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Modelling of Single Transistor Parallel ZVS DC-DC Converter Nikolay Hinov1, Dimitar Arnaudov1, George Kraev1, Bogdan Gilev2 1

Technical University of Sofia, Department of Power Electronics, e-mail: [email protected], [email protected], [email protected] 2 Technical University of Sofia, Department of Stochastics and Optimization, e-mail: [email protected]

Abstract— In present work is carried out modelling of single transistor parallel ZVS DC-DC converter. Main advantage of power circuit is direct work in steady mode. Thus, that allows avoiding a starting procedure and guarantees reliable operation in significant major change in load occur. The model is realised with Matlab. Computer simulations and experiments in lab model are performed by which the model is confirmed. Version of controlling the converter is proposed, also practically realised. Keywords—resonant Simulation

I.

DC-DC

converter,

ZVS,

Rm and Lm – active resistance and inductance used for modelling mutual inductance between primary and secondary coil; Here are some additional parameters: L1'= L1 + L; L2'= L2 + LF; D = L1' L2' - L2m;

R

Modelling,

LF D13

INTRODUCTION

Resonant converters with DC output are a natural advance of resonant inverters. They are widespread in industry and households. When low power is needed (up to 2kW) single transistor circuits with soft commutation are broadly used, because of simple topology, few operated devices and high efficiency [1, 2, 3, 4].

D14

D11

D12 Ɍr

L

In this work is examined a single transistor parallel converter with ZVS (zero voltage switching). Typical characteristic of circuit shown in Fig. 1, is the direct steady mode of operation, when some conditions are met. II.

CT

C

Ud

T

CS

D

CONVERTER MODELLING

The circuit is composed by: capacitive load (CT and R), transistor T, resonant device – capacitor C and inductance L, reverse diode D, transformer, full-bridge rectifier (D11÷D14) and filter inductance LF. All considerations are made with following assumptions: all devices in power circuit are ideal; zero internal resistance of power supply.

Circuit operation is described with 5 time intervals matching its 5 states. Thus, is synthesized a general system of equations describing the examined object with variable structure (Fig.2).

The modelling object has variable structure. Its work in the different stages of operation is described with systems of differential equations [5].

For the different stages of operation, the circuit configuration is described by combination of different values of variables İ1, İ2 and İ3, which are displayed in Table 1.

The following names are used: i1 – current through transformer’s primary winding; i2 – current through transformer’s secondary winding; iR – current through load’s resistor; uC – voltage across commutating capacitor; Ud – input voltage; R1 and L1; R2 and L2 – active resistance and inductance of primary and secondary transformer windings respectively;

,(((

Single transistor resonant DC-DC converter

ε1

TABLE I.

VALUE OF VARIABLE FOR DIFFERENT STAGES OF OPERATION

No of stage

ε2

AND

1

2

3

4

5

ε1 ε2

0

1

1

1

0

1

1

-1

1

1

ε3

1

0

0

0

1

ε3,

Then, for the five time intervals defined by the coefficients displayed in Table 1, the system of differential equations is divided into five separate different systems. Entire object

model is obtained by joint resolution of a.m. five systems of

− L2 ' R1 + Lm Rm di1 i1 = dt D di2 L R − L 'R = ε 2 m 1 1 m i1 dt D di R = dt du 1 ε1 C = ε 1 i1 dt C

across parallel capacitor uC; voltage of secondary transformer coil u2; currents flowing through rectifying diodes iD1 and iD2;

− L2 ' R m + L m R 2 i2 D − L1 ' R2 + Lm Rm i2 + D 1 i2 CT R

Lm R iR D L 'R + 1 iR D 1 + iR CT R

+ ε2

L2 ' uC D L + ε 1ε 2 m u C D

−ε2

− ε1

L2 ' Ud D L − ε 3ε 2 m U d D (1) + ε3

System differential equations describing operation of Single transistor resonant DC-DC converter

equations. For this purpose, are assumed: zero initial conditions for the current and that the voltage across commutating capacitor is equal to Ud. Then these systems are solved consecutively, using the time-stitching method. III.

MODELLING RESULTS

Obtained model is realised in the software package MATLAB. Determination of the different stages is made by setting transistor’s switching-off current, the moment when capacitor’s voltage reaches the voltage of DC power supply, as well as moments when input voltage of rectifier crosses the zero value.

i1 [A]

20 0 -20

5.97

5.975

5.98 5.985 Time [s]

5.99

5.98 5.985 Time [s]

5.99

5.98 5.985 Time [s]

5.99

5.98 5.985 Time [s]

5.99

5.995

6

6.005 -3

x 10

uC [V]

100 0 -100

5.965

5.97

5.975

5.995

6

6.005 -3

x 10

u2 [V]

50 0 -50

5.965

5.97

5.975

5.995

6

6.005

These are values of devices in circuit when testing: L=10μH, L1=10μH, C=1μF, L2=10μH, LF=600μH, CT=100μF, R=25Ω. The input voltage is 25V. 15A is set for transistor’s switching-off current. Graphic results are for steady mode of operation, i.e. when processes in rectifier became stable. ZVS for switching transistor is observed from waveforms, where controlling frequency is 25 kHz. Output current and load’s voltage are with insignificant pulsations, which allows use of circuit in different applications as: power supply for LEDs, charging capacitors with large capacitance value and supercapacitors. The possibility of changing transistor’s switch-off current seems as feature for controlling and stabilizing output voltage and current. This possibility is connected with changing of operation frequency, though in narrow limits, and also it is restrained of the need of storing enough energy in commutating inductance, which can provide twice recharging of commutating capacitor and respectively switching on reverse diode. Modelling of the control of converter grants an opportunity to create an efficient Control System that is needed to conduct simulation and experimental studies.

-3

x 10

IV.

2

1 0

1

iD , iD [A]

Δ i2 [A] T

Δ uC [V]

5.965

output current pulsations ǻi2 and output voltage pulsations ǻuCT.

-1

5.965

5.97

5.975

5.995

6

6.005 -3

SIMULATING STUDIES

Simulating studies of parallel DC resonant converter are carried out jointly with Control System.

x 10

0.8 10V

0.6 0.4

5V

5.965

5.97

5.975

5.98 5.985 Time [s]

5.99

5.995

6

6.005 -3

x 10

SEL>> 0V

Load voltage

20A

13.47 13.46 13.45

0A

5.965

5.97

5.975

5.98 5.985 Time [s]

5.99

5.995

6

6.005 -3

x 10

-20A

Current through resonant inductance

100V

50V

0V 9.80ms

Results obtained through analisys of parallel DC converter

In Fig. 3 are presented results from modelling of parallel resonant converter in following order: input current i1; voltage

9.82ms

9.84ms

Voltage on transistor and diode

9.86ms

9.88ms

9.90ms

9.92ms

9.94ms

9.96ms

9.98ms

10.00ms

Time

Steady mode of operation of parallel converter with activecapacitive load

The chosen current of transistor switch off is 15A. The displayed results are with active load and capacitive filter, in Fig.4 are shown detailed results of steady operation mode, while in Fig.5 – the progress of transient process of starting. 10V

Experiment with active load is performed with following values of circuit’s element: filter capacitor CF=2200μF, LF=500μH, load resistance 25, commutating (resonant) capacitor C=1μF, commutating (resonant) inductance L=40μH, voltage of power supply 25V. Measurement are conducted with voltage probe 10:1, while current is measured with current probe.

5V

0V

The results of experiment with lab model are shown in Fig.7, Fig.8, Fig.9 and Fig.10.

Load voltage

20A 10A 0A -10A SEL>> -20A

Current through resonant inductance

100V

50V

0V 0s

1ms

2ms

Voltage on transistor and diode

3ms

4ms

5ms

6ms

7ms

8ms

9ms

10ms

Time

Start procedure of parallel converter with active-capacitive load

Better handling of output voltage is obtained when working with LC filter. In Fig.6 are shown results with such circuit. 7.5V 5.0V 2.5V SEL>> 0V

Load voltage

20A 10A 0A -10A -20A

Current through resonant inductance

Current flowing through commutating inductance and voltage across transistor

100V

50V

0V 9.80ms

9.82ms

9.84ms

9.86ms

9.88ms

Voltage on transistor and diode

9.90ms

9.92ms

9.94ms

9.96ms

9.98ms

10.00ms

Time

Obtained experimental results confirm those mathematic modelling and computer simulations.

from

Simulating results of parallel converter with LC filter

When using LC filter and keeping same values of elements in circuit, the output voltage is lower than when having capacitive load. Simulation are carried out when filter inductance is 600μH. Typic for both options is maintaining ZVS and as can be seen converter starts operating in steady mode, which was the experiments’ aim. Circuit has promising future in charging supercapacitors, which lately are becoming more actual as alternative or for joint work with different type of batteries [6, 8, 9, 10]. V.

EXPERIMENTS

Experimental results of work of serial converter with active-capacitive load are presented. The Control System is fulfilled with graphical programming software LabVIEW. Thus, system allows maximum flexibility and options for obtaining different settings aiming optimization and dynamics. The switch element is IGBT device type G4PC50W and reverse diode RURG3060, rectifying diodes are PK MUR 460 0479. The widespread distribution of single transistor resonant converters, forced electronics makers to develop a special series of transistors designed for this type of circuit [4]. They take into account the loads and operating modes of these power circuits, thus increasing the reliability of power electronics. Such IGBT was also used in the experimental model.

Current throug primary winding and output current of rectifier (before filter)

From completed experiments is founded that circuit is efficient with different loads. Using combined control, which grants at once maximum current through transistor and ZVS operation gives good results and guarantees indicators at output of the parallel single transistor converter of direct voltage. Characteristic of the research converter is that its power scheme is made up of a few elements. In addition, the use of galvanic separation is not necessary for the realization of the

transistor control. These features lead to a low cost of end products and help to penetrate the electronics market. Therefore, single transistor converters is a very good solution for a variety of industrial and household applications.

electric generation from unconventional sources of energy, charging stations for different type of batteries and supercapacitors, for the realization of high efficient light sources and many other applications. Improved qualities of studied converter is due to features of power circuit of converter, while on Control and Safeguard System are assigned additional roles related to the reliable work of electronic devices.

ACKNOWLEDGMENT The carried out research is realized in the framework of the project ''Model based design of power electronic devices with guaranteed parameters'', Ⱦɇ07/06/15.12.2016, Bulgarian National Scientific Fund. REFERENCES [1] [2] Current pulsation through filter inductance and pulsation of output voltage

On the other hand operation with soft commutation are retained, which improves efficiency of studied converter. Disadvantage of single transistor circuits is the increased load in magnetic components, which reflects in their design [7, 10, 11, 12, 13].

[3] [4]

[5]

[6]

[7] [8] [9]

[10]

[11] Current through resonant capacitor and voltage across transistor

VI.

CONCLUSION

In the paper is offered mathematic model and simulating & experimental studies are conducted with single transistor parallel converter with direct work in steady mode. The tested converter of DC energy thanks to its improved characteristics could find application in electric transport,

[12]

[13]

Erickson R. W., D. Maksimovic, “Fundamentals of Power Electronics”, Second Edition, Kluwer Academic Publishers Group, 2001. Mohan, Ned; Undeland, Tore M.; Robbins, William P., „Power Electronics - Converters, Applications, and Design (3rd Edition)”, © 2003 John Wiley & Sons. Rashid M. H., “Power Electronics handbook: devices, circuits, and applications”, Academic Press, 2007. Cerezo Jorge, „IGBT Definition for Single Ended Induction Heating Cookers”, Bodo´s Power Systems® - Electronics in Motion and Conversion, April 2012, ISSN: 1863-5598, 22-30. Ivensky G., Arkady Kats, Sam Ben-Yaakov, “An RC load model of parallel and series-parallel resonant DC-DC converters with capacitive output filter” , IEEE Trans. Power Electron. Vol.14 pp. 515-521 May 1999. Kraev, G., N. Hinov, D. Arnaudov, N. Ranguelov and N. Gradinarov, Multiphase DC-DC Converter with Improved Characteristics for Charging Supercapacitors and Capacitors with Large Capaci-tance, Annual Journal of Electronics, Volume 6, Book 1, Technical University of Sofia, Faculty of Electronic Engineering and Technologies, ISSN 1314-0078, pp.128-131, 2012. J. Holtz, Pulse width modulation electronic power conversion, Proceedings of the IEEE, vol.82, no. 8, pp. 1194-1214, August 1994. M. P. Kazmierkowski, R. Krishnan, and F. Blaabjerg, Control in power electronics. Academic Press, 2002. C. Cutrona, C. Di Miceli, “A unified approach to series, parallel and series-parallel resonant converters”, Telecommunications Energy Conference, INTELEC '92, pp. 139 – 146, 1992. B. L. Dokiü, B. Blanuša, “Power Electronics Converters and Regulators” - Third Edition, © Springer International Publishing, Switzerland 2015, ISBN 978-3-319-09401-4. G.T. Nikolov, Valchev, V.C., “Nanocrystalline magnetic materials versus ferrites in power electronics”, Procedia Earth and Planetary Science, pp. 1357-1361, 2009. Van Den Bossche, A., Valchev, V.C., Barudov, S.T., “Practical wide frequency approach for calculating eddy current losses in transformer windings”, IEEE International Symposium on Industrial Electronics, pp. 1070-1074, 2006. V.C. Valchev, Van Den Bossche, A.P., Van De Sype, D.M., “Ferrite losses of cores with square wave voltage and DC bias”, IECON Proceedings (Industrial Electronics Conference), 2005, pp. 837-841.

Modelling of Single Transistor Parallel ZVS DC-DC Converter Nikolay Hinov1, Dimitar Arnaudov1, George Kraev1, Bogdan Gilev2 1

Technical University of Sofia, Department of Power Electronics, e-mail: [email protected], [email protected], [email protected] 2 Technical University of Sofia, Department of Stochastics and Optimization, e-mail: [email protected]

Abstract— In present work is carried out modelling of single transistor parallel ZVS DC-DC converter. Main advantage of power circuit is direct work in steady mode. Thus, that allows avoiding a starting procedure and guarantees reliable operation in significant major change in load occur. The model is realised with Matlab. Computer simulations and experiments in lab model are performed by which the model is confirmed. Version of controlling the converter is proposed, also practically realised. Keywords—resonant Simulation

I.

DC-DC

converter,

ZVS,

Rm and Lm – active resistance and inductance used for modelling mutual inductance between primary and secondary coil; Here are some additional parameters: L1'= L1 + L; L2'= L2 + LF; D = L1' L2' - L2m;

R

Modelling,

LF D13

INTRODUCTION

Resonant converters with DC output are a natural advance of resonant inverters. They are widespread in industry and households. When low power is needed (up to 2kW) single transistor circuits with soft commutation are broadly used, because of simple topology, few operated devices and high efficiency [1, 2, 3, 4].

D14

D11

D12 Ɍr

L

In this work is examined a single transistor parallel converter with ZVS (zero voltage switching). Typical characteristic of circuit shown in Fig. 1, is the direct steady mode of operation, when some conditions are met. II.

CT

C

Ud

T

CS

D

CONVERTER MODELLING

The circuit is composed by: capacitive load (CT and R), transistor T, resonant device – capacitor C and inductance L, reverse diode D, transformer, full-bridge rectifier (D11÷D14) and filter inductance LF. All considerations are made with following assumptions: all devices in power circuit are ideal; zero internal resistance of power supply.

Circuit operation is described with 5 time intervals matching its 5 states. Thus, is synthesized a general system of equations describing the examined object with variable structure (Fig.2).

The modelling object has variable structure. Its work in the different stages of operation is described with systems of differential equations [5].

For the different stages of operation, the circuit configuration is described by combination of different values of variables İ1, İ2 and İ3, which are displayed in Table 1.

The following names are used: i1 – current through transformer’s primary winding; i2 – current through transformer’s secondary winding; iR – current through load’s resistor; uC – voltage across commutating capacitor; Ud – input voltage; R1 and L1; R2 and L2 – active resistance and inductance of primary and secondary transformer windings respectively;

,(((

Single transistor resonant DC-DC converter

ε1

TABLE I.

VALUE OF VARIABLE FOR DIFFERENT STAGES OF OPERATION

No of stage

ε2

AND

1

2

3

4

5

ε1 ε2

0

1

1

1

0

1

1

-1

1

1

ε3

1

0

0

0

1

ε3,

Then, for the five time intervals defined by the coefficients displayed in Table 1, the system of differential equations is divided into five separate different systems. Entire object

model is obtained by joint resolution of a.m. five systems of

− L2 ' R1 + Lm Rm di1 i1 = dt D di2 L R − L 'R = ε 2 m 1 1 m i1 dt D di R = dt du 1 ε1 C = ε 1 i1 dt C

across parallel capacitor uC; voltage of secondary transformer coil u2; currents flowing through rectifying diodes iD1 and iD2;

− L2 ' R m + L m R 2 i2 D − L1 ' R2 + Lm Rm i2 + D 1 i2 CT R

Lm R iR D L 'R + 1 iR D 1 + iR CT R

+ ε2

L2 ' uC D L + ε 1ε 2 m u C D

−ε2

− ε1

L2 ' Ud D L − ε 3ε 2 m U d D (1) + ε3

System differential equations describing operation of Single transistor resonant DC-DC converter

equations. For this purpose, are assumed: zero initial conditions for the current and that the voltage across commutating capacitor is equal to Ud. Then these systems are solved consecutively, using the time-stitching method. III.

MODELLING RESULTS

Obtained model is realised in the software package MATLAB. Determination of the different stages is made by setting transistor’s switching-off current, the moment when capacitor’s voltage reaches the voltage of DC power supply, as well as moments when input voltage of rectifier crosses the zero value.

i1 [A]

20 0 -20

5.97

5.975

5.98 5.985 Time [s]

5.99

5.98 5.985 Time [s]

5.99

5.98 5.985 Time [s]

5.99

5.98 5.985 Time [s]

5.99

5.995

6

6.005 -3

x 10

uC [V]

100 0 -100

5.965

5.97

5.975

5.995

6

6.005 -3

x 10

u2 [V]

50 0 -50

5.965

5.97

5.975

5.995

6

6.005

These are values of devices in circuit when testing: L=10μH, L1=10μH, C=1μF, L2=10μH, LF=600μH, CT=100μF, R=25Ω. The input voltage is 25V. 15A is set for transistor’s switching-off current. Graphic results are for steady mode of operation, i.e. when processes in rectifier became stable. ZVS for switching transistor is observed from waveforms, where controlling frequency is 25 kHz. Output current and load’s voltage are with insignificant pulsations, which allows use of circuit in different applications as: power supply for LEDs, charging capacitors with large capacitance value and supercapacitors. The possibility of changing transistor’s switch-off current seems as feature for controlling and stabilizing output voltage and current. This possibility is connected with changing of operation frequency, though in narrow limits, and also it is restrained of the need of storing enough energy in commutating inductance, which can provide twice recharging of commutating capacitor and respectively switching on reverse diode. Modelling of the control of converter grants an opportunity to create an efficient Control System that is needed to conduct simulation and experimental studies.

-3

x 10

IV.

2

1 0

1

iD , iD [A]

Δ i2 [A] T

Δ uC [V]

5.965

output current pulsations ǻi2 and output voltage pulsations ǻuCT.

-1

5.965

5.97

5.975

5.995

6

6.005 -3

SIMULATING STUDIES

Simulating studies of parallel DC resonant converter are carried out jointly with Control System.

x 10

0.8 10V

0.6 0.4

5V

5.965

5.97

5.975

5.98 5.985 Time [s]

5.99

5.995

6

6.005 -3

x 10

SEL>> 0V

Load voltage

20A

13.47 13.46 13.45

0A

5.965

5.97

5.975

5.98 5.985 Time [s]

5.99

5.995

6

6.005 -3

x 10

-20A

Current through resonant inductance

100V

50V

0V 9.80ms

Results obtained through analisys of parallel DC converter

In Fig. 3 are presented results from modelling of parallel resonant converter in following order: input current i1; voltage

9.82ms

9.84ms

Voltage on transistor and diode

9.86ms

9.88ms

9.90ms

9.92ms

9.94ms

9.96ms

9.98ms

10.00ms

Time

Steady mode of operation of parallel converter with activecapacitive load

The chosen current of transistor switch off is 15A. The displayed results are with active load and capacitive filter, in Fig.4 are shown detailed results of steady operation mode, while in Fig.5 – the progress of transient process of starting. 10V

Experiment with active load is performed with following values of circuit’s element: filter capacitor CF=2200μF, LF=500μH, load resistance 25, commutating (resonant) capacitor C=1μF, commutating (resonant) inductance L=40μH, voltage of power supply 25V. Measurement are conducted with voltage probe 10:1, while current is measured with current probe.

5V

0V

The results of experiment with lab model are shown in Fig.7, Fig.8, Fig.9 and Fig.10.

Load voltage

20A 10A 0A -10A SEL>> -20A

Current through resonant inductance

100V

50V

0V 0s

1ms

2ms

Voltage on transistor and diode

3ms

4ms

5ms

6ms

7ms

8ms

9ms

10ms

Time

Start procedure of parallel converter with active-capacitive load

Better handling of output voltage is obtained when working with LC filter. In Fig.6 are shown results with such circuit. 7.5V 5.0V 2.5V SEL>> 0V

Load voltage

20A 10A 0A -10A -20A

Current through resonant inductance

Current flowing through commutating inductance and voltage across transistor

100V

50V

0V 9.80ms

9.82ms

9.84ms

9.86ms

9.88ms

Voltage on transistor and diode

9.90ms

9.92ms

9.94ms

9.96ms

9.98ms

10.00ms

Time

Obtained experimental results confirm those mathematic modelling and computer simulations.

from

Simulating results of parallel converter with LC filter

When using LC filter and keeping same values of elements in circuit, the output voltage is lower than when having capacitive load. Simulation are carried out when filter inductance is 600μH. Typic for both options is maintaining ZVS and as can be seen converter starts operating in steady mode, which was the experiments’ aim. Circuit has promising future in charging supercapacitors, which lately are becoming more actual as alternative or for joint work with different type of batteries [6, 8, 9, 10]. V.

EXPERIMENTS

Experimental results of work of serial converter with active-capacitive load are presented. The Control System is fulfilled with graphical programming software LabVIEW. Thus, system allows maximum flexibility and options for obtaining different settings aiming optimization and dynamics. The switch element is IGBT device type G4PC50W and reverse diode RURG3060, rectifying diodes are PK MUR 460 0479. The widespread distribution of single transistor resonant converters, forced electronics makers to develop a special series of transistors designed for this type of circuit [4]. They take into account the loads and operating modes of these power circuits, thus increasing the reliability of power electronics. Such IGBT was also used in the experimental model.

Current throug primary winding and output current of rectifier (before filter)

From completed experiments is founded that circuit is efficient with different loads. Using combined control, which grants at once maximum current through transistor and ZVS operation gives good results and guarantees indicators at output of the parallel single transistor converter of direct voltage. Characteristic of the research converter is that its power scheme is made up of a few elements. In addition, the use of galvanic separation is not necessary for the realization of the

transistor control. These features lead to a low cost of end products and help to penetrate the electronics market. Therefore, single transistor converters is a very good solution for a variety of industrial and household applications.

electric generation from unconventional sources of energy, charging stations for different type of batteries and supercapacitors, for the realization of high efficient light sources and many other applications. Improved qualities of studied converter is due to features of power circuit of converter, while on Control and Safeguard System are assigned additional roles related to the reliable work of electronic devices.

ACKNOWLEDGMENT The carried out research is realized in the framework of the project ''Model based design of power electronic devices with guaranteed parameters'', Ⱦɇ07/06/15.12.2016, Bulgarian National Scientific Fund. REFERENCES [1] [2] Current pulsation through filter inductance and pulsation of output voltage

On the other hand operation with soft commutation are retained, which improves efficiency of studied converter. Disadvantage of single transistor circuits is the increased load in magnetic components, which reflects in their design [7, 10, 11, 12, 13].

[3] [4]

[5]

[6]

[7] [8] [9]

[10]

[11] Current through resonant capacitor and voltage across transistor

VI.

CONCLUSION

In the paper is offered mathematic model and simulating & experimental studies are conducted with single transistor parallel converter with direct work in steady mode. The tested converter of DC energy thanks to its improved characteristics could find application in electric transport,

[12]

[13]

Erickson R. W., D. Maksimovic, “Fundamentals of Power Electronics”, Second Edition, Kluwer Academic Publishers Group, 2001. Mohan, Ned; Undeland, Tore M.; Robbins, William P., „Power Electronics - Converters, Applications, and Design (3rd Edition)”, © 2003 John Wiley & Sons. Rashid M. H., “Power Electronics handbook: devices, circuits, and applications”, Academic Press, 2007. Cerezo Jorge, „IGBT Definition for Single Ended Induction Heating Cookers”, Bodo´s Power Systems® - Electronics in Motion and Conversion, April 2012, ISSN: 1863-5598, 22-30. Ivensky G., Arkady Kats, Sam Ben-Yaakov, “An RC load model of parallel and series-parallel resonant DC-DC converters with capacitive output filter” , IEEE Trans. Power Electron. Vol.14 pp. 515-521 May 1999. Kraev, G., N. Hinov, D. Arnaudov, N. Ranguelov and N. Gradinarov, Multiphase DC-DC Converter with Improved Characteristics for Charging Supercapacitors and Capacitors with Large Capaci-tance, Annual Journal of Electronics, Volume 6, Book 1, Technical University of Sofia, Faculty of Electronic Engineering and Technologies, ISSN 1314-0078, pp.128-131, 2012. J. Holtz, Pulse width modulation electronic power conversion, Proceedings of the IEEE, vol.82, no. 8, pp. 1194-1214, August 1994. M. P. Kazmierkowski, R. Krishnan, and F. Blaabjerg, Control in power electronics. Academic Press, 2002. C. Cutrona, C. Di Miceli, “A unified approach to series, parallel and series-parallel resonant converters”, Telecommunications Energy Conference, INTELEC '92, pp. 139 – 146, 1992. B. L. Dokiü, B. Blanuša, “Power Electronics Converters and Regulators” - Third Edition, © Springer International Publishing, Switzerland 2015, ISBN 978-3-319-09401-4. G.T. Nikolov, Valchev, V.C., “Nanocrystalline magnetic materials versus ferrites in power electronics”, Procedia Earth and Planetary Science, pp. 1357-1361, 2009. Van Den Bossche, A., Valchev, V.C., Barudov, S.T., “Practical wide frequency approach for calculating eddy current losses in transformer windings”, IEEE International Symposium on Industrial Electronics, pp. 1070-1074, 2006. V.C. Valchev, Van Den Bossche, A.P., Van De Sype, D.M., “Ferrite losses of cores with square wave voltage and DC bias”, IECON Proceedings (Industrial Electronics Conference), 2005, pp. 837-841.