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IECON2015-Yokohama November 9-12, 2015 Analysis of Series-Parallel Resonant Inductive Coupling Circuit using the Two-port Network Theory Kunwar Aditya, Student Member, IEEE, Mohamed Youssef, Senior Member, IEEE, and Sheldon S. Williamson, Senior Member, IEEE Smart Transportation Electrification and Energy Research (STEER) Group Advanced Storage Systems and Electric Transportation (ASSET) Laboratory UOIT-Automotive Center of Excellence (UOIT-ACE) Department of Electrical, Computer, and Software Engineering Faculty of Engineering and Applied Science University of Ontario-Institute of Technology ACE-2025, 2000 Simcoe Street North Oshawa, ON L1H 7K4, Canada Tel: +1/ (905) 721-8668, ext. 5744 Fax: +1(905) 721-3178 EML: sheldon.williamson@uoit.ca URL: http://www.engineering.uoit.ca/; http://ace.uoit.ca/ Abstract²In this paper thorough analysis of Series-Parallel resonant inductive coupling (SPRIC) has been presented using two-port network theory. Characteristic of SPRIC has been derived and plotted against load variation and frequency variation in MATLAB. Concept of forced resonance and natural resonance has been discussed. Important expressions such as current transfer ratio, voltage transfer ratio, maximum efficiency etc. has been derived and presented. To study the circuit behavior of the SPRIC a prototype has been built in the lab. Experimental results along with simulation results has been presented to verify the derived expression and characteristics of series-parallel topology. Keywords²Circuit theory, electric vehicles, inductive energy storage, magnetic resonance, quality factor. I. INTRODUCTION Inductive power transfer (IPT), which while theorized in the 1800s, has only seen relatively recent use in the last 15-20 years most notably for EVs battery charging application [1, 2]. Inductive coupling can provide power to the battery and at the same time provide galvanic isolation and hence are advantageous than conventional energy transmission techniques using wires and connectors from the point of safety, reliability, low maintenance, and long product life [3, 4]. It is EDVHG RQ IXQGDPHQWDO SULQFLSOH RI DPSHUH¶V circuital law and IDUDGD\¶V ODZ of electromagnetic induction on which conventional transformers and induction motor works. However, in case of transformer and induction motor primary and secondary are placed in close proximity therefore coupling coefficient is high usually 90-98%. For battery charging application via IPT system the large air gap is desired to minimise protrusion from either side and to allow good clearance between the vehicle base and the ground therefore leakage flux is more and hence is low (usually 10-20%) [5]. Poor coupling in loosely coupled system leads to poor transfer of power. To improve coupling and compensate leakage inductance, capacitive compensation in primary and secondary 978-1-4799-1762-4/15/$31.00 ©2015 IEEE windings is required. For this purpose capacitor can be connected in either series or parallel of both winding giving total of four topology namely Series-Series (SS) topology, Series-Parallel (SP) topology, Parallel-Series (PS) topology and Parallel-Parallel (PP) topology. Since capacitor is made to resonate with self-inductance of primary and secondary coil it is more appropriate to call it resonant inductive coupling rather than merely inductive coupling. All topology has certain advantage and disadvantage and their choice mainly depends on type of application. There has been research to find out most suitable topology for particular application such as battery charging [6-10]. While there are other topologies that may be used, the parallel secondary architecture is beneficial for battery charging because of its constant current source characteristics, which occurs if primary current is maintained constant [11-13]. In this paper analysis of Series-Parallel resonant inductive coupling has been done using two port network with the aim of understanding its circuit behavior when fed from constant input voltage. For the analysis a prototype of SP resonant coils has been built in the lab. Absence of core makes it similar to linear transformer and therefore linear circuit theory can be applied for the analysis. Paper has been arranged in following manner: Section II explains the concept of forced resonance and natural resonance, in section III characteristics of SP topology has been derived, In section IV efficiency and condition for maximum efficiency has been discussed, in section V simulation and experimental results has been presented, Section VI concludes the paper. II. FORCED RESONANCE AND NATURAL RESONANCE Secondary compensation is done to improve the power transfer capability of the system and Primary capacitor is so chosen that the impedance as seen from the source side is purely resistive in nature so as to ensure that the input current and voltage are in phase, and therefore it reduces the VA 005402 rating of supply [14]. Fig. 1. Shows the equivalent circuit of series-parallel topology. Ip Rp Cp Is Rs Ls z12Is Ic z21Ip Cs RL D R1 Fig.2. Series-Parallel RC equivalent circuits Here, ߱ ൌ ଵ (7) ඥೞ ೄ (1) ଵାఠమ ோಽమ ೞమ ଵାఠమ ோಽమ ೞమ Resonant frequency for loop ABCD is given by [15]: ߱ is the natural resonant frequency of the secondary. This resonant frequency is independent of load and is fixed for selected frequency. To analyze the behavior of this circuit it can be seen as a series RLC circuit operating at resonance in which output is picked off the secondary capacitance, Cs in order to take advantage of the fact that at resonance the amplitude of the voltage across the capacitor is Q (= ఠோೞ ) ೞ times the amplitude of the source voltage . Fig. 4 (a) and (b) gives the phasor diagram of Fig. 5 with and without capacitor Cs for same load. Secondary resistance has been neglected in drawing these phasor for clear understanding of circuit. Rs ோಽ C Fig. 3. Secondary equivalent circuit of SP topology C1 Cs Vs Cs RL Vs Parallel Cs and RL can be replaced by its series equivalent of C1 and R1 as shown in fig.2: ఠమ ோಽమ ೞమ Io IR Fig.1. Equivalent circuit of Series-Parallel Topology ܥଵ ൌ Is Rs j߱MIp Vp ܴଵ ൌ B A j߱IsLs ܥ௦ Ic Is V (2) Then total impedance of secondary is given by: ்ݖൌ ଵ ఠభ ݆߱ܮ௦ ܴ௦ ܴଵ (3) At forced resonance frequency ߱oF, ்ݖshould be purely resistive. µ)¶ VXEscript is used to denote values at forced resonant frequency. Equating imaginary terms in eq. (3) gives the forced resonant frequency as: ߱ி ൌ ට ଵ ೞ ೞ െ ଵ ோಽమ ೞమ ݖோିி ൌ ோభ ାோೞ (5) Since reflected impedance is purely resistive, capacitive compensation in primary, Cp-F is independent of either Mutual inductance or load and is given by: ܥିி ൌ ଵ మ ఠಷ (a) j߱IsLs V (4) This type of tuning can be used in case of fixed load system. With ߱oF, reflected impedance is purely resistive and is given by: మ ఠಷ ெమ Vs Io (6) Therefore tuning in primary is not disrupted with the variation of mutual inductance or load and system remains perfectly tuned. However, one can observe from eq. (4) ߱ி is dependent on load so value of Cs need to be adjusted each time load changes and therefore to eliminate this problem, instead of tuning entire secondary circuit we tune the loop ABCD as shown in fig.3. Io Vs (b) Fig. 4. Phasor diagram of Fig. 3, (a) with capacitor Cs (b) without capacitor Cs V is input voltage of circuit i.e. j߱MIp in this case. From phasor diagram one can observe that for given load ȁܸȁ ൏ ȁܸ௦ ȁ for compensated secondary (with Cs) and ȁܸȁ ȁܸ௦ ȁ for uncompensated secondary (without Cs). This means that for a given load, addition of Cs reduces the input voltage. From practical point of view this means if load increases parallel compensation can be done to maintain the same load voltage without increasing the stress on input voltage. Value of primary capacitance at natural resonance frequency, ߱ is given as: ܥ ൌ మೞ ೞ ൫ ೞ ିெమ ൯ ൌ ଵ మ ሺଵି మ ሻ ఠ ು (8) In the derivation of ܥ effect of primary and secondary coil resistance has been neglected. 005403 However since ଵ ඥೞ ೞ ଵ ටೞ ೞ ିோಽమ ೞమ secondary will not operate at unity factor. ோಽ ߱ is selected as resonant frequency rather than ߱ி . Values of Primary and Secondary Capacitance Natural Forced resonance Resonance ͳ ߱ி ܮ ͳ ߱ଶ ܮ ሺͳ െ ݇ ଶ ሻ ଶ ଶ ͳ ܮ௦ Ͷ߱ி ቌͳ ඨͳ െ ቍ ଶ ܴଶ ʹ߱ி ܮ௦ ͳ ߱ଶ ܮ௦ Primary Capacitance Secondary Capacitance III. CHARACTERISITCS OF SERIES-PARALLEL TOPOLOGY Fig.5 gives the two-port model for SP topology shown in Fig.1 in terms of z parameters. Ip z11 ܴଵ ൌ ܥଵ ൌ ோಽ ଵାఠమ ோಽమ ೞమ ଵାఠమ ோಽమ ೞమ ఠమ ோಽమ ೞమ z12Is' Vp z21Ip Vs R1 Two Port Network Fig.5. Two-Port model of SP topology ଵ ఠ ݆߱ܮ ܴ (9) ݖଵଶ ൌ ݖଶଵ ൌ ݆߱ܯ (10) TABLE II. (14) ܥ௦ (15) Using z-parameters, equations that describes the circuit in Fig.5 can be written as: (16) ܸ ൌ ݖଵଵ ܫ ݖଵଶ ܫ௦ᇱ (17) ܸ௦ ൌ ݖଶଵ ܫ ݖଶଶ ܫ௦ᇱ (18) ܸ௦ ൌ െܫ௦ᇱ ܼ Solving equations (16) to (18) gives the characteristics of SP topology. These characteristics at any frequency ߱ and at resonance frequency ߱o has been arranged in Table 2. To minimize the losses, the IPT coils are designed to have winding resistance as low as possible and are usually in the milliohm range. If that is the case then their product ܴ௦ ܴ ̱Ͳ and can be neglected. So from table 2 one can write: ݖ ൌ ݖ ൌ Is' z22 Here, C1 and R1 are series equivalent of parallel Cs and RL and is given by: ଵ ݆߱ ܮ ఠ ெమ ோಽ C1 Here, ݖଵଵ ൌ (11) (12) (13) ఠభ Table 1 gives the value of primary and secondary capacitance at forced resonance frequency ߱ி and natural resonance frequency, ߱ . From Table 1 one can observe that at forced resonance frequency value of primary capacitance can become imaginary at certain combination of frequency and load ఠ resistance i.e. at certain load quality factor ( ೞ ). Therefore TABLE I. Values of Capacitors ݖଶଶ ൌ ݆߱ܮ௦ ܴ௦ ܫ௦ᇱ ൌ െܫ௦ ଵ ݖ ൌ ܴଵ మೞ ൌ ݇ଶ ು ೄ ெమ ோಽ మೞ െ ఠ ெమ ܴ (19(a)) ఠ ெమ RHIOHFWHG LPSHGDQFHDW UHVRQDQFH LH µ- Ԣ component in ೞ Zin is capacitive component. 0RUHRYHU WKLV FDSDFLWLYH UHDFWDQFH ZLOO RSSRVH WKH SULPDU\ LQGXFWDQFH /S FDXVLQJ D UHGXFWLRQLQ=LQWKLVPHDQVDODUJHUFDSDFLWRUZLOOEHUHTXLUHG WR WXQH RXW WKH FDSDFLWLYH 9$5 ORDGLQJ RQ WKH SULPDU\ E\ VHFRQGDU\ FRLO DV FRPSDUHG WR WKH YDOXH UHTXLUHG LQ VHULHV VHULHVFRPSHQVDWLRQ>@ After neglecting winding resistance and using value of primary and secondary capacitance at resonance voltagetransfer characteristics ݄௩ି௦ of SP topology at resonance can Characteristics of Series-Parallel topology Characteristic At generalized frequency, ߱ At resonance frequency, ߱o hi-sp = Is/Ip ݖଶଵ ݖ ݖଶଶ ݆߱ ܯሺͳ ݆߱ ܴ ܥ௦ ሻ ܴ ሺܴ௦ ݆߱ ܮ௦ ሻሺͳ ݆߱ ܴ ܥ௦ ሻ hv-p=Vs/Vp ݖଶଵ ܼ ݖ ݖଵଵ ݖଶଶ ݖଵଵ െ ݖଵଶ ݖଶଵ Zin=Vp/Ip Gsp=Is/Vp ݖଵଵ െ ݖଵଶ ݖଶଵ ݖ ݖଶଶ ݖଶଵ ݖ ݖଵଵ ݖଶଶ ݖଵଵ െ ݖଵଶ ݖଶଵ (19) ೞ ݆߱ ܯ ሺ݆߱ ܮ ଵ ఠ ܴ ݆߱ ܮ ܴ ሻሺ ோಽ ଵାఠ ೞ ோಽ ோಽ ଵାఠ ೞ ோಽ ܴ௦ ݆߱ ܮ௦ ሻ ߱ଶ ܯଶ ͳ ߱ଶ ܯଶ ሺͳ ݆߱ ܥ௦ ܴ ሻ ݆߱ ܥ ܴ ሺܴ௦ ݆߱ ܮ௦ ሻሺͳ ݆߱ ܥ௦ ܴ ሻ ݆߱ ܯ ሺ݆߱ ܮ 005404 ଵ ఠ ܴ ሻሺ ோಽ ଵାఠ ೞ ோಽ ܴ௦ ݆߱ ܮ௦ ሻ ߱ଶ ܯଶ be simplified as: ݄௩ି௦ ൌ ೞ ൌ ೞ ெ (20) From equation (20) one can observe that output voltage is independent of load resistance RL and will be constant as long as mutual inductance M is maintained constant and therefore SP topology acts as ideal voltage source. Constant voltage characteristics is useful in multiple outlets system and those system which has intermediate dc bus such as modern variable speed ac drives or for multiple outlet [12]. As opposed to SS topology SP topology is not short circuit proof. A. Characteristics with respect to frequency: This subsection study the variation of characteristics w.r.t. frequency for different load. As load resistance increases (i.e. load decreases) load quality factor QL increases and primary quality factor QP decreases as seen in eq. (21) and (22) [12]. ܳ ൌ ܳ ൌ ோಽ (21) ఠ ೞ ఠ ು మೄ ெమ ோಽ (22) Fig.6 shows the voltage transfer characteristics plotted w.r.t. frequency. For low value of QL, characteristics has only one peak since primary quality factor dominates and SP topology behaves as single tuned circuit and is more sensitive to frequency change. As the load quality factor increases characteristics acquires U shape because topology behaves as double tuned circuit. In the middle of U region, characteristics becomes more sensitive to variation in load but less sensitive to variation in frequency. If resonant tank are properly designed it can give stable gain over a large frequency band (bandwidth) as is evident from curve for QL =1. This also shows that at resonant frequency voltage transfer characteristics is almost insensitive to load variation which was established earlier through equation (20). Fig.7. Current Transfer Characteristics Vs Frequency Fig.7 shows the plot of current transfer characteristics w.r.t. frequency. It shows that at resonance frequency, hi-sp has maximum value and is sensitive to load variation however as frequency increases away from resonant frequency hi-sp becomes independent of load variation. Fig.8 shows the plot of total input impedance w.r.t. frequency. From the plot one can observe that for low value of QL plot is similar to any series RLC circuit where total impedance decreases as frequency increases and becomes minimum at resonance frequency, and henceforth increases as frequency increases. This is due to the fact that at low QL primary quality factor dominates and circuit behaves as single tuned circuit. However as QL increases Qp decreases and circuit behaves as double-tuned circuit which explain the nonlinear nature of plot around resonant frequency for high QL. Fig.8. Input impedance Vs Frequency Fig.6. Voltage transfer characteristics Vs Frequency B. Characteristics with respect to load: In this subsection characteristics variation w.r.t. load variation has been discussed. 005405 M (μH) 31.4 Rp 0.2244 Rs 0.1174 Cp (μF) 0.56026 Cs (μF) 2.468 fo(kHz) 21.5 Air-gap(mm) 25 Fig.9. Characteristics Vs load Ip From Fig.9 one can observe that hv-p is almost constant irrespective of change in load resistance as was established earlier in eq. (20). Zin and hi-sp increases as load increases. Vdc Efficiency of Series-Parallel resonant inductive coupling is given by [3]: ߟൌ ூమ ோಽ (21) మ ሺ௦௦௧௬ା௧ௗ௦௦௧ሻ ூು ோಽ మ మ మ మ (22) మ ೃು ೃ ಽ ೃು ೃು ೃ ೃ ೃ ೃ మೃ ೃ ೃ ோೄ ାோಽ ା మ ೄమ ା ೄ మ ା ర మೄ ಽమ ା ಽమ ೄ మ ು ା ಽమ ೄ ಾ ഘ ಾ ഘ ಾ ഘ ಽ ಾ ഘ ಽమ ೄ ೄ Value of load resistance which gives maximum coupling efficiency and value of maximum efficiency is given by eq. (23) and (24) respectively. ܴି௫ ൌ ߱ ܮௌ ට ଵାொమమ ோಽషೌೣ ೃು ೃమ ಽమ ೃು మೃ ೃ ೄ ೄ ଶቆ మ మ ା మ ቇା൬ మೄ మು ାଵ൰ோಽషೌೣ ାோೄ ಾ ഘ ಾ ഘ ಾ In eq. (23) K = Lp M Ls Cs RL Vs ெ ඥೄ ು , Q1 = Fig. 10. Circuit diagram for simulation and experiment. To verify the voltage-source characteristics of Series-Parallel topology the SP topology was simulated in MATLAB and Simulation results were experimentally verified. Fig. 10 gives the circuit diagram for the circuit used in Simulation and experiment and Fig.11 shows the experimental setup. (23) ଵାொమ ொభ ߟ௫ ൌ Cp VP IR Rs EFFICIENCY OF SP TOPOLOGY IV. ߟ ൌ Is Rp ఠ ು ோು (24) and Q2 = ఠ ೞ ோೄ are coupling coefficient, intrinsic quality factor of primary circuit and intrinsic quality factor of secondary circuit respectively. V. SIMULATION AND EXPERIMENTAL RESULTS For the proof of concept of theories presented in this paper a prototype of air cored IPT coils to deliver 30 watt at 20 kHz and 20mm air gap, has been built in the lab. At high frequency operation equivalent series resistance (ESR) of wire and therefore copper loss increases due to the skin effect and proximity effects. To mitigate this loss, braided enamelled conductors known as Litz wire must be used for making coils. Therefore to build coils, Litz wire 90/38 SPN SN (90 strands, 38 awg each strand) has been used. For the experiment, compensation capacitors with very low ESR Polypropylene film type have been selected. Parameters of the coils along with compensating capacitors has been shown in table 3. TABLE III. Fig.11. Experimental setup. To show that SP topology gives load independent output voltage when fed from constant input voltage and fixed frequency supply, load was varied in the sequence [1.2, 2.21, 3.45, 4.57, 5.34, and 6.23] ohm while dc link of inverter was fixed at 15 volts and switching frequency was kept 15 kHz. Due to practical availability, value of capacitor used in experimental setup were adjusted to Cp=0.56 μF and Cs=2.7 μF. Output voltage by input voltage (Vs/Vp) for simulation and experiment has been plotted in fig.13. Parameters of SP Topology Parameters Values Lp (μH) 142.12 Ls (μH) 22.25 005406 Simulation Experiment Ideal 0,7 0,6 Vs/VP 0,5 0,4 0,3 0,2 0,1 0 Ϭ Ϯ ϰ ϲ ϴ Load Resistance RL (ȍ) Fig. 12. Voltage transfer characteristics of SP topology In Fig.12 Ideal plot shows the voltage transfer characteristics for the case when winding resistance has been neglected. It shows that when winding resistance are neglected SP topology behaves as ideal voltage source. However due to coil resistance characteristics deviates from ideal condition as is evident from simulation and experimental plot in Fig. 13. There is difference in magnitude of voltage transfer characteristics for simulation and experiment. This is due to the fact that in simulation tuning was perfect in primary and secondary coil however in experiment ideal tuning was not possible due to practical availability of capacitors. Moreover switching time delay, parasitic elements such as switches and connecting wires inductance were not included in simulations however they are unavoidable during experiment. VI. CONCLUSIONS In this paper characteristics of Series-Parallel topology has been studied when primary is fed from constant input voltage. To simplify the analysis concept of two-port network has been utilized for deriving the characteristics of circuits. All the characteristics has been plotted and discussed with respect to load as well as frequency variations. One of the important characteristics of Series-parallel topology has been found to give constant-voltage output. This particular characteristics has been validated using MATLAB simulation and hardware implementation. Simulation and experimental results were found to be in close agreement. Limiting factor to the theory was found to be parasitic resistance such as coil resistance and connecting wire resistance, as well as deviation form ideal tuning condition. Characteristics not only establish the concept about circuit but also help in establishing the behaviour of the topology which in turn help in selecting the topology for a particular application. REFERENCES [1] J. Y. Lee, H. Y. Shen, K. C. Chan, "Design and implementation of removable and closed-shape dual ring pickup for contactless linear inductive power track system," in Proc. IEEE Energy Conversion Congress and Exposition (ECCE), Denver, CO, 15-19 Sept 2013, pp. 2219-2226. [2] .$GLW\D6:LOOLDPVRQ³$GYDQFHGFRQWUROOHUGHVLJQIRUDVHULHV-series compensated inductive power transfer charging infrastructure using DV\PPHWULFDO FODPSHG PRGH FRQWURO´ LQ Proc. IEEE Applied Power Electronics Conference and Exposition (APEC), Chalotte, NC, USA, 1519 March 2015, pp.2718-2724. [3] K. Aditya, S. Williamson, "Design considerations for loosely coupled inductive power transfer (IPT) system for electric vehicle battery charging - A comprehensive review," in Proc.Transportation Electrification Conference and Expo (ITEC), Dearborn ,Michigan , pp.16, 15-18 June 2014. [4] .$GLW\DDQG66:LOOLDPVRQ³&RPSDUDWLYHVWXG\RIVHULHV-series and series-parallel compensation topology for electric vehicles charging DSSOLFDWLRQ´ LQ Proc. IEEE International Symposium on Industrial Electronics, Istanbul, Turkey, pp. 426-430, 1-4 June 2014. [5] . $GLW\D % 3HVFKLHUD 0 <RXVVHI 6 6 :LOOLDPVRQ ³0RGHOOLQJ DQG calculation of key design parameters for an Inductive Power Transfer system using Finite Element Analysis- D FRPSUHKHQVLYH GLVFXVVLRQ´ LQ proc. IEEE Transportation Electrification Conference and Expo, 14-17 June 2015, Detroit, MI, pp.1-6. [6] - 6DOODQ - / 9LOOD $ /ORPEDUW DQG - ) 6DQ] ³2SWLPDO GHVLJQ RI ,&37 V\VWHPV DSSOLHG WR HOHFWULF YHKLFOH EDWWHU\ FKDUJH´ Trans. IEEE Industrial Electronics, vol. 56, no. 6, pp. 2140±2149, Jun. 2009. [7] & 6 :DQJ * $ &RYLF DQG 2 + 6WLHODX ³3RZHU transfer capability and bifurcation phenomena of loosely coupled inductive power transfer V\VWHPV´IEEE Trans. Industrial Electronics, vol. 51, no. 1, pp. 148±156, Feb. 2004. [8] = 3DQWLF 6 %DL DQG 6 0 /XNLF ³=&6 /&&-compensated resonant inverter for inductive-power-WUDQVIHU DSSOLFDWLRQ´ IEEE Trans. on Industrial Electronics, vol. 58, no. 8, pp. 3500±3510, Aug. 2011. [9] 03LQXHOD'&<DWHV6/XF\V]\QDQG3'0LWFKHVRQ³0D[LPL]LQJ dc-to-ORDG HIILFLHQF\ IRU LQGXFWLYH SRZHU WUDQVIHU´ IEEE Trans. on Power Electronics, vol. 28, no. 5, pp. 2437±2447, 2013. [10] + / /L $ 3 +X DQG * $ &RYLF ³$ GLUHFW DF-ac converter for inductive power-WUDQVIHU V\VWHPV´ IEEE Trans. on Power Electronics, vol. 27, no. 2, pp. 661±668, Feb. 2012. [11] N. Keeling, G. Covic, anG-%R\V³A unity-power-factor ipt pickup for high-SRZHUDSSOLFDWLRQV´ IEEE Trans. On Industrial Electronics, vol. 57, no. 2, pp. 744-751, Feb 2010. [12] C.-S. WDQJ 2 6WLHODX DQG * &RYLF ³Design considerations for a contactless electric vehicle EDWWHU\ FKDUJHU´ IEEE Trans. on Industrial Electronics, vol. 52, no. 5, pp. 1308-1314, Oct 2005. [13] G. Covic, J. T. Boys, A. 0:7DPDQG-&+3HQJ³Self tuning pickups for inductive power transfer," in Proc. IEEE Power Electronics Specialists Conference, June 2008, pp. 3489-3494. [14] K. Aditya, S. Williamson, "Comparative study of series-series and seriesparallel topology for long track EV charging application," in Proc. IEEE Transportation Electrification Conference and Expo (ITEC), Dearborn, Michigan, pp.1-5, 15-18 June 2014. [15] J. M. Burdio, F. Canales, P. M. Barbosa, F. C. Lee, "Comparison study of fixed-frequency control strategies for ZVS DC/DC series resonant converters," in proc. IEEE conference on Power Electronics Specialists Conference, vol. no.1, pp.427-432, 2001 APPENDIX A NOMENCLATURE : Vp= Primary voltage LP= Primary Self inductance CP= Primary capacitor LS= Secondary Self inductance CS= Secondary capacitor RL= Load resistance Ip= Primary current Is= Secondary current Io= Output current Vs= Secondary voltage hi-sp= Current transfer ratio of series-parallel topology hv-sp= Voltage Transfer ratio of Series-Parallel topology Zin= Total input impedance seen by source fo = Resonant frequency fsw = Switching frequency ߱ = Angular frequency ߱o= Angular resonant frequency k=Coupling Coefficient 005407