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

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1 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: ext Fax : sekiya@sasase.ics.keio.ac.jp zdept. of Electrical & Engineering, Nippon Institute of Technology, 41 Gakuendai, Miyashiro, MinamiSaitama, Saitama, JAPAN Abstract The control of class DE amplier is the important probrem since the switching of the amplier does not satisfy class E switching condition when load resistance varies from initial designed values. Therefore, some of control schemes of class DE amplier were proposed. However, the changes of the output voltage and power conversion ef ciency by using the controls can be measured only experimentally and thus, they cannot be found theoretically. In this paper, an exact analysis of class DE amplier with FM and PWM control schemes is presented. From the analysis, we can derive the output voltage, power conversion eciency and so on theoretically. Moreover, the frequency and duty ratio to keep the constant output voltage can be found when the load resistance and input voltage vary. We indicate that the theoretical predictions agree with the experimental results qualitatively. Measured eciency is over 93 percent with 1. MHz and 1.W output. V cc i S v S i Cs S Cs D r L o C o i v i Cs1 Cs1 D r1 v i o v o Fig. 1. Circuit topology of class DE amplier. I. introduction Class DE amplier[1]{[5] has become increasingly valuable highdensity power source. The power conversion ef ciency of class DE amplier is high under highfrequency (MHz order) operation. In addition, the switch current is small and both of the two switches satisfy class E switching condition[6], [7]. That is achieved by the shunt capacitor added to each switch and the deadtime which is given between the switchontimes on the driving pattern. One of the disadvantage of class DE amplier is that the output control with keeping the high conversion eciency is dicult. Because the switching of the amplier does not satisfy class E switching condition when load resistance varies from initial designed values. Therefore, in order to control output power of class DE amplier, several schemes of the control were proposed, i.e., frequency modulated (FM)[1], phase modulated (PM)[5],pulse wides modulated (PWM)[] control and so on. FM control generates a wide and unpredictable noise spectrum, making EMI control more dicult, and poor utilization of magnetic components. As a remedy of these problems, PM control was applied to Class DE amplier. In PM control, the amplier keeps Class E switching conditions so high eciency can be obtained. However, the circuit topology consists of two inverters so its volume becomes as large as twice of the basic one. So far only PWM controlled class DE amplier has cleared above problems. However, the changes of the output voltage and power conversion eciency of class DE amplier by using the PWM control can be measured only experimentally and thus, they cannot be found theoretically. FM control scheme also has the same problem. We recognize that it is a very important and useful theme to clarify the changes of them theoretically since that has a great advantage to design Class DE amplier with these controls. In this paper, we present the exact analysis of class DE amplier with FM and PWM control schemes. From the analysis, we can derive the output voltage, power conversion eciency and so on theoretically. Moreover, the frequency and duty ratio to keep the constant output voltage can be found when the load resistance and input voltage vary. We indicate that the theoretical predictions are similar to experimental results qualitatively. Measured eciency is over 93 percent with 1. MHz and 1.W output.

2 πd πddt D v V cc io ON π π OFF πθ θ π π θ π Fig.. A. Class DE amplier D v V cc OFF π π πds ON πddt πθ π θ Optimum waveforms of class DE amplier. II. Circuit Description Figure 1 depicts the circuit topology of class DE amplier[1]. It is composed of two switches S 1 and S,two capacitors C and C S shunting each switch, and a seriesresonant circuit L C. The waveforms of class DE amplier are shown in Fig. when switch on duty ratio of each switch is.5. The on duty ratio of class DE amplier can be specied to any values from zero to.5. However, the smaller we specify it, the larger power output capability we can obtain[9]. In this paper, we specify switch on duty ratios (D and D S ) and dead time duty ratio (D DT ) of.5. The switches are driven by a driving pattern of D r1 and D r in Fig., where D r1 and D r drive S 1 and S respectively. The driving pattern generates a dead time during the period when one switch has turned o before the other switch has turned on. During dead time, the sinusoidal output current i o charges one shunt capacitor and discharges the other shunt capacitor, and the midpoint voltage between two switches, namely, v becomes V CC or zero at the end of the dead time, allowing class E switching conditions [6], [7]. In this paper, this state is named as \optimum state". B. Class DE amplier with FM control FM control scheme is applied to class DE amplier[1]. If the switching frequency is larger than that of the optimum switching frequency, the switches S 1 and S turn on before the voltage v becomes V CC or zero. Therefore, the input current from the direct voltage source decreases by increasing the switching frequency. Consequently, the amplier can regulate the output voltage by applying FM control. However, this control scheme has some problems. FM control generates a wide and unpredictable noise, making EMI control more dicult, and poor utilization of magnetic components. There is an important class of applications in which the xedfrequency converters are preferable to the variablefrequency converters. One example is the highsensitivity F receiver used in communications and radar. In this case, the detrimental eects of EMI can be minimized if the switching frequency can be xed at a value, and the harmonics of the switching frequency, the beat frequencies between them and other frequencies (e.g., local oscillators) are outside of the passbands of the receiver circuits. Therefore, It is expected to control class DE amplier under the constant switching frequency. C. Class DE amplier with PWM control PWM control scheme is achieved by changing pulsewidth of driving signals D r1 and D r, namely, duty ratio[]. By increasing D DT, both D and D S are decreased, and the midpoint voltage v between S 1 and S is changed. As a result, the fundamental frequency component ofv is reduced, decreasing the current i through L o C o,and regulating the output voltage v o of the amplier. Consequently, the amplier can regulate the output voltage by applying PWM control to drive voltages of class DE amplier. PWM control scheme has the following advantages. First, a wide line and load regulation region are obtained with a xed switching frequency. Second, this scheme can be realized by only adding external gatedrive circuits to drive the class DE conventional amplier. Hence the circuit scale of PWM controlled amplier is lighter and smaller than PM controlled class DE amplier[5]. A. Assumptions III. Exact Analysis The analysis given below is based on the following assumptions. 1) The active devices act as ideal switches. ) The shunt capacitance includes the switch device capacitance. 3) All circuit elements are ideal. 4) The loaded Qfactor is high enough to consider the underdamped case. B. Parameters The following parameters of the circuit are dened below. 1)! =1= p L C : The resonant frequency when each switch is on. ) Q =! L = = 1=! C : The loaded Qfactor when each switch is on. 3) A = f =f : The ratio of the resonant frequency and the operating (switching) frequency. 4) B = C =C = C =(C s1 C s ) : The ratio of the capacitance of resonant circuit capacitor and shunt capacitor. 5) D = D = D S : The switch on duty ratio.

3 Vcc Vcc i S 1 S i S 11 S i S 13 v S 1 v S 11 v S 13 v S 3 CS i Cs 1 Lo i Cs 11 i S 3 i Cs 3 Fig. 3. C CS Lo i Cs 13 C Co v Co v 3 i i 3 VO VO Vcc Vcc i S S i S 1 i S 4 S i S 14 v S v S 1 v S 4 v S 14 CS C i Cs Lo i Cs 1 i Cs 4 CS Lo i Cs 14 C Co v Co v 4 i i 4 Equivalent circuit of class DE amplier. (a) Case 1. (b) Case. (c) Case 3. (d) Case 4. C. Waveforms The analytical procedure is closely related to the work by Albulet et.al [1]{[13]. The range of < is considered since the output voltage is periodic with. We consider four cases due to the state of the switches S 1 and S. Case 1 The range of is < 1 in Case 1, where 1 is dened as follows. 1 =D (1) In this range, the switch S 1 is on and S is o. Fig. 3(a) shows the equivalent circuit in this case. Case The range of is 1 < in Case, where is dened as follows. = D () When S 1 turns o, the state changes from Case 1 to. In this range, the both of switches S 1 and S are o. Fig. 3(b) shows the equivalent circuit in this case. Case 3 The range of is 1 < 3 in Case 3, where 3 is dened as follows. 3 =D (3) In this range, the switch S 1 is o and S is on. Fig. 3(c) shows the equivalent circuit in this case. Case 4 VO VO The range of is 1 3 < 4 in Case 4, where 4 is dened as follows. 4 = D (4) When S turns o, the state changes from Case 3 to Case 4. In this range, the both of switches S 1 and S are o. Fig. 3(b) shows the equivalent circuit in this case. The circuit equations in each case are given like in [14]. From the circuit equations, we can determine for the waveform equations as follows : i o () = i o1 ( a )=i o1 () exp( a ) cos( a ) 1 fi o1() A! C v 1 ()g exp( a ) sin( a ); for a < 1 i o ( b )=i o () exp( b ) cos( b ) 1 fi o() A! C v j ()g exp( b ) sin( b ); for b < i o3 ( c )=i o3 () exp( c ) cos( c ) 1 fi o3() A! C (v 3 () V CC )g exp( c ) sin( c ); for c < 3 i o4 ( d )=i o4 () exp( d ) cos( d ) 1 fi o4() A! C (v 4j () V CC )g exp( d )sin( d ); for d < 4 (5) i 1 ( a )=i o1 ( a ); for a < 1 i i () = ( b )=; for b < (6) i 3 ( c )=; for c < 3 i 4 ( d )=; for d < 4 i ( a )=; for a < 1 i i S () = S ( b )=; for b < (7) i S3 ( c )=i o3 ( c ); for c < 3 i S4 ( d )=; for d < 4 i C11 ( a )=; for a < 1 i C1 ( b )= C i C1 () = C i o( b ); for b < i C13 ( c )=; for c < 3 i C () = i C14 ( d )= C C i o4( d ); for d < 4 () i C1 ( a )=; for a < 1 i C ( b )= C S C i o( b ); for b < i C3 ( c )=; for c < 3 i C4 ( d )= C S C i o4( d ); for d < 4 (9)

4 v() = v 1 ( a )=v 1 () exp( a ) cos( a ) Q i o1 () v 1 () exp( a ) sin( a );! C for a < 1 v ( b )= Bv () v () 1B 1B exp( b)cos( b ) Q i o () v () exp( b ) sin( b );! C 1B for b < v 3 ( c )=V CC (v 3 () V CC )exp( c ) cos( c ) Q i o3 () v 3 () V CC exp( c )sin( c );! C for c < 3 v 4 ( d )= Bv 4() V CC 1B v 4() V CC exp( d ) cos( d ) 1B Q i o4 () v 4() V CC exp( d )sin( d );! C 1B for d < 4 v () = v 1 ( a )=; for a < 1 v ( b )= Bv () Bv () 1B 1B exp( b)cos( b ) B Q i o () v () exp( b ) sin( b );! C 1B for b < v 3 ( c )=; for c < 3 v 4 ( d )= Bv 4() V CC 1B B(v 4() V CC ) exp( d ) cos( d ) B v S () = (1) 1B Q i o4 () v 4() V CC exp( d )sin( d );! C 1B for d < 4 (11) v ( a )=; for a < 1 v S ( b )=V CC v ( b ); for b < v S3 ( c )=; for c < 3 v S4 ( d )=V CC v 4 ( d ): for d < 4 (1) For example, i ok means i o in Case k. Inabove equations, i o1 (), and v 1 () are initial values in Case 1, i o (), and v () are initial values in Case, i o3 () and v 3 () are initial values in Case 3, i o4 () and v 4 () are initial values in Case 4, and =! Q ; (13) vo v V π π Fig. 4. (a) vo v VCC π π (b) Theoretical waveforms for V D =: V,=1.. (a) Waveforms with FM control for f =1: MHz and D =:5. (b) Waveforms with PWM control for f =1:MHz and D =:1. r =! 1 1 4Q ; (14) =! Q ; (15) r =! 1B 1 4Q ; (16) a = ; (17) b = 1 ; (1) c = 1 ; (19) d = 1 3 : () Using above waveform equations with the following transient conditions: i o1 ( 1 ) = i o (); (1) i o ( ) = i o3 (); () i o3 ( 3 ) = i o4 (); (3) i o4 ( 4 ) = i o1 (); (4) v 1 ( 1 ) = v (); (5) v ( ) = v 3 (); (6) v 3 ( 3 ) = v 4 (); (7) v 4 ( 4 ) = v 1 (); () the theoretical optimum waveforms as shown in Fig. are given. The waveforms of FM and PWM controled amplier are derived by changing f and D respectively from optimum states. Figure 4 (a) shows the example waveforms of FM controled class DE amplier and Fig. 4 (b) is that of PWM controled one. D. Output voltage and output power Using (5), the root mean square output voltage V o is s 1 V o = I o = fi o ()g d; (9)

5 where I o expresses the root mean square output current. Similarly, the output power can be derived by the following equation: P o = I o = where I o is given analytically from (5). E. Power losses and eciency fi o ()g d; (3) The conduction loss P in the switch S 1 is calculated as P = r fi ()g d (31) where r is the onresistance of S 1. Similarly, conduction losses P CS 1 in the shunt capacitor C, P CS in C S,and in the resonant circuit L o C o are P LCo P S = r S n P C = r C P CS = r C S P LCo = r LC o fi S ()g d (3) fi C1 ()g d (33) fi C ()g d (34) fi o ()g d (35) where r S is on resistance of S, r C and r CS are the equivalent series resistances (ES's) of C and C S,and r LCo is the ES's of L o C o.from (5) { (9), (31) { (35) can be derived analytically. By using FM and PWM control scheme, the switchings do not achieve class E switching conditions and turnon switching loss occurs. The turnon switching losses P to1 and P to in the switches S 1 and S are P to1 = 1 C ffv 4 ( 4 )g (36) P to = 1 C Sffv S ( )g (37) where v 4j ( 4 ) and v Sj ( ) are the switch voltages at turnon instants. From (31) to (37), the total power loss P total is P total = P P S P C P CS P LCo P to1 P to : (3) Neglecting drive power, one can nd the amplier eciency P o = P o P : (39) total vo v Fig. 5. vo v 5V/div V/div (a) vo v 5V/div V/div 5V/div 5V/div V/div (b) 5V/div (c) 5V/div Experimental waveforms for V D =: V,=1.. (a) Optimum waveforms for f =1:3 MHz and D =:5. (b) Waveforms with FM control for f =1: MHz and D = :5. (c) Waveforms with PWM control for f = 1:MHz and D =:1. TABLE I Circuit parameters for optimum operation. Calculated Measured Dierence L o 17.4 H H.5 % C o 1.7 nf 1.71 nf.5 % C.5 nf.53 nf.4 % C S.5 nf.54 nf.79 % opt % f opt 1. MHz 1.1 MHz 1. % V CCopt. V. V. % V oopt 4.37 V 4.6 V.5 % 94.6 % 93. %.3 % IV. Experimental results A. Design procedure of class DE amplier At rst, the following parameters are given; the operating frequency f, the input voltage V CC, the output resistor, loaded quality factor Q, and switchon duty ratio D. Then the unknown parameters are i o11 (), i o1 (), i o31 (), i o41 (), v 11 (), v 1 (), v 31 (), v 41 (), A and B. From the transient conditions (1) { () and class E switching con TABLE II Element values of ES's. r r S r C r CS r CLCo

6 Output Voltage Vo [V] output voltage efficiency calculated calculated measured measured Operating frequency f [MHz] Power conversion efficiency η [%] Operating frequency f [MHz] frequency efficiency Load resistance [Ω] Fig. 7. The switching frequency f and the power conversion eciency as function of load resistance for V o = V oopt, V D = V Dopt and D=.5. Power conversion efficiency η [%] Fig. 6. Output voltage and power conversion eciency as function of switching frequency for V D = V Dopt, = opt and D=.5. ditions; dv S ( ) =; d v S ( )=; (4) we can get 1 equations and unknown parameters are calculated by using Newton method. Switchign frequency f [MHz] frequency efficiency Input voltage VCC [V] Power conversion efficiency η [%] B. Design example At rst, the following specications are given; operating frequency f =1: MHz, input voltage V CC =: V, output resistor =1:, loaded quality factor Q = 1., and D =:5. Then unknown parameters A and B are given by design procedure as A =:93, B =:341. Therefore, components of circuit are found as shown in Table I. IF53 MOSFET's are used as switch S 1 and S. Moreover, the measured element values of ES's are shown in Table II. In theoretical calculation, the element values of ES's are the same as measured ones. Figure 5 (a) depicts an experimental waveforms for f = 1:1 MHz. In this state, the conversion eciency is highest, that is 93.%. We dene that the operation in Fig. 5 (a) is optimum. The conditions for optimum operation are shown in Table I. From Figs. and 5 (a), and Table I, we can nd that the experimental results of optimum operation are similar to the theoretical ones qualitatively. We control the amplier from optimum operation by using the FM and PWM control scheme. Fig.. The switching frequency f and the power conversion eciency as function of input voltage V D for V o = V oopt, = Dopt and f = f opt. C. Class DE amplier with FM control An example of experimental waveforms of FM controlled amplier is shown in Fig. 5 (a) for f = 1:3. Compared with the waveforms in Fig. 4 (a), we can conrm the waveforms are similar to the theoretical predictions qualitatively. Figure 6 shows the output voltage and the power conversion eciency as a function of the switching frequency f at V D = V Dopt and D =:5. They express the output characteristics of class DE amplier with FM control. The experimental results agree with theoretical ones over 1. MHz. However, there is a dierence between them when f is below 1. MHz. In the analysis, MOSFETs are assumed as ideal switches. Theoretically, v s1 and v s have negative voltage below 1.MHz though v s1 and v s are never negative in practical circuit because of antiparallel diode in

7 MOSFETs. The dierence between ideal and practical cases appears below 1 MHz in Fig. 6. In the circuit experiment, we can observe that the amplier can change the output voltage from.45 V to 5.5 V over 75 percent eciency. Moreover, we can nd the switching frequency f for keeping a constant output when load resistance or input voltage vary. In order to derive f theoretically, we need to solve the 9 algebraic equations (1){(), and P o = V oopt : (41) From these equation, unknown initial values and f is derived by using Newton Method. Figure 7 shows the switching frequency f and power conversion eciency as function of load resistance for V = V oopt and D =:5. In this case, the FM control scheme can keep the constant output voltage V o = V oopt for varying the load resistance from 1 to with keeping over 75% eciency. If is smaller than opt, the theoretical control value cannot obtained because of negative voltage v and v S. Figure shows the switching frequency f and power conversion eciency as function of input voltage V D for V = V oopt and D =:5. In this case, the FM control scheme can keep the constant output voltage V o = V oopt for varying the input voltage from 15 V to 35 V with keeping over % eciency. D. Class DE amplier with PWM control Figure 5 (b) shows an example of experimental waveforms of PWM controlled amplier for D = :1. From Figs. 4 (b) and 5 (b), we can see the waveforms are similar to theoretical predictions. Figure 1 shows the output voltage as a function of the duty ratio D at V D = V Dopt and f = f opt. It expresses the output characteristic of class DE amplier with PWM control. The experimental results agree with theoretical ones below D=.5. However, there is also a dierence between them over.5 because of negative voltage at theoretical vs 1 and vs. In the circuit experiment, we can observe that the amplier can change output voltage from 3.4 V to 4.31 V over percent eciency. Further, when load resistance or input voltage vary, the on time duty ratio D to keep a constant output voltage is derived. In order to nd D, we use 9 equations (1) { (), and 41. From theses equations, unknown initial values and D is founded. Figure 1 shows the on time duty ratio D and power conversion eciency as function of load resistance for V = V oopt and f = f opt. In this case, the FM control scheme can keep the constant output voltage V o = V oopt for varying the load resistance from 1 to 17.5 with keeping over 77% eciency. Figure 11 shows the on time duty ratio D and power conversion eciency as function of input voltage V D for Output Voltage Vo [V] output voltage calculated measured efficiency calculated measured Switch on duty ratio D Fig. 9. Output voltage and power conversion eciency as function of on time duty ratio for V D = V Dopt, = opt and f = f opt. Ω Fig. 1. The switch on duty ratio D and the power conversion eciency as function of load resistance for V o = V oopt, V D = V Dopt and f = f opt. V = V oopt and f = f opt. In this case, the FM control scheme can keep the constant output voltage V o = V oopt for varying the input voltage from V to 7 V with keeping over 7% eciency. Moreover, from this gure, we can recognize that PWM controlled class DE amplier cannot keep the output voltage V o = V oopt when V D is larger than 37. V. V. Conclusion In this paper, we have presented an exact analysis of class DE amplier with FM and PWM control schemes. From the analysis, we can derive the output voltage, power conversion eciency and so on theoretically. Moreover, Power conversion efficiency η [%] η

8 Fig. 11. The switch on duty ratio D and the power conversion eciency as function of input voltage V D for V o = V oopt, = Dopt and f = f opt. the frequency and duty ratio to keep the constant output voltage can be found when the load resistance and input voltage vary. We indicate that the theoretical predictions agree with the experimental results qualitatively. Measured eciency is over 93 percent with 1. MHz and 1.W output. The future problem is the analysis of class DE amplier with FM and PWM control with considering antipararell diode of MOSFETs. η [9] K. Shinoda, T. Suetsugu, M. Matsuo and S. Mori, \Analysis of phasecontrolled resonant DCAC inverters with class E ampli er and frequency multipliers, " IEEE Trans. Indust. Electron., vol. 45, no. 3, pp. 414, June 199. [1] M. Albulet, \Analysis and design of the class E frequency multipliers with F choke," IEEE Trans. Circuits. syst. I, vol. 4, pp. 9514, Feb [11] M. Albulet and S. adu, \Analysis and design of class E frequency multiplier taking into account the Q factor," Int. J. Electron. Commun. (AE U), vol. 49, No. 4 pp. 1316, Mar [1] M. Albulet and S. adu, \Exact analysis of class E frequency multiplier with nite dcfeed inductance at any output Q," Int. J. Electron. Commun. (AE U), vol. 5, No. 4 pp. 151, May [13] M. Albulet and. E. Zulinski, \Eect on switch duty ratio on the performance of class E ampliers and frequency multipliers" IEEE Trans. Circuits syst., vol. CAS45, No. 4 pp , Apr [14] H. Sekiya, M. Matsuo, H. Koizumi, T. Suetsugu, S. Mori, and I. Sasase, \New control method of class DE inverter { class DE thinning out inverter" Proc. of INTELEC '9, pp. 374, Oct eferences [1] H. Koizumi, T. Suetsugu, M. Fujii, K. Shinoda, S. Mori and K. Ikeda, \Class DE higheciency tuned power amplier, " IEEE Trans. Circuits syst., vol. 43, No. 1 pp. 51{6, Jan [] S. A. ElHamamsy, \Design of higheciency F classd power amplier," IEEE Trans. Power Electron., vol. 9, pp. 97{3, May [3] D. C. Hamill \Class DE inverters and rectiers for DC DC converter," Proc. IEEE Power Electron. Specialist Conf. (PESC'96), pp. 54{6, June [4] S. A. Zhukov and V. B. Kozyrev, \Doubleended switching generator without communicating loss," Poluprovodnilovye Pribory v Tekhnike Elektrosvyazi, vol. 15, pp. 95{17, Svyaz (in ussian) [5] H. Koizumi, M. Fujii, K. Shinoda, T. Suetsugu and S. Mori, \Phasecontrolled class DE inverter," Proc IEEE International Telecommunications Energy Conference (INTELEC '95), Hague, Oct [6] N. O. Sokal and A. D. Sokal, \Class E { A new class of higheciency tuned singleended switching power ampliers, " IEEE J. of solidstate circuits, vol. SC1, No. 3 pp , June [7] F. H. aab, \Idealized operation of the class E tuned power amplier, " IEEE Trans. Circuits syst., vol. CAS4, No. 1 pp , Dec [] K. Shinoda, M. Matsuo, T. Suetsugu and S. Mori, \PWM control scheme of resonant dc/dc converter with class DE inverter and class E rectier, " Proc. of INTELEC '96, pp. 93, Oct 1996.