An Improvement on Extended Impedance Method towards Efficient Steady-State Analysis of High-Frequency Class-E Resonant Inverters

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1 An Improvement on Extene Impeance Metho towars Efficient Steay-State Analysis of Hih-Frequency Class-E Resonant Inverters Junrui Lian School of Information Science an Technoloy ShanhaiTech University, Shanhai 11, China The 18 International Power Electronics Conference Abstract The extene impeance metho (EIM) is an efficient an intuitive tool for the esin an analysis of power switchin circuits with active components, nonlinear parasitic components, etc., uner hih-frequency operation. In this paper, we propose an improvement to the EIM by simplifyin the constitutive circuit equations in the iterative calculation, such that to spee up the steay-state searchin process. Takin the class-e resonant inverter, which is riven by a practical MOSFET switch, as the testbe, we comparatively evaluate the computational efforts of this improve EIM, the conventional EIM, an PSpice simulation, uner the same circuit conitions. The result shows that the propose metho can reatly reuce the simulation time. Therefore, the improve EIM can be use as an efficient tool for the analysis, esin, an optimization of hih-frequency power electronics. I. INTRODUCTION Nowaays, in ifferent stuies of power electronics, there is an obvious tren workin towars the hih-frequency an hih-power-ensity power converters. One of the rivin force is the popularization of the hih-spee switchin semiconuctors, such as SiC an GaN; the other is the eman of more compact an more capable power converters. For example, in the wireless power transfer systems, by raisin the switchin frequency from several hunre khz 1] to several MHz ] 4], the power ensity of the circuit can be increase; the size of passive components can be reuce. The sinle-ene class-e resonant inverter, which is also calle class-e power amplifier (PA), was propose in ], 6]. It is known for its simple structure, hih efficiency, an hih-frequency operation. In the ol ays, when the switchin frequency is as low as several hunre khz, the nonlinearity of the parasitic components can be nelecte. The esin of class-e PA usually starts with the circuit equations an waveform equations in the time-omain 7] 1]. As the switchin frequency ets hiher, the electrical susceptance of the nonlinear junction capacitor in the switchin transistor becomes larer an ominant in the shunt capacitance. In such hih-frequency scenario, the effect of nonlinear parasitic components, as well as other lare-sinal behavior, must receive sufficient consieration in the esin an analysis of power converters 11]. Analytical waveform solutions provie intuitions towars the conceptual esin 5]. However, the close-form equations i v sw (t) R i s C s v C r sw (v, v ) i MOSFET switch equivalent c s (v ) r b (v ) v C L 1 C 1 R 1 L i sw i s i b i C i 1 i L c Components iscusse in the basic EIM base class-e PA analysis 1]. Nonlinear components inclue in 13]. More MOSFET equivalent components an revise switch moel inclue in 14]. Fi. 1. Equivalent moel of the class-e resonant inverter consierin the MOSFET parasitic components. can harly express all the etails in practical circuits, in particular, when the effect of parasitic components in semiconuctor becomes sinificant an non-inorable. Numeric simulation or analysis are complementary esin techniques towars the etaile performance. The numeric esin methos can be classifie into time-omain methos 1] an frequency-omain methos 1] 14]. The extene impeance metho (EIM) 1] 14] was propose for realizin the steay-state esin numerically aroun the linear specifications, which are obtaine with the establishe metho 6]. The harmonic base EIM is ifferent from the conventional harmonic balance metho. In EIM, the characteristics of all time-varyin an nonlinear components are reare as special impeances, whose alorithm of combination is compatible with the funamental linear ac circuit analysis. Therefore, it is more intuitive towars nonlinear circuit analysis. This paper iscusses an improvement to the EIM base on the moifie noal analysis (MNA) towars a more efficient esin process of class-e PA uner hih-frequency operation. II. THEORY The equivalent moel of class-e PA usin practical MOS- FET switch is shown in Fi. 1 14]. c, L, an C are the c supply voltae, choke inuctance, an shunt capacitance, 18 IEEJ 41

2 The 18 International Power Electronics Conference I 1 sw Y Y s Y Y L (n) Y Σ I L c Fi.. Simplifie circuit network for moifie noal analysis (MNA). respectively. R 1, C 1, an L 1 form the resonant branch. The components within the ashe frame forms the equivalent of a practical MOSFET. c s an r b represent the nonlinear rainto-source junction capacitance an boy ioe, respectively, whose effects uner ifferent operations were iscusse in 13]; R, C s, C, an r sw are the ate resistance, ate-tosource capacitance, ate-to-rain capacitance, an switchin resistance, respectively, whose effects uner ifferent ate rive voltae were iscusse in 14]. sw provies the ate rive to the MOSFET. Since all the components in Fi. 1 can be reare as impeances, whose expressions have been extene from complex scalars to complex matrices 1]. The constitutive circuit equations can be obtaine accorin to the conventional laws in linear network analysis, e.., KCL, KL. Such network was successfully formulate by usin the loop current metho in 14]. After 14], research effort has been continuously mae with the followin two purposes: 1) Make the EIM base analysis more compatible to the establishe circuit simulation proram, e.., SPICE, such that the netlist file SPICE can be reaily reuse in the EIM base solver. ) Further reuce the computational effort of EIM, in particular, in the iterative state-upate proceure. As the moifie noal analysis (MNA) is extensively use in formulatin the circuit network in most existin simulation proram 15], such as SPICE, this paper aopts the MNA for formulatin the constitutive circuit equations as follows 1 I I L = Y Y E Y Y Σ Y Y Y (n) Σ Y L Y L Y L E E E 1 sw c, (1) where TABLE I PARAMETERS IN THE CASE STUDIES OF CLASS-E PA. Parameter alue c sw amp. of v sw(t)] ±7. f sw freq. of v sw(t)] 6.78 MHz D 5 % L 1 µh C pf L µh C pf R Ω MOSFET IRF51 The circuit parameters are obtaine base on the revise esin equations in 6]. The IRF51 specifications were provie in 14]. Y (n) Σ Y Σ = Y + Y s + Y, () Y (n) Σ = Y + Y (n) Σ + Y L, (3) = Y(n) sw + Y (n) s + Y(n) b + Y C +(Z R1 + Z C1 + Z L1 ) 1. The relations are obtaine accorin to the simplifie circuit network shown in Fi.. The four voltae noes other than the roun are enote as the vectors 1,,, an. The c supply an ate rive voltae are taken as two eneral voltae sources, whose flowin-throuh currents are I an I L, respectively. The superscript (n + 1) enote the results in the (n + 1)th iteration. The corresponin amittance in the matrix of (1) is shown in Fi.. The voltae an current vectors, whose imensions are (K+1) 1, express the circuit states in the frequency omain with a harmonic number of K; the matrices, whose imensions are (K + 1), express the characteristics of the components in the form of amittances. Within those amittances, Y Σ is the only nonlinear one. As we can observe from Fi. 1 an (4), the nonlinearities come from the MOSFET switchin, whose characteristic is enote by Y sw, rain-to-source junction capacitance enote by Y s, an boy ioe enote by Y b. With (1), the four voltae an two current unknowns can be immeiately solve by oin the matrix inversion. A Matlab proram is evelope base on the sparse matrix manipulations for simulatin a class-e PA, whose specifications are liste in Table I. The execution summary by usin full matrix inversion alorithm is shown in Fi. 3. It can be observe from Fi. 3 that the matrix inversion takes the lonest time to execute compare to other lines. On the other han, in the six unknown voltaes an currents, actually only an upatin the amittance matrices Y sw (4) are useful information for, Y s, an Y b in the next roun of calculation. A approximatin solution can be formulate by takin the matrix in (1) as a 6 6 block 413

3 The 18 International Power Electronics Conference Fi. 3. The execution time of the Matlab proram (harmonic no. K = ) base on the full matrix inversion EIM alorithm (the eeper re the hihlihtin line, the more time for the execution). Fi. 4. The execution time of the Matlab proram (harmonic no. K = ) base on the improve EIM alorithm (the eeper re the hihlihtin line, the more time for the execution). matrix. By solvin the six symbolic linear equations, we can obtain ] 1 Y (n) Σ Y Σ Y ] (5) Y L Y c + Y Y (n) Σ sw ] 1 Y (n) Σ Y Σ Y (Y L Y Σ c + Y Y sw ) It shoul point out that, from the funamental principle of linear alebra, the expressions of (5) an (6) o not riorously hol, as there miht be cross-couplin terms across the blocks. Here, we o the approximation an try to use such metho to reuce the computational effort towars the circuit optimization. The case stuy with the class-e specifications of Table I by usin such approximation actually ives a similar voltaes as that usin the full matrix inversion (will be shown in the next section). On the other han, as shown in Fi. 4, the most critical line in the new alorithm is also the matrix inversion step. Yet, the computational time of this critical line in the new proram is about 58% of that in the full matrix solution. Therefore, by only calculatin the useful information, the computation time can be further reuce towars more efficient EIM base analysis. (6) III. DESIGN PROCEDURES An example esin case is provie here for valiatin the waveform analysis of EIM, an also escribin the esin proceures with EIM base circuit optimization. Fi. 5 shows four testin circuits in the esin proceures. 1) Obtain the component values with the conventional esin laws or uielines consierin an ieal switch an linear components. ) Replace the ieal switch with practical transistor moel. Simulate the chanes in operation. 3) Desin the EIM base numerical optimization to approach the esin objective, e.., ZS an ZCS in the class-e circuit, constant output power, or hihest conversion efficiency. 4) Simulate the re-tune parameters an comparatively evaluate the esin feasibility. The results in Fi. 6 show that the class-e nominal ZCS conition is violate when the ieal switch moel (whose result is shown by the ash-ot-ot lines) is replace by a practical MOSFET moel (whose result is shown by the soli lines). Owin to the hih computational efficiency of EIM, we manae to evelop a erivative-free alorithm 1], 13] for re-tunin the nonlinear class-e circuit back to the vicinity of the nominal conitions. The full matrix optimization leas to the optimize capacitance pairs C = 411 pf an C 1 =

4 The 18 International Power Electronics Conference Linear shunt capacitance (base on the Sokal's equations) With practical MOSFET moel (nonlinear capacitance Cj=366.5pF) L6 L e-6 L L e-6 c3 8 S Sbreak C e-1 R4 1.3 c1 7 sw1 IRF51 C e-1 R1 1.3 (a) 1 = -7 1 = -7 = 7 C7 = 7 TD = e-1 TD = TR = TR = TF = TF = (b) PW = e-8 PW = e-8 PER = e-7 PER = e-7 C e-1 With practical MOSFET moel (Optimize with full matrix EIM, K=1) L8 L e-6 With practical MOSFET moel (Optimize with reuce matrix EIM, K=1) L1 L e-6 c4 9 sw3 IRF51 C e-1 R5 1.3 c5 C1 R e sw4 IRF51 (c) 1 = -7 = 7 TD = TR = TF = PW = e-8 PER = e-7 C e-1 () 1 = -7 = 7 TD = TR = TF = PW = e-8 PER = e-7 C e-1 Fi. 5. Four testin conitions for the comparative stuy. (a) Class-E circuit with linear shunt capacitance an ieal switch. (b) The ieal switch is replace by a MOSFET moel. (c) Re-tune circuit with full matrix EIM (C = 411 pf, C 1 = 819 pf). () Re-tune circuit with reuce matrix EIM (C = 43 pf, C 1 = 796 pf). oltae (volt) oltae (volt) Ieal switch w/ linear cap. MOSFET before optimization MOSFET w/ full matrix opt. MOSFET w/ reuce matrix opt. Time (µs) (a) PSpice results Ieal switch w/ linear cap. MOSFET before optimization MOSFET w/ full matrix opt. MOSFET w/ reuce matrix opt..5 1 Cycle (b) EIM results Fi. 6. Simulation waveforms obtaine with PSpice an EIM. waveforms obtaine uner the two optimization alorithm are very close an almost overlap in Fi. 6(b). Both the full matrix an reuce matrix can successfully fulfill the esine task. I. CONCLUSION This paper has consiere a possible improvement on the extene impeance metho (EIM) towars the more efficient frequency-omain simulation an optimize esin of hihfrequency an hih-power-ensity power electronics. In the class-e circuit example, since only the ate an rain voltaes have an effect over the nonlinear parasitic components in the switchin transistor, the execution time can be reuce by eliminatin the other useless information urin the iterative state upate process. The performance evaluator in Matlab shows the superiority of the propose improvement towars the customize analyses an esins of power electronics.. ACKNOWLEDGMENT The work escribe in this paper was supporte by the rants from National Natural Science Founation of China (Project No ) an ShanhaiTech University (Project No. F ). REFERENCES pf, as illustrate in Fi. 5(c); the waveform result is shown by the ash lines in Fi. 6. It shows that the ZS an ZDS conitions are recovere uner the tune C an C 1. The reuce matrix optimization leas to another optimize capacitance pair C = 43 pf an C 1 = 796 pf, which are very close to the full matrix optimization results. The 1] P. C. K. Luk, S. Alhaher, W. Fei, an J. F. Whiborne, State-space moelin of a class E converter for inuctive links, IEEE Transactions on Power Electronics, vol. 3, no. 6, pp , June 15. ] T. Naashima, X. Wei, E. Bou, E. Alarc?n, M. K. Kazimierczuk, an H. Sekiya, Analysis an esin of loosely inuctive couple wireless power transfer system base on class-e c-c converter for efficiency enhancement, IEEE Transactions on Circuits an Systems I: Reular Papers, vol. 6, no. 11, pp , Nov

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