Study of LCC Resonant Transistor DC / DC Converter with Capacitive Output Filter

The transistor LCC resonant DC/DC converters of electrical energy, working at frequencies higher than the resonant one, have found application in building powerful energy supplying equipment for various electrical technologies (Cheron et al., 1985; Malesani et al., 1995; Jyothi & Jaison, 2009). To a great extent, this is due to their remarkable power and mass-dimension parameters, as well as, to their high operating reliability. Besides, in a very wide-working field, the LCC resonant converters behave like current sources with big internal impedance. These converters are entirely fit for work in the whole range from no-load to short circuit while retaining the conditions for soft commutation of the controllable switches. There is a multitude of publications, dedicated to the theoretical investigation of the LCC resonant converters working at a frequency higher than their resonant one (Malesani et al., 1995; Ivensky et al. 1999). In their studies most often the first harmonic analysis is used, which is practically precise enough only in the field of high loads of the converter. With the decrease in the load the mistakes related to using the method of the first harmonic could obtain fairly considerable values. During the analysis, the influence of the auxiliary (snubber) capacitors on the controllable switches is usually neglected, and in case of availability of a matching transformer, only its transformation ratio is taken into account. Thus, a very precise description of the converter operation in a wide range of load changes is achieved. However, when the load resistance has a considerable value, the models created following the method mentioned above are not correct. They cannot be used to explain what the permissible limitations of load change depend on in case of retaining the conditions for soft commutation at zero voltage of the controllable switches – zero voltage switching (ZVS). The aim of the present work is the study of a transistor LCC resonant DC/DC converter of electrical energy, working at frequencies higher than the resonant one. The possible operation modes of the converter with accounting the influence of the damping capacitors and the parameters of the matching transformer are of interest as well. Building the output characteristics based on the results from a state plane analysis and suggesting a methodology for designing, the converter is to be done. Drawing the boundary curves between the different operating modes of the converter in the plane of the output characteristics, as well as outlining the area of natural commutation of the controllable switches are also among the aims of this work. Last but not least, the work aims at designing and experimental investigating a laboratory prototype of the LCC resonant converter under consideration.


Introduction
The transistor LCC resonant DC/DC converters of electrical energy, working at frequencies higher than the resonant one, have found application in building powerful energy supplying equipment for various electrical technologies (Cheron et al., 1985;Malesani et al., 1995;Jyothi & Jaison, 2009). To a great extent, this is due to their remarkable power and mass-dimension parameters, as well as, to their high operating reliability. Besides, in a very wide-working field, the LCC resonant converters behave like current sources with big internal impedance. These converters are entirely fit for work in the whole range from no-load to short circuit while retaining the conditions for soft commutation of the controllable switches. There is a multitude of publications, dedicated to the theoretical investigation of the LCC resonant converters working at a frequency higher than their resonant one (Malesani et al., 1995;Ivensky et al. 1999). In their studies most often the first harmonic analysis is used, which is practically precise enough only in the field of high loads of the converter. With the decrease in the load the mistakes related to using the method of the first harmonic could obtain fairly considerable values. During the analysis, the influence of the auxiliary (snubber) capacitors on the controllable switches is usually neglected, and in case of availability of a matching transformer, only its transformation ratio is taken into account. Thus, a very precise description of the converter operation in a wide range of load changes is achieved. However, when the load resistance has a considerable value, the models created following the method mentioned above are not correct. They cannot be used to explain what the permissible limitations of load change depend on in case of retaining the conditions for soft commutation at zero voltage of the controllable switches -zero voltage switching (ZVS). The aim of the present work is the study of a transistor LCC resonant DC/DC converter of electrical energy, working at frequencies higher than the resonant one. The possible operation modes of the converter with accounting the influence of the damping capacitors and the parameters of the matching transformer are of interest as well. Building the output characteristics based on the results from a state plane analysis and suggesting a methodology for designing, the converter is to be done. Drawing the boundary curves between the different operating modes of the converter in the plane of the output characteristics, as well as outlining the area of natural commutation of the controllable switches are also among the aims of this work. Last but not least, the work aims at designing and experimental investigating a laboratory prototype of the LCC resonant converter under consideration.

Modes of operation of the converter
The circuit diagram of the LCC transistor resonant DC/DC converter under investigation is shown in figure 1. It consists of an inverter (controllable switches constructed on base of the transistors Q 1 ÷Q 4 with freewheeling diodes D 1 ÷D 4 ), a resonant circuit (L, С), a matching transformer Tr, an uncontrollable rectifier (D 5 ÷D 8 ), capacitive input and output filters (C F1 и C F2 ) and a load resistor (R 0 ). The snubber capacitors (C 1 ÷C 4 ) are connected with the transistors in parallel. The output power of the converter is controlled by changing the operating frequency, which is higher than the resonant frequency of the resonant circuit. It is assumed that all the elements in the converter circuit (except for the matching transformer) are ideal, and the pulsations of the input and output voltages can be neglected. All snubber capacitors C 1 ÷C 4 are equivalent in practice to just a single capacitor C S (dotted line in fig.1), connected in parallel to the output of the inverter. The capacity of the capacitor C S is equal to the capacity of each of the snubber capacitors C 1 ÷C 4 . The matching transformer Tr is shown in fig.1 together with its simplified equivalent circuit under the condition that the magnetizing current of the transformer is negligible with respect to the current in the resonant circuit. Then this transformer comprises both the full leakage inductance L S and the natural capacity of the windings 0 , reduced to the primary winding, as well as an ideal transformer with its transformation ratio equal to k. The leakage inductance L S is connected in series with the inductance of the resonant circuit L and can be regarded as part of it . The natural capacity C 0 takes into account the capacity between the windings and the different layers in each winding of the matching transformer. C 0 can has an essential value, especially with stepping up transformers (Liu et al., 2009). Together with the capacity 0 the resonant circuit becomes a circuit of the third order (L, C and 0 ), while the converter could be regarded as LCC resonant DC/DC converter with a capacitive output filter. The parasitic parameters of the matching transformer -leakage inductance and natural capacity of the windings -should be taken into account only at high voltages and high operating frequencies of the converter. At voltages lower than 1000 V and frequencies lower than 100 kHz they can be neglected, and the capacitor 0 should be placed additionally (Liu et al., 2009). Because of the availability of the capacitor C S , the commutations in the output voltage of the inverter (u a ) are not instantaneous. They start with switching off the transistors Q 1 /Q 3 or Q 2 /Q 4 and end up when the equivalent snubber capacitor is recharged from +U d to -U d or backwards and the freewheeling diodes D 2 /D 4 or D 1 /D 3 start conducting. In practice the capacitors 2 and 4 discharge from +U d to 0, while 1 and 3 recharge from 0 to+U d or backwards. During these commutations, any of the transistors and freewheeling diodes of the inverter does not conduct and the current flowed through the resonant circuit is closed through the capacitor S . Because of the availability of the capacitor C 0 , the commutations in the input voltage of the rectifier (u b ) are not instantaneous either. They start when the diode pairs (D 5 /D 7 or D 6 /D 8 ) stop conducting at the moments of setting the current to zero through the resonant circuit and end up with the other diode pair (D 6 /D 8 или D 5 /D 7 ) start conducting, when the capacitor 0 recharges from +kU 0 to -kU 0 or backwards. During these commutations, any of the diodes of the rectifier does not conduct and the current flowed through the resonant circuit is closed through the capacitor 0 . The condition for natural switching on of the controllable switches at zero voltage (ZVS) is fulfilled if the equivalent snubber capacitor S always manages to recharge from +U d to -U d or backwards. At modes, close to no-load, the recharging of S is possible due to the availability of the capacitor C 0 . It ensures the flow of current through the resonant circuit, even when the diodes of the rectifier do not conduct. When the load and the operating frequency are deeply changed, three different operation modes of the converter can be observed. It is characteristic for the first mode that the commutations in the rectifier occur entirely in the intervals for conducting of the transistors in the inverter. This mode is the main operation mode of the converter. It is observed at comparatively small values of the load resistor R 0 . At the second mode the commutation in the rectifier ends during the commutation in the inverter, i.e., the rectifier diodes start conducting when both the transistors and the freewheeling diodes of the inverter are closed. This is the medial operation mode and it is only observed in a narrow zone, defined by the change of the load resistor value which is however not immediate to no-load. At modes, which are very close to no-load the third case is observed. The commutations in the rectifier now complete after the ones in the inverter, i.e. the rectifier diodes start conducting after the conduction beginning of the corresponding inverter's freewheeling diodes. This mode is the boundary operation mode with respect to no-load.

Analysis of the converter
In order to obtain general results, it is necessary to normalize all quantities characterizing the converter's state. The following quantities are included into relative units: where ω is the operating frequency and LC 1 ω 0 = and 0 ZL C = are the resonant frequency and the characteristic impedance of the resonant circuit L-C correspondingly.

Analysis at the main operation mode of the converter
Considering the influence of the capacitors S and 0 , the main operation mode of the converter can be divided into eight consecutive intervals, whose equivalent circuits are shown in fig. 2. By the trajectory of the depicting point in the state plane ( )    It has been proved in (Cheron, 1989;Bankov, 2009) that in the state plane ( fig. 3) the points, corresponding to the beginning (p.М 2 ) and the end (p.М 3 ) of the commutation in the inverter belong to the same arc with its centre in point ( ) . It can be proved the same way that the points, corresponding to the beginning (p.М 8 ) and the end (p.М 1 ) of the commutation in the rectifier belong to an arc with its centre in point ( ) 1;0 . It is important to note that only the end points are of importance on these arcs. The central angles of these arcs do not matter either, because as during the commutations in the inverter and rectifier the electric quantities change correspondingly with angular frequencies 0 ω′ and 0 ω′′ , not with 0 ω .
The following designations are made: (2) From the existing symmetry with respect to the origin of the coordinate system of the state plane it follows: The equations (4)÷(11) allow for calculating the coordinates of the points М 1 ÷М 4 in the state plane, which are the starting values of the current through the inductor L and the voltage of the commutating capacitor C in relative units for each interval of converter operation. The expressions for the coordinates are in function of 0 U′ , Cm U′ , 1 a and 2 a : For converters with only two reactive elements (L and C) in the resonant circuit the expression for its output current 0 I′ is known from (Al Haddad et al., 1986;Cheron, 1989): The LCC converter under consideration has three reactive elements in its resonant circuit (L, С и C 0 ). From fig.4 it can be seen that its output current 0 I′ decreases by the value The following equation is known: where the times t 1 ÷t 4 represent the durations of the different stages -from 1 to 4. For the times of the four intervals at the main mode of operation of the converter within a half-cycle the following equations hold: It should be taken into consideration that for stages 1 and 3 the electric quantities change with angular frequency 0 ω , while for stages 2 and 4 -the angular frequencies are respectively 01 0 ωω n ′ = and 02 0 ωω n ′′ = .

Analysis at the boundary operation mode of the converter
At this mode, the operation of the converter for a cycle can be divided into eight consecutive stages (intervals), whose equivalent circuits are shown in fig. 5. It makes impression that the sinusoidal quantities in the different equivalent circuits have three different angular frequencies: for stages 4 and 8; , for stages 1, 3, 5 and 7; , for stages 2 and 6. xyxy +=+ From the principle of continuity of the current through the inductor L and of the voltage in the capacitor C it follows: At the boundary operation mode the output current 0 I′ is defined by expression (21)

Output characteristics and boundary curves
On the basis of the analysis results, equations for the output characteristics are obtained individually for both the main and the boundary modes of the converter operation. Besides, expressions for the boundary curves of the separate modes are also derived.

Output characteristics and boundary curves at the main operation mode
From equation (21)  The comparison of these characteristics to the known ones from (Al Haddad et al., 1986;Cheron, 1989) shows the entire influence of the capacitors S и 0 . It can be seen that the output characteristics become more vertical and the converter can be regarded to a great extent as a source of current, stable at operation even at short circuit. Besides, an area of operation is noticeable, in which 0 1 U′ > .
At the main operation mode the commutations in the rectifier (stages 4 and 8) must always end before the commutations in the inverter have started. This is guaranteed if the following condition is fulfilled: In order to enable natural switching of the controllable switches at zero voltage (ZVS), the commutations in the inverter (stages 2 and 6) should always end before the current in the resonant circuit becomes zero. This is guaranteed if the following condition is fulfilled: If the condition (49) is not fulfilled, then switching a pair of controllable switches off does not lead to natural switching the other pair of controllable switches on at zero voltage and then the converter stops working. It should be emphasized that these commutation mistakes do not lead to emergency modes and they are not dangerous to the converter. When it "misses", all the semiconductor switches stop conducting and the converter just stops working. This is one of the big advantages of the resonant converters working at frequencies higher than the resonant one.  (12), (14) and (47) are substituted in condition (48), while equations (16) and (47) are substituted in condition (49). Then the inequalities (48) and (49) fig. 7-b) in the plane of the output characteristics. It can be seen that the area of the main operation mode of the converter is limited within the boundary curves А and L 3 or L 4 . The bigger the capacity of the capacitors www.intechopen.com S and 0 , the smaller this area is. However, the increase of the snubber capacitors leads to a decrease in the commutation losses in the transistors as well as to limiting the electromagnetic interferences in the converter. The expression (51) defines the borders, beyond which the converter stops working because of the breakage in the conditions for natural switching the controllable switches on at zero voltage (ZVS). Exemplary boundary curves have been drawn in the plane of the output characteristics ( fig.8) at а 1 =0.10. Four values have been chosen for the other parameter: а 2 = 0.05; 0.1; 0.2 and 1.0. When the capacity of the capacitor 0 is smaller or equal to that of the snubber capacitors C S ( ) 12 aa ≥ , then the converter is fit for work in the area between the curve L 1 or L 2 and the x-axis (the abscissa). Only the main operation mode of the converter is possible in this area. The increase in the load resistance or in the operating frequency leads to stopping the operation of the converter before it has accomplished a transition towards the medial and the boundary modes of work.
When C 0 has a higher value than the value of C S ( ) 12 aa < then the boundary curve of the area of converter operation with ZVS is displaced upward (curve L 3 or L 4 ). It is possible now to achieve even a no-load mode. ,, , U f Ia a ν ′′ = is obtained. Its solving enables with a possibility to build the outer (output) characteristics of the converter in relative units at the boundary operation mode under consideration and at regulation by changing the operating frequency. Such characteristics are shown in fig. 7-а for ν = 3.0; 3.3165; 3.6 and а 1 =0.1; а 2 =0.2 and in fig. 7-b for ν =1.5; 1.6; 1.8 and а 1 =0.1; а 2 =1.0. At the boundary operation mode, the diodes of the rectifier have to start conducting after opening the freewheeling diodes of the inverter. This is guaranteed if the following condition is fulfilled: Condition (53) gives the possibility to define the area of the boundary operation mode of the converter in the plane of the output characteristics ( fig. 7-а and fig. 7-b). It is limited between the y-axis (the ordinate) and the boundary curve B. It can be seen that the converter stays absolutely fit for work at high-Ohm loads, including at a no-load mode. It is due mainly to the capacitor 0 . With the increase in its capacity (increase of a 2 ) the area of the boundary operation mode can also be increased.

Medial operation mode of the converter
At this operation mode, the diodes of the rectifier start conducting during the commutation in the inverter. The equivalent circuits, corresponding to this mode for a cycle, are shown in fig.9. In this case, the sinusoidal quantities have four different angular frequencies:  . Equivalent circuits at the medial operation mode of the converter Therefore, the analysis of the medial operation mode is considerably more complex. The area in the plane of the output characteristics, within which this mode appears, however, is completely defined by the boundary curves A and B for the main and the boundary modes respectively. Having in mind the monotonous character of the output characteristics for the other two modes, their building for the mode under consideration is possible through linear interpolation. It is shown in fig. 7-а for ν = 3.0; 3.3165 as well as in fig. 7-b for ν=1.5; 1.6; 1.8. The larger area of this mode corresponds to the higher capacity of the snubber capacitors C S and the smaller capacity of the capacitor C 0 .

Methododlogy for designing the converter
During the process of designing the LCC resonant DC/DC converter under consideration, the following parameters are usually predetermined: power in the load 0, output voltage U 0 and operating frequency f. Very often, the value of the power supply voltage U d is predefined and the desired output voltage is obtained by adding a matching transformer. The design of the converter could be carried out in the following order: 1. Choice of the frequency distraction ν For converters, operating at frequencies higher than the resonant one, the frequency distraction is usually chosen in the interval ν =1.1÷1.3. It should be noted here that if a higher value of ν is chosen, the exchange of reactive energy between the resonant circuit and the energy source increases, i.e., the current load on the elements in the circuit increases while the stability of the output characteristics decreases. It is due to the increase in the impedance of the resonant circuit of the converter.

Choice of the parameter 20 aC C =
The choice of this parameter is a compromise to a great extent. On the one hand, to avoid the considerable change of the operating frequency during the operation of the converter from nominal load to no-load mode, the capacitor 0 (respectively the parameter а 2 ) should be big enough. On the other hand, with the increase of 0 the current in the resonant circuit increases and the efficiency of the converter decreases. It should be noted that a significant increase in the current load on the elements in the scheme occurs at 2 1 a > (Malesani et al., 1995).
3. Choice of the parameter 1

S aC C =
The parameter а 1 is usually chosen in the interval а 1 = 0.02÷0.20. The higher the value of а 1 (the bigger the capacity of the damping capacitors), the smaller the area of natural commutation of the transistors in the plane of the output characteristics is. However, the increase in the capacity of the snubber capacitors leads to a decrease in the commutation losses and limitation of the electromagnetic interferences in the converter.

Choice of the coordinates of a nominal operating point
The values of the parameters а 1 and а 2 fully define the form of the output characteristics of the converter. The nominal operating point with coordinates 0 I′ and 0 U′ lies on the characteristic, corresponding to the chosen frequency distraction ν. It is desirable that this point is close to the boundary of natural commutation so that the inverter could operate at a high power factor (minimal exchange of reactive energy). If the converter operates with sharp changes in the load and control frequency, however, it is then preferable to choose the operating point in an area, relatively distant from the border of natural commutation. Thus, automatic switching the converter off is avoided -a function, integrated in the control drivers of power transistors.

Defining the transformation ratio of the matching transformer
The transformation ratio is defined, so that the required output voltage of the converter at minimal power supply voltage and at maximum load is guaranteed: where U d min is the minimal permissible value of the input voltage U d .

Calculating the parameters of the resonant circuit
The values of the elements in the resonant circuit L and C are defined by the expressions related to the frequency distraction and the output current in relative units: ). The dotted curve in fig.10 shows the theoretical output characteristics, while the continuous curve shows the output characteristics, obtained in result of the experiments. A good match between the theoretical results and the ones from the experimental investigation can be noted. The small differences between them are mostly due to the losses in the semiconductor switches in their open state and the active losses in the elements of the resonant circuit. Oscillograms, illustrating respectively the main and the boundary operation modes of the converter are shown in fig. 11 and fig. 12. These modes are obtained at a stable operating frequency 61.54 f = kHz ( 1.6 = ) and at certain change of the load resistor. In the oscillograms the following quantities in various combinations are shown: output voltage (u a ) and output current (i) of the inverter, input voltage (u b ) and output current (i 0 ) of the rectifier. From fig. 11-b and fig. 12-b the difference between the main and the boundary operation mode of the converter can be seen. In the first case, the commutations in the rectifier (the process of recharging the capacitor 0 ) end before the commutations in the inverter (the process of recharging the capacitor S ). In the second case, the commutations in the rectifier complete after the ones in the inverter. In both cases during the commutations in the rectifier, all of its diodes are closed and the output current i 0 is equal to zero ( fig. 11-с and  fig. 12-с). Fig. 12-b confirms the fact that at certain conditions the output voltage can become higher than the power supply voltage without using a matching transformer. At no-load mode, the converter operation is shown in fig. 13. In this case, the output voltage is more than two times higher than the power supply one. u a 500V/div; u b 500V/div; х=5µs/div Fig. 13. Oscillograms, illustrating no-load mode of the converter

Conclusions
The operation of an LCC transistor resonant DC/DC converter with a capacitive output filter and working above the resonant frequency has been investigated, taking into account the influence of snubber capacitors and a matching transformer. The particular operation modes of the converter have been considered, and the conditions under which they are obtained have been described. The output characteristics for all operation modes of the converter have been built including at regulation by means of changing the operating frequency. The boundary curves between the different operation modes of the converter as well as the area of natural commutation of the controllable switches have been shown in the plane of the output characteristics. Results from investigations carried out by means of a laboratory prototype of the converter have been obtained and these results confirm the ones from the analysis. The theoretical investigations show that the conditions for ZVS can be kept the same for high-Ohm loads and the converter can stay fit for work even at a no-load mode. For the purpose, it is necessary to have the natural capacity of the matching transformer bigger than the one of the snubber capacitors.
The output characteristics show that in the zone of small loads the value of the normalized output voltage increases to reach a value higher than unit what is characteristic for www.intechopen.com converters with controllable rectifying. This can be explained by the similar mechanism of the rectifier operation in the investigated converter. The results from the investigation can be used for more precise designing of LCC converters used as power supplies for electric arc welding aggregates, powerful lasers, luminescent lamps etc. Nowadays, the increasing use of power electronics equipment origins important distortions. The perfect AC power systems are a pure sinusoidal wave, both voltage and current, but the ever-increasing existence of nonlinear loads modify the characteristics of voltage and current from the ideal sinusoidal wave. This deviation from the ideal wave is reflected by the harmonics and, although its effects vary depending on the type of load, it affects the efficiency of an electrical system and can cause considerable damage to the systems and infrastructures. Ensuring optimal power quality after a good design and devices means productivity, efficiency, competitiveness and profitability. Nevertheless, nobody can assure the optimal power quality when there is a good design if the correct testing and working process from the obtained data is not properly assured at every instant; this entails processing the real data correctly. In this book the reader will be introduced to the harmonics analysis from the real measurement data and to the study of different industrial environments and electronic devices.

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