Stability and Performance Investigation of a Fuzzy-Controlled LCL

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Stability and Performance Investigation of a Fuzzy-Controlled LCL Resonant Converter in an RTOS Environment S. Selvaperumal, C. Christober Asir Rajan, and S. Muralidharan

Abstract—The resonant converter (RC) is finding wide applications in many space and radar power supplies. Among various RCs LCL, LCC, and LCL-T topologies are broadly used. This manuscript presents a comparative evaluation of steady-state stability of LCL, LCC, and LCL-T resonant configurations. Careful analysis favors LCL RC among the aforementioned three configurations since the stability region is good for the LCL RC over the other configurations. Also, this paper presents a comparative evaluation of proportional integral (PI) controller and fuzzy logic controller for a modified LCL RC. The aforementioned controllers are simulated using MATLAB and their performance is analyzed. The outcome of the analysis shows the superiority of fuzzy control over the conventional PI control method. The LCL RC is proposed for applications in many space and radar power supplies. Design, simulation, and experimental results for a 133-W, 50-kHz LCL RC are presented in this manuscript which provide high efficiency (greater than 89%) even for 50% of load. Efficiencies greater than 80% are obtained at significantly reduced loads (11%). In this paper, the applicability of the Philips advanced RISC machine processor LPC 2148 is also investigated for implementing the controller for an RC. Index Terms—DC–DC power converters, fuzzy control, power electronics, proportional integral (PI) control, resonant power converters, stability analysis.

I. INTRODUCTION N RECENT years, the design and development of various dc–dc resonant converters (RCs) have been focused for telecommunication and aerospace applications. It has been found that these converters experience high switching losses, reduced reliability, electromagnetic interference (EMI), and acoustic noise at high frequencies. The series and parallel resonant converter (SRC and PRC, respectively) circuits are the basic RC topologies with two reactive elements. A series parallel resonant converter (SPRC) is found to be suitable, due to various inherent advantages. The merits of SRC include better load ef-

I

Manuscript received March 28, 2012; revised May 22, 2012 and July 10, 2012; accepted August 8, 2012. Date of current version October 26, 2012. Recommended for publication by Associate Editor S. K. Mazumder. S. Selvaperumal and S. Muralidharan are with the Department of Electrical and Electronics Engineering, Mepco Schlenk Engineering College, Sivakasi 626005, Tamil Nadu, India (e-mail: [email protected]; [email protected]). C. C. A. Rajan is with the Department of Electrical and Electronics Engineering, Pondicherry Engineering College, Pillaichavadi 605014, Pondicherry, India (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2012.2214236

ficiency due to the series capacitor in the resonant network and the inherent dc blocking capability of the isolation transformer. However, the load regulation is poor and output-voltage regulation at no load is not possible by switching frequency variations. On the other hand, PRC offers better no-load regulation but suffers from poor load efficiency due to the lack of dc blocking for the isolation transformer. Hence, an RC with three reactive components is suggested for better regulation. The LCL tank circuit-based dc–dc RC has been experimentally demonstrated and reported by many researchers [1]–[12]. Borage et al. have experimentally demonstrated it with independent load when operated at a resonant frequency, making it attractive for an application as a constant voltage (CV) power supply. It has been found from the literature that the LCL tank circuit connected in series–parallel with the load and operated in above resonant frequency improves the load efficiency and independent operation [13]. Later, Borage et al. [14] have demonstrated an LCL-T half-bridge RC with clamp diodes. The output current or voltage is sensed for every change in load because the output voltage or current increases linearly. The feedback control circuit has not been provided. LCL-T RC with constant current supply operated at resonant frequency is presented [15]. The parallel operation is simple without any complex control circuit which increases the ripple frequency. Using human linguistic terms and common sense, several fuzzy logic controllers (FLCs) have been developed and implemented for dc–dc converters [16]–[18]. These controllers have shown promise in dealing with nonlinear systems and achieving voltage regulation in buck converters. FLC uses human-like linguistic terms in the form of IF–THEN rules to capture the nonlinear system dynamics. Qiu et al. [19] have demonstrated different approaches which offer the FLC. This control technique relies on the human capability to understand the system’s behavior and is based on qualitative control rules. The FLC approach with same control rules can be applied to several dc–dc converters. However, some scale factors must be tuned according to converter topology and parameters. The authors utilized the proposed control technique for a buck–boost converter and demonstrated. Tschirhart and Jain [20] have demonstrated a dc/ac SRC with fixed load value considering two control approaches. Sivakumaran and Natarajan [21] have demonstrated a CLC SPRC using FLC for load regulation and line regulation. The performance of controller has been evaluated and found that the load independent operation may not be possible. The FLC-based zero-voltage switching quasi-RC has been demonstrated by Borage et al. [22]. The load independent

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operation was not realized and power handling capacity of the converter is found to be poor. The evaluation of DSP-based PID and fuzzy controllers for dc–dc converters has been demonstrated by Guo et al. [23] and found that under load changes, the fuzzy controller resulted in less oscillation, but the peak variation was higher in a few cases (decreasing load). A control method and soft-switching operation of the SCRC were explained by Sano and Fujita [24] and a wide-adjustable-range LLC RC has been analyzed by Beiranvand et al. [25]. Later, Yepes et al. [26] have demonstrated digital controller implementation of resonant controllers using two integrators. A costeffective secondary-side LLC resonant controller IC for an LED back light units (BLU) has been discussed by Hong et al. [27]. Beiranvand et al. [28] have designed and analyzed an LLC RC with a wide output voltage range. Feng et al. [29] have demonstrated a universal adaptive SR driving scheme for LLC RCs. A general model for a nonisolated resonant gate driver circuit has been devolved by Anthony et al. [30]. Shamsi et al. [31] have presented a very high frequency (VHF) dc–dc boost converter and found that the converter performed VHF switching with low switch voltage stress. Alonso et al. [32] have suggested a constant-frequency dc–dc RC with a magnetic control. Cho et al. [33] have studied a new transformerintegrated RC with the additional resonant inductor. The LLC– LC RC using the first harmonic approximation (FHA) approach has been introduced by Beiranvand et al. [34]. Krihely and BenYaakov [35] briefed resonant rectifier circuit based on highly efficient resonant energy transfer between a capacitive source such as a piezoelectric generators (PZG) and an inductor. A commonduty-ratio control scheme was presented by Shi et al. [36] for a new input-parallel output-parallel converter system, which consists of two dual-active half-bridge converters as the constituent modules. Wen and Trescases [37] discussed the advantages and drawbacks of the existing analog and digital nonlinear control techniques. Stability analysis of fuel cell powered dc–dc converters was discussed in [38]. Kang et al. [39] presented an approach for efficiency optimization in digitally controlled flyback dc–dc converters over wide ranges of operating conditions. Park et al. [40] proposed nonisolated step-up dc–dc converters with an improved switching method. Lee and Moon [41] proposed state trajectory analysis which is used to find the optimum design of converter parameters. This manuscript uses state-space analysis for the aforementioned purpose. It is obvious from the aforementioned literature works that the output voltage regulation of the converter against load and supply voltage fluctuations has an important role in designing high-density power supplies. The LCL RC is expected to have fast response, better voltage regulation, and improved load independent operation. Motivated by these facts, the LCL RC has been modeled and analyzed for estimating various responses. The closed-loop models have been simulated using MATLAB/Simulink for comparing the performance with existing control techniques. Study of various literature works reveals that a microcontroller is being used to develop the two inverts required for driving the switches of the two legs of the singlephase inverter. In a closed-loop system, an ADC is required to monitor at least any one of the system parameters—usually

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 4, APRIL 2013

Fig. 1.

Block diagram of the proposed LCL RC.

the terminal voltage on the output dc side. Hence, with a single microcontroller it is impossible to carry out the two jobs of monitoring and controlling independently. With the advent of the advanced RISC microprocessors [42], we can do many tasks in parallel. The operating system (RTOS) loaded onto the aforementioned processor has a facility that makes it possible to carry out multiple tasks. RTOS stands for real-time operating system. It overcomes, to a certain extent, the drawback of the Von Newman architecture that uses the fetch– decode–execute sequence. With RTOS a number of tasks can be carried out in a “near” simultaneous manner. The processor LPC 2148 has two sets of ADCs: ADC 0 and ADC 1. ADC 0 can handle eight channels while ADC 1 can handle six channels. All these channels can work independently. Typically, a multiple-input multiple-output (MIMO) system can, therefore, be controlled by LPC2148. With LPC 2148, the gap between the MIMO system design and implementation is reduced.

II. PROPOSED LCL RC The block diagram of an LCL RC is shown in Fig. 1. Unlike the conventional RC that has only two elements, the proposed resonant tank has three reactive energy storage elements (LCL). The first stage converts a dc voltage to a high-frequency ac voltage. The second stage converts the ac power to dc power by a suitable high-frequency rectifier and a filter circuit. Power from the resonant circuit is taken either through a transformer in series with the resonant circuit or in parallel with the capacitor comprising the resonant circuit. In both cases, the high-frequency feature of the link allows the use of a high-frequency transformer to provide voltage transformation and ohmic isolation between the dc source and the load. In an LCL RC, the load voltage can be controlled by varying the switching frequency or by varying the phase difference between the inverters. The phase-domain control scheme is suitable for a wide variation of load condition because the output voltage is independent of load. The absence of dc current in the primary side of the transformer does not necessitate the requirement of current balancing. Another advantage of this circuit is that the device currents are proportional to load current. This increases the efficiency of the converter at light loads because there is the reduction in device losses.

SELVAPERUMAL et al.: STABILITY AND PERFORMANCE INVESTIGATION OF A FUZZY-CONTROLLED LLC RESONANT CONVERTER

n diL 2 = Vo . dt L2

Fig. 2.

Equivalent circuit model of the LCL RC.

The resonant circuit consists of a series inductance L1 , a parallel capacitor C, and a series inductance L2 . S1–S4 are switching devices having base/gate turn-on and turn-off capabilities. The gate pulses for S1 and S4 are in phase but are at 180◦ out of phase with respect to the gate pulses for S2 and S3. The positive portion of the current flows through the MOSFET switch and the negative portion flows through the antiparallel diode. The load, resistive or inductive or capacitive or RLE, is connected across bridge rectifier via chopper, L0 and C0 . The voltage across the point AB is rectified and fed to a chopper circuit. Load is connected through L0 and C0 . In the analysis that follows, it is assumed that the converter operates in the continuous conduction mode and the semiconductors have ideal characteristics. III. TRANSFER FUNCTION MODEL OF LCL, LCC, AND LCL-T RCS A. State-Space Analysis The following assumptions are made while doing the statespace analysis of the LCL, LCC, and LCL-T RCs: 1) the switches, diodes, inductors, and capacitors used are ideal; 2) the effects of snubber capacitors are neglected; 3) losses in the tank circuit are neglected; 4) DC supply used is assumed to be smooth; 5) only fundamental components of the waveforms are used in the analysis; 6) ideal high-frequency transformer with turns ratio n = 1; 7) the state equation describes the state during the period tp−1 < t < tp where tp−1 is the starting instant of continuous conduction mode and tp is the instant when the continuous conduction mode ends. The equivalent circuit shown in Fig. 2 is used for the analysis. The vector space equation for the converter is

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From the aforementioned equations, we get ⎤ ⎡ −1 ⎡ ⎡ ⎤ ⎤ 0 0 iL 1 (t) iL 1 L1 ⎥ ⎢ d ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎣ Vc (t) ⎦ ⎣ Vc ⎦ = ⎢ 1 0 0⎦ dt ⎣ C iL 2 (t) iL 2 0 0 0 ⎡ m −n ⎤  ⎢ L1 L1 ⎥ V ⎥ ⎢ (5) +⎣ 0 0 ⎦ V . o n 0 L2 The sum of the zero-input response and the zero-state response for the LCL RC is given by X(t) = [Φ(t)[X(0)]] + L−1 [(Φ(s)B[U (s)]].

(6)

Solving (6), the current and voltage components can be related ⎤ ⎡ Sin ω(t − tp−1 ) ⎡ ⎤ 0 ) Cos ω(t − t p−1 iL 1 (t) ⎥ ⎢ L1 ω ⎥ ⎢ ⎥ ⎢ ⎥ −Sin ω(t − t ) ⎣ Vc (t) ⎦ = ⎢ p−1 ⎢ Cos ω(t − tp−1 ) 0 ⎥ ⎦ ⎣ Cω iL 2 (t) 0 0 1 ⎤ ⎡ mV − nVo ⎡ ⎤ iL 1 (tp−1 ) ZosSin ω(t − tp−1 ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ [1 − Cos ω(t − t )] mV − nV × ⎣ Vc (tp−1 ) ⎦ + ⎢ o p−1 ⎥. (7) ⎢ ⎦ ⎣ nVo iL 2 (tp−1 ) L2 (t − tp−1 ) From the aforementioned equation, it can be concluded that the output voltage is not dependent on the load resistance and converter gain follows a sine function. In order to maintain the output voltage constant at a desired value against the variations in the load resistance and supply voltage, the pulse width has to be changed in a closed-loop manner. However, the required change in pulse width is very small. B. Design of an RC

The state-space equation for the LCL converter is obtained from Fig. 2 and given as

In order to pursue the analysis, a converter with the following specification is designed: power output = 133 W, minimum input voltage = 125 V, minimum output voltage = 100 V, maximum load current = 1.33 A, maximum overload current = 4 A, inductance ratio KL = 1. The high-frequency transformer turns ratio is assumed to be unity. The input RMS voltage to the diode bridge is √ 2 2 v0 . (8) (D1 − D4 ) (VL 2 ) = π The input RMS current at the input of the diode bridge is

m n 1 diL 1 = V − Vo − Vc dt L1 L1 L1

π Io Id = √ . 2 2

x˙ = Ax + Bu.

dVc 1 = iL 1 dt C

(1)

(2) (3)

100 = 75.18 Ω. The load impedance ZL = 1.33

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Fig. 3.

Equivalent circuit model of the LCC RC.

The reflected ac resistance on the input side of the diode bridge is Rac

VL 2 = Id √ 8V 2 2 V0 /π √ = 20 = π πI0 /2 2

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The quality factor Q and resonant frequency fo of the LCL RC are given as

Lr /C 1

and Q = fo = ZL 2π (L1 + L2 )C where Lr = L1 + L2 and Cr = C.

DC gain characteristics and operating region of the LCC RC.

(10)

VL 2 (11) Id √ L 8 × 75 8Z √ 1 = 2L = = 60.8 Ω. (12) π π2 C Since the switching frequency is 50 kHz

√ 1 √ = 50 × 103 , L1 = 60.8 C 2π L1 C √ √ L1 (13) and C= 60.8 1 60.8 √ = 50 × 103 and 2 π L1 2π L1 × L1 /60.8 = 50 × 103 .

Fig. 4.

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From the aforementioned equation, we get the values of L1 and L2 as 185 μH and C as 0.052 μF. The equivalent circuit of the LCC RC is shown in Fig. 3. The quality factor and resonant frequency of the LCC RC are given as 1 ZL and fo = . (16) Q=  C2 L 2π L CC1 1+C C1C2 2 C1+C2

Fig. 4 shows the dc characteristic of the LCC RC. The major problems of the SRC are light-load regulation, high circulating energy, and high turn-off current at nominal input voltage. The major problems of the PRC are high circulating energy and high turn-off current under the high input-voltage condition. When compared with the SRC, the operating region is much smaller. At light load, the frequency does not need to change too much to keep output voltage regulated. So a light-load regulation problem does not exist in the PRC. The LCC resonant tank can be

considered as the combination of SRC and PRC thereby combining the advantages of both. Fig. 4 shows that the maximum gain, which is determined by Q, affects the operating range of the switching frequency regulating the output voltage under line variation. From the graph, it can be observed that LCC RCs can achieve a narrow switching frequency range with load change. Under light-load conditions, the circulating energy is smaller. The LCC has two resonant frequencies: series resonant frequency and parallel resonant frequency. Although operating at the series resonant frequency is desirable for high efficiency, when doing so, ZVS is lost for certain load conditions. Thus, the operating region is designed to be on the right-hand side of the parallel resonant frequency to achieve ZVS under all load conditions. Unfortunately, like the PRC and SRC, for the LCC, the circulating energy and the turn-off current of the device also increase at the nominal input voltage Vin m ax . In sum, to deal with a wide input-voltage range, all these traditional RCs encounter some problems. To achieve higher efficiency, LCL resonant topologies should be considered. The voltage gain of the LCL RC is drawn in Fig. 5. LCL-T and LCC RCs encounter problems while dealing with a wide inputvoltage range. Meanwhile, under the nominal condition, the LCL RC operates very close to the resonant frequency, which is the best operation point to accomplish high efficiency. In addition, voltage gains of different Q converge at the series resonant point. The LCL resonant tank parameters can be optimized to achieve high efficiency for a wide load range. As a result, holdup time extension capability is accomplished without sacrificing the efficiency under the nominal condition. The LCL RC is considered as one of the most desirable topologies for a wide input-voltage range.

C. Transfer Function of the LCL RC From Fig. 2, applying the Kirchhoff voltage law, we get the transfer function of the LCL RC as ZL CS nVo (S) = 2 . mV (S) S (L1 + L2 )C + ZL CS + 1

(17)

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E. Transfer Function of the LCL-T RC Applying the Kirchhoff voltage law in Fig. 6, we get the transfer function ZL nVo (S) = 3 . mV (S) S CL1 L2 + S 2 CL1 ZL + S(L1 + L2 ) + ZL (21) Apply the values L1 , L2 , and C in (21) nVo (S) = mV (S) 75 S 3 (0.178×10−18 )+S 2 (7.22×10−12 )+S(3.7

× 10−6 )+75

.

(22)

Fig. 5.

The transfer functions of LCL and LCC RCs are of order 2 and that of LCL-T is of order 3. From the basic principles, we know that if order of the system increases, the system will go to an unstable region. We feel that it is very essential to study the performance of RC when the load varies. Hence, the following discussion briefs the steady-state analysis of LCL, LCC, and LCL-T RC to determine their ability.

DC gain characteristics and operating region of the LCL RC.

IV. STABILITY ANALYSIS OF RCS A. Lyapunov Stability Analysis According to Lyapunov [43], the system is asymptotically stable if and only if for any Q = QT > 0 there exists a unique P = P T > 0 such that (23) is satisfied AT P + P A = −Q, Fig. 6.

Equivalent circuit model of the LCL-T RC.

where Q is identity that is Q = I3 .

Apply the values L1 , L2 , and C in (17) (3.9 × 10−6 )S nVo (S) = 2 . mV (S) S (9.62 × 10−12 ) + (3.9 × 10−6 )S + 1

(18)

The equivalent circuits of the LCC and LCL-T RCs are shown in Figs. 3 and 6, respectively. The mathematical modeling using the state-space technique can be obtained assuming all the components to be ideal. The transfer functions of the LCC and LCL-T RCs are given below from Figs. 3 and 6. Applying the Kirchhoff voltage law in Fig. 3 (19)

Apply the values of C1 = C2 = 0.052 × 10−6 and L = 185 × 10−6 in (19), we get 0.052 × 10−6 nVo (S) = 2 . mV (S) S (0.5 × 10−18 ) + 0.104 × 10−6

(20)

(23)

While converting transfer function equations into state-space model using MATLAB coding, we got the values of A, B, C, and D matrices as listed next. While using (18) for the LCL RC   −405405 −103950103950 1 A= , B= 1 0 0 C = [ 405405

D. Transfer Function of the LCC RC

C1 nVo (S) = 2 . mV (S) S C1 C2 L + C1 + C2

Q = QT > 0

0],

D = 0.

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While using (20) for the LCC RC  0 −208 × 109 A= , 1 0 C = [ 0 104 × 109 ] ,

 1 B= 0

D = 0.

(25)

While using (21) for the LCL-T RC ⎡ ⎤ −40561797 −20786516853932 −4 × 1020 ⎦ A=⎣ 1 0 0 0 1 0 ⎡ ⎤ 1 B = ⎣ 0 ⎦ C = [ 0 0 4 × 1020 ] , D = 0. (26) 0 With the help of the MATLAB function “lyap” (used for solving the algebraic Lyapunov equation), we attained the following

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solutions: for the LCL resonant converter:  1.2 × 10−6 4.8 × 10−12 P = 4.8 × 10−12 128205 for the LCC resonant converter:  −325 × 107 −0.5 × 1014 P = −0.5 × 1014 −6.76 × 1032 for the LCL − T resonant converter: ⎤ ⎡ 0.5 −9 × 10−10 2 × 10−8 0.5 2 × 107 10 × 1012 ⎦ . P =⎣ −10 −9 × 10 10 × 1012 2 × 1034 Examining the positive definiteness of the matrix P , the eigenvalues λ of this matrix are obtained as follows.

 1.23 × 10−6 For the LCL resonant converter: λ = . 128205 Hence, P is positive definite and the Lyapunov test indicates that the (LCL RC) system under consideration is stable.

 −6.76 × 1032 For the LCC resonant converter: λ = . −325 × 107 Hence, P is negative definite and the Lyapunov test indicates that the (LCC RC) system under consideration is unstable. ⎞ ⎛ 1.23 × 10−8 For the LCL-T resonant converter: λ = ⎝ 2.02 × 107 ⎠ . 2.1 × 1020 Hence, P is positive definite and the Lyapunov test indicates that the (LCL-T RC) system under consideration is stable. The above discussion confirms that for LCL and LCL-T RCs for any Q = QT > 0 there exists a unique P = P T > 0 which satisfies (23) and there by confirming its stable operation. An adaptive fuzzy control strategy does not depend on accurate mathematical models, and the stability and convergence of the overall system can be proved by the Lyapunov method [44]. B. Stability Analysis Using Root Locus For the transfer function (18), (20), and (22), the Nyquist and root locus plots are prepared using MATLAB coding. In Fig. 7(a), the locus path is lying on the left half of the S-plane. So, the LCL RC is a stable system. In Fig. 7(b), the locus path is lying on the imaginary axis of the S-plane, and hence, the LCC RC is a marginally stable system. In Fig. 7(c), the locus paths of LCL-T are lying on the real axis of the S-plane and the right half of the S-plane. Hence, the LCL-T RC could be considered as an unstable system. C. Stability Analysis Using the Nyquist Plot The conclusion regarding the stability of the LCL RC can be proved from the Nyquist plot in Fig. 7(d) where the locus of roots does not encircle −1+j0. But for the Nyquist plot for LCC shown in Fig. 7(e) it encircles −1+j0 which is one of the criteria for an unstable system. The Nyquist plot of the LCL-T

converter in Fig. 7(f) encircles −1+j0 twice; hence, the system is said to be unstable. For determining the stability of the three types of converters, all possible studies were undertaken. Since LCC fails in all the stability studies, it has not been opted for further analysis. Moreover, LCL-T passes though Lyapunov and root locus stability analysis, but its failure in Nyquists forces us to come to a conclusion that in real cases the stability margin will be less than the LCL converter. Hence, only LCL RC has been opted for further analysis in this research manuscript. D. State Plane Analysis of the LCL RC Phase plane (state plane) analysis is one of the most important techniques for studying the behavior of nonlinear systems, since there is usually no analytical solution for a nonlinear system. A phase-plane plot for a two-state variable system consists of curves of one state variable versus the other state variable (x1 (t) versus x2 (t)), where each curve is based on a different initial condition. PPLANE is a tool for phase plane analysis of a system of differential equations [45] of the form x• =

dx = f (x, y) dt

and

y• =

dy = g(x, y). dt

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At each point,(x, y) of a grid, PPLANE draws an arrow indicating the direction and magnitude of the vector(x• , y • ). This dy /dt dy = dx and is independent of t; therefore, it vector equals dx/dt must be tangent to any solution curve through (x, y). Solutions can be viewed in three dimensions. Since point (0, 0) is a stable equilibrium point for the system, if the arrow in a solution plot is pointing toward it, the system is assumed to be stable. PPLANE also graphs the linearization about equilibrium points, and displays eigen values, eigenvectors, nullclines, and stable and unstable orbits. The LCL RC has two state variables: one is iL 1 (t) and another one is VC (t). From (5), we got the state plane of the LCL RC using the PPLANE tool shown in the figure. For the LCL RC, the solution plot (see Fig. 8) converges to (0, 0) for all initial conditions and λ1 , λ2 < 0; we could conclude that the solution of (5) approaches to the origin as t increases so thereby said to be asymptotically stable. Stability analysis of a FLC and stability analysis of an FLC-based LCL RC are briefly discussed in Sections VI-C and VI-D, respectively. At this juncture, further discussions in this paper focus on the study of PI controller and fuzzy controller-based LCL RC. The simulated results are compared with the hardware results to validate the same. V. DESIGN OF THE PI CONTROLLER Controllers based on the PI approach are commonly used for a dc–dc converter [46], [47] applications. Power converters have relatively low-order dynamics that can be well controlled by the PI method. PI-based closed-loop Simulink diagram of LCL is shown in Fig. 9. Ziegler and Nichols [48] conducted numerous experiments and proposed rules for determining values of KP and KI based on the transient step response of a system. It applies to an RC with neither integrator nor dominant complex-conjugate poles,

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Fig. 7. (a) Root locus plot for the LCL RC. (b) Root locus plot for the LCC RC. (c) Root locus plot for the LCL-T RC. (d) Nyquist plot for the LCL RC. (e). Nyquist Plot for the LCC RC. (f) Nyquist plot for the LCL-T RC.

whose unit-step response resemble an S-shaped curve with no overshoot. This S-shaped curve is called the reaction curve. The S-shaped reaction curve (shown in Fig. 10) can be characterized by two constants, delay time L and time constant T , which are determined by drawing a tangent line at the inflection point of the curve and finding the intersections of the tangent line with the time axis and the steady-state level line. Using the parameters L and T , we can set the values of KP , KI , and KD according to the formula shown in Table I. These parameters will typically give a response with an overshoot about 25% and good settling time. Based on the Ziegler–Nichols tun-

ing rule, proportional gain constant KP = 0.05 and integral time constant KI = 25 are obtained for the converter under study. The system is simulated with a switching frequency of 50 kHz. It is observed that the PI controller for the LCL configuration regulates the output voltage with a settling time of 0.1 ms. VI. DESIGN OF A FUZZY CONTROLLER The FLC provides an adaptive control for better system performance. Fuzzy logic is aimed to provide a solution for controlling nonlinear processes and to handle ambiguous and uncertain

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TABLE I FUZZY RULE MATRIX

Fig. 8.

State plane analysis of the LCL RC.

Fig. 9.

Closed-loop Simulink diagram of the LCL RC using PI controller.

Fig. 11.

Fuzzy membership functions used for the LCL RC.

verter and the results were recorded. The duty cycle d(k), at the kth sampling time, is determined by adding the previous duty cycle [d(k−1)] to the calculated change in duty cycle D d(k) = d (k − 1) + D.

Fig. 10.

S-shaped reaction curve.

situations. Fuzzy control is based on the fundamentals of fuzzy sets. The fuzzy controller suggested in this manuscript takes error “e” and change in error “ce” as its input parameter and change in duty ratio D of the converter as its output parameter. The fuzzy control scheme is implemented in this dc–dc con-

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LCL is modeled by using the MATLAB software. Fuzzy control is developed using the fuzzy toolbox. The fuzzy variables “e,” “ce,” and “D” are described by triangular membership functions. Seven triangular membership functions (see Fig. 11) are chosen and Table I shows the fuzzy rule base considered in this manuscript based on intuitive reasoning and experience. Fuzzy memberships NB, NM, NS, Z, PS, PM, PB are defined as negative big, negative medium, negative small, zero and positive small, positive medium, and positive big. A. Rule Table and Inference Engine The rule firing contains the following steps:

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1) From the membership functions of input 1, the lingual under whose range the actual values is present is found. Also, its decimal height degree of belief (DOB) is obtained. 2) From the membership functions of input 2, the corresponding lingual is also found along with the DOB. 3) For these two inputs, the output is fetched from the rule matrix. B. Development of the FLC At every sampling interval, the instantaneous RMS values of the sinusoidal reference voltage and load voltage are used to calculate the error e and change in error ce signals that act as the input to the FLC. The stages of fuzzification, fuzzy inference, and defuzzification are then performed by the program as described in the flowchart of Fig. 12 e = Vr − VL

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ce = e − pe

(30)

where Vr is the reference or the desired output voltage and VL is the actual output voltage. The duty ratio of the converter will be determined by the fuzzy inference. For instance, if the output voltage continues to increase gradually while the current is low during the charging process, the fuzzy controller will maintain the increase in voltage to reach the set point. A drop in the output voltage level triggers the fuzzy controller to increase the output voltage of the converter by modifying the duty cycle of the converter. The resolution of an FLC system relies on the fuzziness of the control variables while the fuzziness of the control variables depends on the fuzziness of their membership functions. The closed-loop simulation using FLC for the LCL is carried out using the MATLAB/Simulink software. Depending on error and the change in error, the value of change of duty cycle is calculated. Set parameter instruction and function blocks available in MATLAB are used to update the new duty cycle of the pulse generators. The closed-loop Simulink diagram of the LCL RC using an FLC is shown in Fig. 13. The entire system is simulated with a switching frequency of 50 kHz. It is observed that the FLC for LCL regulates the output voltage with a settling time of 0.06 ms. C. Stability Analysis of an FLC Table II indicates the set of 49 rules which makes LCL RC stable. On evaluating the rules, the relational matrix corresponding to the output D, representing membership values, is obtained as ⎡

1 0.4 ⎢ 0.1 1 ⎢ ⎢ 0.4 0.4 ⎢ D = ⎢ 0.3 0.3 ⎢ ⎢ 0.2 0.3 ⎣ 0.1 0.2 0 0.1

0.4 0.4 0.3 0.5 0.8 0.3 1 0.3 0.3 0.3 1 0.3 0.3 0.4 1 0.2 0.4 0.8 0.3 0.4 0.4

⎤

0.1 0 0.4 0.2 ⎥ ⎥ 0.2 0.1 ⎥ ⎥ 0.3 0.3 ⎥ . ⎥ 0.4 0.1 ⎥ ⎦ 1 0.3 0.4 1

(31)

Fig. 12.

Flowchart.

The defuzzified output D from FLC is given to the LCL RC as shown in Fig. 13 D2 = D ◦ D D3 = D2 ◦ D . . . . Dn = Dn −1 ◦ D.

(32)

The FLC was designed in a MATLAB environment using a Mamdani fuzzy inference system editor and the control surface of the fuzzy controller is shown in Fig. 14. The surface represents the information about the output D based on e and ce in the fuzzy

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relational matrix D given in (31), we get ⎡ 1 0.4 0.4 0.4 0.3 ⎢ 0.4 1 0.5 0.8 0.4 ⎢ ⎢ 0.4 0.4 1 0.4 0.3 ⎢ 2 D = ⎢ 0.3 0.3 0.3 1 0.3 ⎢ ⎢ 0.3 0.3 0.3 0.4 1 ⎣ 0.3 0.3 0.3 .4 .8 0.3 0.3 0.3 .4 .4

Fig. 13.

Closed-loop Simulink diagram of the LCL RC using an FLC.

TABLE II SUMMARY OF DYNAMIC RESPONSE (SIMULATED) FOR THE LCL RC

Fig. 14.

Control surface for the designed FLC.

controller. This output-controlled voltage matrix D obtained from an FLC is further used to analyze the nature of stability of FLC-based LCL RC. A method is proposed in this section using the output relational matrix D obtained from the FLCbased LCL RC. This method utilizes the compositional matrices obtained using D for declaring the nature of stability. To assess the periodic or aperiodic stability, the following composition matrices are formed using D. The composition is performed using the max–min principle [49]. Performing the operation mentioned in (32) for the output

⎤ 0.4 0.3 0.4 0.3 ⎥ ⎥ 0.4 0.3 ⎥ ⎥ 0.3 0.3 ⎥ . ⎥ 0.4 0.3 ⎥ ⎦ 1 0.3 .4 1

On performing the next composition over D2 , then ⎤ ⎡ 1 0.4 0.4 0.4 0.3 0.4 0.3 ⎢ 0.4 1 0.5 0.8 0.4 0.4 0.3 ⎥ ⎥ ⎢ ⎢ 0.4 0.4 1 0.4 0.3 0.4 0.3 ⎥ ⎢ ⎥ 3 D = ⎢ 0.3 0.3 0.3 1 0.3 0.3 0.3 ⎥ . ⎢ ⎥ ⎢ 0.3 0.3 0.3 0.4 1 0.4 0.3 ⎥ ⎣ ⎦ 0.3 0.3 0.3 0.4 0.8 1 0.3 0.3 0.3 0.3 0.4 0.4 0.4 1 Further composition over D3 , we get ⎡ 1 0.4 0.4 0.4 0.3 ⎢ 0.4 1 0.5 0.8 0.4 ⎢ ⎢ 0.4 0.4 1 0.4 0.3 ⎢ 4 D = ⎢ 0.3 0.3 0.3 1 0.3 ⎢ ⎢ 0.3 0.3 0.3 0.4 1 ⎣ 0.3 0.3 0.3 0.4 0.8 0.3 0.3 0.3 0.4 0.4

⎤ 0.4 0.3 0.4 0.3 ⎥ ⎥ 0.4 0.3 ⎥ ⎥ 0.3 0.3 ⎥ . ⎥ 0.4 0.3 ⎥ ⎦ 1 0.3 0.4 1

(33)

(34)

(35)

This composition can be carried out up to the required observation. An intuitive criterion for aperiodic and periodic nature of stability for the given system based on these compositional matrices is dictated as given below and the proof for the given procedure is provided in [50]: “Any row or any column of respective D has same elemental value then due to max–min composition; the same value will appear in all other compositional matrices. This implies that there is no oscillation in the given fuzzy dynamic system; otherwise, if there are variations with repetition (in a regular order), then the system has periodic oscillations.” Applying the aforementioned discussion to the matrices given in (33)–(35), it can be observed that elemental membership values remain the same in all entries from compositional matrix D2 onward without any variations. It indicates the stable aperiodic nature of the designed FLC-based LCL RC. On repeating further compositions up to the required observation, it has been noted that the same observations are resulted. D. Stability Analysis of an FLC-Based LCL RC Fig. 15 shows the closed-loop control system block diagram of the LCL RC G(s) with a fuzzy controller. The LCL RC G(s) is a second-order hold (SOH) with the sampling interval T to simulate the dc–dc converters. The feedback network is assumed as a unity element. The system closed-loop transfer function in the s-domain is Gc (s) =

F (s)G(s) . 1 + F (s)G(s)

(36)

SELVAPERUMAL et al.: STABILITY AND PERFORMANCE INVESTIGATION OF A FUZZY-CONTROLLED LLC RESONANT CONVERTER

Fig. 15.

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Fuzzy-controlled closed-loop control system of an LCL RC.

Fig. 17.

Simulated output voltage and current waveform with 100% load.

and the settling time are very high, and also the response is oscillatory. B. PI Closed-Loop Response

Fig. 16. Locations of the zero and pole of the fuzzy-controlled closed-loop control system (SOH) in the z-plane.

By using a bilinear transformation technique, we get the system closed-loop transfer function in the Z-domain as Gc (z) = Z[Gc (s)] =

(z − 1)(z + 1) . (z + 0.99)(z + 0.75)

(37)

The pole of this system is inside the unit circle. Therefore, this closed-loop system is stable [51]. The locations of the zero and pole are shown in Fig. 16. Comparing the result obtained using inspection test with that of the unit step response of the closed-loop FLC-based RC model given in Fig. 15, the aperiodic stability of the system is validated. This proposed methodology is simple and straightforward in application as well as the stability information is depicted easily compared to the energy stability criterion given in [50]. VII. RESULTS AND DISCUSSION The performance of a fuzzy controller was compared with that of a PI controller for identifying the better one. The controllers were implemented using MATLAB/Simulink and analyzed. A. Open-Loop Response Under the open-loop condition with manually adjusted pulse widths of the converter switching, for a reference voltage of 100 V the output voltage of the system was 95 V. The overshoot

With the PI control model, for the same reference set point voltage the output voltage was observed as 99.2 V. In the closed loop using a PI controller, the overshoot and settling time are less compared to the open loop, and also less oscillatory. The slight drop in the resonant characteristics is due to the increase in conduction losses in the bridge inverter and resonant network. C. FLC Closed-Loop Response The output voltage while using an FLC closed-loop system for a set voltage of 100 V is 99.6 V. While using this controller, the overshoot and settling time are less compared to open-loop and PI controllers, and the response is less oscillatory. The harmonic performance of the ac components present in an output voltage is significantly improved compared to that of the PI controller model. The simulated values for open-loop LCL and closed-loop RC have been estimated and provided in Table II. The result of simulation as shown in Fig. 17 has been verified and validated using the experimental setup. D. Hardware Implementation As mentioned earlier, a real-time operating system-based microcontroller that can undertake multiple tasks is employed in this attempt. The three tasks to be carried out independently are: 1) to monitor the output voltage continuously; 2) to send the switching pulses with appropriate duty cycle for the inverter; and 3) to send the switching pulses with appropriate duty cycle for the chopper. The large memory capacity of the processor LPC2148 enables the FLC and the PI controller programs to be accommodated in the same processor memory. In this study, the applicability of the Philips advanced RISC machine (ARM) processor LPC 2148, shown in Fig. 18(a), is investigated as the controller for

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Fig. 19. The experimental DSO waveforms of V rm s (Y -axis 100 V/div) and Irm s (Y -axis 1.5 A/div) with X -axis 5 ms at Q = 1 with full-input dc voltage demonstrating when (a) D = 0.5 and (b) D = 0.25.

TABLE III SUMMARY OF DYNAMIC RESPONSE (EXPERIMENTAL RESULTS) FOR THE LCL RC

Fig. 18.

(a). ARM processor LPC 2148. (b). Prototype model.

the LCL RC. The time-sharing feature of the LPC2148 offers ample possibility for its use in the designed LCL RC which has a resonance frequency of 50 kHz. A 50-kHz, 133-W, 125–100 V prototype, shown in Fig. 18(b), is built to verify the proposed LCL RCs. L1 is chosen to be 185 μH, L2 is chosen to be 185 μH, and the resonant capacitor C is chosen to be 0.052 μF. The transformer turns ratio is unity. The primary- and secondary-side switches are selected to be IRF 540. The secondary-side inductor Lo is chosen to be 202 μH. The transformer core is chosen to be EER4242. E. Performance Evaluation of a Hardware Setup It is vivid from Fig. 19 obtained from a hardware setup that soft switching happens by way of zero-voltage switching. The open-loop LCL and closed-loop RC experimental results have been estimated and provided in Table III. From Table III, it could be concluded that the FLC exhibits better performance in terms of reduced settling time, rise time, and peak overshoot. Fig. 20(a) shows that the output voltage is 90 V, the settling time is 0.8 ms, and overshoot also occurred in this control, but the required output voltage is 100 V. Fig. 20(b) shows the output voltage of 100 V obtained using a PI controller with the settling time 0.065 ms and small overshoot occurred in this control. If the input voltage is suddenly varied from 125 to 160 V at the time 0.3 ms, the output voltage varies from 100 to 110 V and settled after 0.2 ms.

Fig. 20. (a) Output voltage for the open-loop control of the LCL RC (Y -axis 30 V/div, X -axis 1 ms/div). (b) and (c) Output voltage for PI and fuzzy control-based LCL RC (Y -axis 100 V/div, X -axis 0.1 ms/div) respectively.

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Fig. 22. Experimental result showing the start-up response and transient response for 50% load using a PI controller (Y -axis 50 V/div, X -axis 0.05 ms/div).

Fig. 23. Experimental result showing the start-up response and transient response for 50% load using the FLC (Y -axis 50 V/div, X-axis 0.05 ms/div). Fig. 21. (a). Input voltage is 125 V. (b) Input voltage is 110 V. Settling time comparison for both PI- and fuzzy controller-based LCL RC (Y -axis 20 V/div, X -axis 0.02 ms/div).

Fig. 20(c) shows that the fuzzy controller obtained the output voltage (100 V) at the (settling) time 0.05 ms and no overshoot in this control. If the input voltage is suddenly varied from 125 to 160 V at the time 0.3 ms, the output voltage varies from 100 to 110 V and settled after 0.12 ms. A settling time comparison between the PI controller and the fuzzy controller is made in Fig. 21(a). The settling times of the PI controller and the fuzzy controller are 0.06 and 0.05 ms, respectively. For the input-voltage variation case Fig. 21(b), the settling times of the fuzzy controller and the PI controller are 0.04 and 0.07 ms, respectively, and also 10% of overshoot occurred in a PI controller-based LCL RC. Fig. 22 shows the start-up response and transient response for 50% load using a PI controller. It shows a settling time during the start-up response of 0.058 ms and when suddenly loaded (at the time period 0.1 ms), the system settles at 0.18 ms. Fig. 23 shows the start-up response and transient response for 50% load using an FLC. It shows a settling time during the start-up response of 0.05 ms and when suddenly loaded (at the time period 0.1 ms), the transient response gets settled in

0.15 ms. The aforementioned responses (see Figs. 22 and 23) clearly favor for an FLC-based LCL RC. Fig. 24(a) shows the start-up response and transient response for 75% load using a PI controller. It shows a settling time during the start-up response of 0.057 ms and when suddenly loaded (at the time period 0.1 ms) the system settles at 0.148 ms. Fig. 24(b) shows the start-up response and transient response for 75% load using an FLC. It shows a settling time during the start-up response of 0.05 ms and when suddenly loaded (at the time period 0.1 ms), the transient response gets settled in 0.14 ms. Experimental results show that the fuzzy controller-based LCL RC is more robust against input voltage and load resistance variations than the PI controller. In conclusion, the fuzzy controller can achieve a better performance than the PI controller. Thus, the Fuzzy controller design method is highly suitable for application to LCL dc–dc RCs. The results obtained indicate that the FLC is an effective approach for the output voltage regulation of the LCL dc–dc RC. Fig. 25 shows the variation in percentage efficiency with output power in watts. It is interesting to note that the maximum efficiency occurs at a load of 133 W for which it has been

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designed. Efficiency calculation for 100% load is given next VDD = drain source voltage = 50 V ID = drain current = 10 A TON = ON time period = T /2 TR = rise time = 21 ns Tfall = fall time = 48 ns VON = ON state voltage drop = 2 V IOFF = OFF state current = 250 μA PT = Total losses = PON + POFF + PSW

(38)

where PON = average power lost during conduction in the switch, POFF = average power lost in the switch when it is OFF, PSW = the switching loss is given by the sum of losses during rise time and fall time of the switch PSW = Pturnon + Pturnoff =

VDD × ID × TF × FSW VDD × ID × TR × FSW + 6 6

=

50 × 10 × 21 × 10−9 × 50 × 103 6

50 × 10 × 48 × 10−9 × 50 × 103 6 = 0.0875 W + 0.2 W = 0.2875 W +

PON = POFF =

(39)

2 × 10 × 0.5T VON × ION × TON = = 10 W (40) T T 50 × 250 × 10−6 × 0.5T VDD × IOFF × TOFF = T T

= 6.25 × 10−3 W.

(41)

Fig. 24. (a) and (b) Experimental result showing the start-up response and transient response for 75% load using the PI and Fuzzy controllers, respectively (Y -axis 50 V/div, X -axis 0.05 ms/div).

Substitute (39), (40), and (41) into (38), we get ∴ PT = 10 + 6.25 × 10−3 + 0.2875 = 10.29 W

(42)

Output Power (POUT ) = 100 × 1.33 = 133 W

(43)

Efficiency (η) =

Output Power (Pout ) . Output Power (Pout )+Total Losses (PT ) (44) Fig. 25.

Substitute (42) and (43) into (44), we get Efficiency (η) = 133 133+10.29 = 92.83%. Based on the results obtained in the present investigations, the following line of research seems to be worth pursuing for further research work: 1) meta-heuristic-based search techniques such as GA, PSO, etc., can be used to find the modulation index of the LCL RC to improve the performance (efficiency) at low load; 2) an adaptive control law can be extended to study the stability and dynamic characteristics of the LCL RC.

Plot for % efficiency versus output power in watts.

VIII. CONCLUSION The LCL RC has a better stability margin compared to LCC and LCL-T converters after a detailed stability studies using Lyapunov, root locus, Nyquist, and state plane analysis. Fuzzycontrolled closed-loop control system stability was also analyzed using in the z-plane. Both FLC-based and PI-based LCL RC circuits were simulated using MATLAB/Simulink and also experimentally verified by implementing the ARM processor.

SELVAPERUMAL et al.: STABILITY AND PERFORMANCE INVESTIGATION OF A FUZZY-CONTROLLED LLC RESONANT CONVERTER

The FLC provides better voltage regulation even under sudden variation of load. The reduction in switching power losses enhances the efficiency. The dynamic response and steady-state errors of FLC as compared with the PI controller and open loop were determined by simulation and verified by experimental studies. The LCL RC can be used for applications such as space and radar with high-voltage power supplies with the appropriate turn’s ratio of the HF transformer. The targeted design constraints, including efficiency, output voltage, and output power, were met successfully. ARM processor LPC 2148 has been introduced as the controller for both PI- and Fuzzy-based RCs. At full load, an efficiency of 92.83% was obtained. As the load was decreased, efficiencies remained high. However, at 11% load the efficiency of the circuit was measured as 79.9%. An added advantage of the FLC is that, with minor modification, the same controller coding could be extended to control any type of dc–dc converters since their behavior can also be described by the similar set of linguistic rules. REFERENCES [1] J. A. K. S. Bhat, “Fixed frequency PWM series-parallel resonant converter,” IEEE Trans. Ind. Electron., vol. 28, no. 5, pp. 1002–1009, Sep. 1992. [2] J. A. Sabate, M. M. Jovanic, F. C. Lee, and R. T. Gean, “Analysis and design-optimization of LCC resonant inverter for high-frequency ac distributed power system,” IEEE Trans. Ind. Electron., vol. 42, no. 1, pp. 63– 71, Feb. 1995. [3] A. K. S. Bhat and S. B. Dewan, “A generalized approach for the steadystate analysis of resonant inverters,” IEEE Trans. Ind. Electron., vol. 25, no. 2, pp. 326–338, Mar./Apr. 1989. [4] C. Q. Lee, R. Liu, and S. Sooksatra, “Nonresonant and resonant coupled zero voltage switching converters,” IEEE Trans. Power Electron., vol. 5, no. 4, pp. 404–412, Oct. 1990. [5] Z. Ye, J. C. W. Lam, P. K. Jain, and P. C. Sen, “A robust one-cycle controlled full-bridge series-parallel resonant inverter for a high-frequency AC (HFAC) distribution system,” IEEE Trans. Power Electron., vol. 22, no. 6, pp. 2331–2343, Nov. 2007. [6] T.-J. Liang, R.-Y. Chen, and J.-F. Chen, “Current-fed parallel-resonant DC–AC inverter for cold-cathode fluorescent lamps with zero-current switching,” IEEE Trans. Power Electron., vol. 23, no. 4, pp. 2206–2210, Jul. 2008. [7] M. S. Agamy and P. K. Jain, “A variable frequency phase-shift modulated three-level resonant single-stage power factor correction converter,” IEEE Trans. Power Electron., vol. 23, no. 5, pp. 2290–2300, Sep. 2008. [8] D. Fu, F. C. Lee, Y. Qiu, and F. Wang, “A novel high-power-density three-level LCC resonant converter with constant-power-factor- control for charging applications,” IEEE Trans. Power Electron., vol. 23, no. 5, pp. 2411–2420, Sep. 2008. [9] H. Pollock, “Simple constant frequency constant current load-resonant power supply under variable load conditions,” Inst. Electr. Eng. Electron. Lett., vol. 33, no. 18, pp. 1505–1506, Aug. 1997. [10] H. Seidel, “A high power factor tuned class D converter,” in Proc. IEEE Power Electron. Spec. Conf., Apr. 1988, pp. 1038–1042. [11] H. Irie and H. Yamana, “Immittance converter suitable for power electronics,” Trans. Inst. Electr. Eng. Jpn., vol. 11, no. 8, pp. 962–969, 1997. [12] M. Borage, S. Tiwari, and S. Kotaiah, “Analysis and design of LCL-T resonant converter as a constant-current power supply,” IEEE Trans. Ind. Electron., vol. 52, no. 6, pp. 1547–1554, Dec. 2005. [13] M. Borage, S. Tiwari, and S. Kotaiah, “LCL-T resonant converter with clamp diodes: A novel constant-current power supply with inherent constant-voltage limit,” IEEE Trans. Ind. Electron., vol. 54, no. 2, pp. 741– 746, Apr. 2007. [14] M. Borage, S. Tiwari, and S. Kotaiah, “A constant-current, constantvoltage half-bridge resonant power supply for capacitor charging,” Proc. Inst. Electr. Eng. Electr. Power. Appl., vol. 153, no. 3, pp. 343–347, May 2006. [15] M. Borage, K. V. Nagesh, M. S. Bhatia, and S. Tiwari, “Design of LCL-T resonant converter including the effect of transformer winding

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S. Selvaperumal was born in Virudhunagr, Tamil Nadu, India, in 1977. He received the B.Eng. degree in electrical and electronics engineering and the M.E. degree in power electronics and drives both from Anna University, Chennai, Tamil Nadu, in 2005 and 2007, respectively. He is currently working toward the Ph.D. degree in the Department of Electrical and Electronics Engineering, Anna University, Tiruchirappalli, Tamil Nadu. He has published three technical papers in a refereed international journal and two technical papers in IEEE international conference proceedings. He is currently an Assistant Professor in the Department of Electrical and Electronics Engineering, Mepco Schelenk Engineering College, Sivakasi, Tamil Nadu. His research interest includes resonant converter.

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 4, APRIL 2013

C. Christober Asir Rajan was born in 1970. He received the B.E. degree (Distn.) in electrical and electronics and the M.E. degree (Distn.) in power system from the Madurai Kamaraj University, Madurai, India, in 1991 and 1993, respectively. He also received the postgraduate degree (Distn.) in DI.S. from the Annamalai University, Chidambaram, Tamil Nadu, India, and the Ph.D. degree in power system from the College of Engineering, Guindy, Anna University, Chennai, Tamil Nadu, during 2001–2004. He published technical papers in international and national journals and conferences. He is currently an Associate Professor in the Department of Electrical and Electronics Engineering, Pondicherry Engineering College, Pondicherry. His research interests include optimization, operational planning, and control.

S. Muralidharan was born in 1973. He received the B.E. degree in electrical engineering from Madurai Kamaraj University, Madurai, India, the M.S. degree in software systems from the Birla Institute of Technology and Science, Pilani, Rajasthan, India, and the Ph.D. degree from SASTRA University, Thanjavur, Tamil Nadu, India, in 1994, 1997, and 2009, respectively. He is serving as a Professor in the Department of Electrical and Electronics Engineering, MEPCO Schlenk Engineering College, Sivakasi, Tamil Nadu. He has published several technical papers in refereed international journals and international conferences. His research interests include special electrical machines and soft computing applications in electrical power system.