of the joint connected Ts. The ethodology, applied in this study, follows the concept of Generic Model to represent the behavior of the cobined Ts in the ost general way [3]. The influence of each paraeter of T on the operation of the parallelparallel connected Ts is exained and tested by siulation and experient. II. ARAMETERS OF A SINGLE T The equivalent circuit of the T, reflected to the priary side, is given in Fig., where C in is the input capacitance of L T, Co R are the reflected values of the T output capacitance and the load resistance, Cr Lr R are the equivalent electrical paraeters that represent the echanical resonant circuit of the T. This study applies the principles of the Generic Model. All paraeters of the odel are noralized. It is assued that the T inverter operates at the frequency of the axiu output voltage [3-4]. The noralized paraeters of the base T are defined as follows: Co a Cr ωr LrCr Q ωrcor L Q ωrcrr where ( ω r is the echanical resonant frequency of the T, Q is the noralized load factor, and Q is the echanical quality factor. Following [3-4], the frequency corresponding to the axiu output voltage ω, the efficiency η, the voltage transfer ratio k, and the output power o are obtained as: ( ω ωr + ( a + Q η (3 a + + Q Q Q k (4 a ωr a ω a + ax + x + QQ ω Q ωr Q Vo V in o k RL RL where x ( /04/$ IEEE. 779

2 004 35th Annual IEEE ower Electronics Specialists Conference Aachen, Gerany, 004 III. OSSIBLE CONNECTION OF TS FOR COMBINED OERATION (IDENTICAL TS Generally, the T can be considered as a two port network. There are four possible ways of interconnection Ts for cobined operation [5]: parallel input parallel output, series input series output, series input parallel output, and parallel input series output (Fig., a-d. It should be noted that last two types of connections are realizable only in the case of ground isolated Ts. Otherwise input or output circuit will be shorted [5]. We consider these ways of connection of N Ts under the assuption that all Ts are identical. In order to achieve the ultiple power in the output, the following conditions for the load resistance T, input and output voltages of the total (cobined circuit were derived for each type of connection: a arallel-parallel connection (Fig., a. In this case V in V inn and V o V on. The voltage gain k T k. The load resistance and the output power of the cobined circuit: R LT N (6 ( N V o ot No (7 b Series-series connection (Fig., b. The input and output voltages of the total circuit: and, and the load resistance T are distributed equally between the Ts. Therefore: VinT NVin VoT NVo (8 T N The voltage gain of the total circuit k T is equal to k : VoT NV k o T k (9 VinT N The output power of the cobined circuit: ( NV o ot No (0 N c Series-parallel connection (Fig., c. In a siilar way as above, the input voltage for N Ts has to be N ties higher than of single unit, the output voltage is the sae as for a single T, and the load resistance will be R T L. N d arallel-series connection (Fig., d. The input conditions in this case are the sae as for parallel-parallel connection and the output are the sae as in series-series. Table suarizes results indicated above. L r C r R V in C in C o R L Fig.. The equivalent circuit of a T after reflecting the secondary to the priary side. Vo TABLE : LOAD AND VOLTAGE RATIO CONDITIONS FOR DIFFERENT CON- NECTIONS OF COMBINED T INVERTERS. T Topology Figure RL Vin Vo arallel-arallel a /N Series-Series b N N N Series-arallel c /N N arallel-series d N N V in V in V in V in V in V in V in V in T T (a T T (b T T (c T T (d Fig.. ossible ways of Ts interconnection (illustrated for two units: a parallel-parallel, b series-series, c series-parallel, d parallelseries IV. ARALLEL-ARALLEL CONNECTION OF NON-IDENTICAL TS In the detailed analysis of this study we consider the cobined operation of two Ts. The equivalent circuit of two parallel-parallel connected Ts, after reflecting the secondary to the priary, is given in Fig. 3a. If the cobined network operates at the frequency closed to its echanical resonant frequency this circuit can be siplified to the one shown in Fig. 3b replacing the two networks by a single one. In the following analysis we replace the two parallel branches and (that is L r -C r -R networks by an equivalent ipedance T and derive the paraeters of the new T fro the expression of T. This is used to quantify the effect of non-equality in the T paraeters on the overall operation of the parallel-connected Ts. V o V o V o V o V o V o V o V o 780

3 004 35th Annual IEEE ower Electronics Specialists Conference Aachen, Gerany, 004 V in L r C r R L r C in +C in C r R (a L rt C rt R T C o +C o C int C ot R LT R L VoT (b Fig. 3. The equivalent circuits of parallel-parallel connected Ts: (a coon case, (b operation close to the resonant frequency. We copare the output power ot and efficiency η T of the cobined T with the output power o and efficiency η of a single T. A. The Influence of the Non-Equality of C o The deviation in C o is presented as follows: C o Co Co Co + Co Co( + δc ( δ C C + C C + c ot o o o ( where C o and Co are the output capacitances of the parallel Ts reflected to the priary side, C o and δ c C o /C o are the absolute and relative deviation of the paraeter C o fro the noinal value, and C ot is the reflected capacitance of total T. We analyze the influence of δ c in the case when the ipedances of the RLC networks and are the sae: R + jωlr + R + jωlr + jωcr jωcr (3 η T η V o Q Co Co (% Fig. 4. The relation of efficiency of the cobined circuit to the noinal efficiency as a function of the relative deviation of the output capacitance C o for Q range and Q 900. If the capacitance deviation is not too great, one can assue that the reflected load resistance of the circuit under consideration is the sae as in ideal case (Table : T 0.5R L. The ipedance and the paraeters of the total network are: T 0.5 (4 RT 0.5R CrT Cr (5 LrT 0.5Lr The noralized paraeters ( for this case are: C δ a ot + c T a CrT ωrt ωr (6 δ Q ω + c T rtcotr LT Q QT Q By substituting these noralized paraeters into equations (-(5 we calculate the optial operating frequency for axiu power transfer ω T, the efficiency η Τ, the voltage transfer ratio k T, and the output power ot. Fig. 4 represents the relation of the efficiencies (η T /η of the cobined and single Ts as a function of the noralized non-equality C o /C o for different values of the noralized load paraeter Q while the echanical quality factor is assued to be Q 900. Fro these plots one can see that the efficiency is decreasing with an increase in C o /C o This can be explained by the following: an increase of the output capacitance C o causes a larger current in the L r -C r -R network, which in turn, causes a rise of the losses in R and hence decreasing the efficiency. This effect is siilar to case of a single T with larger C o. The larger the output capacitance the lower will be the T s efficiency. This study shows, however, that substantial deviation in the values of the output capacitances of the Ts (up to 30% results in inor change in the overall efficiency. B. The Influence of Non-Equality of R The deviation in R is presented as follows: R R (7 R R + R R( + δ where R and δ R /R are the absolute and the relative deviation of the losses resistance fro its noinal value. Assuing the rest of paraeters to be identical: C o Co Lr Lr (8 C r Cr we find the total ipedance of the network T : R δ ( T + + jωlr + + ϕ R,Cr, ω, δ (9 jωcr The function ϕ (R, C r, ω,δ in (9 has a relatively sall value and can be neglected. Therefore the equivalent series coponents of the total T are: 78

4 004 35th Annual IEEE ower Electronics Specialists Conference Aachen, Gerany, 004 ( + 0.5δ RT 0.5R LrT 0.5Lr (0 CrT Cr The total output capacitance C ot C o. The load resistance is the sae as in 4.. Using these paraeters, we recalculate the noralized paraeters of the total circuit: at a ωrt ωr LrTCrT ( QT ωrtcrtrlt Q Q Q T ωrtcrtrt + 0.5δ Substituting these noralized paraeters into (-(5, we obtain the expressions for the operating frequency, efficiency, voltage transfer ratio, and output power of the equivalent inverter. The plots of the output power and efficiency ratios of the cobined T to the single T ot / o and η T /η (Fig. 5, a, b as a function of the non-equality R /R for different values of load resistance (paraeter Q show that fairly large changes in the paraeter R (up to 0% cause rather sall changes in the output characteristics. Since the ipedances of the parallel networks are different, we copare the power losses in the networks and. The loss relation Loss / Loss can be calculated using the Ts paraeters: Vz Loss I R R ( V ( z Loss IR + δ R( + δ where I and I are the rs currents, V z and V z are the rs voltages across the series parts ( and of Ts. Since V z Vz, the losses relation λ is: where: ω A Q r (6 ω r ω It should be noted that a change in R has practically no ipact on the axiu output voltage frequency [3-4]. Hence, A is independent of δ. The sign of the derivative λ, which is defined by the sign of the expression δ ( ω ωr + δ Q, deterines the slope of the ω losses ratio curves. η T η ot o..9 56Ω 00Ω R R (% 56Ω (a 00Ω [%] R R Loss λ Loss + δ (3 where: ( ωr Q + δ + ω r ω (4 ω Q r + ω r ω Loss Loss. 0.9 (b 56Ω 80Ω 00Ω When R increases the current through RLC network decreases. Hence, the effect on the product I R (that is, the losses could be positive or negative. In order to exaine the behavior of the loss sharing, we take the derivative of (3 with respect to δ : ( ( { λ + A + δ A} δ ( + δ ( + A ( R (% R (c Fig. 5. The efficiency, (a, output power, (b, and losses ratio, (c as a function of the relative deviation of the loss resistance R for different values of the load resistances. Solid line theoretical prediction, dots SICE siulation results. Siulation was carried out assuing that Ts noinal paraeters are identical to those of T-, Face Co., VA, USA. 78

5 004 35th Annual IEEE ower Electronics Specialists Conference Aachen, Gerany, 004 If Q is very high, then the derivative is negative and the plots of losses ratio will go down. If Q is very low, then the derivative is positive and the plots of losses ratio will go up. Otherwise, the slope of the losses ratio dependent on the load factor (the load resistance and can even change the polarity for a given δ. lots of the losses ratio as a function of the ratio R /R for different values of the load resistance (paraeters Q varied fro to 0. and Q 370 (Fig. 5, c show that the losses relation in the parallel parts varies less than 5-0% due to ore than 5% of change in the ratio R /R. Since the echanical quality factor of the T is interediate (370 one can see that the slopes of the curves change polarity fro one load resistance to another. The boundaries of losses ratio deviation can be obtained by analyzing (3: R ( ( + δ + R + δ + X X + δ R + X R + X X ωlr. ωcr R <<, then Loss. X Loss + δ R >>, then Loss + δ. X Loss λ (7 where If If Therefore: < Loss < + δ + δ Loss ( + δ (8 C. The Influence of Non-Equality of ω r The ain proble of this case is that the currents in the parallel branches of the Ts are strongly unequal, causing a large difference in the losses. The losses ratio λ ω : Loss Loss Q ωr (% ωr Fig. 6. The losses ratio in the parallel connected Ts as a function of the relative change of the echanical resonant frequencies of Ts for different values of Q while Q 900. ( + δω ωr + Q ω ( + δω λω (9 ω Q r + ω r ω where δ ω is the noralized echanical resonant frequency deviation (δ ω ω r /ω r. Fig. 6 shows the losses ratio as a function of the frequency deviation for different values of the paraeter Q while Q 900. One can see that even inor deviation in the resonant frequency of the Ts (less than -3% causes great difference in the losses (up to ten ties and ore. That renders the cobined operation ipractical. D. The Influence of Non-Equality of C r and L r while ω r is Kept Constant In order to exaine how non-equality of the paraeters C r and L r in parallel parts (when ω r are the sae ipacts the output characteristics of the cobined operated Ts we assue the following conditions: the series capacitance and inductance of the second T are different fro the paraeters of the first (noinal T while the series resonant frequency of the parallel parts reains constant (δ Cr and δ Lr are relative variations of C r and L r : Cr Cr( δcr (30 Lr Lr( + δlr while: ω r ω r, which eans: C rl r CrLr (3 or: δ δ Cr Lr (3 δcr The total series ipedance is: ( + δ + δ Lr Lr R T + ( + δ jωcr ( 0.5δCr Lr (33 + δ + jωl Lr r + 0.5δLr which iplies that: RT 0.5R LrT 0.5Lr ( + 0.5δLr (34 CrT Cr ( 0.5δCr The losses ratio λ r : ω ωr ωr ω + Q δ Cr λ r (35 ω ωr + Q ωr ω 783

6 004 35th Annual IEEE ower Electronics Specialists Conference Aachen, Gerany, 004 Analyzing (33 we obtain the boundary of the losses ratio deviation: If If Q Q ω ωr ωr ω <<, then ω ωr ωr ω Loss Loss ( δcr (36 >>, then Loss (37 Loss Losses relation as a function of relative non-equality of C r and L r for two load resistances is shown in Fig. 7. The plots reveal that the losses ratios is lower than.5 ties for a nonequality of the paraeters C r and L r of up to 5%. V. SIMULATION AND EXERIMENT The equivalent circuit paraeters of the experiental Ts, (T- Face Co., VA, USA were easured [6] to be as follows: R Ω, C r 7.5pF, L r 0.5H, C o.33nf, C in.78nf, n.08. (Q 370, a8.33. The influence of deviation of different T paraeters on the output power, efficiency and losses sharing in the cobined circuit were tested by tie doain SICE (siulation and found to be in a good agreeent with the theoretical predictions (Figs. 5, 7. Experients were carried out on two radial vibration ode Ts. The echanical resonant frequencies of Ts were found to be 8.43kHz and 8.38kHz respectively (less than 0.06% difference. The axiu efficiency achieved in both individual and cobined operation of the Ts was 96.5%. arallel operation provided double output power as copared to the single T. For the experiental verification of the theoretical study, the iitation of two Ts was built in the for of two LCC series-parallel resonant networks with no output transforer. The experiental set up coprised the following data: Loss Loss Ω 00Ω Cr Cr ( ω const r (% Fig. 7. The losses ratio in the parallel-connected T as a function of the relative change of the paraeters C r (copleentary to L r while echanical resonant frequencies ω r 8.4kHz are identical.. Solid line theory, dots siulation results. Siulation was carried out assuing that Ts noinal paraeters are identical to those of T-, Face Co., VA, USA The series inductances and capacitances were: C r.0nf, L r.087h, C r 0.945nF, L r.5h. Measured series resonant frequencies were: f r 08.6kHz and f r 09.3kHz. Thus, the paraeters deviations were δ Cr º 6.5%, δ Lr º7.8%, and echanical frequency deviation δ ω was about 0.65%. The parallel capacitances were equal: C o C o 0nF (capacitances ratio a The easured ac resistances of the inductances and the capacitances in the experiental setup were: R 3.67Ω, R 7Ω which eans the paraeter deviation δ 4.4%. 4 Mechanical quality factor was Q º05. 5 Experiental resistances were 56Ω and 00Ω (load factor Q was equal to 0.77 and.75, the capacitances ratio, a, was about 0 (siilar to the one of the experiental T. The experient involved low input power (up to 5W. The two LCC networks were connected in parallel-parallel schee. The results of the easureents were as follows:. 56Ω. Currents in the RLC networks were: I r 85A, I r 53A, that caused the losses Loss 0.468W and Loss 0.4W, and hence the losses ratio λ was.7. 00Ω. The easured currents were I r 5A, I r 3A, losses: Loss 0.37W, Loss 0.96W, and losses ratio λ.07. It can be shown that for sall deviations, λ, the total losses ratio λ T can be approxiated by: λt ( λ ( λω ( λcr (38 Using expressions (3, (9 and (35 the theoretical predictions were found to be as following: For 56Ω: λ Τ (.37 ( ( For 00Ω: λ T (. ( 0.85 ( As can be seen, the experiental results are in good agreeent with theoretical prediction. These results also deonstrate the high sensitivity to δ ω, especially when the load factor Q is sall (Fig. 6. VI. DISCUSSION AND CONCLUSIONS This study carried out detailed analysis of the influence of each T paraeter deviation on the effectiveness of parallelparallel operation of Ts. It was found that the requireents for equality of R and C o in two parallel-connected Ts are not strong. The ost critical paraeter that can har the parallel operation is ω r the echanical resonant frequencies of Ts. For good current sharing the echanical resonance of the parallel-connected Ts has to be very close while C r and L r could be slightly different. The ain conclusion of this study is that effective parallel connection of Ts can be achieved if the echanical resonant frequencies of the parallel units are atched. This could be explained by the fact that current sharing in parallel 784

7 004 35th Annual IEEE ower Electronics Specialists Conference Aachen, Gerany, 004 parts is dependent on the ipedances of the parallel branches. The ipedance of the L r -C r -R network,, is a function of the characteristic ipedance r of the network and its series resonant frequency: ω R + r (39 L where ωω ω r and r r Cr r in practical Ts is very high. For the experiental units r is about 7.5kΩ while in others it ay reach 00kΩ [3] and above. Consequently, a sall deviation in ω r, that will cause a corresponding shift in ω, will result in a large difference in and hence in the losses ratio. As discussed in [7] the equivalent paraeters of T are dependent on the transforer diensions and ceraic constants. When two T are produced fro the sae ceraic under the sae technological process, their diensions are equal with high accuracy, the sound velocity in the Ts are practically the sae (that is, the sae echanical resonant frequencies, and their asses and all echanical properties are the sae, thus their equivalent paraeters L r C r are probably very close (which suggest equal characteristic ipedances. That eans that the ain requireent for proper cobined operation could be fulfilled in practical piezoelectric transforers. Since discrete resonant LCC type inverters follow the sae network relationships as the T equivalent circuit, the results of this study are relevant to the case of parallel connected LCC inverters. ACKNOWLEDGMENT This research was supported by THE ISRAEL SCIENCE FOUNDATION (grant No. 3/0 and by the aul Ivanier Center for Robotics and roduction anageent. REFERENCES [] H. Kakehashi, T. Hidaka, T. Ninoiya, M. Shoyaa, H. Ogasawara, and Y. Ohta, Electric ballast using piezoelectric transforers for fluorescent laps, IEEE ESC Record, pp. 9-35, 998. [] C. Y. Lin and F. C. Lee, iezoelectric transforer and its applications, roc. of VEC Seinar, pp. 9-36, Sep [3] G. Ivensky, I. afrany, and S. Ben-Yaakov, Generic operational characteristics of piezoelectric transforers, IEEE Trans. on ower Electronics, Nov. 00, vol. 7, no 6, pp [4] S. Bronstein, and S. Ben-Yaakov, Design considerations for achieving VS in a half bridge inverter that drives a piezoelectric transforer with no series inductor, IEEE ESC 00 Record, pp [5] L. Weinberg, Network analysis and synthesis, McGraw-Hill Book Copany, Inc. 96. [6] G. Ivensky, S. Bronstein, and S. Ben-Yaakov, A coparison of piezoelectric transforer AC/DC converters with current doubler and voltage doubler rectifiers, to be published in IEEE Trans. on ower Electronics. [7] R. Lin, iezoelectric transforer characterization and application of electronic ballast, h. D. Dissertation, Virginia olytechnic Institute,