LOW PEAK CURRENT CLASS E RESONANT FULL-WAVE LOW dv/dt RECTIFIER DRIVEN BY A VOLTAGE GENERATOR

Électronique et transmission de l information LOW PEAK CURRENT CLASS E RESONANT FULL-WAVE LOW dv/dt RECTIFIER DRIVEN BY A VOLTAGE GENERATOR ŞERBAN BÎRCĂ-GĂLĂŢEANU 1 Key words : Power Electronics, Rectifiers, Class E, Resonant Rectifiers. Abstract - Full-wave rectifiers allow to halve the diode peak voltages (or currents), but these values may still be high at duty factor values far from 0.5. A new class E resonant full-wave rectifier is proposed and analyzed, whose peak diode current is equal to the load current. Diode voltage slopes are ideally zero after switch-off only for duty factors larger than 0.5. The circuit contains two rectifier diodes but a single LC circuit. All major parasitic reactive elements are included in the rectifier topology. Basic parameters of the circuit are derived as functions of the diodes' duty factor. This allows the user to find the optimum component values for a given application. Experimental waveforms and measurements agree with simulations and calculations. 1. INTRODUCTION To reduce switching losses at higher frequencies, resonant and pseudoresonant circuits have been developed, for instance Class E [7] rectifiers [1, 2]. After the half-wave rectifiers, full-wave circuits are proposed [4] for further reduction of the transformer size, of the diode peak voltages (or currents) and of the input current (or voltage) harmonics. This paper aims to present the analysis of a resonant full-wave voltage-driven low dv/dt rectifier whose peak diode current equals the load current (Fig. 1a, b). This allows to find component values for optimum operation in a given application. Diode voltage slopes are ideally zero after switch-off only for duty factors larger than 0.5. The basic circuit contains two rectifier diodes, two inductors, but a single capacitor C across the two rectifier diodes (Fig. 1a). Voltages on the two inductances are identical, hence they may be coupled, which reduces circuit cost and radiated EMI. Output current is almost constant due to the filter inductor L f and a constant current produces no voltage drop across an inductance, hence the load can be connected at any terminal of the inductance L 1, without affecting circuit operation. Consequently, inductances L and L 1 may be concentrated in a single inductance: L (Fig. 1b) or L 1. The L C circuit has the resonant frequency f 0. It now appears like a second-order low-pass filter. 1 I.U.F.M. of the Academy of Nantes, France. Rev. Roum. Sci. Techn. Électrotechn. et Énerg., 52, 3, p. 331 342, Bucarest, 2007

332 Şerban Bîrcă-Gălăţeanu 2 ~ v L D C v D i D i D1 vd1 D 1 L f I o V o R L L 1 a L D i D i v D1 L D f I o ~ v v V D1 o R C D L 1 b D L i D L f I o v C v ~ D1 V o R L v D L L f I o ~ v C D 1 i D1 V o R L c D L i D L f I o ~ v D 1 i D1 V o R L d Fig. 1 The resonant full-wave voltage-driven low dv/dt rectifier with low peak diode current: a) basic circuit diagram; b) single L circuit diagram; c) equivalent circuit when diode D is off; d) equivalent circuit when diode D 1 is off; e) equivalent circuit when both diodes are on. Rectifier diodes may never be both off, because there is no other current path towards the output inductor L f which acts as a current sink. At light loads, diodes conduct only one at a time (Fig. 1c and d). The duty cycle D is 0.5 (Fig. 2b). At heavy loads, there are intervals when both diodes are on (D 0.5, see Fig. 1e and Fig. 2a). The circuit has been analyzed and measured in both operating modes, at frequencies around f 0. At the resonant frequency the input reactive current may be very high and the voltage transfer factor very low. e 2. ANALYSIS OF OPERATION The circuit is symmetrical in ac, that is the waveforms of the voltages and currents on D 1 are identical with the waveforms of the voltages and currents on D with a delay of half a period. Also, the waveform of the input current i has two identical half-waves plus a continuous component equal to half the output current I o.

3 Low peak current class E resonant full-wave low dv/dt rectifier 333 All the circuit components are assumed ideal. Diode capacitances are absorbed into the resonant capacitor C and isolation transformer leakage inductance is included in the inductance L. Hence, the rectifier is especially suitable for highfrequency applications such as resonant dc-dc converters. The inductance L f is large enough for the output current have negligible ripple; thus the L f R L circuit can be replaced by a dc current sink I o = V o / R L. The rectifier is fed by an ideal sinusoidal voltage source v = V m sin (ωt ϕ), (1) where ϕ is the phase angle of the input voltage when the series diode D turns off (Figs.2, a and b). Notations are, as in [2 6, 9 and 10], for Fig.1.b: 2 ω 0 = 1 LC, Q = RL ω L, M = Vo V, A = ω m ω. (2) 0 Q has only the form of a quality factor, but is in fact the rated load resistance. From the continuity conditions for the capacitor voltages and inductance currents, a system of two equations is obtained which gives sin ϕ and cos ϕ of the form sin ϕ = M Q (A2 1) NS DΦ, cos ϕ = M Q (A2 1) NC DΦ, (3) tan ϕ = NS/NC. The average value of the voltage v is zero, therefore the average value of v D1 and v D is V o. From this condition we get the equation to derive the Q - factor, hence M = Q DΦ / ( A ) 2 1 NS 2 NC 2. (4) Circuit parameters are given by the following expressions when D 0.5, where d = 2 π D a = (2 π d)/a, a d a d DΦ = 4 Acos sin sin cos, 2 2 2 2 NS = A sin a sin d, NC = cos d cos a, 2 (5) Q = 1 2 π tan a tan d A A tan d tan a 2 2 2 2. (6) 2

334 Şerban Bîrcă-Gălăţeanu 4 We have noticed that the expression of the normalized load resistance Q as a function of the diode conduction duty factor D can always be brought to a quadratic form [6]. This one can be written as a transcendental equation for D, with Q as a parameter. It also has been demonstrated [6] and checked for this circuit that the normalized start-up diode current h = i D ON /I o = A 2 π Q. The input voltage being sinusoidal, the input power of the rectifier is fed only by the fundamental component of the input current i. At the operating frequency, the rectifier has the input impedance of a parallel R i L i circuit. By equating the input and the output power, one obtains R i / R L = 1 / (2 M 2 ). (7) The equivalent input inductance L i at the operating frequency is obtained from the reactive component of the input current. L i /L = π (A 2 1) / DL, (8) DL = A 2 π /2 (A sin a cos d cos a sin d) A 2 /(A 2 1) A sin a sin ϕ sin(ϕ d) [π d sin d cos(d 2ϕ)]/2. The rectifier has the following parameters for D = 0.5 : a = π/a DΦ = 2 A 2 (1 cos a) NS = A sin a NC = (1 cos a) tan ϕ = NS/NC = A tan(a/2) Q = 1/(2 π A 2 ) M = 1 / ( A ) 2 1 1 A 2 tan 2 ( a/2) (9) 2 L i L = A 1 1 2 1 A tan( a/2) 2 2 2. A 2 π A 1 1 [ A tan( a/2)] (10) Parameters ϕ, Q, M, V DRM /V o and L i (for A 1) (Figs. 3, 4, 5, 6) are monotonic functions of D: ϕ, V DRM /V o and L i increase, Q and M decrease when D increases. For D > 0.7, L i /L is close to 1. For D close to 0.5 and A < 0.8, L i has very high values, hence reactive currents are very low. Operating frequencies close to f 0 must be avoided, because M is very low. For any load value, the voltage transfer factor M is low and strongly dependent on the load, which is an important drawback of the class E rectifiers. Diode peak voltage is almost independent of the operating frequency. When D = 0.5, d = π = constant (a = π/a). Circuit parameters may no longer be expressed and plotted as functions of D, but as functions of Q. In order to have a limited range for the x-axis, ϕ, L i /L and M are plotted as functions of the auxiliar parameter q = 1/(2 π A 2 Q), q = 0 to 1.

5 Low peak current class E resonant full-wave low dv/dt rectifier 335 a) b) Fig. 2 Key waveforms of the ideal rectifier, when: a) D 0.5; b) D = 0.5.

336 Şerban Bîrcă-Gălăţeanu 6 Circuit operation has been simulated using OrCAD Capture version 9.1. Waveforms show no ringing, because there are no parasitics in simulation models: DΦ = 2 [(1 cos a) A 2 (2/π π/2) A sin a (A 2 1)], NS = (1 cos a) A 2 (1 2 π Q) A sin a (4 Q A 2 π/2) 2 (A 2 1), (11) NC = (1 cos a)(4 Q A 2 π/2) A sin a (1 2 π Q). The voltage transfer factor M is still given by expression (4): L i /L = π (A 2 1)/(A 2 DL), (12) π Asin a DL = 2 2 A 1 2 1 sinϕ [A sin a cos ϕ (1 cos a) sin ϕ] π sin ϕ (cos ϕ sin ϕ sin a / A). (13) At D = 0.5, maximum point phase grows from 90 to 120 degrees and V DRM /V o from 3.2 to 4.08 when A decreases from 2 to 0.5 (Fig. 10) and q increases from 0 to 1. For A = 1, V DRM /V o grows rapidly when q approaches 1. Two resonances take place at D = 0.5, at frequencies between f 0 /2 and 1.2 f 0. A resonance has already been noticed for other Class E rectifiers operating at frequencies different from f 0 [5]. At frequencies close to f 0, the input equivalent inductance Li is almost zero in a large range of load values (Figs. 7 and 8). The circuit then has a very high input current. Voltage transfer factor M has much higher values than for D > 0.5. When considering all circuit parameters, f = f 0 / 2 seems to be a good compromise (Fig. 9). Operation at D = 0.5 still reduces switching losses because dv/dt is low, even if it is not zero after turn-off. This mode is a good choice because M is relatively high, V DRM /V o is low and both are only slightly dependent on the load (on parameter q ). For q = 0.9 to 1, L i /L >> 1, hence the reactive current is very low. Derive 5 has been used to calculate and draw parameters as functions of the duty factor D, because this was the only way to have explicit expressions only. 3. EXPERIMENTAL RESULTS A circuit of the full-wave rectifier of Fig. 1a was constructed with the resonant frequency f 0 = 1 MHz, the output voltage V o = 5 V and the duty factor D = 0.5 when the output current I o = 50 ma (R L = 100 Ω). It follows L = L 1 = 20.5 µh (8/9 IF220 fixed inductors from Affero S.A.) and C = C 1 = 1.2 nf ceramic capacitors, as in [5], to allow a direct comparison. C f = 470 nf is a multilayer ceramic capacitor. D and D 1 are Motorola 1N5817 (or ST Semi BYV10-20A). Current and voltage waveforms have been obtained with Tektronix 7613 oscilloscope and P6022 current probe. Waveforms are presented for the operating

7 Low peak current class E resonant full-wave low dv/dt rectifier 337 frequency f = f 0 = 1 MHz and load resistance of 25 Ω (Fig.11). Waveforms at the frequency f 0 / 2 = 707 khz are not significantly different. One may note that Fig. 3 Initial phase angle ϕ of the input voltage, as a function of D. Fig. 4 Normalized load resistance Q, as a function of D.

338 Şerban Bîrcă-Gălăţeanu 8 Fig. 5 Ac to dc voltage transfer factor M, as a function of D. Fig. 6 L i /L versus diodes duty ratio D.

9 Low peak current class E resonant full-wave low dv/dt rectifier 339 Fig. 7 D = 0.5. Initial phase angle ϕ of the input current, as a function of q. Fig. 8 D = 0.5. L i /L versus parameter q.

340 Şerban Bîrcă-Gălăţeanu 10 Fig. 9 D = 0.5. Ac to dc voltage transfer factor M, as a function of parameter q. Fig. 10 V DRM /V o versus diodes' duty ratio D. ringing is always present on the diode and capacitor current waveforms but not on the inductance current waveforms. This ringing is mainly due to the inductance introduced by the current probe itself and should be much lower in a short-leads

11 Low peak current class E resonant full-wave low dv/dt rectifier 341 operational circuit. Ringing is, proportionally, larger when R L value is larger, especially on the input voltage v. The input voltage has very low harmonics. Even harmonics are absent due to the half-wave symmetry. Input voltage rms value was measured with a Rohde & Schwarz wide-band voltmeter. The normalized output-power capability is : c p = I o V o /(I DM V DRM ) = V o / V DRM, (14) because I DM = I o. Peak reverse diode voltage V DRM is almost half as high as in the half-wave low dv/dt rectifier [3, 5, 9, 10]. Lower V DRM allow the use of Schottky diodes, which are, generally, low-voltage devices. As in the other Class E rectifiers, using D 0.5 is an optimum choice in order to have low diode peak voltage. If the input reactive current is to be minimized, the rectifier is to be driven at frequencies lower than f 0 / 2 [11]. a b 4. CONCLUSIONS A class E resonant full-wave voltage-driven low dv/dt rectifier has been analyzed. Its peak diode current is always equal to the load current. Peak diode reverse voltage is almost half as high as in half-wave rectifiers. Major parasitic reactive elements are included in the rectifier topology. Basic parameters of the circuit were derived as functions of the diodes' duty factor, using the time-domain analysis. c Fig. 11 Current and voltage waveforms (R L = 25 Ω): a) i D (200 ma/div) and v D (5 V/div); b) i D1 (200 ma/div) and v D1 (10 V/div); c) i (100 ma/div) and v (20 V/div).

342 Şerban Bîrcă-Gălăţeanu 12 Input current has low level odd harmonics only. When D 0.5, normalized load resistance Q and the voltage transfer factor M have lower values and the input inductance has values closer to 1 as the duty factor is higher. Best parameter values are obtained at f around f 0 / 2, for D = 0.5 0.6 or D = 0.5 and parameter q = 0.8 to 1, that is around the boundary between the two operating modes. The user must choose D value according to the application constraints, then find the other parameters from the graphs. Component values follow. Experimental waveforms and measurements agree with simulations and calculations. Received on 11 October, 2006 REFERENCES 1. W.A. Nitz et al., A new family of resonant rectifier circuits for high-frequency DC-DC converter applications, Proc. Applied Power Electronics Conference APEC, New Orleans, LA, 1988, pp. 12 22. 2. M.K. Kazimierczuk, J.Józwik, Class E zero-voltage-switching and zero-current-switching rectifiers, IEEE Trans. Circuits Syst., 37, 3, pp. 436 444, 1990. 3. A. Ivascu, M.K. Kazimierczuk and S. Bîrcă-Gălăteanu, Class E resonant low dv/dt rectifier, IEEE Trans. Circuits Syst., 39, 8, pp. 604 613, 1992. 4. A. Reatti, M.K. Kazimierczuk, R. Redl, Class E full-wave low dv/dt rectifier, IEEE Trans. Circuits Syst., 40, 2, pp. 73 84, 1993. 5. S. Bîrcă-Gălăteanu, J.-L. Cocquerelle, Class E half-wave low dv/dt rectifier operating in a range of frequencies around resonance, IEEE Trans. Circuits Syst., 42, 2, pp. 83-94, 1995. 6. S. Bîrcă-Gălăteanu, A. Ivascu, Class E low dv/dt and low di/dt rectifiers, IEEE Trans. on Circuits and Systems 1 : Fundamental Theory and Appl-s, 48, 9, pp. 1065 1074, 2001. 7. N.O. Sokal, A.D. Sokal, Class E a New Class of High-Efficiency Tuned Single-Ended Switching Power Amplifiers, IEEE Journal of Solid-State Circuits, 10, 3, pp. 168 176, 1965. 8. J.J. Jòzwik, M.K. Kazimierczuk, Analysis and Design of Class-E 2 dc/dc Converter, IEEE Trans. Ind. Electronics, 37, 2, pp. 173 183, 1990. 9. A. Ivascu, S. Bîrcă-Gălăteanu, Class E resonant full-wave low dv/dt rectifier driven by a current generator, Proc. Power Conversion and Intelligent Motion Conf. PCIM, pp. 465 474, Nürnberg, Germany, 7 9 June 1997. 10. S. Bîrcă-Gălăteanu, A. Ivascu, Class E resonant full-wave low dv/dt rectifier driven by a voltage generator, Power Conversion and Intelligent Motion Conf. PCIM, paper PC 6.3, Nürnberg, Germany, 22 24 June 1999. 11. S. Bîrcă-Gălăteanu, Low peak current class E resonant full-wave low dv/dt rectifier driven by a voltage generator, 30 th annual Power Electronics Specialists Conference PESC 99, Conference Record 1, pp. 469 474, Charleston, SC, 27 30 June 1999.