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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 3, MARCH 2012 1369 Analysis, Design, and Experimentation on Constant-Frequency DC-DC Resonant Converters With Magnetic Control J. Marcos Alonso, Senior Member, IEEE, Marina S. Perdigão, Student Member, IEEE, David Gacio Vaquero, Student Member, IEEE, Antonio J. Calleja, Member, IEEE, and Eduardo Sousa Saraiva, Member, IEEE Abstract—In this paper, a new technique for controlling dc-dc resonant converters is investigated. A variable inductance is used to control and regulate the dc output voltage maintaining constant switching frequency for the half-bridge transistors. The output voltage characteristics of the series resonant and parallel resonant converters under the proposed magnetic control are obtained and analyzed. In order to evaluate the proposed technique, a laboratory prototype for a 48 V-input 5 V/10 A-output 500 kHz parallel resonant converter is presented. A methodology for obtaining the converter dynamic response using a step response test is carried out. From the dynamic response, a compensator for operating the converter at closed loop is developed and tested in the laboratory. The results prove that the proposed technique is suitable for controling resonant inverters at constant frequency using a low-cost half-bridge inverter. Index Terms—Closed loop operation, constant switching frequency, dc-dc converters, low EMI, magnetic control, magnetic regulator, resonant conversion, variable inductance. I. INTRODUCTION C-DC converters are used in many applications including telecommunications, battery chargers, fuel cells, ultracapacitors, electric vehicles, etc. [1]–[16]. DC-DC resonant converters were introduced in an effort to reduce switching losses, and in turn boost the operating frequency ultimately achieving a lower profile, a major requirement of today’s appliances. Typical modes of operation are: 1) zero voltage switching (ZVS) at turn-ON, where the switches do not present losses during the turn-ON process but do during the turning OFF, and 2) zero cur- D Manuscript received April 1, 2011; revised June 22, 2011; accepted July 30, 2011. Date of current version February 7, 2012. This work was supported in part by the Spanish Government, Education and Science Office, under research grants DPI2007-61267 and DPI2010-15889 and in part by the Government of the Portuguese Republic, FCT-MCTES, under research grants PTDC/EEAENE/66859/2006 and SFRH/BD/36143/2007. Recommended for publication by Associate Editor S. Williamson. J. M. Alonso, D. G. Vaquero, and A. J. Calleja are with the Universidad de Oviedo, Electrical and Electronics Engineering Department, 33204-Gijón, Asturias, Spain (e-mail: marcos@uniovi.es). M. S. Perdigão is with the Instituto de Telecomunicações, Pólo II, 3030-290, Coimbra, Portugal. She is also with the DEE, ISEC, IPC, R. Pedro Nunes, 3030-199 Coimbra, Portugal. E. S. Saraiva is with the DEEC, University of Coimbra, Portugal. He is also with the Instituto de Telecomunicações, Pólo II, 3030-290, Coimbra, Portugal, and with the DEE, ISEC, IPC, R. Pedro Nunes, 3030-199 Coimbra, Portugal. 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.2011.2165083 rent switching (ZCS) at turn-OFF, where the losses are negligible during the turn-OFF process but are still present during turning ON [17]. Resonant converters usually employ the switching frequency as a control parameter to regulate the output voltage. Changes in the operating frequency vary the impedances of the reactive elements used in the resonant tank, so that the output voltage is modified. In some cases, depending on the input-output voltage ratio and the power handled by the resonant tank, the frequency excursion can be high. This complicates the design and construction of the EMI filter used to avoid electromagnetic interference with other equipment, which today is a basic requirement because of the high number of electronic equipment operating simultaneously in a particular installation. Moreover, the EMI filter needs to be designed for the lowest frequency of operation, which in many cases coincides with the maximum power handled by the converter [18]. This is the case, for example, with the parallel resonant converter operating above resonant frequency in ZVS mode. In order to avoid variable frequency operation in dc-dc resonant converters, several approaches have been proposed in the literature. The most common one makes use of a full-bridge inverter with phase-shift control, where the gate signals of each branch are shifted in time to obtain a quasisquare output voltage waveform with adjustable rms value [19]. This solution can also be implemented by coupling two inverters in series using an output transformer. However, this solution requires some extra components, power switches, transformers and associated control, and driving circuitry, making it costly and not appropriate for many cost-conscious applications. Another possibility is the use of a two-stage configuration in which the first stage is used to modify the input voltage to the second stage [20], [21]. The main drawbacks of this solution are the higher cost and the lower efficiency as a result of the two-stage configuration. In this paper, an alternative solution to implement a singlestage two-switch constant-frequency dc-dc resonant converter is presented. The new approach is based on the use of a variable inductance to include an additional control parameter, namely the converter resonant inductance. Variable inductors have extensively been used in power applications in recent years [22]–[30], [32] and provide a means to changing the inductance that exists in a resonant tank by using a dc-control current circulating through an additional winding. In Section II, the magnetic control is introduced and the characteristics of two basic dc-dc 0885-8993/$26.00 © 2011 IEEE 1370 Fig. 1. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 3, MARCH 2012 Proposed constant frequency dc-dc resonant converter. Fig. 3. Equivalent circuit of the dc-dc series resonant converter. between the inductance and the capacitance, it is usually referred to as a parallel resonant converter because, in this case, the effective load of the resonant tank is connected in parallel with the resonant capacitor [31]. A. Analysis of the DC-DC Series Resonant Converter with Magnetic Control It is well known that the series resonant converter can be modeled with the circuit shown in Fig. 3 [31]. The voltage source V1 represents the fundamental component of the square waveform generated by the half-bridge transistors. The peak value of V1 is given by V1 = Fig. 2. Basic dc-dc converters with magnetic control. (a) Series resonant tank. (b) Parallel resonant tank. 2VCC 4VCC /2 = . π π (1) The converter net load formed by the rectifier, filter and load can be modeled as an effective resistance, RES , with its value given by [31] 8 R (2) π2 where R is the converter load resistance. In this way, it is simple to calculate the current passing through the resonant tank V1 I¯R E S = RES + jωL − j(1/ωCS ) RES = resonant converters are presented. In Section III, the design and implementation of the laboratory prototype is shown. Experimental results are provided in Section IV. Finally, the paper is concluded in Section V. II. MAGNETIC CONTROL OF DC-DC RESONANT CONVERTERS Fig. 1 illustrates the block diagram of the proposed converter. The basic structure is similar to any dc-dc resonant converter. What differs is that the proposed converter uses a variable inductance in the resonant tank circuit. The variable inductance is controlled by means of a dc current that makes it possible to change the effective inductance value by taking certain parts of the magnetic core to a close-to-saturation region. The effective magnetic reluctance of the arrangement is, therefore, increased resulting in a lower value for the inductance [25]. Fig. 2 shows the two most common resonant inverters in which this technique could be applied. Fig. 2(a) shows the series resonant converter (SRC), which corresponds to a resonant tank in which the effective load (given by the arrangement rectifier, filter capacitor and load resistance) is connected in series with the resonant elements. In Fig. 2(b) the parallel resonant converter (PRC) is shown. Even though this converter makes use of a series resonance = V1 (8R/π 2 ) + jωL − j(1/ωCS ) (3) where ω is the angular frequency of the input voltage ω = 2πf . This current is rectified and then filtered by the output capacitance, CO . Thus, the dc output voltage VOS will be given by the rectified mean output current multiplied by the load resistance. This is 2||I¯R E S || R VOS = π 1 2V1 = . (4) 2 2 2 π (8/π ) + (1/R )(ωL − (1/ωCS ))2 Using (4), the dc output voltage can be plotted for different load resistances as a function of the resonant inductance, which is the control parameter in the proposed technique. Fig. 4 illustrates the output voltage characteristics of the series resonant converter with magnetic control. ALONSO et al.: ANALYSIS, DESIGN, AND EXPERIMENTATION ON CONSTANT-FREQUENCY DC-DC RESONANT CONVERTERS Fig. 4. Output voltage characteristics of the series resonant converter with magnetic control. 1371 Fig. 6. Output voltage characteristics of a parallel resonant converter with magnetic control. Using (5) in (7), the following expression is obtained for the dc output voltage VO = Fig. 5. Equivalent circuit of the dc-dc parallel resonant converter. As can be seen in Fig. 4, in the case of the series resonant inverter, a high inductance excursion might be necessary if the converter must operate with very light loads. In fact, for no load operation, the required inductance value would be infinite in the ideal case. This effect may limit the use of this type of converter in applications in which a wide load range is required. B. Analysis of the DC-DC Parallel Resonant Converter with Magnetic Control Fig. 5 illustrates the equivalent circuit of the dc-dc parallel resonant converter. It is well known that for this converter, the effective load given by the rectifier, filter, and load resistance is given by the following expression [31] REP π2 R. = 8 (5) The voltage across the effective resistance can then be easily calculated by V̄R E P = V1 −(jREP /ωCP ) . (L/CP ) + jREP (ωL − (1/ωCP )) (6) 1 2V1 . π ((8/π 2 R)ωL)2 + 1(1 − ω 2 LCP )2 (8) Fig. 6 illustrates the dc output voltage characteristic of the parallel resonant converter obtained by plotting (8). As can be seen, in the case of the parallel converter, it is possible to attain operation from full load to open load maintaining a reasonable excursion of the resonant inductance. Therefore, this topology would be ideal for applications in which operation at light loads is required. III. LABORATORY PROTOTYPE In order to test the possibilities of the dc-dc resonant converters with magnetic control, an experimental prototype has been developed. The prototype has been designed to operate with a 48 V input voltage and a 5 V output voltage. The maximum output current is 10 A. Switching frequency has been selected to be 500 kHz. The complete electric diagram of the prototype can be seen in Fig. 7. A step-down transformer is used between the resonant capacitor and the output rectifier to achieve the necessary step-down ratio. A. Resonant Tank The first step for designing the converter is the selection of the transformer turns ratio. In this prototype, the transformer is designed so that the resonant tank operates with an input-output voltage ratio close to unity. This gives the following value for the transformer secondary to primary turns ratio n= π 2 VO πVO /2 = 2VCC /π 4VCC (9) where VO is the dc output voltage and VCC is the total dc input voltage that supplies the converter. Using the values previously mentioned, a turns ratio n = 0.26 is obtained. The transformer was implemented using an EFD20 core with N87 magnetic 2V1 (REP /ωCP ) 2||V̄R E P || = . material, with nine turns in the primary side and two turns in VO = 2 (ωL − (1/ωC ))2 π π each secondary winding. This gives a final turns ratio of n = (L/CP )2 + EEP P (7) 0.23. Additional details are shown in Table I. This voltage is rectified and filtered, so that the dc output voltage applied to the converter load is equal to the rectified mean voltage, which can be calculated as follows 1372 Fig. 7. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 3, MARCH 2012 Closed loop diagram of the dc-dc resonant converter with magnetic control. TABLE I EXPERIMENTAL PROTOTYPE MATERIAL LIST Fig. 8. Inductance as a function of load resistance for different values of tank capacitance. Once the turns ratio has been obtained, it is possible to calculate the effective load resistance of the resonant tank transferred to the primary side REP = π2 R . 8n2 (10) Using the values for the present prototype R = 0.5 Ω and n = 0.23, in (10), a value REP = 11.7 Ω is obtained. The output voltage can be calculated as a function of the inductance by using (8) and taking into account the transformer turns ratio, as follows: VO = 1 2nV1 . (11) 2 2 2 π ((8n /π R)ωL) + 1(1 − ω 2 LCP )2 L= 2ω 2 CP ± Solving (11) for the inductance L, the necessary inductance for a given output voltage and load resistance is obtained as shown in (12), at the bottom of this page. There are two solutions corresponding to the operation below and above resonance. In this prototype, operation above resonance is preferred so that ZVS can be achieved. Thus, Fig. 8 illustrates the inductance as a function of the load resistance for different values of the resonant tank capacitance, obtained by plotting (12). It is seen how for high values of the load resistance, the inductance remains nearly constant because the operation is close to the no load condition. It can also be seen how, as long as the capacitance decreases, the inductance excursion increases. Thus, in order to avoid an excessive inductance excursion, a capacitance CP = 45 nF has been selected for this prototype. (2ω 2 CP )2 − 4[(8n2 ω/π 2 R)2 + (ω 2 CP )2 ][1 − (2nV1 /πVO )2 ] . 2[(8n2 ω/π 2 R)2 + (ω 2 CP )2 ] (12) ALONSO et al.: ANALYSIS, DESIGN, AND EXPERIMENTATION ON CONSTANT-FREQUENCY DC-DC RESONANT CONVERTERS 1373 The inductor must be designed to operate at its maximum value when no dc-bias current is injected through the control winding. This condition gives the number of turns necessary for the variable inductor. Then, with the calculated inductance factor and the maximum inductance value, which in this case is 4.5 μH, and using (14), a number of turns equal to 6 is obtained. The expected peak value of the flux density for the ac component can be calculated as follows: B̂ = Fig. 9. Output voltage characteristics of the designed prototype. Therefore, the inductance will vary between approximately 3 μH and 4.3 μH. In order to check these results, Fig. 9 illustrates the output voltage characteristics obtained by plotting (11) for the nominal input voltage. As can be seen, operation from no load to nominal load is achieved. To conclude the design, the values of the output filter inductance and capacitance must be obtained. These components are used to filter the rectified sinusoidal output voltage. The design of these components follows the typical methodology for a low pass filter, given the inductance current ripple and the output voltage ripple. In this case, the calculated values were L = 10μH and C = 6.8 μF. In order to check the design, a simulation was carried out using the PSIM simulator. Fig. 10 illustrates the circuit used for the simulation. The results are shown in Figs. 11 and 12. As can be seen, the output voltage reaches the nominal level of approximately 5.0 V in 50 μs. The peak inductance current is around 6.0 A and the transformer primary current is a square waveform with a peak value of 2.32 A. All simulation results are in good agreement with the theoretical design. B. Variable Inductor An EFD25 core with N87 material by Siemens was selected to implement the variable inductor. It was built following the structure shown in Fig. 13(a) [25]–[30]. The selected cores are 0.3 mm gapped each, giving a total gap g = 0.6 mm. (Note that this selection was made based on laboratory availability. Other cores and especially other magnetic materials would surely be more adequate, since the N87 material is in the limit for 500 kHz operation.) Then, the inductance factor AL for this core can be calculated as follows: AL = μ0 Ae g (13) where μ0 is the vacuum permeability and Ae is the effective area of the core. Using the values for the EFD25, an inductance factor equal to 121.5 nH is obtained. With this value it is possible to calculate the number of turns necessary for a given inductance L . (14) N= AL N AL ˆ I Ae (15) where Iˆ is the peak value of the ac current through the inductance. In the design, the peak current is nearly 6.0 A for the nominal operating point. Employing (15), the calculated ac peak flux density is 75 mT. Using this value in the losses characteristics given by the manufacturer, the estimated loss per volume is 400 kW/m3 . This means that the total loss for the core set is 1.3 W. To this should be added the losses in the winding. This calculation is not however obvious and is therefore out of the scope of this paper. These losses will be evaluated experimentally as shown in next section. The control winding was implemented using NDC = 65 turns and with a 0.15 mm diameter wire. Fig. 13(b) shows a photograph of the variable inductor. The measured inductance as a function of the dc control current applied through the control windings is illustrated in Fig. 14. As can be seen, an inductance variation from 3.0 to 4.5 μH is achieved with a maximum control current of 180 mA. This current level is perfectly handled by a linear regulator without excessive efficiency penalty. The details of the laboratory prototype are shown in Table I and the electric diagram of the prototype for open-loop operation is illustrated in Fig. 15. C. Dynamic Response The dynamic response of the laboratory prototype was obtained by means of a step-response test. For this, a dc voltage with a 100 mV peak-to-peak superposed square waveform was applied to the control input shown in Fig. 15. The result is illustrated in Fig. 16. From the obtained waveform, it is clear that the system follows a second-order response that can be expressed as kωn2 vo (s) = 2 (16) G(s) = vc (s) s + 2ξωn s + ωn2 where vo and vc are the small-signal perturbations of the output voltage and control voltage, respectively, and k, ξ, and ω n are the dc gain, the damping factor and the angular frequency of the response, respectively. These values can be calculated from the waveforms shown in Fig. 16 using traditional control theory [33], as shown in Appendix I. The values obtained for the different parameters in (16) are shown in Table II. Using these parameters, the Bode diagram illustrated in Fig. 17 can be represented. In order to test the converter in closed loop, a compensator based on the electric diagram shown in Fig. 18 was designed. The dynamic response of the compensator can be expressed as 1374 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 3, MARCH 2012 Fig. 10. Electric diagram of the prototype for simulation under PSIM. Fig. 11. Simulation results at nominal load. (a) Output voltage. (b) Resonant tank input voltage and resonant tank current. ALONSO et al.: ANALYSIS, DESIGN, AND EXPERIMENTATION ON CONSTANT-FREQUENCY DC-DC RESONANT CONVERTERS Fig. 12. Fig. 13. 1375 Simulation results at nominal load. (Top) Transformer primary voltage. (Bottom) current. Developed variable inductor. (a) Schematic diagram. (b) Photograph. The compensator was designed using traditional techniques, obtaining the values for the constant K, and the zero and pole frequencies ω z and ω p , respectively. Capacitor CA is used to avoid high frequency noise at the input of the error amplifier. Therefore, the frequency of the pole is selected to be around 1 kHz. The frequency of the zero is selected to compensate the effect of the filter pole and it is placed around 10 kHz. In this manner, the compensator behaves mainly as an integral action, thus making zero the steady state error. The exact parameters used in the compensator are shown in Table III. The Bode diagram of the compensator is illustrated in Fig. 19 and the system open-loop Bode diagram is shown in Fig. 20. With the values shown in Table III, a 75◦ phase margin is achieved. The obtained gain margin was 24 dB and the 0 dB bandwidth was 350 Hz. Thus, the theoretical step response of the converter operating in closed loop is illustrated in Fig. 21. As can be seen, an over damped response is obtained with a settling time around 1 ms. IV. EXPERIMENTAL RESULTS Fig. 14. Inductance value as a function of the dc control current. follows: C(s) = K 1 + s/ωz s(1 + s/ωp ) (17) where K = RA [R1 C2 (RA + RB )]−1 , ωz = (R2 C2 )−1 , ωp = (Rp CA )−1 and RP = RA RB /(RA + RB ). Fig. 22 illustrates the complete electric diagram of the laboratory prototype, including the error amplifier circuit and the voltage-controlled current source used to generate the dc control current for the variable inductance. Fig. 23 shows the gate-to-source driving signals of the bridge transistors. As can be seen, the operating frequency is 500 kHz with a dead time of approximately 250 ns, which is used to assure ZVS operation. Fig. 24 illustrates the waveforms at the input of the resonant tank, i.e., the square input voltage and the resonant current, for 48 V input voltage and 50 W output power. Note that the voltage was measured using an isolated voltage probe with a 5:1 attenuation. It can be observed a ZVS operation because the voltage transition is smooth with no voltage spikes. A good agreement is found with the simulation results shown in Fig. 11. Fig. 25 shows the voltage and current waveforms in the 1376 Fig. 15. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 3, MARCH 2012 Electric diagram of the prototype for open-loop operation. Fig. 17. Bode diagram of the open-loop control-to-output voltage response of the laboratory prototype. TABLE III PARAMETERS OF THE COMPENSATOR Fig. 16. Experimental dynamic response of the laboratory prototype. (Bottom) Reference voltage step (1 V/div). (Top) Output voltage (200 mV/div). Horiz. scale: 500 μs/div. TABLE II PARAMETERS OF THE DYNAMIC RESPONSE OF THE LABORATORY PROTOTYPE half-bridge low transistor. Fig. 26 illustrates the transformer input voltage and current. These waveforms also match quite well the theoretical analysis and the simulation results illustrated in Fig. 12. Fig. 18. Electric diagram of the compensator. ALONSO et al.: ANALYSIS, DESIGN, AND EXPERIMENTATION ON CONSTANT-FREQUENCY DC-DC RESONANT CONVERTERS Fig. 19. Bode diagram of the compensator. Fig. 20. Bode diagram of the system in open loop. Fig. 27 shows the measured output voltage as a function of the resonant inductance for several load resistances when operating at 48 V input voltage. The voltage response is as expected from the theoretical analysis and demonstrates that the inductance can be used as a control parameter for the output voltage. It must be noted that in this figure, the normal operation range for which the converter was designed goes from approximately 3 to 4.5 μH. This can be seen by the fact that the 5 V output for the 0.5 Ω load is achieved for an inductance of nearly 3 μH. This point represents the full load operating point. The rest of the data represented out of the normal operation point were measured in order to check the static response of the converter. The closed-loop operation results are illustrated in Fig. 28. As can be seen, a good voltage regulation is achieved as long as there is margin for the dc control current to control the inductance. The regulation is limited in the higher input voltage range by Fig. 21. Step response of the converter in closed loop operation. 1377 1378 Fig. 22. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 3, MARCH 2012 Schematic diagram of the complete prototype. Fig. 23. Gate-to-source voltage in the bridge transistors illustrating the dead time between the two driving signals. 5 V/div, 500 ns/div. Fig. 24. Resonant tank waveforms: input voltage, input current and input power (top). 25 V/div, 10 A/div, 500 ns/div. the dc control current reaching zero and in the lower range by the ZVS-ZCS boundary. When ZCS is reached, the switching losses of the bridge transistors increase greatly, mainly due to the poor behavior or the MOSFETs body diodes. This reduces efficiency and could even destroy the switches due to excessive junction temperature. Therefore, operation in this region was avoided. Fig. 29(a) shows the experimental results for a load step from 5 to 10 A. The converter output voltage and the compensator ALONSO et al.: ANALYSIS, DESIGN, AND EXPERIMENTATION ON CONSTANT-FREQUENCY DC-DC RESONANT CONVERTERS 1379 Fig. 27. Experimental results for open-loop operation at 48 V input voltage. Measured dc output voltage as a function of the variable inductance for different loads. Fig. 25. Current and voltage waveforms in the half-bridge low transistor. 20 V/div, 5 A/div, 500 ns/div. Fig. 28. Experimental results for closed-loop operation. Measured output voltage and inductance control current as a function of the dc input voltage. TABLE IV ESTIMATED LOSSES FOR THE LABORATORY PROTOTYPE Fig. 26. Transformer input voltage and current at nominal operation. 20 V/div, 5 A/div, 1 μs/div. output voltage are shown. Similarly, Fig. 29(b) shows the same waveforms for a load step from 10 to 5 A. As can be seen, in both tests, the output voltage is compensated in around 1 ms, which is in good agreement with the theoretical design. One of the main drawbacks of the developed prototype was found in the low efficiency attained, which reached a maximum of only 82% at full load (48 V input, 5 V-10 A output). Even though synchronous rectification was implemented using IRFZ48 transistors, the efficiency was not improved significantly. In order to investigate the low efficiency attained, the losses in the variable inductor were measured using the oscilloscope method [34]. This is not the most accurate method for measuring inductor losses but it is admissible in this case in order to determine if an excessive efficiency penalty is being caused by this component. Thus, Fig. 30 illustrates the inductor voltage, current, and power waveforms. The power waveform is calculated by the digital scope from the inductor voltage and current, and then averaged to obtain the losses. Taking into account the voltage probe attenuation the measured inductor losses are 5·0.768 = 3.84 W. Since the estimated core losses were 1.3 W, this leaves 2.54 W for the winding losses. Table IV gathers the losses in each component, which were estimated by using both electrical and thermal procedures. 1380 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 3, MARCH 2012 Fig. 29. Experimental results for closed-loop operation. CH1: Output voltage (2 V/div). CH2: Compensator output (0.5 V/div). (a) Load step from 5 A to 10 A. (b) Load step from 10 A to 5 A. Horiz. scale: 1 ms/div. energy. To improve the efficiency a new design with a closer operation to resonance, especially at full load, could be carried out. Future research will include the application of the magnetic control to other resonant converter topologies such as the LLC resonant converter. The LLC converter has proven to be highly efficient under variable frequency control, achieving ZVS over the entire operating range with small variation of the switching frequency and low component count [35]. It is expected that using magnetic control same efficiency could be attained and, therefore, it makes a good objective for a future work. APPENDIX I Fig. 30. Voltage, current and power waveforms in the variable inductance at nominal current. 50 V/div, 10 A/div, 500 ns/div. V. CONCLUSION A new methodology for controlling dc-dc resonant converters has been proposed in this paper. The proposed control method is based on the use of a variable inductor in the resonant tank so the inductance can be used as the control parameter to regulate the converter output voltage. The proposed solution allows the converter to operate at constant frequency, thus reducing the size of the EMI filter, which otherwise would be high in the case of using variable frequency control. A 48 V/5 V/10 A prototype operating at 500 kHz was developed showing the correct operation of the proposed control methodology. The one weakness of the prototype was the attainment of only 82% efficiency. This may be caused by the operation of the converter far from resonance, as can be seen in Fig. 9, thus making use of a higher reactive In order to calculate the different parameters of a second order response as that shown in (18), the following expressions can be found in the literature (e.g., [33]) DC gain: Δvo (I.1) k= Δvc where Δvo and Δvc are the dc values of the output voltage and control voltage, respectively. Damping factor: −Ln Mp ξ= . (I.2) π 2 + (Ln Mp )2 Mp being the per-unit overshoot. Angular frequency: ωn = tp π 1 − ξ2 (I.3) where tp is the time instant in which the overshoot takes place. REFERENCES [1] S. Dwari and L. Parsa, “An efficient high-step-up interleaved DC–DC converter with a common active clamp,” IEEE Trans. Power Electron., vol. 26, no. 1, pp. 66–78, Jan. 2011. ALONSO et al.: ANALYSIS, DESIGN, AND EXPERIMENTATION ON CONSTANT-FREQUENCY DC-DC RESONANT CONVERTERS [2] R.-L. Lin, C.-C. Hsu, and S.-K. 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Saraiva, “Magnetically-controlled dimming technique with isolated output,” IET Electron. Lett., vol. 45, no. 14, pp. 756–758, Jul. 2009. [31] R. W. Erikson and D. Maksimovic, Fundamentals of Power Electronics, New York: Kluwer Acad. Pub., 2001. [32] A. S. Kislovski, “Quasi-linear controllable inductor,” Proc. IEEE, vol. 75, no. 2, pp. 267–269, Feb. 1987. [33] K. Ogata, Modern Control Engineering. Upper Saddle River, NJ: Pearson, 1998. [34] A. Van den Bossche and V. C. Valchev, Inductors and Transformers for Power Electronics. New York: Taylor and Francis, 2005. [35] H. Huang, “Designing an LLC resonant half-bridge power converter,” Texas Instruments 2010–2011 Power Supply Design Seminar, SEM1900, 2011. J. Marcos Alonso (S’94–M’98–SM’03) received the M.Sc. and Ph.D. degrees, both in electrical engineering, from the University of Oviedo, Spain, in 1990 and 1994, respectively. From 1990 to 1999, he was an Assistant Professor in the Electrical and Electronic Department of the University of Oviedo. From 1999 to 2007, he was an Associate Professor in the same department, and then in 2007 he was appointed a Full Professor. He is the primary author of more than 70 journal and international conference papers in power and industrial electronics, and has coauthored more than 200. He was the advisor of six Ph.D. theses in the field of power electronics. He is the holder of six Spanish patents. He has been awarded national funding for the intensification of research activity, for the period 2008–2012. His research interests include high-frequency electronic ballasts, discharge lamp modeling, power converters for ozone generation, power converters for electrostatic applications, power factor correction topologies, and high frequency switching converters in general. Dr. Alonso is a member of the European Power Electronics Association and belongs to the International Steering Committee of the European Conference on Power Electronics and Applications (EPE), where he usually collaborates as Topic Co-Chair. He is also a member of the International Ozone Association (IOA). In 2006, he was awarded the Early Career Award of the IEEE Industrial Electronics Society. He received the Best Paper Award of the 2009 IEEE International Symposium on Industrial Electronics, the Second Prize Paper Award of the 2005 IEEE Industry Applications Society Meeting, Production and Application of Light Committee, and the IEEE Industrial Electronics Society Meritorious Paper Award for 1996. Since October 2002, he has served as an Associate Editor of the IEEE TRANSACTIONS ON POWER ELECTRONICS in the field of lighting applications. He has served as a Guest Editor for the Special Issue on Lighting Applications, published in IEEE TRANSACTIONS ON POWER ELECTRONICS, in May 2007. 1382 Marina S. Perdigão (S’06) was born in Coimbra, Portugal, in 1978. She received the M.Sc. degree in electrical engineering from the University of Coimbra, Portugal, in 2004. Since 2006, she has been working toward the Ph.D. degree at the University of Coimbra in cooperation with the University of Oviedo, Spain. Since 2002, she is an Assistant Professor in the Electrical Department of the Superior Institute of Engineering of Coimbra, Portugal,. She is also a Researcher at the Instituto de Telecomunicações, Coimbra, Portugal, since 2001. Her research interests include high-frequency electronic ballasts, discharge lamp modeling, high-frequency switching converters, resonant converters, power electronics for renewable energies, and computer simulation applications. Prof. Perdigão received the Best Paper Award of the 2009 IEEE International Symposium on Industrial Electronics. She also collaborates as a reviewer of the IEEE transaction papers. David Gacio Vaquero (S’08) received the M.Sc. degree in electrical engineering from the University of Oviedo, Spain, in December 2007. He has been working toward the Ph.D. degree in the field of LED lighting in the Electrical and Electronics Department of the University of Oviedo, Spain, since July 2008. He has been a Lecturer and Researcher in the Electrical and Electronics Department of the University of Oviedo, Spain, since March 2008. He has authored or coauthored more than sixteen international conference papers in power and industrial electronics. His research interests include power factor correction topologies, electronic ballasts, power converter modeling, LED modeling, and solid state lighting in general. He collaborates as a transactions’ paper reviewer with the IEEE. Mr. Vaquero received the Best Paper Award of the 2009 IEEE International Symposium on Industrial Electronics. IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 27, NO. 3, MARCH 2012 Antonio J. Calleja (S’96–A’98–M’04) was born in Leon, Spain, in 1964. He received the B.Sc., M.Sc., and Ph.D. degrees from the University of Oviedo, Gijon, Spain in 1987, 1995, and 2000, respectively. From 1995 to 2001, he was an Assistant Professor. Since 2002, he has been an Associate Professor. His research interests include areas of high frequency electronic ballast, discharge lamps modeling, power factor correction topologies, and industrial control system. Eduardo Sousa Saraiva (M’79) received the Degree of Electrical Engineer from the University of Oporto, Portugal, in 1970, and the Ph.D. degree in electrical engineering from the University of London, in 1979. He received the Aggregation in Electrical Engineering from the University of Coimbra, Portugal, in 1985. He worked at the Faculty of Science and Technology of the University of Coimbra, Portugal, as Assistant Eventual from July 1973 to July 1975, as Assistant from July 1975 to July 1979, as Auxiliary Professor from July 1979 to November 1979, as Associated Professor from December 1979 to November 1986, and since from November 1986 to July 2011, as a Full Professor. He retired on August 1, 2011. He is the coauthor of approximately one hundred papers in national and international journals and conferences. His research interests included electric machines, power electronics, the influence of the electromagnetic field in the human body, and most recently, electronic ballasts for fluorescent lamps. He served as advisor or coadvisor to over a dozen Masters and Ph.D. candidates. He has been collaborating as reviewer of the IEEE papers. He is one of the founders of the IEEE Portugal Chapter. Dr. Saraiva is a member of the Portuguese “Ordem dos Engenheiros,” having in the past been elected to positions in the Central Region and on National Boards.