THE CLASS DE inverter [1] [8] has become an increasingly

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1 1250 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 51, NO. 7, JULY 2004 FM/PWM Control Scheme in Class DE Inverter Hiroo Sekiya, Member, IEEE, Hirotaka Koizumi, Member, IEEE, Shinsaku Mori, Member, IEEE, Iwao Sasase, Senior Member, IEEE, Jianming Lu, Member, IEEE, and Takashi Yahagi, Member, IEEE Abstract This paper presents a new control scheme a Class DE inverter, that is, frequency modulation/pulsewidth modulation (FM/PWM) control. Further, the FM/PWM controlled Class DE inverter is analyzed and we clarify permance characteristics. Since the FM/PWM controlled inverter has two control parameters, namely, the switching frequency and the switch-on duty ratio, it has one more degree of freedom the control than the inverter with the conventional control scheme. The increased degree of freedom is used to minimize the switching losses. Theree, it is possible to control the output power with high power-conversion efficiency wide-range control. Carrying out the circuit experiments, we confirm that the experimental results agree well with the theoretical predictions quantitatively. For example, the proposed controlled inverter can control the output voltage from 56% to 191% of the optimum one, which is designed 1.8 W at 1.0 MHz, with maintaining over 90% power-conversion efficiency. Index Terms Class DE inverter, Class E switching conditions, frequency modulation (FM) control, power-conversion efficiency, pulsewidth modulation (PWM) control, switching losses. I. INTRODUCTION THE CLASS DE inverter [1] [8] has become an increasingly valuable high-density power source. In the Class DE inverter, the switch current is small and both the switches satisfy Class E switching conditions [9], [10]. That is achieved by the shunt capacitor added to each switch and the dead-time which is given between the switch-on-times on the driving pattern. Class E switching conditions mean that both the switching voltage and the slope of switching voltage ( ) are zero when each switch turns on. Because of zero slope voltage switching, the inverter satisfied with Class E switching conditions has low sensitivity to component tolerances, which is one of features of Class E switching conditions. Theree, the power-conversion efficiency of the Class DE inverter is high under high-frequency (megahertz order) operation. Manuscript received May 21, 2001; revised November 29, This work was supported in part by the Support Center Advanced Telecommunications Techonology Research (SCAT), The Telecommunications Advancement Foundation (TAF), and The Yazaki Memorial Foundation Science and Technology. This paper was presented in part at the IEEE Symposium on Circuits and Systems, This paper was recommended by Associate Editor D. Czarkowski. H. Sekiya, J. Lu, and T. Yahagi are with the Graduate School of Science and Technology, Chiba University, Chiba, , Japan ( sekiya/jmlu/ yahagi@faculty.chiba-u.jp). H. Koizumi is with the Department of Electrical and Electronic Engineering, Tokyo University of Agriculture and Technology, Koganei , Japan ( koizumih@cc.tuat.ac.jp). S. Mori, retired, was with the Department of Electrical and Electronics Engineering, Nippon Institute of Technology, Miyashiro, Saitama, Japan. I. Sasase is with the Department of Inmation and Computer Science, Keio University, Yokohama , Japan ( sasase@ics.keio.ac.jp). Digital Object Identifier /TCSI One of the disadvantages of the Class DE inverter is that the output control with keeping high power-conversion efficiency is difficult because the switching of the inverter never satisfies the Class E switching conditions when the load resistance varies from initial designed values. Several control schemes of the Class DE inverter have been proposed, e.g., frequency modulation (FM) control [1], [8], pulsewidth modulation (PWM) control [5], [6], and so on [7], [8]. These control schemes make use of the above feature of the Class E switching conditions. Namely, a change of the control parameter is regarded as a criterion of component tolerances when the change of the control parameter is small. Theree, in case of the narrow control range, the output power can be controlled with high power-conversion efficiency. Under wide control range, however, the high power-conversion efficiency cannot be kept because of the increase of switching losses. The increase of switching losses is a common problem of the control schemes the Class DE inverter. The minimization of the switching losses is a main problem in order to improve the power-conversion efficiency wide-range control. The above control schemes have only one control parameter. Since the parameter is used in order to control only output power, the current and voltage wavems of the switches cannot be controlled. Theree, there is no room to improve the powerconversion efficiency. If the inverter has two control parameters, both the output power and the switching losses may be controlled. That is because it has one more degree of freedom control than that of the above control. As a result, we expect that the inverter achieves high power-conversion efficiency the wide-range control. This paper presents a new control scheme on the Class DE inverter keeping high power-conversion efficiency in wide range. The proposed control is FM/PWM control. Further, the FM/PWM controlled Class DE inverter is analyzed and we clarify permance characteristics. Since the FM/PWM controlled inverter has two control parameters, namely, the switching frequency and the switch-on duty ratio, it has one more degree of freedom the control than the inverter with FM or PWM control scheme. The increased degree of freedom is used to minimize the switching losses. From the analytical results, we clarify that the power-conversion efficiency keeps high wide range when the output power is controlled so as to achieve zero slope of the switchin voltage by varying the switching frequency and on time duty ratio. For example, the proposed controlled inverter can control the output voltage from 56% to 191% of the optimum one, which is designed 1.8 W at 1.0 MHz, while maintaining over 90% power-conversion efficiency. Carrying out the circuit experiments, we confirm that the experimental results agree well with the theoretical predictions quantitatively /04$ IEEE

2 SEKIYA et al.: FM/PWM CONTROL SCHEME IN CLASS DE INVERTER 1251 TABLE I CIRCUIT PARAMETERS FOR OPTIMUM OPERATION Fig. 1. Circuit topology of the Class DE inverter. TABLE II MEASURED ELEMENT VALUES OF ESRs Fig. 2. Optimum wavems of the Class DE inverter D = 0:25. The parameters are the same as the ones in Table I. II. CIRCUIT DESCRIPTION A. Class DE Inverter Fig. 1 depicts the circuit topology of Class DE inverter [1]. It is composed of two switches and, two capacitors and shunting each switch, and a series-resonant circuit. The wavems of the Class DE inverter are shown in Fig. 2, when the switch-on duty ratio of each switch is 25%. The switches are driven by a driving pattern of and in Fig. 2, where and drive and, respectively. The parameters are the same as the ones in Table I. Table II gives the measured element values of ESRs. The driving pattern generates a dead time during the period when one switch has turned off bee the other switch has turned on. During the dead time, the sinusoidal output current charges one shunt capacitor and discharges the other shunt capacitor. When the inverter allows Class E switching conditions [9], [10], the midpoint voltage between two switches, namely becomes or zero, and the slope of the voltage, namely becomes zero at the end of the dead time. In this paper, this state is named as optimum state. B. Class DE Inverter with FM or PWM Control The FM control scheme is applied to the Class DE inverter [1], [6]. If the switching frequency is larger than that of the optimum one, the switch or turns on bee the voltage becomes or zero. Theree, the input current from the direct voltage source decreases by increasing the switching frequency. Consequently, the inverter can regulate the output voltage by applying FM control. The PWM control scheme is achieved by changing the pulsewidth of driving signals and, namely, the switch-on duty ratio [5], [6]. By increasing, both and are decreased, the midpoint voltage between and is changed. As a result, the fundamental frequency component of is reduced. Consequently, the inverter can regulate the output voltage by applying PWM control to Class DE inverter. The Class DE inverter with FM or PWM control is analyzed exactly in [6]. One of features of Class E switching conditions is a low sensitivity to component tolerances. Hence, in case of narrow range control, the output power can be controlled with high power-conversion efficiency since the low switching losses are achieved. From this point of view, FM and PWM control are suitable the Class DE inverter. In wide-range control, however, the power-conversion efficiency decreases rapidly because of the increase of switching losses. The suppression of the switching losses is important in order to keep high power-conversion efficiency wide-range control. C. Class DE Inverter with FM/PWM Control We propose the FM/PWM control scheme the Class DE inverter. The FM or PWM control scheme has only one control parameter that is used to control the output power. Theree, there is no room to decrease switching losses and to improve the power-conversion efficiency. The proposed control scheme, however, has two control parameters which are the switching frequency and the switch-on duty ratio. Theree, both the output power and the switching losses can be controlled. The principle operation of FM/PWM controlled Class DE inverter is same as that of FM or PWM controlled one. Namely, by varying the switching frequency and, the midpoint voltage between and is changed. As a result, the fundamental frequency component of is changed and the inverter regulates the output power. Moreover, if the control parameters are set up desired output power and minimum switching voltage, we can control the output power with improvement of the power-conversion efficiency. If zero voltage switching is achieved, there is no switching loss. Theree, the control parameters are determined desired the output power and the zero voltage switching. However, it is often happened that no value of the parameters zero voltage switchings can be found. In this case, the control parameters are determined so as to achieve zero slope of switching

3 1252 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 51, NO. 7, JULY 2004 B. Parameters The following parameters of the circuit are defined below. 1) : The resonant frequency when each switch is on. 2) : The loaded -factor when each switch is on. 3) : The ratio of the resonant frequency and the operating (switching) frequency. 4) : The ratio of the capacitance of the resonant circuit capacitor and the shunt capacitor. 5) : The switch-on duty ratio. Fig. 3. V Theoretical wavems of FM/PWM controlled Class DE inverter = 3:0 V. The parameters are same as ones in Table III. TABLE III EXAMPLE OF CONTROL FOR DESIRED OUTPUT VOLTAGE C. Wavems The analytical procedure is closely related to the work by Albulet et al. [12] [15]. The range of is considered since the output voltage is periodic with. We consider four cases due to the states of the switches and. The classification of the four cases and circuit equations in each case are given in Appendix I. From the circuit equations, we can determine the wavem equations as shown in (1) (8) at the bottom of the next page. For example, means in Case. In the above equations,, and are initial values in Case 1,, and are initial values in Case 2, and are initial values in Case 3, and are initial values in Case 4, and (9) (10) (11) (12) The element values which are not in this table are same as ones in Table I. voltage as shown in Fig. 3, where the parameters are same as ones in Table III. Because the zero slope of the switching voltage means that voltage of the switches is local minimum at the turn-on instant, and the suppression of the switching losses can be achieved. III. ANALYSIS OF PROPOSED CONTROL A. Assumptions The analysis given below is based on the following assumptions. 1) The active devices act as ideal switches. 2) The shunt capacitance includes the switch device capacitance. 3) All circuit elements are ideal. 4) The loaded -factor is high enough to consider the underdumped case. (13) (14) (15) (16) (17) (18) D. Output Voltage and Output Power Using (1), the root mean-square output voltage is (19) where expresses the root mean square output current. Similarly, the output power can be derived by the following equation: (20) where is given analytically in Appendix II.

4 SEKIYA et al.: FM/PWM CONTROL SCHEME IN CLASS DE INVERTER 1253 E. Power Losses and Efficiency The conduction loss in the switch is calculated as (21) where is the on-resistance of. Similarly, the conduction losses in the shunt capacitor, in, and in the resonant circuit are (22) (1) (2) (3) (4) (5) (6) (7) (8)

5 1254 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 51, NO. 7, JULY 2004 (23) (24) (25) where is the on resistance of, and are the equivalent series resistances (ESRs) of and, and is the ESRs of. From (1) (5), (21) (25) can be derived analytically as shown in Appendix III. By using the FM/PWM control scheme, the switchings often do not achieve Class E switching conditions, and turn-on switching losses occur in this case. The turn-on switching losses and in the switches and are (26) (27) where, and they are the voltage of the switches at turn-on instants. From (21) to (27), the total power loss is (28) Neglecting drive power, one can find the amplifier efficiency (the power-conversion efficiency) IV. CIRCUIT DESIGN (29) A. Design Procedure of Class DE Inverter At first, the following parameters are given; the operating frequency, the input voltage, the output resistor, loaded quality factor, and the switch-on duty ratio. Then, the unknown parameters are,,,,,,,,, and. We get 12 equations from the transient conditions and Class E switching conditions, namely (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) Fig. 4. Experimental wavems. (a) Optimum wavems f = 1.00 MHz, D =0:25, and V = 4.7 V. (b) Wavems of FM/PWM control f = 1.03 MHz, D =0:20, and V = 3.0 V. (40) (41) If (38) and (39) are satisfied, (40) and (41) are also satisfied. Thus, ten unknown parameters are founded by using the Newton method from the ten equations, namely (30) (39). B. Design Example At first, the following specifications are given; the operating frequency 1.0 MHz, the input voltage 20.0 V, the output resistor 10.0, the loaded quality factor, and the switching-on duty ratio. Then, the unknown parameters and are given by the design procedure as and. Theree, components of the circuit are found as shown in Table I. IRF530 MOSFETs are used as switches and. Moreover, the measured element values of ESRs are shown in Table II. In theoretical calculations, the element values of ESRs are same as measured ones. Fig. 4(a) depicts experimental wavems 1.00 MHz. In this state, the conversion efficiency is highest, that is 93.6%. We define that the operation in Fig. 4(a) is optimum. The conditions optimum operation are shown in Table I. From Figs. 2 and 4(a), and Table I, we can find that the experimental results of optimum operation agree well with the theoretical ones quantitatively. The inverter is controlled from the optimum state by using the proposed scheme. V. PERFORMANCE OF FM/PWM CONTROLLED CLASS DE INVERTER A. Analysis of FM/PWM Control The parameters corresponding to the switching frequency and the switch-on duty ratio are and, respectively. When the FM/PWM control scheme is applied to the Class DE inverter, the parameters and are varied from the optimum states. In

6 SEKIYA et al.: FM/PWM CONTROL SCHEME IN CLASS DE INVERTER 1255 Fig. 5. Values of control parameters, power-conversion efficiency, and switching voltage as a function of desired output voltage. The element values are same as ones in Table I. (a) Value of control parameters. (b) Powerconversion efficiency and switching voltage. Fig. 6. Values of control parameters, power-conversion efficiency, and switching voltage as a function of input voltage V = V. The element values are same as ones in Table I. (a) Value of control parameters. (b) Powerconversion efficiency and switching voltage. case of the desired output voltage equation is given:, the following (42) Moreover, the zero slope of the switching voltage has a feature of low sensitivity to component tolerances. If there is no solution (42) and (43), we use (44) instead of (43). Since FM/PWM control has two control parameters, we can give one more equation to suppress the switching losses. If zero voltage switching can be achieved, there is no switching loss. Hence, we give the condition zero voltage switching as (43) From (42) and (43), we find control parameters and theoretical wavems. However, it has often happened that no solution of these equations can be found. In this case, zero voltage switching is never achieved any control parameters. In this paper, we propose the following condition: (44) Since this equation means the voltage of the switches is local minimum at turn on instant, the switching losses are suppressed. B. Output Characteristics Fig. 5 shows values of control parameters, the power-conversion efficiency, and switching voltage as a function of desired output voltage. We notice a discontinuous variation of the slope of the switch-on duty ratio in Fig. 5(a). This means that (43) is used from 4.37 V to 5.05 V, and (44) is used the outer region. From Fig. 5(a), we find that the changes of two parameters of FM/PWM control are narrower than those of FM or PWM control. As shown in Fig. 5(b), the power-conversion efficiency of FM/PWM control improves compared with FM or PWM control schemes. For example, the power-conversion efficiency of FM/PWM control is 91% though that of FM control is 82% 2.5 V. The proposed control scheme can control in the range from 56% to 191% of the optimum output voltage with maintaining over 90% of the power-conversion efficiency.

7 1256 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 51, NO. 7, JULY 2004 TABLE IV EXAMPLE OF CONTROL FOR CHANGE OF INPUT VOLTAGE The element values which are not in this table are same as ones in Table I. Table III shows the comparison between the circuit experiment and theoretical calculation and Fig. 4(b) shows the experimental wavem of FM/PWM control. From Figs. 3 and 4(b) and Table III, we confirm that the experimental results agree well with the theoretical predictions quantitatively. C. Control Change of Input Voltage and Load Resistance Fig. 6 shows values of the control parameters, the power-conversion efficiency, and the switching voltage as functions of input voltage. From Fig. 6(b), we find the power-conversion efficiency of FM/PWM control improves as compared with that of FM or PWM control, that is, because the switching losses are suppressed. The circuit with FM/PWM control maintains the output voltage varying the input voltage from 52% to 179% of the optimum output voltage with maintaining over 90% of the power-conversion efficiency. Table IV shows the comparison between the circuit experiments and theoretical calculations. From Table IV, we confirm that the experimental results agree with the theoretical predictions quantitatively. Fig. 7 shows values of the control parameters, the power-conversion efficiency, and the switching voltage as functions of load resistance. From Fig. 7(b), zero voltage switching is always achieved in case of. The circuit with FM/PWM control maintains the output voltage varying the load resistance from 50% to 150% of the optimum output voltage with maintaining over 90% of the power-conversion efficiency. Table V shows the comparison between the circuit experiments and theoretical calculations. From Table V, we confirm that the experimental results agree with the theoretical predictions quantitatively. Fig. 7. Values of control parameters, power-conversion efficiency, and switching voltage as a function of load resistance V = V. The element values are same as ones in Table I. (a) Value of control parameters. (b) Powerconversion efficiency and switching voltage. TABLE V EXAMPLE OF CONTROL FOR CHANGE OF LOAD RESISTANCE D. Discussion When the control range is narrow around the optimum point, there is no difference of the switching frequency between FM/PWM and FM control as shown in Figs This means that the switch-on duty ratio affects only suppression of the switching losses in this range. Theree, the circuit should control the output power by the switching frequency and suppress the switching voltage by the switch-on duty ratio if the narrow range of the control, e.g., the control of the output from 75% to 117% of in case of Fig. 5, is required. However, a wide range of the control, we must consider that both the switching frequency and switch-on duty ratio affects the output power and the switching voltage. The element values which are not in this table are same as ones in Table I. Moreover, the switch-on duty ratio is always satisfied with when zero voltage switching is achieved. This means that it is unnecessary to give (43), and we always consider (44) to find the control parameters. Because (44) also means that anti-parallel diode is on at. When the diode is on, zero voltage switching is always achieved. As a result, we pay attention only the output voltage and the slope of the switching voltage to realize the control circuit.

8 SEKIYA et al.: FM/PWM CONTROL SCHEME IN CLASS DE INVERTER 1257 Fig. 8. Equivalent circuits of Class DE inverter. (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4. VI. CONCLUSION This paper has presented a new control scheme Class DE inverter which is able to keep high power-conversion efficiency in wide range. The proposed control is FM/PWM control. Further, FM/PWM controlled Class DE inverter is analyzed and we clarify permance characteristics. Since the FM/PWM controlled inverter has two control parameters, namely, switching frequency and the switch-on duty ratio, it has one more degree of freedom design than the inverter with conventional control scheme. The increased degree of freedom is used to minimize the switching losses. From the analytical results, we clarify the power-conversion efficiency is kept high wide range when the inverter controls the output power so as to achieve zero slope of the switching voltage by varying the switching frequency and on time duty ratio. Carrying out the circuit experiments, we confirm that the experimental results agree well with the theoretical predictions qualitatively. The equivalent circuit in Fig. 8(a) is expressed by following differential equations: (46) The underdumped case, namely, is considered in this case. Case 2: The range of is in Case 2. When turns off, the state changes from Case 1 to Case 2. In this range, the both of switches and are off. Fig. 8(b) shows the equivalent circuit in this case. The basic equations of the inverter are given by the assumptions APPENDIX I CLASSIFICATION OF FOUR CASES AND CIRCUIT EQUATIONS Case 1: The range of is in Case 1. In this range, the switch is on and is off. Fig. 8(a) shows the equivalent circuit in this case. The basic equations of the inverter are given by the assumptions (47) The circuit is expressed by the following differential equations: (45) (48), is consid- The underdumped case, namely, ered in this case.

9 1258 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 51, NO. 7, JULY 2004 Case 3: The range of is in Case 3. In this range, the switch is off and is on. Fig. 8(c) shows the equivalent circuit in this case. The basic equations of the inverter are given by the assumptions Using (1), is rewritten as (49) (54) The circuit is expressed by following differential equations: where (50) The underdumped case, namely, is considered in this case. Case 4: The range of is in Case 4. When turns off, the state changes from Case 3 to Case 4. In this range, both switches and are off. Fig. 8(d) shows the equivalent circuit in this case. The basic equations of the inverter are given by the assumptions (55) (56) (51) Theree, the circuit equations are (52) (57) The underdumped case, namely is considered in this case. Since (46), (48), (50), and (52) are linear differential equations, they are solved strictly and (1) (8) are obtained. (58) From (19), APPENDIX II RESULTING EQUATION OF is given by (59) (60) (61) (62) (53) (63)

10 SEKIYA et al.: FM/PWM CONTROL SCHEME IN CLASS DE INVERTER 1259 From (53) to (58), we can derive APPENDIX III POWER LOSSES analytically. (64) (65) From (1) to (5), (21) to (25) are rewritten as the following equations: (66) (67) [6] H. Sekiya, H. Koizumi, S. Mori, and I. Sasase, Exact analysis of Class DE inverter with FM and PWM control scheme, IEICE Trans. Commun., vol. E86-B, pp , Oct [7] H. Koizumi, M. Fujii, K. Shinoda, T. Suetsugu, and S. Mori, Phasecontrolled Class DE inverter, in Proc. Int. Telecommunication Energy Conf. (INTELEC 95), Hague, The Netherlands, Oct [8] H. Sekiya, M. Matsuo, H. Koizumi, S. Mori, and I. Sasase, New control scheme Class DE inverter by varying driving signals, IEEE Trans. Ind. Electron., vol. 47, pp , Dec [9] N. O. Sokal and A. D. Sokal, Class E A new class of high-efficiency tuned single-ended switching power amplifiers, IEEE J. Solid-State Circuits, vol. SC-10, pp , June [10] F. H. Raab, Idealized operation of the Class E tuned power amplifier, IEEE Trans. Circuits Syst., vol. CAS-24, pp , Dec [11] M. Albulet, An exact analysis of class-de amplifier at any output Q, IEEE Trans. Circuits. Syst. I, vol. 46, pp , Oct [12], Analysis and design of the Class E frequency multipliers with RF choke, IEEE Trans. Circuits. Syst. I, vol. 42, pp , Feb [13] M. Albulet and S. Radu, Analysis and design of Class E frequency multiplier taking into account the Q factor, Int. J. Electron. Commun. (AEÜ), vol. 49, no. 4, pp , [14], Exact analysis of Class E frequency multiplier with finite dc-feed inductance at any output Q, Int. J. Electron. Commun. (AEÜ), vol. 50, no. 4, pp , [15] M. Albulet and R. E. Zulinski, Effect on switch duty ratio on the permance of Class E amplifiers and frequency multipliers, IEEE Trans. Circuits Syst. I, vol. 45, pp , Apr (68) (69) (70) Substituting (53) (58) into (66) (70),,,, and are derived analytically. Hiroo Sekiya (S 97 M 01) was born in Tokyo, Japan, on July 5, He received the B.E., M.E., and Ph.D. degrees in electrical engineering from Keio University, Yokohama, Japan, in 1996, 1998, and 2001, respectively. Since April 2001, he has been with Graduate School of Science and Technology, Chiba University, Chiba, Japan where he is a Research Associate. His research interests include high-frequency high-efficiency tuned power amplifiers, frequency multipliers, resonant dc/dc power converter, dc/ac inverters, bifurcation and chaotic phenomena in nonlinear electrical circuits, and digital signal processing speech, image and communication. Dr. Sekiya is a member of the Institute of Electronics, Inmation and Communication Engineers (IEICE) of Japan, and the Research Institute of Signal Processing (RISP), Japan. REFERENCES [1] H. Koizumi, T. Suetsugu, M. Fujii, K. Shinoda, S. Mori, and K. Ikeda, Class DE high-efficiency tuned power amplifier, IEEE Trans. Circuits Syst. I, vol. 43, pp , Jan [2] S. A. El-Hamamsy, Design of high-efficiency RF class-d power amplifier, IEEE Trans. Power Electron., vol. 9, pp , May [3] D. C. Hamill, Class DE inverters and rectifiers DC-DC converter, in Proc. IEEE Power Electron. Specialist Conf. (PESC 96), June 1996, pp [4] S. A. Zhukov and V. B. Kozyrev, Double-ended switching generator without communicating loss, in Poluprovodnilovye Pribory v Tekhnike Elektrosvyazi. Moscow, Russia: Svyaz, 1975, vol. 15, pp [5] K. Shinoda, M. Matsuo, T. Suetsugu, and S. Mori, PWM control scheme of resonant dc/dc converter with Class DE inverter and Class E rectifier, in Proc. 18th Int. Telecommunication Energy Conf. (INTELEC 96), Oct 1996, pp Hirotaka Koizumi (S 98 M 01) was born in Tokyo, Japan, in He received the B.E, M.E., and Ph.D. degrees in electrical engineering from Keio University, Yokohama, Japan in 1993, 1995, and 2001 respectively. From 1995 to 2001, he was an Electrical Engineer with Tokyo Electric Power Company Inc., Tokyo, Japan. From 1998 to 2001, he was also with the Graduate School, Keio University. He has been a Research Assistant in the Department of Electrical and Electronic Engineering, Faculty of Technology, Tokyo University of Agriculture and Technology, since His research interests are in high-frequency high-efficiency tuned power amplifiers, resonant dc/dc power converters, dc/ac inverters, and high-frequency rectifiers, especially Class D, Class E, and Class DE circuits. His current research interests include photovoltaic solar energy systems. Dr. Koizumi is a member of the Institute of Electronics, Inmation, and Communication Engineers (IEICE) of Japan.

11 1260 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 51, NO. 7, JULY 2004 Shinsaku Mori (M 80) was born in Kagoshima, Japan, on August 19, He received the B.E., M.E., and Ph.D. degrees in electrical engineering from Keio University, Yokohama, Japan, in 1957, 1959, and 1965, respectively. From 1957 to 1997, he was with the Department of Electrical Engineering, Keio University. From 1978 to 1979, he was a Visiting Professor of Electrical Engineering at the University of Wisconsin, Madison. From April 1997 until his retirement in April 2004, he was with the Department of Electrical and Electronics Engineering, Nippon Institute of Technology, Miyashiro, Saitama, Japan. His research interests include circuit theory, communication engineering, synchronization, inmation theory, and medical engineering, especially nonlinear circuits, power electronics, chaos, phase-locked loops, modulation and coding, and hyperthermia. Dr. Mori is a Fellow of the Institute of Electronics, Inmation, and Communication Engineers (IEICE) of Japan, the Japan Society Simulation Technology, the Society of Instrument and Control Engineers, the Society of Inmation Theory and Its Applications (SITA), and the Japanese Society of Hyperthermic Oncology. Jianming Lu (M 93) received the M.S. and Ph.D. degrees from Chiba University, Chiba, Japan, in 1990 and 1993, respectively. In 1993, he joined Chiba University, as an Associate in the Department of Inmation and Computer Sciences. Since 1994 he has been with the Graduate School of Science and Technology, Chiba University, where in 1998, he was promoted to Associate Professor in the Graduate School of Science and Technology. His current research interests are in the theory and applications of digital signal processing and control theory. Dr. Lu is a member of IEICE (Japan), SICE (Japan), IEEJ (Japan), and JSME (Japan). Iwao Sasase (S 80 M 84 SM 03) was born in Osaka, Japan in He received the B.E., M.E., and Ph.D. degrees in electrical engineering from Keio University, Yokohama, Japan, in 1979, 1981, and 1984, respectively. From 1984 to 1986, he was a Post-Doctoral Fellow and Lecturer of Electrical Engineering at the University of Ottawa, Ottawa, ON, Canada. He is currently a Professor of Inmation and Computer Science at Keio University, Yokohama, Japan. His research interests include modulation and coding, mobile and wireless communications, optical communications, communication networks, and inmation theory. He has authored more than 200 journal papers and 290 international conference papers. Dr. Sasase received the 1984 IEEE Communications Society Student Paper Award (Region 10), the 1986 Inoue Memorial Young Engineer Award, the 1988 Hiroshi Ando Memorial Young Engineer Award, the 1988 Shinohara Memorial Young Engineer Award, and the 1996 Institute of Electronics, Inmation, and Communication Engineers of Japan Switching System Technical Group Best Paper Award. He served as the IEEE Communication Society (ComSoc) Satellite and Space Communications Technical Committee Chair during 2001 and He now serves as the IEEE ComSoc Asia and Pacific Board Vice Director. He is a member of the Institute of Electronics, Inmation and Communication Engineers of Japan, Inmation Processing Society of Japan, and the Society of Inmation Theory and Its Applications, Japan, respectively. Takashi Yahagi (M 78) received the B.S., M.S., and Ph.D. degrees, all in electronics engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1966, 1968, and 1971, respectively. In 1971, he joined Chiba University, Chiba, Japan, as a Lecturer in the Department of Electronics Engineering. From 1974 to 1984, he was an Associate Professor, and in 1984, he was promoted to Professor in the Department of Electrical Engineering. From 1989 to 1998, he was with the Department of Inmation and Computer Sciences. Since 1998, he has been with the Graduate School of Science and Technology, Chiba University. His current research interests are in the theory and applications of digital signal processing and other related areas. He is the editor of the Library of Digital Signal Processing (Tokyo, Japan: Corona). Dr. Yahagi is a member of IEICE (Japan), The New York Academy of Sciences (USA), and ISCIE (Japan). Since 1999 he has been Chairman of the IEEE Japan Chapter of Signal Processing Society.

with FM and PWM Control Hiroo Sekiyay, Shinsaku Moriz and Iwao Sasasey y Dept. of Electrical Engineering, Keio Univercity,

with FM and PWM Control Hiroo Sekiyay, Shinsaku Moriz and Iwao Sasasey y Dept. of Electrical Engineering, Keio Univercity, Exact Analysis of Class DE Amplier with FM and PWM Control Hiroo Sekiyay, Shinsaku Moriz and Iwao Sasasey y Dept. of Electrical Engineering, Keio Univercity, 3141, Hiyoshi, Kohoku, Yokohama, 35 JAPAN Phone:

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